--- /srv/rebuilderd/tmp/rebuilderdrj3qB7/inputs/macaulay2-common_1.25.11+ds-2_all.deb +++ /srv/rebuilderd/tmp/rebuilderdrj3qB7/out/macaulay2-common_1.25.11+ds-2_all.deb ├── file list │ @@ -1,3 +1,3 @@ │ -rw-r--r-- 0 0 0 4 2025-12-14 14:09:53.000000 debian-binary │ --rw-r--r-- 0 0 0 540512 2025-12-14 14:09:53.000000 control.tar.xz │ --rw-r--r-- 0 0 0 31297916 2025-12-14 14:09:53.000000 data.tar.xz │ +-rw-r--r-- 0 0 0 540532 2025-12-14 14:09:53.000000 control.tar.xz │ +-rw-r--r-- 0 0 0 31297284 2025-12-14 14:09:53.000000 data.tar.xz ├── control.tar.xz │ ├── control.tar │ │ ├── ./control │ │ │ @@ -1,13 +1,13 @@ │ │ │ Package: macaulay2-common │ │ │ Source: macaulay2 │ │ │ Version: 1.25.11+ds-2 │ │ │ Architecture: all │ │ │ Maintainer: Debian Math Team │ │ │ -Installed-Size: 305306 │ │ │ +Installed-Size: 305291 │ │ │ Depends: fonts-katex (>= 0.16.10+~cs6.1.0), libjs-bootsidemenu (>= 1.0.0), libjs-bootstrap5 (>= 5.3.8+dfsg), libjs-d3 (>= 3.5.17), libjs-jquery (>= 3.7.1+dfsg+~3.5.33), libjs-katex (>= 0.16.10+~cs6.1.0), libjs-nouislider (>= 15.8.1+ds), libjs-three (>= 111+dfsg1), node-clipboard (>= 2.0.11+ds+~cs9.6.11), node-fortawesome-fontawesome-free (>= 6.7.2+ds1) │ │ │ Section: math │ │ │ Priority: optional │ │ │ Multi-Arch: foreign │ │ │ Homepage: http://macaulay2.com │ │ │ Description: Software system for algebraic geometry research (common files) │ │ │ Macaulay 2 is a software system for algebraic geometry research, written by │ │ ├── ./md5sums │ │ │ ├── ./md5sums │ │ │ │┄ Files differ ├── data.tar.xz │ ├── data.tar │ │ ├── file list │ │ │ @@ -3723,18 +3723,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 76682 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BeginningMacaulay2/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4226 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BeginningMacaulay2/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 2909 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BeginningMacaulay2/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Benchmark/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Benchmark/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2927 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Benchmark/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Benchmark/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 423 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Benchmark/example-output/_run__Benchmarks.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 433 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Benchmark/example-output/_run__Benchmarks.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Benchmark/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 29 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Benchmark/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 5574 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Benchmark/html/_run__Benchmarks.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5584 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Benchmark/html/_run__Benchmarks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5233 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Benchmark/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4242 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Benchmark/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 2912 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Benchmark/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BernsteinSato/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BernsteinSato/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 289851 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BernsteinSato/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BernsteinSato/example-output/ │ │ │ @@ -3998,15 +3998,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10474 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bertini/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BettiCharacters/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BettiCharacters/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 141706 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BettiCharacters/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 6933 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp1.out │ │ │ -rw-r--r-- 0 root (0) root (0) 8038 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp2.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6928 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp3.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6929 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp3.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2267 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp4.out │ │ │ -rw-r--r-- 0 root (0) root (0) 758 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Character_sp__Array.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1576 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Equality_spchecks.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1927 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Labels.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2080 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Sub.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3164 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 62 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 7189 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Action.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6096 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Action__On__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8643 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Action__On__Graded__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15258 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp1.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14939 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp2.html │ │ │ --rw-r--r-- 0 root (0) root (0) 18503 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp3.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 18504 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp3.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8439 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp4.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9931 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Character.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6339 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Character__Decomposition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6812 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Character__Table.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5714 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Character_sp__Array.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4366 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Character_spoperations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6855 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Equality_spchecks.html │ │ │ @@ -4315,22 +4315,22 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5291 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Browse/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4374 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Browse/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3057 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Browse/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 19549 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 4572 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/example-output/_bruns.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4571 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/example-output/_bruns.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1717 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/example-output/_bruns__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2579 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/example-output/_elementary.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1678 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/example-output/_evans__Griffith.out │ │ │ -rw-r--r-- 0 root (0) root (0) 570 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/example-output/_is__Syzygy.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 49 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 14041 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/html/_bruns.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14040 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/html/_bruns.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8831 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/html/_bruns__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10497 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/html/_elementary.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8129 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/html/_evans__Griffith.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6572 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/html/_is__Syzygy.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9591 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6809 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4075 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Bruns/html/toc.html │ │ │ @@ -4365,15 +4365,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 422 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_is__Minimal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 217 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_is__Simplex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 505 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_is__Well__Defined_lp__Cell__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 595 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_is__Well__Defined_lp__Cell_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 363 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_max__Cells.out │ │ │ -rw-r--r-- 0 root (0) root (0) 275 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_new__Cell.out │ │ │ -rw-r--r-- 0 root (0) root (0) 228 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_new__Simplex__Cell.out │ │ │ --rw-r--r-- 0 root (0) root (0) 733 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_relabel__Cell__Complex.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 738 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_relabel__Cell__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 249 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_ring_lp__Cell__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 520 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_scarf__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2473 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_skeleton_lp__Z__Z_cm__Cell__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1367 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_subcomplex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 821 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_taylor__Complex.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 39 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/.Headline │ │ │ @@ -4406,15 +4406,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 7399 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_is__Minimal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5693 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_is__Simplex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8187 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_is__Well__Defined_lp__Cell__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8974 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_is__Well__Defined_lp__Cell_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6426 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_max__Cells.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8432 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_new__Cell.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7358 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_new__Simplex__Cell.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8605 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_relabel__Cell__Complex.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8610 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_relabel__Cell__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5874 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_ring_lp__Cell__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6841 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_scarf__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8430 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_skeleton_lp__Z__Z_cm__Cell__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9468 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_subcomplex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6493 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_taylor__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 23005 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19645 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/master.html │ │ │ @@ -4429,19 +4429,19 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 2151 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_is__Chain__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1447 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_is__Chain__Complex__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 721 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_is__Exact_lp__Chain__Complex_cm__Length__Limit_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 578 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_is__Exact_lp__Chain__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1448 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_is__Quasi__Isomorphism.out │ │ │ -rw-r--r-- 0 root (0) root (0) 771 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_is__Quasi__Isomorphism_lp..._cm__Length__Limit_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 278 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_koszul__Complex.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1962 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_minimize_lp__Chain__Complex_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1960 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_minimize_lp__Chain__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 694 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_nonzero__Max.out │ │ │ -rw-r--r-- 0 root (0) root (0) 684 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_prepend__Zero__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 899 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_remove__Zero__Trailing__Terms.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3453 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_resolution__Of__Chain__Complex.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3450 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_resolution__Of__Chain__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 541 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_resolution_lp__Chain__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2570 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_scarf__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 537 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_substitute_lp__Chain__Complex_cm__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 672 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_taylor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1333 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_taylor__Resolution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1351 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_trivial__Homological__Truncation.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/ │ │ │ @@ -4459,20 +4459,20 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9376 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_is__Exact_lp__Chain__Complex_cm__Length__Limit_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6452 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_is__Exact_lp__Chain__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4630 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_is__Minimal__Chain__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7551 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_is__Quasi__Isomorphism.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9544 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_is__Quasi__Isomorphism_lp..._cm__Length__Limit_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5473 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_koszul__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7220 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_koszul__Complex_lp..._cm__Length__Limit_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10096 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_minimize_lp__Chain__Complex_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10094 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_minimize_lp__Chain__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6612 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_nonzero__Max.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6519 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_nonzero__Min.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5900 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_prepend__Zero__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6665 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_remove__Zero__Trailing__Terms.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12265 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_resolution__Of__Chain__Complex.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12262 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_resolution__Of__Chain__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7373 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_resolution__Of__Chain__Complex_lp..._cm__Length__Limit_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9135 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_resolution_lp__Chain__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10433 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_scarf__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5755 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_substitute_lp__Chain__Complex_cm__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5714 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_taylor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6750 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_taylor__Resolution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7255 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_taylor__Resolution_lp..._cm__Length__Limit_eq_gt..._rp.html │ │ │ @@ -4501,49 +4501,49 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 12258 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexOperations/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8418 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexOperations/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4775 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ChainComplexOperations/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 123136 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 4377 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___C__S__M.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4376 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___C__S__M.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1503 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Check__Smooth.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3419 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Check__Toric__Variety__Valid.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3442 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Chern.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2404 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Chow__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 265 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Class__In__Chow__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 775 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Class__In__Toric__Chow__Ring.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2017 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Comp__Method.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4380 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2013 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Comp__Method.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4378 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler.out │ │ │ -rw-r--r-- 0 root (0) root (0) 342 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler__Affine.out │ │ │ --rw-r--r-- 0 root (0) root (0) 759 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Inds__Of__Smooth.out │ │ │ --rw-r--r-- 0 root (0) root (0) 667 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Input__Is__Smooth.out │ │ │ --rw-r--r-- 0 root (0) root (0) 624 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Method.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 760 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Inds__Of__Smooth.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 664 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Input__Is__Smooth.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 621 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Method.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1049 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Multi__Proj__Coord__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6662 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Output.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3327 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Segre.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1666 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Toric__Chow__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 570 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/_is__Multi__Homogeneous.out │ │ │ -rw-r--r-- 0 root (0) root (0) 802 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/_probabilistic_spalgorithm.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 665 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 55 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 23694 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___C__S__M.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 23693 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___C__S__M.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6377 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Check__Smooth.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11239 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Check__Toric__Variety__Valid.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18179 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Chern.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9577 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Chow__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6409 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Class__In__Chow__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7339 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Class__In__Toric__Chow__Ring.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10385 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Comp__Method.html │ │ │ --rw-r--r-- 0 root (0) root (0) 18563 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10381 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Comp__Method.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 18561 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5955 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler__Affine.html │ │ │ --rw-r--r-- 0 root (0) root (0) 5902 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Inds__Of__Smooth.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6013 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Input__Is__Smooth.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6466 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Method.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5903 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Inds__Of__Smooth.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6010 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Input__Is__Smooth.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6463 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Method.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8377 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Multi__Proj__Coord__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16398 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Output.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16710 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Segre.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8690 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Toric__Chow__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4910 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/_bertini__Check.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5128 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/_configuring_sp__Bertini.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7468 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/_is__Multi__Homogeneous.html │ │ │ @@ -4558,15 +4558,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 2624 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/___Chordal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 304 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/___Ring__Element_sp_pc_sp__Chordal__Net.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1273 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/___Ring__Map_sp__Chordal__Net.out │ │ │ -rw-r--r-- 0 root (0) root (0) 178 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_adjacent__Minors__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2151 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal__Elim.out │ │ │ -rw-r--r-- 0 root (0) root (0) 969 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal__Graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1787 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal__Net.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1202 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal__Net_lp__Hash__Table_cm__Hash__Table_cm__Elim__Tree_cm__Digraph_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1196 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal__Net_lp__Hash__Table_cm__Hash__Table_cm__Elim__Tree_cm__Digraph_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1633 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal__Tria.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2559 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal_spnetworks_spexamples.out │ │ │ -rw-r--r-- 0 root (0) root (0) 429 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chromatic__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 237 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_codim__Count.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1454 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_components_lp__Chordal__Net_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 414 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_constraint__Graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 726 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_digraph_lp__Chordal__Net_rp.out │ │ │ @@ -4599,15 +4599,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 3907 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/___Get__Table.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7598 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/___Ring__Element_sp_pc_sp__Chordal__Net.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7248 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/___Ring__Map_sp__Chordal__Net.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5071 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/_adjacent__Minors__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10905 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chordal__Elim.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7288 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chordal__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9374 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chordal__Net.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8344 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chordal__Net_lp__Hash__Table_cm__Hash__Table_cm__Elim__Tree_cm__Digraph_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8338 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chordal__Net_lp__Hash__Table_cm__Hash__Table_cm__Elim__Tree_cm__Digraph_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8861 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chordal__Tria.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10852 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chordal_spnetworks_spexamples.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5434 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chromatic__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7090 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/_codim__Count.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8141 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/_components_lp__Chordal__Net_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5721 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/_constraint__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7695 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Chordal/html/_digraph_lp__Chordal__Net_rp.html │ │ │ @@ -4899,15 +4899,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 231378 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 650 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___B__G__G__L.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1959 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___B__Ranks.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3164 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Complete__Intersection__Resolutions.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4598 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4595 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4116 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash__Total.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2743 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Ext__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1023 2025-12-14 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./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___A__Ranks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4935 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Augmentation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6307 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___B__G__G__L.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9094 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___B__Ranks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5966 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Check.html │ │ │ --rw-r--r-- 0 root (0) root (0) 15259 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Eisenbud__Shamash.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 15256 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Eisenbud__Shamash.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14302 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Eisenbud__Shamash__Total.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11218 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Ext__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10327 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Ext__Module__Data.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5284 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Grading.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5531 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Hom__With__Components.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5130 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Layered.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4742 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Lift.html │ │ │ @@ -4993,18 +4993,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10902 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_module__As__Ext.html │ │ │ -rw-r--r-- 0 root (0) root (0) 23042 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_new__Ext.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8736 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_odd__Ext__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5888 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_psi__Maps.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7031 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_regularity__Sequence.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7383 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_splittings.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5224 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_stable__Hom.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6048 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_sum__Two__Monomials.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6046 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_sum__Two__Monomials.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5573 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_tensor__With__Components.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4869 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_to__Array.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6366 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_two__Monomials.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6364 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_two__Monomials.html │ │ │ -rw-r--r-- 0 root (0) root (0) 56031 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 35373 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15036 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Complexes/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Complexes/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 706402 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Complexes/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Complexes/example-output/ │ │ │ @@ -5342,29 +5342,29 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9518 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConformalBlocks/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 50933 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 36623 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Cosmological_spcorrelator_spfor_spthe_sp2-site_spchain.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1233 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Gauss_sq_sphypergeometric_spfunction.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3314 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Massless_spone-loop_sptriangle_sp__Feynman_spdiagram.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3313 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Massless_spone-loop_sptriangle_sp__Feynman_spdiagram.out │ │ │ -rw-r--r-- 0 root (0) root (0) 240 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/_base__Fraction__Field.out │ │ │ -rw-r--r-- 0 root (0) root (0) 424 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/_connection__Matrices.out │ │ │ -rw-r--r-- 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./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/_standard__Monomials.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 56 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 46479 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Cosmological_spcorrelator_spfor_spthe_sp2-site_spchain.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7850 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Gauss_sq_sphypergeometric_spfunction.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9213 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Massless_spone-loop_sptriangle_sp__Feynman_spdiagram.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9212 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Massless_spone-loop_sptriangle_sp__Feynman_spdiagram.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5611 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_base__Fraction__Field.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6834 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_connection__Matrices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6189 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_connection__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7537 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_gauge__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7266 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_gauge__Transform.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7099 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_is__Epsilon__Factorized.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7183 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_is__Integrable.html │ │ │ @@ -5584,136 +5584,136 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 38506 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CpMackeyFunctors/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 35258 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CpMackeyFunctors/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16572 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/CpMackeyFunctors/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Cremona/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Cremona/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 239171 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Cremona/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/ │ │ │ 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./usr/share/doc/Macaulay2/FastMinors/example-output/___Regular__In__Codimension__Tutorial.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 14279 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/___Regular__In__Codimension__Tutorial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1054 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/___Strategy__Default.out │ │ │ -rw-r--r-- 0 root (0) root (0) 337 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_choose__Good__Minors.out │ │ │ -rw-r--r-- 0 root (0) root (0) 246 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_choose__Random__Submatrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 307 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_choose__Submatrix__Largest__Degree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 308 2025-12-14 14:09:53.000000 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14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_is__Dim__At__Most.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9506 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_is__Rank__At__Least.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5364 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_is__Rank__At__Least_lp..._cm__Threads_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10205 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_proj__Dim.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10204 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_proj__Dim.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7912 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_recursive__Minors.html │ │ │ --rw-r--r-- 0 root (0) root (0) 43514 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_regular__In__Codimension.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 43511 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_regular__In__Codimension.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6458 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_reorder__Polynomial__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 24783 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 26704 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7533 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FastMinors/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 25936 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 1849 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/___Fitting_spideals_spof_spfinite_spmodules.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1848 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/___Fitting_spideals_spof_spfinite_spmodules.out │ │ │ -rw-r--r-- 0 root (0) root (0) 334 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/_affine__Part.out │ │ │ -rw-r--r-- 0 root (0) root (0) 694 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/_co1__Fitting.out │ │ │ -rw-r--r-- 0 root (0) root (0) 848 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/_gauss__Col.out │ │ │ -rw-r--r-- 0 root (0) root (0) 365 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/_gotzmann__Test.out │ │ │ -rw-r--r-- 0 root (0) root (0) 610 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/_next__Degree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 197 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/_quot__Scheme.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 32 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 10957 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/___Fitting_spideals_spof_spfinite_spmodules.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10956 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/___Fitting_spideals_spof_spfinite_spmodules.html │ │ │ -rw-r--r-- 0 root (0) root (0) 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105269 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 338 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/___Bounds.out │ │ │ -rw-r--r-- 0 root (0) root (0) 318 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/___Frobenius__Thresholds.out │ │ │ -rw-r--r-- 0 root (0) root (0) 793 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/___Guess__Strategy.out │ │ │ -rw-r--r-- 0 root (0) root (0) 866 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_compare__F__P__T.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4034 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_fpt.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2459 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_frobenius__Nu.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2456 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_frobenius__Nu.out │ │ │ -rw-r--r-- 0 root (0) root (0) 760 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_is__F__Jumping__Exponent.out │ │ │ -rw-r--r-- 0 root (0) root (0) 552 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_is__F__P__T.out │ │ │ -rw-r--r-- 0 root (0) root (0) 828 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_is__Simple__Normal__Crossing.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 617 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 12 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5582 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Bounds.html │ │ │ @@ -7057,15 +7057,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10580 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Guess__Strategy.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4665 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Return__List.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4769 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Search.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4735 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Standard__Power.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5945 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Use__Special__Algorithms.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14756 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_compare__F__P__T.html │ │ │ -rw-r--r-- 0 root (0) root (0) 26000 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_fpt.html │ │ │ --rw-r--r-- 0 root (0) root (0) 24542 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_frobenius__Nu.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 24539 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_frobenius__Nu.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13295 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_is__F__Jumping__Exponent.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12519 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_is__F__P__T.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9887 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_is__Simple__Normal__Crossing.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19600 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20028 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7092 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/FunctionFieldDesingularization/ │ │ │ @@ -7117,15 +7117,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 614 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_make__K__Class_lp__G__K__M__Variety_cm__Flag__Matroid_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1001 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_make__K__Class_lp__G__K__M__Variety_cm__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 339 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_map_lp__G__K__M__Variety_cm__G__K__M__Variety_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1060 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_moment__Graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 732 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_moment__Graph_lp__G__K__M__Variety_cm__Moment__Graph_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 170 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_moment__Graph_lp__G__K__M__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 235 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_normal__Toric__Variety_lp__G__K__M__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 8064 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_orbit__Closure.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 8063 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_orbit__Closure.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1060 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_projective__Space.out │ │ │ -rw-r--r-- 0 root (0) root (0) 612 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_pullback_lp__Equivariant__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 615 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_pushforward.out │ │ │ -rw-r--r-- 0 root (0) root (0) 220 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_set__Indicator.out │ │ │ -rw-r--r-- 0 root (0) root (0) 437 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_trivial__K__Class.out │ │ │ -rw-r--r-- 0 root (0) root (0) 401 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_underlying__Graph_lp__Moment__Graph_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/ │ │ │ @@ -7168,15 +7168,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 7893 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_make__K__Class_lp__G__K__M__Variety_cm__Flag__Matroid_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8393 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_make__K__Class_lp__G__K__M__Variety_cm__Toric__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9167 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_map_lp__G__K__M__Variety_cm__G__K__M__Variety_cm__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8448 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_moment__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7456 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_moment__Graph_lp__G__K__M__Variety_cm__Moment__Graph_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5700 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_moment__Graph_lp__G__K__M__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7617 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_normal__Toric__Variety_lp__G__K__M__Variety_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 19815 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_orbit__Closure.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 19814 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_orbit__Closure.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7021 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_projective__Space.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7191 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_pullback_lp__Equivariant__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7746 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_pushforward.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6964 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_set__Indicator.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5716 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_trivial__K__Class.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5537 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_underlying__Graph_lp__Moment__Graph_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 26820 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/index.html │ │ │ @@ -8244,32 +8244,32 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 33164 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/___Groebner__Strata.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1505 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_find__Weight__Constraints.out │ │ │ -rw-r--r-- 0 root (0) root (0) 598 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_find__Weight__Vector.out │ │ │ -rw-r--r-- 0 root (0) root (0) 30368 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_groebner__Family.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2709 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_groebner__Stratum.out │ │ │ -rw-r--r-- 0 root (0) root (0) 328 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_linear__Part.out │ │ │ --rw-r--r-- 0 root (0) root (0) 13267 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_nonminimal__Maps.out │ │ │ --rw-r--r-- 0 root 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./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_smaller__Monomials.out │ │ │ -rw-r--r-- 0 root (0) root (0) 960 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_standard__Monomials.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1344 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_tail__Monomials.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 42 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6030 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/___All__Standard.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4977 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/___Minimalize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9811 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_find__Weight__Constraints.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8464 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_find__Weight__Vector.html │ │ │ -rw-r--r-- 0 root (0) root (0) 42133 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_groebner__Family.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10343 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_groebner__Stratum.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6000 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_linear__Part.html │ │ │ --rw-r--r-- 0 root (0) root (0) 23321 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_nonminimal__Maps.html │ │ │ --rw-r--r-- 0 root (0) root (0) 23549 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_random__Point__On__Rational__Variety_lp__Ideal_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 14869 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_random__Points__On__Rational__Variety_lp__Ideal_cm__Z__Z_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 23098 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_nonminimal__Maps.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 23971 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_random__Point__On__Rational__Variety_lp__Ideal_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14890 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_random__Points__On__Rational__Variety_lp__Ideal_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7289 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_smaller__Monomials.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7660 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_standard__Monomials.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8494 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_tail__Monomials.html │ │ │ -rw-r--r-- 0 root (0) root (0) 50235 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11279 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5699 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/ │ │ │ @@ -8293,28 +8293,28 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Hadamard/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Hadamard/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 18704 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Hadamard/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Hadamard/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 123 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Hadamard/example-output/___Point_sp_eq_eq_sp__Point.out │ │ │ -rw-r--r-- 0 root (0) root (0) 153 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Hadamard/example-output/___Point_sp_st_sp__Point.out │ │ │ -rw-r--r-- 0 root (0) root (0) 585 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Hadamard/example-output/_hadamard__Power_lp__Ideal_cm__Z__Z_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 860 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Hadamard/example-output/_hadamard__Power_lp__List_cm__Z__Z_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 934 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Hadamard/example-output/_hadamard__Power_lp__List_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1188 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Hadamard/example-output/_hadamard__Product_lp__Ideal_cm__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 210 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Hadamard/example-output/_hadamard__Product_lp__List_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1084 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Hadamard/example-output/_hadamard__Product_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1159 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Hadamard/example-output/_ideal__Of__Projective__Points.out │ │ │ -rw-r--r-- 0 root (0) root (0) 160 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Hadamard/example-output/_point.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Hadamard/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 44 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Hadamard/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4648 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Hadamard/html/___Point.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5014 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Hadamard/html/___Point_sp_eq_eq_sp__Point.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5344 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Hadamard/html/___Point_sp_st_sp__Point.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4254 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Hadamard/html/_hadamard__Power.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5986 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Hadamard/html/_hadamard__Power_lp__Ideal_cm__Z__Z_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6146 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Hadamard/html/_hadamard__Power_lp__List_cm__Z__Z_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6220 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Hadamard/html/_hadamard__Power_lp__List_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4559 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Hadamard/html/_hadamard__Product.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8096 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Hadamard/html/_hadamard__Product_lp__Ideal_cm__Ideal_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5757 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Hadamard/html/_hadamard__Product_lp__List_cm__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7389 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Hadamard/html/_hadamard__Product_lp__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6927 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Hadamard/html/_ideal__Of__Projective__Points.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5174 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Hadamard/html/_point.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8397 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Hadamard/html/index.html │ │ │ @@ -8417,61 +8417,61 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 50644 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 920 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/___Appell__F1.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2967 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/___Canonical_sp__Series_sp__Tutorial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 630 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_css__Expts.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1007 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_css__Expts__Mult.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4893 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_css__Lead__Term.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 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./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp..._cm__Use__Linear__Algebra_eq_gt..._rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1158 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp..._cm__Use__Linear__Algebra_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1326 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp__Diagonal__Action_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 560 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp__Finite__Group__Action_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2647 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp__Finite__Group__Action_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 773 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp__Linearly__Reductive__Action_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 798 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp__Linearly__Reductive__Action_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1070 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_is__Abelian.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2617 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_is__Invariant.out │ │ │ @@ -8761,32 +8761,32 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6600 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_action__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5896 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_ambient_lp__Ring__Of__Invariants_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6404 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_cyclic__Factors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6918 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_defining__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6562 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_degrees__Ring_lp__Diagonal__Action_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8613 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_diagonal__Action.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5362 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_dim_lp__Group__Action_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6792 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_equivariant__Hilbert.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6791 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_equivariant__Hilbert.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10166 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_equivariant__Hilbert__Series.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7032 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_finite__Action.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6553 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_generators_lp__Finite__Group__Action_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6507 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_generators_lp__Ring__Of__Invariants_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6798 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_group.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6565 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_group__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10288 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_hilbert__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6898 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_hilbert__Series_lp__Ring__Of__Invariants_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11550 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_hironaka__Decomposition.html │ │ │ --rw-r--r-- 0 root (0) root (0) 23420 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_hsop_spalgorithms.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 23419 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_hsop_spalgorithms.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8819 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_invariant__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11242 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8843 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp..._cm__Degree__Bound_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8291 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp..._cm__Degree__Limit_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6139 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp..._cm__Subring__Limit_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8197 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp..._cm__Use__Coefficient__Ring_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8939 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp..._cm__Use__Linear__Algebra_eq_gt..._rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8940 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp..._cm__Use__Linear__Algebra_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13243 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp__Diagonal__Action_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10068 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp__Finite__Group__Action_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11208 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp__Finite__Group__Action_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10290 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp__Linearly__Reductive__Action_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11099 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp__Linearly__Reductive__Action_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7319 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_is__Abelian.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12644 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_is__Invariant.html │ │ │ @@ -8873,21 +8873,21 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6395 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/InvolutiveBases/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Isomorphism/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Isomorphism/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 24878 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Isomorphism/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Isomorphism/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 823 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Isomorphism/example-output/_check__Degrees.out │ │ │ -rw-r--r-- 0 root (0) root (0) 531 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Isomorphism/example-output/_check__Degrees_lp..._cm__Strict_eq_gt..._rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 53645 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Isomorphism/example-output/_is__Isomorphic.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 53647 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Isomorphism/example-output/_is__Isomorphic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 557 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Isomorphism/example-output/_isomorphism.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Isomorphism/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 49 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Isomorphism/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 9558 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Isomorphism/html/_check__Degrees.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6908 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Isomorphism/html/_check__Degrees_lp..._cm__Strict_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 66341 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Isomorphism/html/_is__Isomorphic.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 66343 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Isomorphism/html/_is__Isomorphic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8559 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Isomorphism/html/_isomorphism.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8412 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Isomorphism/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7777 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Isomorphism/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3610 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Isomorphism/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/JSON/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/JSON/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 17304 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/JSON/dump/rawdocumentation.dump │ │ │ @@ -8983,62 +8983,62 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9828 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Jets/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7299 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Jets/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 101880 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1949 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_all__Gradings.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2783 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_analyze__Strand.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2786 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_analyze__Strand.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5835 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_canonical__Homotopies.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1036 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2262 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Table.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3425 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Tables.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1005 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Det.out │ │ │ --rw-r--r-- 0 root (0) root (0) 269 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_compute__Bound.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2261 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Table.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3426 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Tables.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1001 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Det.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 268 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_compute__Bound.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6823 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_correspondence__Scroll.out │ │ │ -rw-r--r-- 0 root (0) root (0) 804 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_cox__Matrices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1634 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_degenerate__K3.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6875 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_degenerate__K3__Betti__Tables.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6874 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_degenerate__K3__Betti__Tables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2293 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_homotopy__Ranks.out │ │ │ -rw-r--r-- 0 root (0) root (0) 720 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_irrelevant__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 551 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_product__Of__Projective__Spaces.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1031 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_relative__Equations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1075 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_relative__Resolution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1638 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_relative__Resolution__Twists.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2221 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_resonance__Det.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2218 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_resonance__Det.out │ │ │ -rw-r--r-- 0 root (0) root (0) 129 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_resonance__Scroll.out │ │ │ -rw-r--r-- 0 root (0) root (0) 758 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_scheme__In__Product.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3037 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_schreyer__Name.out │ │ │ -rw-r--r-- 0 root (0) root (0) 813 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_small__Diagonal.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 30 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4913 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/___Fine__Grading.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4489 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/___Scrolls.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7927 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_all__Gradings.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9534 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_analyze__Strand.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9537 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_analyze__Strand.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6565 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_canonical__Carpet.html │ │ │ 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./usr/share/doc/Macaulay2/K3Carpets/html/_degenerate__K3__Betti__Tables.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5106 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_gorenstein__Double.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8478 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_hankel__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8111 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_homotopy__Ranks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6177 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_irrelevant__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8020 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_product__Of__Projective__Spaces.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7118 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_relative__Equations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7204 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_relative__Resolution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8344 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_relative__Resolution__Twists.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8321 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_resonance__Det.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8318 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_resonance__Det.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5851 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_resonance__Scroll.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7523 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_scheme__In__Product.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10854 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_schreyer__Name.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6384 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_small__Diagonal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 31911 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 22352 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8781 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/toc.html │ │ │ @@ -9162,27 +9162,27 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1849 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LAYOUT.gz │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LLLBases/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LLLBases/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 33504 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LLLBases/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2858 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/___L__L__L.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2411 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/___L__L__L_lp..._cm__Change__Matrix_eq_gt..._rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1434 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1437 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 476 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/_gcd__L__L__L.out │ │ │ -rw-r--r-- 0 root (0) root (0) 678 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/_is__L__L__L.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 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2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___L__L__L_lp..._cm__Change__Matrix_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 22126 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 22129 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3996 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___N__T__L.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3797 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Real__F__P.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3817 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Real__Q__P.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3999 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Real__Q__P1.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3833 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Real__R__R.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3866 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Real__X__D.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4470 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Threshold.html │ │ │ @@ -9198,15 +9198,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 70564 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 604 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/___Working_spwith_splattice_sppolytopes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 265 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_adjoint__Polytope.out │ │ │ -rw-r--r-- 0 root (0) root (0) 329 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_ambient__Halfspaces.out │ │ │ --rw-r--r-- 0 root (0) root (0) 596 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_are__Isomorphic.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 595 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_are__Isomorphic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 684 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_cayley.out │ │ │ -rw-r--r-- 0 root (0) root (0) 85 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_codegree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 281 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_degree__Of__Jet__Separation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 312 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_epsilon__Bounds.out │ │ │ -rw-r--r-- 0 root (0) root (0) 304 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_gauss__Fiber.out │ │ │ -rw-r--r-- 0 root (0) root (0) 373 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_gauss__Image.out │ │ │ -rw-r--r-- 0 root (0) root (0) 350 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_gaussk__Fiber.out │ │ │ @@ -9223,15 +9223,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 235 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_toric__Div.out │ │ │ -rw-r--r-- 0 root (0) root (0) 167 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_torus__Embedding.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 17 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5005 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/___Working_spwith_splattice_sppolytopes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5659 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_adjoint__Polytope.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5749 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_ambient__Halfspaces.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7712 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_are__Isomorphic.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7711 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_are__Isomorphic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9462 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_cayley.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4957 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_codegree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6907 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_degree__Of__Jet__Separation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6587 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_epsilon__Bounds.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6730 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_gauss__Fiber.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6764 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_gauss__Image.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7032 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LatticePolytopes/html/_gaussk__Fiber.html │ │ │ @@ -9475,30 +9475,30 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 60488 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 790 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/___Linear__Truncations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 231 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_comp__Min.out │ │ │ -rw-r--r-- 0 root (0) root (0) 283 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_diagonal__Multidegrees.out │ │ │ -rw-r--r-- 0 root (0) root (0) 191 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_find__Mins.out │ │ │ --rw-r--r-- 0 root (0) root (0) 701 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_find__Region.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 702 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_find__Region.out │ │ │ -rw-r--r-- 0 root (0) root (0) 754 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_irrelevant__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1046 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_is__Linear__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 904 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_is__Quasi__Linear.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1150 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_linear__Truncations__Bound.out │ │ │ -rw-r--r-- 0 root (0) root (0) 976 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_multigraded__Polynomial__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1807 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_partial__Regularities.out │ │ │ -rw-r--r-- 0 root (0) root (0) 577 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_regularity__Bound.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1734 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_support__Of__Tor.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 604 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 61 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6060 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_comp__Min.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6901 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_diagonal__Multidegrees.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5897 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_find__Mins.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9399 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_find__Region.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9400 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_find__Region.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7158 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_irrelevant__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8139 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_is__Linear__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9029 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_is__Quasi__Linear.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8199 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_linear__Truncations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8381 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_linear__Truncations__Bound.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10456 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_multigraded__Polynomial__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8861 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_partial__Regularities.html │ │ │ @@ -9711,15 +9711,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1999 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Expression.out │ │ │ -rw-r--r-- 0 root (0) root (0) 788 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ext^__Z__Z_lp__Matrix_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1449 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ext^__Z__Z_lp__Module_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 548 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Flat__Monoid.out │ │ │ -rw-r--r-- 0 root (0) root (0) 100 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Function__Closure.out │ │ │ -rw-r--r-- 0 root (0) root (0) 340 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Function_sp_at_at_sp__Function.out │ │ │ -rw-r--r-- 0 root (0) root (0) 836 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Function_sp_us_sp__Thing.out │ │ │ --rw-r--r-- 0 root (0) root (0) 414 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Cstats.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 415 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Cstats.out │ │ │ -rw-r--r-- 0 root (0) root (0) 611 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__F.out │ │ │ -rw-r--r-- 0 root (0) root (0) 194 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Lex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 561 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Rev__Lex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 175 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Gamma.out │ │ │ -rw-r--r-- 0 root (0) root (0) 230 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Global__Assign__Hook.out │ │ │ -rw-r--r-- 0 root (0) root (0) 371 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Global__Release__Hook.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6869 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Grassmannian.out │ │ │ @@ -9965,46 +9965,46 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 128 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_apply_lp__Z__Z_cm__Function_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1347 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_apropos.out │ │ │ -rw-r--r-- 0 root (0) root (0) 180 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_ascii.out │ │ │ -rw-r--r-- 0 root (0) root (0) 112 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_asin.out │ │ │ -rw-r--r-- 0 root (0) root (0) 114 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_asinh.out │ │ │ -rw-r--r-- 0 root (0) root (0) 307 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_assert.out │ │ │ -rw-r--r-- 0 root (0) root (0) 436 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_assigning_spvalues.out │ │ │ --rw-r--r-- 0 root (0) root (0) 235 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_at__End__Of__File_lp__File_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 234 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_at__End__Of__File_lp__File_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 154 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_atan.out │ │ │ -rw-r--r-- 0 root (0) root (0) 245 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_atan2.out │ │ │ -rw-r--r-- 0 root (0) root (0) 207 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_atanh.out │ │ │ -rw-r--r-- 0 root (0) root (0) 624 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_augmented_spassignment.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1258 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_autoload.out │ │ │ -rw-r--r-- 0 root (0) root (0) 137 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_base__Filename.out │ │ │ -rw-r--r-- 0 root (0) root (0) 345 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_base__Name.out │ │ │ -rw-r--r-- 0 root (0) root (0) 418 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_base__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2978 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_basic_sparithmetic_spof_spmatrices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1001 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_basic_spconstruction_cm_spsource_spand_sptarget_spof_spa_spring_spmap.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1118 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_basic_springs_spof_spnumbers.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4584 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_basis.out │ │ │ --rw-r--r-- 0 root (0) root (0) 133 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_benchmark.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 134 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_benchmark.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3673 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_betti.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2269 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_betti_lp__Betti__Tally_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 202 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_between.out │ │ │ -rw-r--r-- 0 root (0) root (0) 332 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_binomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 609 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_block__Matrix__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 223 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_borel_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 725 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_break.out │ │ │ -rw-r--r-- 0 root (0) root (0) 755 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cache.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1307 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_caching_spcomputation_spresults.out │ │ │ --rw-r--r-- 0 root (0) root (0) 591 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cancel__Task_lp__Task_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 592 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cancel__Task_lp__Task_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1803 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_capture.out │ │ │ -rw-r--r-- 0 root (0) root (0) 77 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_ceiling_lp__Number_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 128 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_center__String.out │ │ │ -rw-r--r-- 0 root (0) root (0) 946 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_change__Base.out │ │ │ -rw-r--r-- 0 root (0) root (0) 259 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_change__Directory.out │ │ │ -rw-r--r-- 0 root (0) root (0) 227 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_char.out │ │ │ -rw-r--r-- 0 root (0) root (0) 196 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_characters.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1522 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_check.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1523 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_check.out │ │ │ -rw-r--r-- 0 root (0) root (0) 303 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_class.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1307 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_clean.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2015 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_code.out │ │ │ -rw-r--r-- 0 root (0) root (0) 543 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 241 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 164 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Monomial__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 267 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_column__Swap.out │ │ │ -rw-r--r-- 0 root (0) root (0) 175 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_columnate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1095 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_combine.out │ │ │ -rw-r--r-- 0 root (0) root (0) 198 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_command__Interpreter.out │ │ │ -rw-r--r-- 0 root (0) root (0) 149 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_comments.out │ │ │ -rw-r--r-- 0 root (0) root (0) 331 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_common__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 465 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_commonest.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1535 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_communicating_spwith_spprograms.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1547 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_communicating_spwith_spprograms.out │ │ │ -rw-r--r-- 0 root (0) root (0) 225 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_comodule.out │ │ │ -rw-r--r-- 0 root (0) root (0) 372 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compact__Matrix__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 251 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compare__Exchange.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1085 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compose.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2477 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compositions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 272 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compress.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4287 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_sp__Groebner_spbases.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4286 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_sp__Groebner_spbases.out │ │ │ -rw-r--r-- 0 root (0) root (0) 640 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_spsyzygies.out │ │ │ -rw-r--r-- 0 root (0) root (0) 103 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_concatenate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1767 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_concatenating_spmatrices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 392 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_conditional_spexecution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 148 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_conjugate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 478 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_conjugate_lp__Partition_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 517 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_constructing_spmaps_spbetween_spmodules.out │ │ │ @@ -10061,15 +10061,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 130 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current__Directory.out │ │ │ -rw-r--r-- 0 root (0) root (0) 84 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current__File__Directory.out │ │ │ -rw-r--r-- 0 root (0) root (0) 90 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current__File__Name.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2983 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current__Layout.out │ │ │ -rw-r--r-- 0 root (0) root (0) 193 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current__Package.out │ │ │ -rw-r--r-- 0 root (0) root (0) 180 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current__Position.out │ │ │ -rw-r--r-- 0 root (0) root (0) 81 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current__Row__Number.out │ │ │ --rw-r--r-- 0 root (0) root (0) 330 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current__Time.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 331 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current__Time.out │ │ │ -rw-r--r-- 0 root (0) root (0) 318 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_debug_lp__Package_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 957 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_debug_lp__String_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 691 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_deep__Splice.out │ │ │ -rw-r--r-- 0 root (0) root (0) 201 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_default.out │ │ │ -rw-r--r-- 0 root (0) root (0) 406 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_default__Precision.out │ │ │ -rw-r--r-- 0 root (0) root (0) 443 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_definition_spof_spproduct_sp_lpblock_rp_sporders.out │ │ │ -rw-r--r-- 0 root (0) root (0) 735 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_degree__Group.out │ │ │ @@ -10115,15 +10115,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1720 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eagon__Northcott.out │ │ │ -rw-r--r-- 0 root (0) root (0) 532 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eigenvalues.out │ │ │ -rw-r--r-- 0 root (0) root (0) 717 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eigenvectors.out │ │ │ -rw-r--r-- 0 root (0) root (0) 111 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eint.out │ │ │ -rw-r--r-- 0 root (0) root (0) 103 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elapsed__Time.out │ │ │ -rw-r--r-- 0 root (0) root (0) 162 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elapsed__Timing.out │ │ │ -rw-r--r-- 0 root (0) root (0) 366 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elements.out │ │ │ --rw-r--r-- 0 root (0) root (0) 21211 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elimination_spof_spvariables.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 21210 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elimination_spof_spvariables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 782 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_end.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3683 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_end__Package.out │ │ │ -rw-r--r-- 0 root (0) root (0) 487 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_entries.out │ │ │ -rw-r--r-- 0 root (0) root (0) 185 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_entries_lp__Vector_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 605 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_equality_spand_spcontainment.out │ │ │ -rw-r--r-- 0 root (0) root (0) 111 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_erf.out │ │ │ -rw-r--r-- 0 root (0) root (0) 114 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_erfc.out │ │ │ @@ -10264,15 +10264,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1260 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_induced__Map_lp__Module_cm__Module_cm__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 609 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_induced__Map_lp__Module_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1035 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_inheritance.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1187 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_inputting_spa_spmatrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 592 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_insert.out │ │ │ -rw-r--r-- 0 root (0) root (0) 686 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_installing_spassignment_spmethods.out │ │ │ -rw-r--r-- 0 root (0) root (0) 936 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_installing_spaugmented_spassignment_spmethods.out │ │ │ --rw-r--r-- 0 root (0) root (0) 932 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_instances.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 933 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_instances.out │ │ │ -rw-r--r-- 0 root (0) root (0) 316 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_integers_spmodulo_spa_spprime.out │ │ │ -rw-r--r-- 0 root (0) root (0) 315 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_integrate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1162 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect.out │ │ │ -rw-r--r-- 0 root (0) root (0) 997 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect_lp__Ideal_cm__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 189 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect_lp__Set_cm__Set_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 190 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersection_spof_spideals.out │ │ │ -rw-r--r-- 0 root (0) root (0) 188 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_inverse__Erf.out │ │ │ @@ -10404,18 +10404,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 769 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 424 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Mutable__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 125 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Ring__Element_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 345 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Ring__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 319 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Ring_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 228 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Vector_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 560 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max.out │ │ │ --rw-r--r-- 0 root (0) root (0) 82 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Allowable__Threads.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 83 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Allowable__Threads.out │ │ │ -rw-r--r-- 0 root (0) root (0) 83 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Exponent.out │ │ │ -rw-r--r-- 0 root (0) root (0) 311 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Position.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1637 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_memoize.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1638 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_memoize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1730 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_merge.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2673 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_method.out │ │ │ -rw-r--r-- 0 root (0) root (0) 970 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_method__Options_lp__Function_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6748 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_methods.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2721 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_methods_spfor_spnormal_spforms_spand_spremainder.out │ │ │ -rw-r--r-- 0 root (0) root (0) 551 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_min.out │ │ │ -rw-r--r-- 0 root (0) root (0) 84 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_min__Exponent.out │ │ │ @@ -10501,16 +10501,16 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 217 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_override.out │ │ │ -rw-r--r-- 0 root (0) root (0) 681 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pack.out │ │ │ -rw-r--r-- 0 root (0) root (0) 144 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_package.out │ │ │ -rw-r--r-- 0 root (0) root (0) 187 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_packages.out │ │ │ -rw-r--r-- 0 root (0) root (0) 498 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_packing_spmonomials_spfor_spefficiency.out │ │ │ -rw-r--r-- 0 root (0) root (0) 131 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pad.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1059 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pairs.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1650 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallel_spprogramming_spwith_spthreads_spand_sptasks.out │ │ │ --rw-r--r-- 0 root (0) root (0) 8669 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallelism_spin_spengine_spcomputations.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1651 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallel_spprogramming_spwith_spthreads_spand_sptasks.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 8670 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallelism_spin_spengine_spcomputations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 317 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parsing_spprecedence_cm_spin_spdetail.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3030 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_part.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1297 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_partition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1097 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_partitions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 652 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parts.out │ │ │ -rw-r--r-- 0 root (0) root (0) 277 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_path.out │ │ │ -rw-r--r-- 0 root (0) root (0) 535 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pdim_lp__Module_rp.out │ │ │ @@ -10518,15 +10518,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 232 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_peek_sq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 267 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_permanents.out │ │ │ -rw-r--r-- 0 root (0) root (0) 874 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_permutations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 495 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pfaffian.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2202 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pfaffians.out │ │ │ -rw-r--r-- 0 root (0) root (0) 301 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pi.out │ │ │ -rw-r--r-- 0 root (0) root (0) 304 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pivots_lp__Matrix_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 13708 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_poincare.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 13706 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_poincare.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2331 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_polarize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4055 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_polynomial_springs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 624 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_position.out │ │ │ -rw-r--r-- 0 root (0) root (0) 475 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_positions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 274 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_powermod.out │ │ │ -rw-r--r-- 0 root (0) root (0) 173 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_precision.out │ │ │ -rw-r--r-- 0 root (0) root (0) 657 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_prefix__Path.out │ │ │ @@ -10560,15 +10560,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 3534 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_push__Forward_lp__Ring__Map_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 796 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient__Remainder.out │ │ │ -rw-r--r-- 0 root (0) root (0) 332 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient__Remainder_lp__Ring__Element_cm__Ring__Element_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 825 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient__Remainder_sq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3975 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient_lp__Matrix_cm__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1476 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient_springs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1405 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient_sq.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1250 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random__K__Rational__Point.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1251 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random__K__Rational__Point.out │ │ │ -rw-r--r-- 0 root (0) root (0) 669 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random__Mutable__Matrix_lp__Z__Z_cm__Z__Z_cm__R__R_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 399 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random__Subset.out │ │ │ -rw-r--r-- 0 root (0) root (0) 843 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random_lp__List_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 138 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1109 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random_lp__Module_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 232 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random_lp__Q__Q_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 869 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random_lp__Type_rp.out │ │ │ @@ -10655,15 +10655,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 418 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_show__User__Structure.out │ │ │ -rw-r--r-- 0 root (0) root (0) 175 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_sign.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2452 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_simple_sp__Groebner_spbasis_spcomputations_spover_spvarious_springs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 100 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_sin.out │ │ │ -rw-r--r-- 0 root (0) root (0) 933 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_singular__Locus.out │ │ │ -rw-r--r-- 0 root (0) root (0) 294 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_size2.out │ │ │ 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./usr/share/doc/Macaulay2/Macaulay2Doc/html/___G__Rev__Lex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7031 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Galois__Field.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6301 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Gamma.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4965 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___General__Ordered__Monoid.html │ │ │ @@ -11515,15 +11515,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4087 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_argument.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5900 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_ascii.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5036 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_asin.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4592 2025-12-14 14:09:53.000000 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./usr/share/doc/Macaulay2/Macaulay2Doc/html/_borel_lp__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7894 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_break.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5393 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cache.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7058 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_caching_spcomputation_spresults.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7156 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cancel__Task_lp__Task_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7157 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cancel__Task_lp__Task_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9947 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_capture.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3525 2025-12-14 14:09:53.000000 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./usr/share/doc/Macaulay2/MonodromySolver/example-output/_dynamic__Flower__Solve.out │ │ │ -rw-r--r-- 0 root (0) root (0) 331 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_flower__Graph__Augment.out │ │ │ -rw-r--r-- 0 root (0) root (0) 242 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_flower__Graph__Init.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9570 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_monodromy__Group.out │ │ │ -rw-r--r-- 0 root (0) root (0) 427 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_monodromy__Solve_lp__System_cm__Abstract__Point_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1395 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_monodromy__Solve_lp__System_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 956 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_potential__E.out │ │ │ -rw-r--r-- 0 root (0) root (0) 442 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_potential__Lower__Bound.out │ │ │ @@ -13609,15 +13609,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5869 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/___Point__Array.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4500 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_append__Point.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4552 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_append__Points.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4875 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_complete__Graph__Augment.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4697 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_complete__Graph__Init.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5190 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_compute__Mixed__Volume.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7814 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_create__Seed__Pair.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7624 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_dynamic__Flower__Solve.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7623 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_dynamic__Flower__Solve.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5009 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_flower__Graph__Augment.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4670 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_flower__Graph__Init.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4889 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_get__Track__Time.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6199 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_homotopy__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4606 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_indices_lp__Point__Array_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4847 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_is__Member_lp__Abstract__Point_cm__Point__Array_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4736 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_length_lp__Point__Array_rp.html │ │ │ @@ -13855,28 +13855,28 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 19109 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedBGG/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12849 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedBGG/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7429 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedBGG/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 41382 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 2242 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_components__Of__Kernel.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2241 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_components__Of__Kernel.out │ │ │ -rw-r--r-- 0 root (0) root (0) 719 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_compute__Component.out │ │ │ -rw-r--r-- 0 root (0) root (0) 723 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_interpolate__Component.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1000 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_max__Grading.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1009 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_trim__Basis__In__Degree.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 52 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5318 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Coefficient__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4235 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Grading.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4491 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Previous__Gens.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4750 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Return__Target__Grading.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5012 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Use__Interpolation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4647 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Use__Matroid.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11856 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_components__Of__Kernel.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11855 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_components__Of__Kernel.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6602 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_components__Of__Kernel_lp..._cm__Verbose_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9224 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_compute__Component.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10098 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_interpolate__Component.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8008 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_max__Grading.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9470 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_trim__Basis__In__Degree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14051 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13133 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/master.html │ │ │ @@ -13885,27 +13885,27 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 35822 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 369 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/___N__P.out │ │ │ -rw-r--r-- 0 root (0) root (0) 656 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_gr__Gr.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1249 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_hilbert__Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 413 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_j__Mult.out │ │ │ --rw-r--r-- 0 root (0) root (0) 261 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_mon__Analytic__Spread.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 262 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_mon__Analytic__Spread.out │ │ │ -rw-r--r-- 0 root (0) root (0) 573 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_mon__Reduction.out │ │ │ -rw-r--r-- 0 root (0) root (0) 596 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_monj__Mult.out │ │ │ -rw-r--r-- 0 root (0) root (0) 805 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_multiplicity__Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1239 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_print__Hilbert__Sequence.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 594 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 47 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6190 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/___N__P.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6706 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_gr__Gr.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8885 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_hilbert__Sequence.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6040 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_j__Mult.html │ │ │ --rw-r--r-- 0 root (0) root (0) 5705 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_mon__Analytic__Spread.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5706 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_mon__Analytic__Spread.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6831 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_mon__Reduction.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6478 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_monj__Mult.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9585 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_multiplicity__Sequence.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6921 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_print__Hilbert__Sequence.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15265 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9576 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4371 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/toc.html │ │ │ @@ -13958,15 +13958,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5192 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiplierIdealsDim2/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 379736 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 279 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Embedded__Projective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1954 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Embedded__Projective__Variety_sp!.out │ │ │ --rw-r--r-- 0 root (0) root (0) 20227 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Embedded__Projective__Variety_sp_eq_eq_eq_gt_sp__Embedded__Projective__Variety.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 20228 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Embedded__Projective__Variety_sp_eq_eq_eq_gt_sp__Embedded__Projective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 670 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Embedded__Projective__Variety_sp_pl_pl_sp__Embedded__Projective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1106 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Fano_lp__Z__Z_cm__Embedded__Projective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 374 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___G__G.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5148 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___G__G_lp__Z__Z_cm__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1057 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Hom_lp__Multiprojective__Variety_cm__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 329 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp^_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 951 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_bs_bs_sp__Multiprojective__Variety.out │ │ │ @@ -13974,15 +13974,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 507 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_pc_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 399 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_pl_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 447 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_st_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 663 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_st_st_sp__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1341 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp__Multiprojective__Variety.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1275 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1200 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_st_sp__Multirational__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1848 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_st_st_sp__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 772 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_vb_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 803 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_vb_sp__Multirational__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 763 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_vb_vb_sp__Multirational__Map.out │ │ │ @@ -14002,41 +14002,41 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 3115 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_cone__Of__Lines.out │ │ │ -rw-r--r-- 0 root (0) root (0) 350 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_conormal__Variety_lp__Embedded__Projective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 575 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_cycle__Class.out │ │ │ -rw-r--r-- 0 root (0) root (0) 671 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_decompose_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 324 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree__Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 158 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1006 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multirational__Map_cm__Option_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 427 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 428 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 426 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degrees_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 722 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_describe_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2265 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_describe_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 155 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_dim_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 482 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_dual_lp__Embedded__Projective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 809 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_entries_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 239 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_euler_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2030 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_factor_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1459 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_fiber__Product.out │ │ │ -rw-r--r-- 0 root (0) root (0) 673 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_force__Image_lp__Multirational__Map_cm__Multiprojective__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1857 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_graph_lp__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1856 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_graph_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 324 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_hilbert__Polynomial_lp__Embedded__Projective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 384 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_ideal_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1016 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_image_lp__Multirational__Map_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 801 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_inverse2.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1406 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_inverse_lp__Multirational__Map_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 932 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Isomorphism_lp__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 802 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_inverse2.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1407 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_inverse_lp__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 931 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Isomorphism_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 386 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Member_lp__Multirational__Map_cm__R__A__T_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 753 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Morphism_lp__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 754 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Morphism_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 249 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Subset_lp__Multiprojective__Variety_cm__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 678 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Well__Defined_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 790 2025-12-14 14:09:53.000000 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root (0) 533 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multidegree_lp__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 661 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multidegree_lp__Z__Z_cm__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2393 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multirational__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1182 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multirational__Map_lp__Multiprojective__Variety_cm__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 251 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multirational__Map_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 363 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multirational__Map_lp__Multirational__Map_cm__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2768 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_parametrize_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 490 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_permute_lp__Multiprojective__Variety_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1238 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_point_lp__Multiprojective__Variety_rp.out │ │ │ @@ -14051,15 +14051,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 2154 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_random_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 582 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_rational__Map_lp__Multiprojective__Variety_cm__Tally_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 440 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_ring_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 328 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_schubert__Cycle.out │ │ │ -rw-r--r-- 0 root (0) root (0) 151 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_sectional__Genus.out │ │ │ -rw-r--r-- 0 root (0) root (0) 468 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_segre__Embedding.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6206 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_segre_lp__Multiprojective__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1051 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_segre_lp__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1050 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_segre_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 288 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_shape_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 813 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_shortcuts.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3242 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_show_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 512 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_singular__Locus_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 497 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_super_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 352 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_support_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 590 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_tangent__Cone_lp__Embedded__Projective__Variety_cm__Embedded__Projective__Variety_rp.out │ │ │ @@ -14073,15 +14073,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 631 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_⋂.out │ │ │ -rw-r--r-- 0 root (0) root (0) 443 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_⋃.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 641 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 50 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 14090 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Embedded__Projective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8119 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Embedded__Projective__Variety_sp!.html │ │ │ --rw-r--r-- 0 root (0) root (0) 29042 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Embedded__Projective__Variety_sp_eq_eq_eq_gt_sp__Embedded__Projective__Variety.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 29043 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Embedded__Projective__Variety_sp_eq_eq_eq_gt_sp__Embedded__Projective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6724 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Embedded__Projective__Variety_sp_pl_pl_sp__Embedded__Projective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9902 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Fano_lp__Z__Z_cm__Embedded__Projective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6913 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___G__G.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11424 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___G__G_lp__Z__Z_cm__Multirational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6539 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Grassmannian__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9178 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Hom_lp__Multiprojective__Variety_cm__Multiprojective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 24980 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multiprojective__Variety.html │ │ │ @@ -14092,15 +14092,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6488 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multiprojective__Variety_sp_pc_sp__Multiprojective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7160 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multiprojective__Variety_sp_pl_sp__Multiprojective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6880 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multiprojective__Variety_sp_st_sp__Multiprojective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6950 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multiprojective__Variety_sp_st_st_sp__Multiprojective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7756 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multiprojective__Variety_sp_st_st_sp__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 23929 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8185 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp^_st_st_sp__Multiprojective__Variety.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8286 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp__Multiprojective__Variety.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8284 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp__Multiprojective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5781 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp_eq_eq_sp__Multirational__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5662 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp_lt_eq_eq_gt_sp__Multirational__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8285 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp_lt_lt_sp__Multiprojective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6810 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp_st_sp__Multirational__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8946 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp_st_st_sp__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7803 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp_vb_sp__Multiprojective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8771 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp_vb_sp__Multirational__Map.html │ │ │ @@ -14125,41 +14125,41 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9636 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_cone__Of__Lines.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6858 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_conormal__Variety_lp__Embedded__Projective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7109 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_cycle__Class.html │ │ │ -rw-r--r-- 0 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14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_string__To__Graph.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 575 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 18 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5988 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/___Comparison_spof_sp__Graph6_spand_sp__Sparse6_spformats.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5714 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/___Example_co_sp__Checking_spfor_spisomorphic_spgraphs.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8522 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8521 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10316 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_add__Edges.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6991 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_are__Isomorphic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14195 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_build__Graph__Filter.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7805 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_count__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8103 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_filter__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13715 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_generate__Bipartite__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14088 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_generate__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9658 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_generate__Random__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8443 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_generate__Random__Regular__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6502 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_graph6__To__Sparse6.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8741 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_graph__Complement.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8742 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_graph__Complement.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8943 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_graph__To__String.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5998 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_is__Planar.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8315 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_neighborhood__Complements.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7139 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_new__Edges.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6373 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_only__Planar.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7363 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_relabel__Bipartite.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8940 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Nauty/html/_relabel__Graph.html │ │ │ @@ -14629,15 +14629,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1404 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 826 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_add__Edges.out │ │ │ -rw-r--r-- 0 root (0) root (0) 388 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_are__Isomorphic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1498 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_build__Graph__Filter.out │ │ │ -rw-r--r-- 0 root (0) root (0) 184 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 326 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Random__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 124 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Random__Regular__Graphs.out │ │ │ --rw-r--r-- 0 root (0) root (0) 497 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_graph__Complement.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 498 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_graph__Complement.out │ │ │ -rw-r--r-- 0 root (0) root (0) 323 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_graph__To__String.out │ │ │ -rw-r--r-- 0 root (0) root (0) 217 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_is__Planar.out │ │ │ -rw-r--r-- 0 root (0) root (0) 849 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_neighborhood__Complements.out │ │ │ -rw-r--r-- 0 root (0) root (0) 241 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_new__Edges.out │ │ │ -rw-r--r-- 0 root (0) root (0) 247 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_only__Planar.out │ │ │ -rw-r--r-- 0 root (0) root (0) 322 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_relabel__Bipartite.out │ │ │ -rw-r--r-- 0 root (0) root (0) 381 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_relabel__Graph.out │ │ │ @@ -14655,15 +14655,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 7800 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_count__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8080 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_filter__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9132 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Bipartite__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9062 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8413 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Random__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6700 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Random__Regular__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6470 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_graph6__To__Sparse6.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7731 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_graph__Complement.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7732 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_graph__Complement.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8914 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_graph__To__String.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5929 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_is__Planar.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7363 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_neighborhood__Complements.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6390 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_new__Edges.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6301 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_only__Planar.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6955 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_relabel__Bipartite.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9029 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_relabel__Graph.html │ │ │ @@ -14691,15 +14691,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 179468 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 431 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Dependent__Set.out │ │ │ -rw-r--r-- 0 root (0) root (0) 791 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Diff__Op.out │ │ │ -rw-r--r-- 0 root (0) root (0) 553 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Diff__Op_sp__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 416 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Strategy_sp_eq_gt_sp_dq__Hybrid_dq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 498 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Strategy_sp_eq_gt_sp_dq__Macaulay__Matrix_dq.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2349 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2350 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 614 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_amult.out │ │ │ -rw-r--r-- 0 root (0) root (0) 931 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_coordinate__Change__Ops.out │ │ │ -rw-r--r-- 0 root (0) root (0) 354 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_diff__Op__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 403 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_diff__Op_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 667 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_differential__Primary__Decomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 349 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_eliminating__Dual.out │ │ │ -rw-r--r-- 0 root (0) root (0) 270 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_evaluate_lp__Diff__Op_cm__Abstract__Point_rp.out │ │ │ @@ -14732,15 +14732,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 75 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 8524 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Dependent__Set.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9032 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Diff__Op.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7316 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Diff__Op_sp__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5026 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Sampler.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7406 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Hybrid_dq.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7805 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Macaulay__Matrix_dq.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7814 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7815 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4027 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Tolerance_sp_lp__Noetherian__Operators_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6738 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_amult.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5314 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_colon.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8444 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_coordinate__Change__Ops.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4120 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_diff__Op.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7027 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_diff__Op__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6518 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_diff__Op_lp__Matrix_rp.html │ │ │ @@ -14829,15 +14829,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8147 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NonminimalComplexes/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5466 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NonminimalComplexes/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3915 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NonminimalComplexes/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 608995 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 3509 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Chow_spring.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3510 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Chow_spring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3270 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___H__H^__Z__Z_lp__Normal__Toric__Variety_cm__Coherent__Sheaf_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1905 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp^_sp__Array.out │ │ │ -rw-r--r-- 0 root (0) root (0) 908 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp^_st_st_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 840 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp_st_st_sp__Normal__Toric__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1922 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp_us_sp__Array.out │ │ │ -rw-r--r-- 0 root (0) root (0) 708 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp_us_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1977 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___O__O_sp__Toric__Divisor.out │ │ │ @@ -14901,20 +14901,20 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1271 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_lattice__Points_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1229 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_make__Simplicial_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1568 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_make__Smooth_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 601 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_map_lp__Normal__Toric__Variety_cm__Normal__Toric__Variety_cm__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1011 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_map_lp__Normal__Toric__Variety_cm__Normal__Toric__Variety_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 912 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_matrix_lp__Toric__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 543 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_max_lp__Normal__Toric__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1560 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_monomials_lp__Toric__Divisor_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1561 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_monomials_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2228 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_nef__Generators_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1002 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Fan_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3171 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__List_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1693 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Matrix_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1964 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Polyhedron_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1965 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Polyhedron_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 704 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1599 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_orbits_lp__Normal__Toric__Variety_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1640 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_orbits_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1605 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_picard__Group_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1337 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_picard__Group_lp__Toric__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1807 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_polytope_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2219 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_projective_spspace.out │ │ │ @@ -14939,15 +14939,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 881 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_vector_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1765 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_vertices_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1650 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_weighted__Projective__Space_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 452 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_weil__Divisor__Group_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 985 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_weil__Divisor__Group_lp__Toric__Map_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 68 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 12847 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Chow_spring.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12848 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Chow_spring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11949 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___H__H^__Z__Z_lp__Normal__Toric__Variety_cm__Coherent__Sheaf_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 27601 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14079 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety_sp^_sp__Array.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9111 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety_sp^_st_st_sp__Z__Z.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9410 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety_sp_st_st_sp__Normal__Toric__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13026 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety_sp_us_sp__Array.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8752 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety_sp_us_sp__Z__Z.html │ │ │ @@ -15016,20 +15016,20 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10561 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_make__Simplicial_lp__Normal__Toric__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13649 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_make__Smooth_lp__Normal__Toric__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9577 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_making_spnormal_sptoric_spvarieties.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11442 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_map_lp__Normal__Toric__Variety_cm__Normal__Toric__Variety_cm__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12498 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_map_lp__Normal__Toric__Variety_cm__Normal__Toric__Variety_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11093 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_matrix_lp__Toric__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9104 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_max_lp__Normal__Toric__Variety_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10405 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_monomials_lp__Toric__Divisor_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10406 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_monomials_lp__Toric__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10956 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_nef__Generators_lp__Normal__Toric__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10436 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Fan_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19155 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__List_cm__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13815 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Matrix_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 13553 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Polyhedron_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 13554 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Polyhedron_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9792 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12084 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_orbits_lp__Normal__Toric__Variety_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10736 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_orbits_lp__Normal__Toric__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12153 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_picard__Group_lp__Normal__Toric__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10563 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_picard__Group_lp__Toric__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10843 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_polytope_lp__Toric__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11081 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_projective_spspace.html │ │ │ @@ -15337,40 +15337,40 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 40489 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalCertification/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 24404 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalCertification/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6254 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalCertification/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 146200 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 1059 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/___Convert__To__Cone.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1058 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/___Convert__To__Cone.out │ │ │ -rw-r--r-- 0 root (0) root (0) 451 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/___Numerical__Interpolation__Table.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1473 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/___Pseudo__Witness__Set.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1517 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_extract__Image__Equations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 443 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_is__On__Image.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1223 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Hilbert__Function.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1222 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Hilbert__Function.out │ │ │ -rw-r--r-- 0 root (0) root (0) 240 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Image__Degree.out │ │ │ --rw-r--r-- 0 root (0) root (0) 575 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Image__Dim.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 576 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Image__Dim.out │ │ │ -rw-r--r-- 0 root (0) root (0) 712 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Image__Sample.out │ │ │ -rw-r--r-- 0 root (0) root (0) 184 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Nullity.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1569 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Source__Sample.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1201 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_pseudo__Witness__Set.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1181 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_real__Point.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 594 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 43 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 7153 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/___Convert__To__Cone.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7152 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/___Convert__To__Cone.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6034 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/___Max__Threads.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8844 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/___Numerical__Interpolation__Table.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11290 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/___Pseudo__Witness__Set.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11363 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_extract__Image__Equations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9536 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_is__On__Image.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12837 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Hilbert__Function.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12836 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Hilbert__Function.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10681 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Image__Degree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8623 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Image__Degree_lp..._cm__Verbose_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9337 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Image__Dim.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9338 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Image__Dim.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9832 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Image__Sample.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8015 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Nullity.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12322 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Source__Sample.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18773 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_pseudo__Witness__Set.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10900 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_pseudo__Witness__Set_lp..._cm__Software_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8802 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_real__Point.html │ │ │ -rw-r--r-- 0 root (0) root (0) 34695 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/index.html │ │ │ @@ -15409,15 +15409,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 105 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalSchubertCalculus/example-output/___Pieri__Root__Count.out │ │ │ -rw-r--r-- 0 root (0) root (0) 279 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalSchubertCalculus/example-output/_bracket2partition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3900 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalSchubertCalculus/example-output/_change__Flags.out │ │ │ -rw-r--r-- 0 root (0) root (0) 17148 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalSchubertCalculus/example-output/_parse__Triplet.out │ │ │ -rw-r--r-- 0 root (0) root (0) 237 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalSchubertCalculus/example-output/_partition2bracket.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1565 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalSchubertCalculus/example-output/_print__Statistics.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4092 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/NumericalSchubertCalculus/example-output/_random__Schubert__Problem__Instance.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6402 2025-12-14 14:09:53.000000 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./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_betti_lp__Graded__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 518 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_chain__Complex_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 358 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_chain__Complex_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 832 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_chain__Complex_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 283 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_complete_lp__Chain__Complex_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1284 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_computing_spresolutions.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1283 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_computing_spresolutions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1250 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_cone_lp__Chain__Complex__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 621 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_dd.out │ │ │ -rw-r--r-- 0 root (0) root (0) 536 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_dual_lp__Chain__Complex__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1335 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_extend_lp__Chain__Complex_cm__Chain__Complex_cm__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1130 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_extracting_spinformation_spfrom_spchain_spcomplexes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1095 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_free_spresolutions_spof_spmodules.out │ │ │ -rw-r--r-- 0 root (0) root (0) 338 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_graded__Module__Map.out │ │ │ @@ -15790,15 +15790,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4997 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_chain__Complex_lp__Graded__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6585 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_chain__Complex_lp__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5865 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_chain__Complex_lp__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6228 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_chain__Complex_lp__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5591 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_complete_lp__Chain__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4290 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_complete_lp__Graded__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4214 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_components_lp__Chain__Complex_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7475 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_computing_spresolutions.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7474 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_computing_spresolutions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7151 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_cone_lp__Chain__Complex__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5466 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_dd.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5785 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_dual_lp__Chain__Complex__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3841 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_dual_lp__Chain__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8237 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_extend_lp__Chain__Complex_cm__Chain__Complex_cm__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7530 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_extracting_spinformation_spfrom_spchain_spcomplexes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5900 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_free_spresolutions_spof_spmodules.html │ │ │ @@ -16254,22 +16254,22 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6360 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OpenMath/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4999 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OpenMath/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3143 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/OpenMath/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 84707 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 17045 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 17049 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1080 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.1_co_spunique_spgraph_spon_sp8_spvertices_spwith_spexotic_spsolutions_spand_spno_spinduced_spcycle_spof_splength_spat_spleast_sp5.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2543 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.2_co_spa_sp__K5_spand_sppentagon_spglued_spalong_span_spedge.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2540 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.2_co_spa_sp__K5_spand_sppentagon_spglued_spalong_span_spedge.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2081 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.3_co_spexamples_spof_spgluing_sptwo_spcycles_spalong_span_spedge.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5401 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.4_co_sp__The_spsquare_spwithin_spa_spsquare.out │ │ │ -rw-r--r-- 0 root (0) root (0) 997 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Generation_spof_spall_sp__S__C__T_sp_lpsimple_cm_spconnected_cm_sp2-connected_rp_spgraphs_spon_spsmall_spnumbers_spof_spvertices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 16187 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Oscillators.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3811 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___S__C__T_spgraphs_spwith_spexotic_spsolutions.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3810 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___S__C__T_spgraphs_spwith_spexotic_spsolutions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1682 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_all__Unique__Principal__Minors.out │ │ │ -rw-r--r-- 0 root (0) root (0) 10260 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_find__Real__Solutions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 151 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_get__Angles.out │ │ │ -rw-r--r-- 0 root (0) root (0) 429 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_get__Linearly__Stable__Solutions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2330 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_identify__Stability.out │ │ │ -rw-r--r-- 0 root (0) root (0) 459 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_is__Stable__Solution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 10664 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_osc__Jacobian.out │ │ │ @@ -16277,22 +16277,22 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 625 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_osc__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 23259 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_osc__System.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2051 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_show__Exotic__Solutions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 241 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_standard__Sols.out │ │ │ -rw-r--r-- 0 root (0) root (0) 236 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_vertex__Spanning__Polynomial.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 33 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 25404 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 25408 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6234 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Example_sp4.1_co_spunique_spgraph_spon_sp8_spvertices_spwith_spexotic_spsolutions_spand_spno_spinduced_spcycle_spof_splength_spat_spleast_sp5.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7736 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Example_sp4.2_co_spa_sp__K5_spand_sppentagon_spglued_spalong_span_spedge.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7733 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Example_sp4.2_co_spa_sp__K5_spand_sppentagon_spglued_spalong_span_spedge.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7644 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Example_sp4.3_co_spexamples_spof_spgluing_sptwo_spcycles_spalong_span_spedge.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10645 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Example_sp4.4_co_sp__The_spsquare_spwithin_spa_spsquare.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6691 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Generation_spof_spall_sp__S__C__T_sp_lpsimple_cm_spconnected_cm_sp2-connected_rp_spgraphs_spon_spsmall_spnumbers_spof_spvertices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3563 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Harrington-__Schenck-__Stillman.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8190 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___S__C__T_spgraphs_spwith_spexotic_spsolutions.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8189 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___S__C__T_spgraphs_spwith_spexotic_spsolutions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8807 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_all__Unique__Principal__Minors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19710 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_find__Real__Solutions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6864 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_get__Angles.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6738 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_get__Linearly__Stable__Solutions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10573 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_identify__Stability.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7493 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_is__Stable__Solution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20003 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_osc__Jacobian.html │ │ │ @@ -16524,15 +16524,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 12104 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Parsing/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9154 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Parsing/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7244 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Parsing/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 136324 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 2842 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___A_spfamily_spof_sppaths_spon_spa_spcone.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2840 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___A_spfamily_spof_sppaths_spon_spa_spcone.out │ │ │ -rw-r--r-- 0 root (0) root (0) 456 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___Array_sp_us_sp__N__C__Polynomial__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 863 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___C__Axis__Tensor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1364 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___C__Mon__Tensor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2634 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___Path.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1530 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___Path__Signatures.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1392 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___Polynomial_sppaths_spof_spdegree_spm.out │ │ │ -rw-r--r-- 0 root (0) root (0) 11000 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___The_spuniversal_spvariety_spand_sptoric_spcoordinates.out │ │ │ @@ -16558,15 +16558,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 204 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/_tensor__Log.out │ │ │ -rw-r--r-- 0 root (0) root (0) 886 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/_tensor__Parametrization.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2023 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/_word__Algebra.out │ │ │ -rw-r--r-- 0 root (0) root (0) 387 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/_word__Format.out │ │ │ -rw-r--r-- 0 root (0) root (0) 215 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/_word__String.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 38 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 11402 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___A_spfamily_spof_sppaths_spon_spa_spcone.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11400 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___A_spfamily_spof_sppaths_spon_spa_spcone.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7073 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___Array_sp_us_sp__N__C__Polynomial__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7959 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___C__Axis__Tensor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8285 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___C__Mon__Tensor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3941 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___Computing_sp__Path_sp__Varieties.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7451 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___N__C__Polynomial__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21351 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___N__C__Ring__Element.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15444 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___Path.html │ │ │ @@ -16962,15 +16962,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 14765 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PlaneCurveLinearSeries/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12202 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PlaneCurveLinearSeries/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4887 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PlaneCurveLinearSeries/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Points/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Points/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 47371 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Points/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Points/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 1730 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Points/example-output/_affine__Fat__Points.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1729 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Points/example-output/_affine__Fat__Points.out │ │ │ -rw-r--r-- 0 root (0) root (0) 567 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Points/example-output/_affine__Fat__Points__By__Intersection.out │ │ │ -rw-r--r-- 0 root (0) root (0) 619 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Points/example-output/_affine__Make__Ring__Maps.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1456 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Points/example-output/_affine__Points.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1010 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Points/example-output/_affine__Points__By__Intersection.out │ │ │ -rw-r--r-- 0 root (0) root (0) 463 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Points/example-output/_affine__Points__Mat.out │ │ │ -rw-r--r-- 0 root (0) root (0) 316 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Points/example-output/_expected__Betti.out │ │ │ -rw-r--r-- 0 root (0) root (0) 234 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Points/example-output/_min__Max__Resolution.out │ │ │ @@ -16982,15 +16982,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 307 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Points/example-output/_projective__Points__By__Intersection.out │ │ │ -rw-r--r-- 0 root (0) root (0) 333 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Points/example-output/_random__Points.out │ │ │ -rw-r--r-- 0 root (0) root (0) 651 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Points/example-output/_random__Points__Mat.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Points/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 14 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Points/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4517 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Points/html/___All__Random.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4654 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Points/html/___Verify__Points.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10167 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Points/html/_affine__Fat__Points.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10166 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Points/html/_affine__Fat__Points.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7047 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Points/html/_affine__Fat__Points__By__Intersection.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6451 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Points/html/_affine__Make__Ring__Maps.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8176 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Points/html/_affine__Points.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7009 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Points/html/_affine__Points__By__Intersection.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6970 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Points/html/_affine__Points__Mat.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6070 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Points/html/_expected__Betti.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5252 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Points/html/_min__Max__Resolution.html │ │ │ @@ -17699,15 +17699,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 411 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_irreducible__Decomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 280 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_is__Primary.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2223 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_kernel__Of__Localization.out │ │ │ -rw-r--r-- 0 root (0) root (0) 627 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_localize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1734 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_primary__Decomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7407 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_primary__Decomposition_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 765 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_primary_spdecomposition.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1389 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_reg__Seq__In__Ideal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1390 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_reg__Seq__In__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 423 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_remove__Lowest__Dimension.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2249 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_strategies_spfor_spcomputing_spprimary_spdecomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 242 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_top__Components.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 52 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 17690 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_associated__Primes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4670 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_associated_spprimes.html │ │ │ @@ -17715,15 +17715,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8232 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_is__Primary.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8733 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_kernel__Of__Localization.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10172 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_localize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8209 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_primary__Component.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11813 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_primary__Decomposition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21071 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_primary__Decomposition_lp__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5735 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_primary_spdecomposition.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9885 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_reg__Seq__In__Ideal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9886 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_reg__Seq__In__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7321 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_remove__Lowest__Dimension.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10490 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_strategies_spfor_spcomputing_spprimary_spdecomposition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7361 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_top__Components.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20841 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13241 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5480 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Probability/ │ │ │ @@ -18142,15 +18142,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1962 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Example_sp__Type_sp[300b].out │ │ │ -rw-r--r-- 0 root (0) root (0) 2633 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Example_sp__Type_sp[300c].out │ │ │ -rw-r--r-- 0 root (0) root (0) 2039 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Finding_spall_sppossible_spbetti_sptables_spfor_spquadratic_spcomponent_spof_spinverse_spsystem_spfor_spquartics_spin_sp4_spvariables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9657 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Finding_spthe_sp16_spbetti_sptables_sppossible_spfor_spquartic_spforms_spin_sp4_spvariables_cm_spand_spexamples.out │ │ │ -rw-r--r-- 0 root (0) root (0) 13630 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Finding_spthe_sp__Betti_spstratum_spof_spa_spgiven_spquartic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3258 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Finding_spthe_sppossible_spbetti_sptables_spfor_sppoints_spin_sp__P^3_spwith_spgiven_spgeometry.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1713 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Half_spcanonical_spdegree_sp20.out │ │ │ --rw-r--r-- 0 root (0) root (0) 28147 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 22546 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2741 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Noether-__Lefschetz_spexamples.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7051 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Pfaffians_spon_spquadrics.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1233 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Singularities_spof_splifting_spof_sptype_sp[300b].out │ │ │ -rw-r--r-- 0 root (0) root (0) 4091 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Type_sp[000]_cm_sp__C__Y_spof_spdegree_sp20.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2380 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Type_sp[210]_cm_sp__C__Y_spof_spdegree_sp18_spvia_splinkage.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1645 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Type_sp[310]_cm_sp__C__Y_spof_spdegree_sp17_spvia_splinkage.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2176 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Type_sp[331]_cm_sp__C__Y_spof_spdegree_sp17_spvia_splinkage.out │ │ │ @@ -18184,15 +18184,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9061 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Example_sp__Type_sp[300b].html │ │ │ -rw-r--r-- 0 root (0) root (0) 9848 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Example_sp__Type_sp[300c].html │ │ │ -rw-r--r-- 0 root (0) root (0) 6828 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Finding_spall_sppossible_spbetti_sptables_spfor_spquadratic_spcomponent_spof_spinverse_spsystem_spfor_spquartics_spin_sp4_spvariables.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16172 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Finding_spthe_sp16_spbetti_sptables_sppossible_spfor_spquartic_spforms_spin_sp4_spvariables_cm_spand_spexamples.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19856 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Finding_spthe_sp__Betti_spstratum_spof_spa_spgiven_spquartic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9268 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Finding_spthe_sppossible_spbetti_sptables_spfor_sppoints_spin_sp__P^3_spwith_spgiven_spgeometry.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8693 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Half_spcanonical_spdegree_sp20.html │ │ │ --rw-r--r-- 0 root (0) root (0) 53795 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 48194 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10984 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Noether-__Lefschetz_spexamples.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4123 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Normalize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12440 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Pfaffians_spon_spquadrics.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8586 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Singularities_spof_splifting_spof_sptype_sp[300b].html │ │ │ -rw-r--r-- 0 root (0) root (0) 14692 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Type_sp[000]_cm_sp__C__Y_spof_spdegree_sp20.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12230 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Type_sp[210]_cm_sp__C__Y_spof_spdegree_sp18_spvia_splinkage.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9284 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Type_sp[310]_cm_sp__C__Y_spof_spdegree_sp17_spvia_splinkage.html │ │ │ @@ -18311,44 +18311,44 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 40654 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RInterface/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 31191 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RInterface/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7199 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RInterface/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 4410 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 416 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/example-output/_canonical__Curve.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 417 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/example-output/_canonical__Curve.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 61 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 6065 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/html/_canonical__Curve.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6066 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/html/_canonical__Curve.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5886 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4523 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3193 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomCanonicalCurves/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 27181 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2678 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_disturb.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1358 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_histogram.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1315 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_maximal__Entry.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1522 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_normalize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5055 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_random__Chain__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 510 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_random__Simplicial__Complex.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1159 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_test__Time__For__L__L__Lon__Syzygies.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1158 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_test__Time__For__L__L__Lon__Syzygies.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 44 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 3937 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/___Discrete.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4295 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/___With__L__L__L.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4151 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/___Zero__Mean.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9992 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_disturb.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7241 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_histogram.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7257 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_maximal__Entry.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6973 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_normalize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14308 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_random__Chain__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6662 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_random__Simplicial__Complex.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8070 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_test__Time__For__L__L__Lon__Syzygies.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8069 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_test__Time__For__L__L__Lon__Syzygies.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11390 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9528 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4800 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomCurves/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomCurves/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1330 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomCurves/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomCurves/html/ │ │ │ @@ -18378,50 +18378,50 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 13233 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 276 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/example-output/_canonical__Curve__Genus14.out │ │ │ -rw-r--r-- 0 root (0) root (0) 279 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/example-output/_curve__Genus14__Degree18in__P6.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2925 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/example-output/_random__Canonical__Curve__Genus8with8__Points.out │ │ │ --rw-r--r-- 0 root (0) root (0) 485 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/example-output/_random__Curve__Genus14__Degree18in__P6.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 484 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/example-output/_random__Curve__Genus14__Degree18in__P6.out │ │ │ -rw-r--r-- 0 root (0) root (0) 393 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/example-output/_random__Curve__Genus8__Degree14in__P6.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 32 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5382 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/_canonical__Curve__Genus14.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5485 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/_curve__Genus14__Degree18in__P6.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9332 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/_random__Canonical__Curve__Genus8with8__Points.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7434 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/_random__Curve__Genus14__Degree18in__P6.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7433 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/_random__Curve__Genus14__Degree18in__P6.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6239 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/_random__Curve__Genus8__Degree14in__P6.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8399 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6319 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4166 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 85866 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 604 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Finding_sp__Extreme_sp__Examples.out │ │ │ --rw-r--r-- 0 root (0) root (0) 447 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Random__Ideals.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 429 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Random__Ideals.out │ │ │ -rw-r--r-- 0 root (0) root (0) 481 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_ideal__Chain__From__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 308 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_ideal__From__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 188 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_is__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 250 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Addition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 894 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Binomial__Edge__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 491 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Binomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 564 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Edge__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 479 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Elements__From__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 461 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Ideal.out │ │ │ --rw-r--r-- 0 root (0) root (0) 287 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 285 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 421 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 359 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Pure__Binomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 249 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Shellable__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 733 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Shellable__Ideal__Chain.out │ │ │ -rw-r--r-- 0 root (0) root (0) 752 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 426 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Sparse__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 491 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Monomial__Ideal.out │ │ │ --rw-r--r-- 0 root (0) root (0) 8939 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Step.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 8838 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Step.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1615 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Toric__Edge__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 233 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_reg__Seq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 350 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_square__Free.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 39 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5095 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/___Alexander__Probability.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10300 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/___Finding_sp__Extreme_sp__Examples.html │ │ │ @@ -18430,28 +18430,28 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5789 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_is__Shelling.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7774 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Addition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7286 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Binomial__Edge__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7700 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Binomial__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6565 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Edge__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7793 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Elements__From__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7658 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Ideal.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6274 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Monomial.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6272 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Monomial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7113 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Monomial__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7700 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Pure__Binomial__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7367 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Shellable__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7426 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Shellable__Ideal__Chain.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9696 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Shelling.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8172 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Sparse__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7458 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Square__Free__Monomial__Ideal.html │ │ │ --rw-r--r-- 0 root (0) root (0) 18095 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Square__Free__Step.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 17994 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Square__Free__Step.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8317 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Toric__Edge__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5737 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_reg__Seq.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6021 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_square__Free.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5742 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_square__Free_lp__Z__Z_cm__Ring_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 25638 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 25620 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 17515 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8052 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 151393 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 319 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/example-output/___C__M__Stats.out │ │ │ @@ -18580,40 +18580,40 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 96387 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/ │ │ │ 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./usr/share/doc/Macaulay2/RandomPoints/example-output/_find__A__Non__Zero__Minor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1379 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_generic__Projection.out │ │ │ -rw-r--r-- 0 root (0) root (0) 841 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_get__Random__Linear__Forms.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2154 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_projection__To__Hypersurface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 805 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_random__Coordinate__Change.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1227 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_random__Points.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1226 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_random__Points.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 598 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 51 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4748 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/___Decomposition__Strategy.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5859 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/___Dimension__Function.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6001 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/___Extend__Field.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5743 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/___Max__Coordinates__To__Replace.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4903 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/___Num__Threads__To__Use.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6021 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/___Point__Check__Attempts.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9659 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/___Replacement.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8823 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/_dim__Via__Bezout.html │ │ │ --rw-r--r-- 0 root (0) root (0) 13399 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/_extend__Ideal__By__Non__Zero__Minor.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 13400 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/_extend__Ideal__By__Non__Zero__Minor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10917 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/_find__A__Non__Zero__Minor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11404 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/_generic__Projection.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7899 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/_generic__Projection_lp..._cm__Verbose_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10142 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/_get__Random__Linear__Forms.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10723 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/_projection__To__Hypersurface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9036 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/_random__Coordinate__Change.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12424 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/_random__Points.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12423 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/_random__Points.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13452 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/_random__Points_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19055 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 26109 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6784 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomSpaceCurves/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomSpaceCurves/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 23231 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RandomSpaceCurves/dump/rawdocumentation.dump │ │ │ @@ -18641,15 +18641,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 117469 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1111 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/example-output/___Rational__Mapping.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1086 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/example-output/___Rational__Mapping_sp_st_sp__Rational__Mapping.out │ │ │ -rw-r--r-- 0 root (0) root (0) 608 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/example-output/_base__Locus__Of__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 866 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/example-output/_ideal__Of__Image__Of__Map.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4962 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/example-output/_inverse__Of__Map.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4961 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/example-output/_inverse__Of__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 956 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/example-output/_is__Birational__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1041 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/example-output/_is__Birational__Onto__Image.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1413 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/example-output/_is__Embedding.out │ │ │ -rw-r--r-- 0 root (0) root (0) 346 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/example-output/_is__Regular__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 588 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/example-output/_is__Same__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 613 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/example-output/_jacobian__Dual__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 749 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/example-output/_map__Onto__Image.out │ │ │ @@ -18668,15 +18668,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8727 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/html/___Rational__Mapping_sp_st_sp__Rational__Mapping.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5262 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/html/___Rees__Strategy.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4590 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/html/___Saturate__Output.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5196 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/html/___Saturation__Strategy.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5426 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/html/___Simis__Strategy.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9275 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/html/_base__Locus__Of__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8967 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/html/_ideal__Of__Image__Of__Map.html │ │ │ --rw-r--r-- 0 root (0) root (0) 19627 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/html/_inverse__Of__Map.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 19626 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/html/_inverse__Of__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12452 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/html/_is__Birational__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12159 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/html/_is__Birational__Onto__Image.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15074 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/html/_is__Embedding.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6538 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/html/_is__Regular__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8013 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/html/_is__Same__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9364 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/html/_jacobian__Dual__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8082 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalMaps/html/_map__Onto__Image.html │ │ │ @@ -18705,26 +18705,26 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1462 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalPoints2/example-output/___Rational__Points2.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1247 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalPoints2/example-output/_base__Change.out │ │ │ -rw-r--r-- 0 root (0) root (0) 491 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalPoints2/example-output/_charpoly.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1084 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalPoints2/example-output/_ext__Field.out │ │ │ -rw-r--r-- 0 root (0) root (0) 311 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalPoints2/example-output/_global__Height.out │ │ │ -rw-r--r-- 0 root (0) root (0) 298 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalPoints2/example-output/_hermite__Normal__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 361 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalPoints2/example-output/_integers.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4104 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalPoints2/example-output/_rational__Points.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4102 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalPoints2/example-output/_rational__Points.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1244 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalPoints2/example-output/_zeros.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 37 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5780 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/___Projective__Point.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10338 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/_base__Change.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6609 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/_charpoly.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12341 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/_ext__Field.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5854 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/_global__Height.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5818 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/_hermite__Normal__Form.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6215 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/_integers.html │ │ │ --rw-r--r-- 0 root (0) root (0) 21131 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/_rational__Points.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 21129 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/_rational__Points.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7798 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/_zeros.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19956 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RationalPoints2/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15620 2025-12-14 14:09:53.000000 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(0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 37681 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 415 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_resource_splimits.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6282 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_run__External__M2.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6284 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_run__External__M2.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 61 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5266 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/___Keep__Statistics__Command.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5527 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/___Pre__Run__Script.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5029 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_is__External__M2__Child.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5095 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_is__External__M2__Parent.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7601 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_resource_splimits.html │ │ │ --rw-r--r-- 0 root (0) root (0) 23865 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_run__External__M2.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 23867 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_run__External__M2.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5662 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_run__External__M2__Return__Answer.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6323 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_run__External__M2_lp..._cm__Keep__Files_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7559 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_suggestions_spfor_spusing_sp__Run__External__M2.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9683 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7490 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4878 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SCMAlgebras/ │ │ │ @@ -19203,15 +19203,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8415 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SCSCP/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6794 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SCSCP/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3956 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SCSCP/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 101515 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 2516 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/___S__L__Pexpressions.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2514 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/___S__L__Pexpressions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 525 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/_arithmetic_spwith_spcircuits.out │ │ │ -rw-r--r-- 0 root (0) root (0) 830 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/_compressing_spcircuits.out │ │ │ -rw-r--r-- 0 root (0) root (0) 623 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/_creating_spgates.out │ │ │ -rw-r--r-- 0 root (0) root (0) 142 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/_declare__Variable.out │ │ │ -rw-r--r-- 0 root (0) root (0) 635 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/_differentiating_spcircuits.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1597 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/_evaluate_lp__S__L__Program_cm__Mutable__Matrix_cm__Mutable__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 524 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/_evaluating_spgates.out │ │ │ @@ -19246,15 +19246,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4088 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_make__Compiled__S__L__Program.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7422 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_make__Interpreted__S__L__Program.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4854 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_make__S__L__Program.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8288 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_measuring_spthe_spsize_spof_spcircuits.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6033 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_undeclare__Variable.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6529 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_value__Hash__Table.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8773 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_working_spwith_spgate_spmatrices.html │ │ │ --rw-r--r-- 0 root (0) root (0) 36826 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 36824 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20897 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6550 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLnEquivariantMatrices/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLnEquivariantMatrices/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 42778 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLnEquivariantMatrices/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLnEquivariantMatrices/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 3218 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SLnEquivariantMatrices/example-output/_sl2__Equivariant__Constant__Rank__Matrix.out │ │ │ @@ -19494,29 +19494,29 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 44888 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2826 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 11853 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Complexes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3437 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Homology.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5542 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_are__Pseudo__Inverses.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2088 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_common__Entries.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6983 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_euclidean__Distance.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6984 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_euclidean__Distance.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7308 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_laplacians.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1301 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_numeric__Rank.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9529 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_project__To__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9254 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_pseudo__Inverse.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 84 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 3737 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___Laplacian.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3751 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___Projection.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11976 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13012 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Homology.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15236 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_are__Pseudo__Inverses.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3541 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_check__S__V__D__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10080 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_common__Entries.html │ │ │ --rw-r--r-- 0 root (0) root (0) 14701 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_euclidean__Distance.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14702 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_euclidean__Distance.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14335 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_laplacians.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3882 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_new__Chain__Complex__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9100 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_numeric__Rank.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18251 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_project__To__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18702 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_pseudo__Inverse.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3961 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_pseudo__Inverse1.html │ │ │ -rw-r--r-- 0 root (0) root (0) 28647 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/index.html │ │ │ @@ -19539,24 +19539,24 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 52965 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 859 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_annihilator.out │ │ │ -rw-r--r-- 0 root (0) root (0) 846 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_ideal_spquotients_spand_spsaturation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 259 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_is__Supported__In__Zero__Locus.out │ │ │ -rw-r--r-- 0 root (0) root (0) 807 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_module_spquotients_cm_spsaturation_cm_spand_spannihilator.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1834 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_quotient_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1835 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_quotient_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2582 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_quotient_lp__Module_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 837 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_saturate.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 69 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 8747 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/html/_annihilator.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6560 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/html/_ideal_spquotients_spand_spsaturation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6876 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/html/_is__Supported__In__Zero__Locus.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7263 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/html/_module_spquotients_cm_spsaturation_cm_spand_spannihilator.html │ │ │ --rw-r--r-- 0 root (0) root (0) 17354 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 17355 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15598 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp__Module_cm__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11944 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/html/_saturate.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12655 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/html/_saturate_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13853 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11866 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4273 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Saturation/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Schubert2/ │ │ │ @@ -19997,29 +19997,29 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 435 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_is__Multi__Hom.out │ │ │ -rw-r--r-- 0 root (0) root (0) 277 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_make__Chow__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 345 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_make__Product__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 453 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_multiplicity.out │ │ │ -rw-r--r-- 0 root (0) root (0) 881 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_projective__Degree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 932 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_projective__Degrees.out │ │ │ -rw-r--r-- 0 root (0) root (0) 876 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_segre.out │ │ │ --rw-r--r-- 0 root (0) root (0) 719 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_segre__Dim__X.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 720 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_segre__Dim__X.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 97 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 7697 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_chow__Class.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8412 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_contained__In__Singular__Locus.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9833 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_intersection__Product.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9715 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_is__Component__Contained.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6577 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_is__Multi__Hom.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5950 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_make__Chow__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6488 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_make__Product__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7807 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_multiplicity.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9364 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_projective__Degree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9107 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_projective__Degrees.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8878 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_segre.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8747 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_segre__Dim__X.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8748 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_segre__Dim__X.html │ │ │ -rw-r--r-- 0 root (0) root (0) 17760 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13748 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6810 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SemidefiniteProgramming/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SemidefiniteProgramming/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 44566 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SemidefiniteProgramming/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SemidefiniteProgramming/example-output/ │ │ │ @@ -20348,15 +20348,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 597 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/example-output/_get__Facet__Bases.out │ │ │ -rw-r--r-- 0 root (0) root (0) 967 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/example-output/_graph__From__Slack__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 575 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/example-output/_graphic__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2288 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/example-output/_grassmann__Section__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1617 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/example-output/_reconstruct__Slack__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1436 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/example-output/_reduced__Slack__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1601 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/example-output/_rehomogenize__Ideal.out │ │ │ --rw-r--r-- 0 root (0) root (0) 617 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/example-output/_rehomogenize__Polynomial.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 619 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/example-output/_rehomogenize__Polynomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1011 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/example-output/_set__Ones__Forest.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6891 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/example-output/_slack__From__Gale__Circuits.out │ │ │ -rw-r--r-- 0 root (0) root (0) 852 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/example-output/_slack__From__Gale__Plucker.out │ │ │ -rw-r--r-- 0 root (0) root (0) 802 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/example-output/_slack__From__Plucker.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1105 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/example-output/_slack__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 727 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/example-output/_slack__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 15577 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/example-output/_specific__Slack__Matrix.out │ │ │ @@ -20404,15 +20404,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10463 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_reduced__Slack__Matrix_lp..._cm__Coefficient__Ring_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5375 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_reduced__Slack__Matrix_lp..._cm__Flag__Indices_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7248 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_reduced__Slack__Matrix_lp..._cm__Object_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7130 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_reduced__Slack__Matrix_lp..._cm__Vars_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10128 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_rehomogenize__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6100 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_rehomogenize__Ideal_lp..._cm__Saturate_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16687 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_rehomogenize__Ideal_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8305 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_rehomogenize__Polynomial.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8307 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_rehomogenize__Polynomial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7939 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_set__Ones__Forest.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15037 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_slack__From__Gale__Circuits.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5349 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_slack__From__Gale__Circuits_lp..._cm__Tolerance_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8218 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_slack__From__Gale__Plucker.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8729 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_slack__From__Plucker.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7336 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_slack__From__Plucker_lp..._cm__Object_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11211 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SlackIdeals/html/_slack__Ideal.html │ │ │ @@ -20541,18 +20541,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1122 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/___Multidimensional__Matrix_sp_st_sp__Multidimensional__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 646 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/___Multidimensional__Matrix_sp_us_sp__Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 597 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/___Ring__Element_sp_st_sp__Multidimensional__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 478 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/___Sparse__Discriminant_sp__Thing.out │ │ │ -rw-r--r-- 0 root (0) root (0) 930 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/___Sparse__Resultant_sp__Thing.out │ │ │ -rw-r--r-- 0 root (0) root (0) 326 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_char_lp__Sparse__Discriminant_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 385 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_char_lp__Sparse__Resultant_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1025 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_degree__Determinant.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1021 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_degree__Determinant.out │ │ │ -rw-r--r-- 0 root (0) root (0) 814 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dense__Discriminant.out │ │ │ --rw-r--r-- 0 root (0) root (0) 929 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dense__Resultant.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1364 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_determinant_lp__Multidimensional__Matrix_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 927 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dense__Resultant.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1360 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_determinant_lp__Multidimensional__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 973 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dim_lp__Multidimensional__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 494 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_entries_lp__Multidimensional__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 561 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_exponents__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 394 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_exponents_lp__Sparse__Discriminant_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 471 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_exponents_lp__Sparse__Resultant_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2043 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_flattening.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1250 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_generic__Laurent__Polynomials.out │ │ │ @@ -20566,15 +20566,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 634 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_random__Multidimensional__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2126 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_rank_lp__Multidimensional__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1772 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_reverse__Shape.out │ │ │ -rw-r--r-- 0 root (0) root (0) 537 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_ring_lp__Multidimensional__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 457 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_shape.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2104 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_sort__Shape.out │ │ │ -rw-r--r-- 0 root (0) root (0) 12400 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_sparse__Discriminant.out │ │ │ --rw-r--r-- 0 root (0) root (0) 55253 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_sparse__Resultant.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 55251 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_sparse__Resultant.out │ │ │ -rw-r--r-- 0 root (0) root (0) 911 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_sylvester__Matrix_lp__Multidimensional__Matrix_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 611 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 35 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 10142 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Multidimensional__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6947 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Multidimensional__Matrix_sp-_sp__Multidimensional__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6345 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Multidimensional__Matrix_sp_eq_eq_sp__Multidimensional__Matrix.html │ │ │ @@ -20584,18 +20584,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6195 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Ring__Element_sp_st_sp__Multidimensional__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5440 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Sparse__Discriminant.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5997 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Sparse__Discriminant_sp__Thing.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5468 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Sparse__Resultant.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6526 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Sparse__Resultant_sp__Thing.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5198 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_char_lp__Sparse__Discriminant_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5264 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_char_lp__Sparse__Resultant_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6903 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_degree__Determinant.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6899 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_degree__Determinant.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8190 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_dense__Discriminant.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8266 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_dense__Resultant.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8707 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_determinant_lp__Multidimensional__Matrix_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8264 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_dense__Resultant.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8703 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_determinant_lp__Multidimensional__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6122 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_dim_lp__Multidimensional__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5711 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_entries_lp__Multidimensional__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7003 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_exponents__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5551 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_exponents_lp__Sparse__Discriminant_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5569 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_exponents_lp__Sparse__Resultant_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8550 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_flattening.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7760 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_generic__Laurent__Polynomials.html │ │ │ @@ -20609,15 +20609,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6579 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_random__Multidimensional__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8294 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_rank_lp__Multidimensional__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7610 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_reverse__Shape.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5628 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_ring_lp__Multidimensional__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5589 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_shape.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7887 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sort__Shape.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20153 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Discriminant.html │ │ │ --rw-r--r-- 0 root (0) root (0) 65534 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Resultant.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 65532 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Resultant.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7079 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sylvester__Matrix_lp__Multidimensional__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21933 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20968 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11034 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 178042 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/dump/rawdocumentation.dump │ │ │ @@ -20644,15 +20644,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 210 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_cycle__Decomposition_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 252 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_elementary__Symmetric__Polynomials_lp__Polynomial__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 223 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_entries_lp__Young__Tableau_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 439 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_first__Row__Descent_lp__Young__Tableau_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 505 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_garnir__Element.out │ │ │ -rw-r--r-- 0 root (0) root (0) 557 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_generate__Permutation__Group_lp__List_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3116 2025-12-14 14:09:53.000000 ./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) 3117 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_higher__Specht__Polynomials.out │ │ │ -rw-r--r-- 0 root (0) root (0) 325 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_hook__Length__Formula_lp__Partition_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 538 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_index__Tableau_lp__Young__Tableau_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 272 2025-12-14 14:09:53.000000 ./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 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_list__To__Tableau_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 993 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_matrix__Representation.out │ │ │ @@ -20664,15 +20664,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 556 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_permute__Polynomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 297 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_power__Sum__Symmetric__Polynomials_lp__Polynomial__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 274 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_reading__Word_lp__Young__Tableau_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1936 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_representation__Multiplicity.out │ │ │ -rw-r--r-- 0 root (0) root (0) 410 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_row__Permutation__Tableaux_lp__Young__Tableau_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 335 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_row__Stabilizer_lp__Young__Tableau_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 563 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_schur__Polynomial_lp__List_cm__Partition_cm__Polynomial__Ring_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 45652 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_secondary__Invariants_lp__List_cm__Polynomial__Ring_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 45653 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_secondary__Invariants_lp__List_cm__Polynomial__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1508 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_semistandard__Tableaux_lp__Partition_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 206 2025-12-14 14:09:53.000000 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./usr/share/doc/Macaulay2/SpechtModule/html/_secondary__Invariants_lp__List_cm__Polynomial__Ring_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 53546 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_secondary__Invariants_lp__List_cm__Polynomial__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7236 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_semistandard__Tableaux_lp__Partition_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5183 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_size_lp__Young__Tableau_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6503 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_sort__Columns__Tableau_lp__Specht__Module__Element_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5609 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_sort__Columns__Tableau_lp__Young__Tableau_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7433 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_specht__Module__Element.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7369 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_specht__Polynomial_lp__Young__Tableau_cm__Polynomial__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8731 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_specht__Polynomials_lp__Partition_cm__Polynomial__Ring_rp.html │ │ │ @@ -20755,74 +20755,74 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 203984 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 395 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/___Congruence__Of__Curves_sp__Embedded__Projective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 914 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/___G__Mtables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1325 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_ambient__Fivefold.out │ │ │ --rw-r--r-- 0 root (0) root (0) 954 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__Castelnuovo__Surface.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 952 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__Castelnuovo__Surface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1005 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Special__Cubic__Fourfold_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1191 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Special__Gushel__Mukai__Fourfold_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1190 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Special__Gushel__Mukai__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 614 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_beauville__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 351 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_clean_lp__Hodge__Special__Fourfold_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1173 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Special__Cubic__Fourfold_cm__Z__Z_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1550 2025-12-14 14:09:53.000000 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./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_map_lp__Congruence__Of__Curves_rp.html │ │ │ @@ -20831,29 +20831,29 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9405 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_mirror__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6394 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_normal__Sheaf.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8483 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7955 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Special__Cubic__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8182 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Special__Gushel__Mukai__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8186 2025-12-14 14:09:53.000000 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14:09:53.000000 ./usr/share/doc/Macaulay2/SpectralSequences/dump/ │ │ │ @@ -21367,15 +21367,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 212 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_no__Packed__All__Subs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 180 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_no__Packed__Sub.out │ │ │ -rw-r--r-- 0 root (0) root (0) 174 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_squarefree__Gens.out │ │ │ -rw-r--r-- 0 root (0) root (0) 169 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_squarefree__In__Codim.out │ │ │ -rw-r--r-- 0 root (0) root (0) 651 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symb__Power__Prime__Pos__Char.out │ │ │ -rw-r--r-- 0 root (0) root (0) 184 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symbolic__Defect.out │ │ │ -rw-r--r-- 0 root (0) root (0) 193 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symbolic__Polyhedron.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1212 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symbolic__Power.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1211 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symbolic__Power.out │ │ │ -rw-r--r-- 0 root (0) root (0) 180 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symbolic__Power__Join.out │ │ │ -rw-r--r-- 0 root (0) root (0) 320 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_waldschmidt.out │ │ │ -rw-r--r-- 0 root (0) root (0) 200 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_waldschmidt_lp..._cm__Sample__Size_eq_gt..._rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 593 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 15 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4616 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/___A_spquick_spintroduction_spto_spthis_sppackage.html │ │ │ @@ -21408,15 +21408,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5966 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_no__Packed__All__Subs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5711 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_no__Packed__Sub.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5703 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_squarefree__Gens.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5851 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_squarefree__In__Codim.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6490 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symb__Power__Prime__Pos__Char.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6533 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symbolic__Defect.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6014 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symbolic__Polyhedron.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9673 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symbolic__Power.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9672 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symbolic__Power.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5808 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symbolic__Power__Join.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7926 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_waldschmidt.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7018 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_waldschmidt_lp..._cm__Sample__Size_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21712 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19884 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11090 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/SymmetricPolynomials/ │ │ │ @@ -21787,25 +21787,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 806 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_compatible__Ideals.out │ │ │ -rw-r--r-- 0 root (0) root (0) 378 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_decompose__Fraction.out │ │ │ -rw-r--r-- 0 root (0) root (0) 606 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_descend__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 116 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_floor__Log.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1009 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1518 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Power.out │ │ │ -rw-r--r-- 0 root (0) root (0) 227 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Preimage.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1329 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Root.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1328 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Root.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1117 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Trace__On__Canonical__Module.out │ │ │ --rw-r--r-- 0 root (0) root (0) 487 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__Cohen__Macaulay.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1824 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Injective.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 488 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__Cohen__Macaulay.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1821 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Injective.out │ │ │ -rw-r--r-- 0 root (0) root (0) 637 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Pure.out │ │ │ -rw-r--r-- 0 root (0) root (0) 517 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Rational.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1492 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Regular.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1491 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Regular.out │ │ │ -rw-r--r-- 0 root (0) root (0) 187 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_multiplicative__Order.out │ │ │ -rw-r--r-- 0 root (0) root (0) 416 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_parameter__Test__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 216 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Element.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1514 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Ideal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1512 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2756 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Module.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 590 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 40 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4326 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/___Ascent__Count.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4385 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/___Assume__C__M.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5130 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/___Assume__Domain.html │ │ │ @@ -21838,25 +21838,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9151 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_compatible__Ideals.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7168 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_decompose__Fraction.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9334 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_descend__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5225 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_floor__Log.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10611 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12408 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Power.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5237 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Preimage.html │ │ │ --rw-r--r-- 0 root (0) root (0) 16746 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Root.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 16745 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Root.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8554 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Trace__On__Canonical__Module.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8215 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__Cohen__Macaulay.html │ │ │ --rw-r--r-- 0 root (0) root (0) 15811 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Injective.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8216 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__Cohen__Macaulay.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 15808 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Injective.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10346 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Pure.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9661 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Rational.html │ │ │ --rw-r--r-- 0 root (0) root (0) 16392 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Regular.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 16391 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Regular.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5824 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_multiplicative__Order.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7890 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_parameter__Test__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6302 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Element.html │ │ │ --rw-r--r-- 0 root (0) root (0) 14994 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Ideal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14992 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 17708 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 37577 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 34814 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12484 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Text/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Text/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 136721 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Text/dump/rawdocumentation.dump │ │ │ @@ -22056,30 +22056,30 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 37607 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThinSincereQuivers/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 27702 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThinSincereQuivers/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11513 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThinSincereQuivers/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 24388 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 1472 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Minimal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1429 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Minimal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6776 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Threaded__G__B.out │ │ │ --rw-r--r-- 0 root (0) root (0) 615 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_matrix_lp__Lineage__Table_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 938 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_minimize_lp__Lineage__Table_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1256 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_reduce.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4068 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 535 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_matrix_lp__Lineage__Table_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1147 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_minimize_lp__Lineage__Table_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 934 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_reduce.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2564 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb.out │ │ │ -rw-r--r-- 0 root (0) root (0) 328 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb_lp..._cm__Verbose_eq_gt..._rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 605 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 77 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6425 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/___Lineage__Table.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7059 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/___Minimal.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6670 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_matrix_lp__Lineage__Table_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6880 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_minimize_lp__Lineage__Table_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7259 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_reduce.html │ │ │ --rw-r--r-- 0 root (0) root (0) 14771 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7016 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/___Minimal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6590 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_matrix_lp__Lineage__Table_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7089 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_minimize_lp__Lineage__Table_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6937 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_reduce.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 13267 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7430 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb_lp..._cm__Verbose_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21626 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6806 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4627 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Topcom/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Topcom/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 66598 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Topcom/dump/rawdocumentation.dump │ │ │ @@ -22186,25 +22186,25 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 20320 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 944 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_cm__Class.out │ │ │ -rw-r--r-- 0 root (0) root (0) 567 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_cm__Volumes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 717 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_dual__Deg__Codim.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1536 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_ed__Deg.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1537 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_ed__Deg.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1032 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_polar__Degrees.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 86 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4388 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/___Force__Amat.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4214 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/___Output.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4206 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/___Text__Output.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8847 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_cm__Class.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7510 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_cm__Volumes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7471 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_dual__Deg__Codim.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8817 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_ed__Deg.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8818 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_ed__Deg.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8849 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_polar__Degrees.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9974 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7189 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4667 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricTopology/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricTopology/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 38537 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/ToricTopology/dump/rawdocumentation.dump │ │ │ @@ -22375,15 +22375,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 7217 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/TriangularSets/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 31886 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2276 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/___Triangulations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 18048 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_all__Triangulations_lp__Matrix_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 75224 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_generate__Triangulations.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 75242 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_generate__Triangulations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 714 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_is__Regular__Triangulation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1654 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_triangulation.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 42 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4460 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/___Chirotope.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3951 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/___Cone__Index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7308 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/___Triangulation.html │ │ │ @@ -22391,15 +22391,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 30386 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_all__Triangulations_lp__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3970 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_bistellar__Flip.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3884 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_chirotope.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3827 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_delaunay__Subdivision.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3823 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_delaunay__Weights.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3867 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_fine__Star__Triangulation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3781 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_flips.html │ │ │ --rw-r--r-- 0 root (0) root (0) 86004 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_generate__Triangulations.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 86022 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_generate__Triangulations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3911 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_gkz__Vector.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3873 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_is__Fine.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9223 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_is__Regular__Triangulation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3883 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_is__Star.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3814 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_naive__Chirotope.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4176 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_naive__Is__Triangulation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3807 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_neighbors.html │ │ │ @@ -23373,15 +23373,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 620 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_ceiling_lp__R__Weil__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 264 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_clean__Support.out │ │ │ -rw-r--r-- 0 root (0) root (0) 470 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_clear__Cache.out │ │ │ -rw-r--r-- 0 root (0) root (0) 254 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_coefficient_lp__Basic__List_cm__Basic__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 335 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_coefficient_lp__Ideal_cm__Basic__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 728 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_coefficients_lp__Basic__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2714 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_divisor.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1714 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_dualize.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1715 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_dualize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1174 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_embed__As__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 500 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_find__Element__Of__Degree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 532 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_gbs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 831 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_get__Linear__Diophantine__Solution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 501 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_get__Prime__Count.out │ │ │ -rw-r--r-- 0 root (0) root (0) 357 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_get__Prime__Divisors.out │ │ │ -rw-r--r-- 0 root (0) root (0) 336 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_ideal__Power.out │ │ │ @@ -23405,16 +23405,16 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 441 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_is__Zero__Divisor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 667 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_map__To__Projective__Space.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1342 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_non__Cartier__Locus.out │ │ │ -rw-r--r-- 0 root (0) root (0) 780 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_positive__Part.out │ │ │ -rw-r--r-- 0 root (0) root (0) 604 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_primes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 765 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_pullback_lp__Ring__Map_cm__R__Weil__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 846 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_ramification__Divisor.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4354 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_reflexify.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1095 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_reflexive__Power.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4355 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_reflexify.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1096 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_reflexive__Power.out │ │ │ -rw-r--r-- 0 root (0) root (0) 249 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_ring_lp__Basic__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 375 2025-12-14 14:09:53.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_to__Q__Weil__Divisor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 458 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./usr/share/info/ThinSincereQuivers.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 8461 2025-12-14 14:09:53.000000 ./usr/share/info/ThreadedGB.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 8198 2025-12-14 14:09:53.000000 ./usr/share/info/ThreadedGB.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 13901 2025-12-14 14:09:53.000000 ./usr/share/info/Topcom.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7599 2025-12-14 14:09:53.000000 ./usr/share/info/TorAlgebra.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7449 2025-12-14 14:09:53.000000 ./usr/share/info/ToricHigherDirectImages.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 4015 2025-12-14 14:09:53.000000 ./usr/share/info/ToricInvariants.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 4014 2025-12-14 14:09:53.000000 ./usr/share/info/ToricInvariants.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 6581 2025-12-14 14:09:53.000000 ./usr/share/info/ToricTopology.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 31435 2025-12-14 14:09:53.000000 ./usr/share/info/ToricVectorBundles.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 9584 2025-12-14 14:09:53.000000 ./usr/share/info/TriangularSets.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 13681 2025-12-14 14:09:53.000000 ./usr/share/info/Triangulations.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 13683 2025-12-14 14:09:53.000000 ./usr/share/info/Triangulations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7188 2025-12-14 14:09:53.000000 ./usr/share/info/Triplets.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 11010 2025-12-14 14:09:53.000000 ./usr/share/info/Tropical.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 10698 2025-12-14 14:09:53.000000 ./usr/share/info/TropicalToric.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 5772 2025-12-14 14:09:53.000000 ./usr/share/info/Truncations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8693 2025-12-14 14:09:53.000000 ./usr/share/info/Units.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 4687 2025-12-14 14:09:53.000000 ./usr/share/info/VNumber.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8877 2025-12-14 14:09:53.000000 ./usr/share/info/Valuations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 44204 2025-12-14 14:09:53.000000 ./usr/share/info/Varieties.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 19378 2025-12-14 14:09:53.000000 ./usr/share/info/VectorFields.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 51845 2025-12-14 14:09:53.000000 ./usr/share/info/VectorGraphics.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 41369 2025-12-14 14:09:53.000000 ./usr/share/info/VersalDeformations.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 41370 2025-12-14 14:09:53.000000 ./usr/share/info/VersalDeformations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 12693 2025-12-14 14:09:53.000000 ./usr/share/info/VirtualResolutions.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 10443 2025-12-14 14:09:53.000000 ./usr/share/info/Visualize.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 38082 2025-12-14 14:09:53.000000 ./usr/share/info/WeilDivisors.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 10700 2025-12-14 14:09:53.000000 ./usr/share/info/WeylAlgebras.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 38096 2025-12-14 14:09:53.000000 ./usr/share/info/WeilDivisors.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 10702 2025-12-14 14:09:53.000000 ./usr/share/info/WeylAlgebras.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 33226 2025-12-14 14:09:53.000000 ./usr/share/info/WeylGroups.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 14777 2025-12-14 14:09:53.000000 ./usr/share/info/WhitneyStratifications.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8865 2025-12-14 14:09:53.000000 ./usr/share/info/XML.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 49484 2025-12-14 14:09:53.000000 ./usr/share/info/gfanInterface.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 49483 2025-12-14 14:09:53.000000 ./usr/share/info/gfanInterface.info.gz │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/lintian/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/lintian/overrides/ │ │ │ -rw-r--r-- 0 root (0) root (0) 11489 2025-12-14 14:09:53.000000 ./usr/share/lintian/overrides/macaulay2-common │ │ │ lrwxrwxrwx 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/Macaulay2/Style/katex/contrib/auto-render.min.js -> ../../../../javascript/katex/contrib/auto-render.js │ │ │ lrwxrwxrwx 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/Macaulay2/Style/katex/contrib/copy-tex.min.js -> ../../../../javascript/katex/contrib/copy-tex.js │ │ │ lrwxrwxrwx 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/Macaulay2/Style/katex/contrib/render-a11y-string.min.js -> ../../../../javascript/katex/contrib/render-a11y-string.js │ │ │ lrwxrwxrwx 0 root (0) root (0) 0 2025-12-14 14:09:53.000000 ./usr/share/Macaulay2/Style/katex/fonts/KaTeX_AMS-Regular.ttf -> ../../../../fonts/truetype/katex/KaTeX_AMS-Regular.ttf │ │ ├── ./usr/share/doc/Macaulay2/AInfinity/example-output/___Check.out │ │ │ @@ -10,25 +10,25 @@ │ │ │ │ │ │ o2 = cokernel | a b c | │ │ │ │ │ │ 1 │ │ │ o2 : R-module, quotient of R │ │ │ │ │ │ i3 : elapsedTime burkeResolution(M, 7, Check => false) │ │ │ - -- 1.71138s elapsed │ │ │ + -- 1.32813s elapsed │ │ │ │ │ │ 1 3 9 27 81 243 729 2187 │ │ │ o3 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R │ │ │ │ │ │ 0 1 2 3 4 5 6 7 │ │ │ │ │ │ o3 : Complex │ │ │ │ │ │ i4 : elapsedTime burkeResolution(M, 7, Check => true) │ │ │ - -- 2.15695s elapsed │ │ │ + -- 1.72872s elapsed │ │ │ │ │ │ 1 3 9 27 81 243 729 2187 │ │ │ o4 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R │ │ │ │ │ │ 0 1 2 3 4 5 6 7 │ │ │ │ │ │ o4 : Complex │ │ ├── ./usr/share/doc/Macaulay2/AInfinity/html/___Check.html │ │ │ @@ -90,28 +90,28 @@ │ │ │ 1 │ │ │ o2 : R-module, quotient of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i3 : elapsedTime burkeResolution(M, 7, Check => false)
│ │ │ - -- 1.71138s elapsed
│ │ │ + -- 1.32813s elapsed
│ │ │  
│ │ │        1      3      9      27      81      243      729      2187
│ │ │  o3 = R  <-- R  <-- R  <-- R   <-- R   <-- R    <-- R    <-- R
│ │ │                                                               
│ │ │       0      1      2      3       4       5        6        7
│ │ │  
│ │ │  o3 : Complex
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i4 : elapsedTime burkeResolution(M, 7, Check => true)
│ │ │ - -- 2.15695s elapsed
│ │ │ + -- 1.72872s elapsed
│ │ │  
│ │ │        1      3      9      27      81      243      729      2187
│ │ │  o4 = R  <-- R  <-- R  <-- R   <-- R   <-- R    <-- R    <-- R
│ │ │                                                               
│ │ │       0      1      2      3       4       5        6        7
│ │ │  
│ │ │  o4 : Complex
│ │ │ ├── html2text {} │ │ │ │ @@ -23,24 +23,24 @@ │ │ │ │ i2 : M = coker vars R │ │ │ │ │ │ │ │ o2 = cokernel | a b c | │ │ │ │ │ │ │ │ 1 │ │ │ │ o2 : R-module, quotient of R │ │ │ │ i3 : elapsedTime burkeResolution(M, 7, Check => false) │ │ │ │ - -- 1.71138s elapsed │ │ │ │ + -- 1.32813s elapsed │ │ │ │ │ │ │ │ 1 3 9 27 81 243 729 2187 │ │ │ │ o3 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 7 │ │ │ │ │ │ │ │ o3 : Complex │ │ │ │ i4 : elapsedTime burkeResolution(M, 7, Check => true) │ │ │ │ - -- 2.15695s elapsed │ │ │ │ + -- 1.72872s elapsed │ │ │ │ │ │ │ │ 1 3 9 27 81 243 729 2187 │ │ │ │ o4 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 7 │ │ │ │ │ │ │ │ o4 : Complex │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjoint__Matrix.out │ │ │ @@ -49,15 +49,15 @@ │ │ │ o8 : BettiTally │ │ │ │ │ │ i9 : c=codim I │ │ │ │ │ │ o9 = 4 │ │ │ │ │ │ i10 : elapsedTime fI=res I │ │ │ - -- .0228791s elapsed │ │ │ + -- .0271592s elapsed │ │ │ │ │ │ 1 14 33 28 8 │ │ │ o10 = Pn <-- Pn <-- Pn <-- Pn <-- Pn <-- 0 │ │ │ │ │ │ 0 1 2 3 4 5 │ │ │ │ │ │ o10 : ChainComplex │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjunction__Process.out │ │ │ @@ -87,30 +87,30 @@ │ │ │ o13 : BettiTally │ │ │ │ │ │ i14 : phi=map(P2,Pn,H); │ │ │ │ │ │ o14 : RingMap P2 <-- Pn │ │ │ │ │ │ i15 : elapsedTime betti(I'=trim ker phi) │ │ │ - -- .670263s elapsed │ │ │ + -- .520556s elapsed │ │ │ │ │ │ 0 1 │ │ │ o15 = total: 1 11 │ │ │ 0: 1 . │ │ │ 1: . 3 │ │ │ 2: . 8 │ │ │ │ │ │ o15 : BettiTally │ │ │ │ │ │ i16 : I'== I │ │ │ │ │ │ o16 = true │ │ │ │ │ │ i17 : elapsedTime basePts=primaryDecomposition ideal H; │ │ │ - -- 5.77797s elapsed │ │ │ + -- 4.82354s elapsed │ │ │ │ │ │ i18 : tally apply(basePts,c->(dim c, degree c, betti c)) │ │ │ │ │ │ 0 1 │ │ │ o18 = Tally{(1, 1, total: 1 2) => 5} │ │ │ 0: 1 2 │ │ │ 0 1 │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_parametrization.out │ │ │ @@ -79,40 +79,40 @@ │ │ │ 1: . . │ │ │ 2: . . │ │ │ 3: . 8 │ │ │ │ │ │ o13 : BettiTally │ │ │ │ │ │ i14 : elapsedTime sub(I,H) │ │ │ - -- .0136638s elapsed │ │ │ + -- .0155972s elapsed │ │ │ │ │ │ o14 = ideal (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) │ │ │ │ │ │ o14 : Ideal of P2 │ │ │ │ │ │ i15 : phi=map(P2,Pn,H); │ │ │ │ │ │ o15 : RingMap P2 <-- Pn │ │ │ │ │ │ i16 : elapsedTime betti(I'=trim ker phi) │ │ │ - -- .0563731s elapsed │ │ │ + -- .0668926s elapsed │ │ │ │ │ │ 0 1 │ │ │ o16 = total: 1 12 │ │ │ 0: 1 . │ │ │ 1: . 12 │ │ │ │ │ │ o16 : BettiTally │ │ │ │ │ │ i17 : I'== I │ │ │ │ │ │ o17 = true │ │ │ │ │ │ i18 : elapsedTime basePts=primaryDecomposition ideal H; │ │ │ - -- 1.83465s elapsed │ │ │ + -- 1.49597s elapsed │ │ │ │ │ │ i19 : tally apply(basePts,c->(dim c, degree c, betti c)) │ │ │ │ │ │ 0 1 │ │ │ o19 = Tally{(0, 34, total: 1 15) => 1} │ │ │ 0: 1 . │ │ │ 1: . . │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjoint__Matrix.html │ │ │ @@ -149,15 +149,15 @@ │ │ │ │ │ │ o9 = 4 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i10 : elapsedTime fI=res I
│ │ │ - -- .0228791s elapsed
│ │ │ + -- .0271592s elapsed
│ │ │  
│ │ │          1       14       33       28       8
│ │ │  o10 = Pn  <-- Pn   <-- Pn   <-- Pn   <-- Pn  <-- 0
│ │ │                                                    
│ │ │        0       1        2        3        4       5
│ │ │  
│ │ │  o10 : ChainComplex
│ │ │ ├── html2text {} │ │ │ │ @@ -54,15 +54,15 @@ │ │ │ │ 2: . 12 │ │ │ │ │ │ │ │ o8 : BettiTally │ │ │ │ i9 : c=codim I │ │ │ │ │ │ │ │ o9 = 4 │ │ │ │ i10 : elapsedTime fI=res I │ │ │ │ - -- .0228791s elapsed │ │ │ │ + -- .0271592s elapsed │ │ │ │ │ │ │ │ 1 14 33 28 8 │ │ │ │ o10 = Pn <-- Pn <-- Pn <-- Pn <-- Pn <-- 0 │ │ │ │ │ │ │ │ 0 1 2 3 4 5 │ │ │ │ │ │ │ │ o10 : ChainComplex │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjunction__Process.html │ │ │ @@ -217,15 +217,15 @@ │ │ │ │ │ │ o14 : RingMap P2 <-- Pn │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i15 : elapsedTime betti(I'=trim ker phi)
│ │ │ - -- .670263s elapsed
│ │ │ + -- .520556s elapsed
│ │ │  
│ │ │               0  1
│ │ │  o15 = total: 1 11
│ │ │            0: 1  .
│ │ │            1: .  3
│ │ │            2: .  8
│ │ │  
│ │ │ @@ -238,15 +238,15 @@
│ │ │  
│ │ │  o16 = true
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i17 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ - -- 5.77797s elapsed
│ │ │ + -- 4.82354s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i18 : tally apply(basePts,c->(dim c, degree c, betti c))
│ │ │  
│ │ │                            0 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -110,28 +110,28 @@
│ │ │ │            6: . 7
│ │ │ │  
│ │ │ │  o13 : BettiTally
│ │ │ │  i14 : phi=map(P2,Pn,H);
│ │ │ │  
│ │ │ │  o14 : RingMap P2 <-- Pn
│ │ │ │  i15 : elapsedTime betti(I'=trim ker phi)
│ │ │ │ - -- .670263s elapsed
│ │ │ │ + -- .520556s elapsed
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o15 = total: 1 11
│ │ │ │            0: 1  .
│ │ │ │            1: .  3
│ │ │ │            2: .  8
│ │ │ │  
│ │ │ │  o15 : BettiTally
│ │ │ │  i16 : I'== I
│ │ │ │  
│ │ │ │  o16 = true
│ │ │ │  i17 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ │ - -- 5.77797s elapsed
│ │ │ │ + -- 4.82354s elapsed
│ │ │ │  i18 : tally apply(basePts,c->(dim c, degree c, betti c))
│ │ │ │  
│ │ │ │                            0 1
│ │ │ │  o18 = Tally{(1, 1, total: 1 2) => 5}
│ │ │ │                         0: 1 2
│ │ │ │                            0 1
│ │ │ │              (1, 3, total: 1 3) => 8
│ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_parametrization.html
│ │ │ @@ -193,15 +193,15 @@
│ │ │  
│ │ │  o13 : BettiTally
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i14 : elapsedTime sub(I,H)
│ │ │ - -- .0136638s elapsed
│ │ │ + -- .0155972s elapsed
│ │ │  
│ │ │  o14 = ideal (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
│ │ │  
│ │ │  o14 : Ideal of P2
│ │ │ │ │ │ │ │ │ │ │ │ @@ -210,15 +210,15 @@ │ │ │ │ │ │ o15 : RingMap P2 <-- Pn │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i16 : elapsedTime betti(I'=trim ker phi)
│ │ │ - -- .0563731s elapsed
│ │ │ + -- .0668926s elapsed
│ │ │  
│ │ │               0  1
│ │ │  o16 = total: 1 12
│ │ │            0: 1  .
│ │ │            1: . 12
│ │ │  
│ │ │  o16 : BettiTally
│ │ │ @@ -230,15 +230,15 @@ │ │ │ │ │ │ o17 = true │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i18 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ - -- 1.83465s elapsed
│ │ │ + -- 1.49597s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i19 : tally apply(basePts,c->(dim c, degree c, betti c))
│ │ │  
│ │ │                             0  1
│ │ │ ├── html2text {}
│ │ │ │ @@ -82,36 +82,36 @@
│ │ │ │            0: 1 .
│ │ │ │            1: . .
│ │ │ │            2: . .
│ │ │ │            3: . 8
│ │ │ │  
│ │ │ │  o13 : BettiTally
│ │ │ │  i14 : elapsedTime sub(I,H)
│ │ │ │ - -- .0136638s elapsed
│ │ │ │ + -- .0155972s elapsed
│ │ │ │  
│ │ │ │  o14 = ideal (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
│ │ │ │  
│ │ │ │  o14 : Ideal of P2
│ │ │ │  i15 : phi=map(P2,Pn,H);
│ │ │ │  
│ │ │ │  o15 : RingMap P2 <-- Pn
│ │ │ │  i16 : elapsedTime betti(I'=trim ker phi)
│ │ │ │ - -- .0563731s elapsed
│ │ │ │ + -- .0668926s elapsed
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o16 = total: 1 12
│ │ │ │            0: 1  .
│ │ │ │            1: . 12
│ │ │ │  
│ │ │ │  o16 : BettiTally
│ │ │ │  i17 : I'== I
│ │ │ │  
│ │ │ │  o17 = true
│ │ │ │  i18 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ │ - -- 1.83465s elapsed
│ │ │ │ + -- 1.49597s elapsed
│ │ │ │  i19 : tally apply(basePts,c->(dim c, degree c, betti c))
│ │ │ │  
│ │ │ │                             0  1
│ │ │ │  o19 = Tally{(0, 34, total: 1 15) => 1}
│ │ │ │                          0: 1  .
│ │ │ │                          1: .  .
│ │ │ │                          2: .  .
│ │ ├── ./usr/share/doc/Macaulay2/BGG/example-output/_pure__Resolution.out
│ │ │ @@ -114,26 +114,26 @@
│ │ │        | 19a+19b  -38a-16b -18a-13b 16a+22b  |
│ │ │        | -10a-29b 39a+21b  -43a-15b 45a-34b  |
│ │ │  
│ │ │                4      4
│ │ │  o13 : Matrix A  <-- A
│ │ │  
│ │ │  i14 : time betti (F = pureResolution(M,{0,2,4}))
│ │ │ - -- used 0.440925s (cpu); 0.367004s (thread); 0s (gc)
│ │ │ + -- used 0.496304s (cpu); 0.412543s (thread); 0s (gc)
│ │ │  
│ │ │               0 1 2
│ │ │  o14 = total: 3 6 3
│ │ │            0: 3 . .
│ │ │            1: . 6 .
│ │ │            2: . . 3
│ │ │  
│ │ │  o14 : BettiTally
│ │ │  
│ │ │  i15 : time betti (F = pureResolution(11,4,{0,2,4}))
│ │ │ - -- used 0.477334s (cpu); 0.400537s (thread); 0s (gc)
│ │ │ + -- used 0.548375s (cpu); 0.468363s (thread); 0s (gc)
│ │ │  
│ │ │               0 1 2
│ │ │  o15 = total: 3 6 3
│ │ │            0: 3 . .
│ │ │            1: . 6 .
│ │ │            2: . . 3
│ │ ├── ./usr/share/doc/Macaulay2/BGG/html/_pure__Resolution.html
│ │ │ @@ -253,15 +253,15 @@
│ │ │                4      4
│ │ │  o13 : Matrix A  <-- A
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i14 : time betti (F = pureResolution(M,{0,2,4}))
│ │ │ - -- used 0.440925s (cpu); 0.367004s (thread); 0s (gc)
│ │ │ + -- used 0.496304s (cpu); 0.412543s (thread); 0s (gc)
│ │ │  
│ │ │               0 1 2
│ │ │  o14 = total: 3 6 3
│ │ │            0: 3 . .
│ │ │            1: . 6 .
│ │ │            2: . . 3
│ │ │  
│ │ │ @@ -272,15 +272,15 @@
│ │ │          

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

│ │ │          

│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -192,26 +192,26 @@
│ │ │ │  o18 : ActionOnComplex
│ │ │ │  i19 : A2 = action(RI2,G,Sub=>false)
│ │ │ │  
│ │ │ │  o19 = Complex with 6 actors
│ │ │ │  
│ │ │ │  o19 : ActionOnComplex
│ │ │ │  i20 : elapsedTime a1 = character A1
│ │ │ │ - -- .852762s elapsed
│ │ │ │ + -- .692703s elapsed
│ │ │ │  
│ │ │ │  o20 = Character over R
│ │ │ │  
│ │ │ │        (0, {0}) => | 1 1 1 1 1 1 |
│ │ │ │        (1, {8}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │ │        (2, {11}) => | 1 1 1 1 1 1 |
│ │ │ │        (2, {13}) => | 1 1 1 1 1 1 |
│ │ │ │  
│ │ │ │  o20 : Character
│ │ │ │  i21 : elapsedTime a2 = character A2
│ │ │ │ - -- 34.344s elapsed
│ │ │ │ + -- 25.4401s elapsed
│ │ │ │  
│ │ │ │  o21 = Character over R
│ │ │ │  
│ │ │ │        (0, {0}) => | 1 1 1 1 1 1 |
│ │ │ │        (1, {16}) => | 6 2 0 0 -1 -1 |
│ │ │ │        (2, {19}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │ │        (2, {21}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │ │ @@ -308,15 +308,15 @@
│ │ │ │  i30 : M = Is2 / I2;
│ │ │ │  i31 : B = action(M,G,Sub=>false)
│ │ │ │  
│ │ │ │  o31 = Module with 6 actors
│ │ │ │  
│ │ │ │  o31 : ActionOnGradedModule
│ │ │ │  i32 : elapsedTime b = character(B,21)
│ │ │ │ - -- 14.2617s elapsed
│ │ │ │ + -- 11.4504s elapsed
│ │ │ │  
│ │ │ │  o32 = Character over R
│ │ │ │  
│ │ │ │        (0, {21}) => | 1 1 1 1 1 1 |
│ │ │ │  
│ │ │ │  o32 : Character
│ │ │ │  i33 : b/T
│ │ ├── ./usr/share/doc/Macaulay2/Bruns/example-output/_bruns.out
│ │ │ @@ -230,15 +230,15 @@
│ │ │            0: 1 . . . .
│ │ │            1: . 4 2 . .
│ │ │            2: . 1 6 5 1
│ │ │  
│ │ │  o22 : BettiTally
│ │ │  
│ │ │  i23 : time j=bruns F.dd_3;
│ │ │ - -- used 0.354218s (cpu); 0.289007s (thread); 0s (gc)
│ │ │ + -- used 0.265543s (cpu); 0.19889s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of S
│ │ │  
│ │ │  i24 : betti res j
│ │ │  
│ │ │               0 1 2 3 4
│ │ │  o24 = total: 1 3 6 5 1
│ │ ├── ./usr/share/doc/Macaulay2/Bruns/html/_bruns.html
│ │ │ @@ -380,15 +380,15 @@
│ │ │  
│ │ │  o22 : BettiTally
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │              
│ │ │            
│ │ │          

│ │ │                
i15 : time betti (F = pureResolution(11,4,{0,2,4}))
│ │ │ - -- used 0.477334s (cpu); 0.400537s (thread); 0s (gc)
│ │ │ + -- used 0.548375s (cpu); 0.468363s (thread); 0s (gc)
│ │ │  
│ │ │               0 1 2
│ │ │  o15 = total: 3 6 3
│ │ │            0: 3 . .
│ │ │            1: . 6 .
│ │ │            2: . . 3
│ │ │ ├── html2text {}
│ │ │ │ @@ -161,30 +161,30 @@
│ │ │ │        | -30a-29b -29a-24b -47a-39b 38a+2b   |
│ │ │ │        | 19a+19b  -38a-16b -18a-13b 16a+22b  |
│ │ │ │        | -10a-29b 39a+21b  -43a-15b 45a-34b  |
│ │ │ │  
│ │ │ │                4      4
│ │ │ │  o13 : Matrix A  <-- A
│ │ │ │  i14 : time betti (F = pureResolution(M,{0,2,4}))
│ │ │ │ - -- used 0.440925s (cpu); 0.367004s (thread); 0s (gc)
│ │ │ │ + -- used 0.496304s (cpu); 0.412543s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               0 1 2
│ │ │ │  o14 = total: 3 6 3
│ │ │ │            0: 3 . .
│ │ │ │            1: . 6 .
│ │ │ │            2: . . 3
│ │ │ │  
│ │ │ │  o14 : BettiTally
│ │ │ │  With the form pureResolution(p,q,D) we can directly create the situation of
│ │ │ │  pureResolution(M,D) where M is generic product(m_i+1) x #D-1+sum(m_i) matrix of
│ │ │ │  linear forms defined over a ring with product(m_i+1) * #D-1+sum(m_i) variables
│ │ │ │  of characteristic p, created by the script. For a given number of variables in
│ │ │ │  A this runs much faster than taking a random matrix M.
│ │ │ │  i15 : time betti (F = pureResolution(11,4,{0,2,4}))
│ │ │ │ - -- used 0.477334s (cpu); 0.400537s (thread); 0s (gc)
│ │ │ │ + -- used 0.548375s (cpu); 0.468363s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               0 1 2
│ │ │ │  o15 = total: 3 6 3
│ │ │ │            0: 3 . .
│ │ │ │            1: . 6 .
│ │ │ │            2: . . 3
│ │ ├── ./usr/share/doc/Macaulay2/Benchmark/example-output/_run__Benchmarks.out
│ │ │ @@ -1,10 +1,10 @@
│ │ │  -- -*- M2-comint -*- hash: 1330545576567
│ │ │  
│ │ │  i1 : runBenchmarks "res39"
│ │ │ --- beginning computation Sun Dec 14 15:31:42 UTC 2025
│ │ │ --- Linux sbuild 6.12.57+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.57-1 (2025-11-05) x86_64 GNU/Linux
│ │ │ --- AMD EPYC 7702P 64-Core Processor  AuthenticAMD  cpu MHz 1996.249  
│ │ │ +-- beginning computation Thu Jan  1 11:06:24 UTC 2026
│ │ │ +-- Linux sbuild 6.12.57+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.57-1 (2025-11-05) x86_64 GNU/Linux
│ │ │ +-- Intel Xeon Processor (Skylake, IBRS)  GenuineIntel  cpu MHz 2099.998  
│ │ │  -- Macaulay2 1.25.11, compiled with gcc 15.2.0
│ │ │ --- res39: res of a generic 3 by 9 matrix over ZZ/101: .153215 seconds
│ │ │ +-- res39: res of a generic 3 by 9 matrix over ZZ/101: .192783 seconds
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Benchmark/html/_run__Benchmarks.html
│ │ │ @@ -75,19 +75,19 @@
│ │ │          

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

│ │ │          

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

│ │ │                
i1 : runBenchmarks "res39"
│ │ │ --- beginning computation Sun Dec 14 15:31:42 UTC 2025
│ │ │ --- Linux sbuild 6.12.57+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.57-1 (2025-11-05) x86_64 GNU/Linux
│ │ │ --- AMD EPYC 7702P 64-Core Processor  AuthenticAMD  cpu MHz 1996.249  
│ │ │ +-- beginning computation Thu Jan  1 11:06:24 UTC 2026
│ │ │ +-- Linux sbuild 6.12.57+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.57-1 (2025-11-05) x86_64 GNU/Linux
│ │ │ +-- Intel Xeon Processor (Skylake, IBRS)  GenuineIntel  cpu MHz 2099.998  
│ │ │  -- Macaulay2 1.25.11, compiled with gcc 15.2.0
│ │ │ --- res39: res of a generic 3 by 9 matrix over ZZ/101: .153215 seconds

│ │ │ +-- res39: res of a generic 3 by 9 matrix over ZZ/101: .192783 seconds
│ │ │              

│ │ │        
│ │ │        

│ │ │          

│ │ │            
For the programmer

│ │ │ ├── html2text {}
│ │ │ │ @@ -23,18 +23,18 @@
│ │ │ │  "gb3445" -- gb of an ideal with elements of degree 3,4,4,5 in 8 variables
│ │ │ │  "gbB148" -- gb of Bayesian graph ideal #148
│ │ │ │  "res39" -- res of a generic 3 by 9 matrix over ZZ/101
│ │ │ │  "resG25" -- res of the coordinate ring of Grassmannian(2,5)
│ │ │ │  "yang-gb1" -- an example of Yang-Hui He arising in string theory
│ │ │ │  "yang-subring" -- an example of Yang-Hui He
│ │ │ │  i1 : runBenchmarks "res39"
│ │ │ │ --- beginning computation Sun Dec 14 15:31:42 UTC 2025
│ │ │ │ --- Linux sbuild 6.12.57+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.57-1
│ │ │ │ -(2025-11-05) x86_64 GNU/Linux
│ │ │ │ --- AMD EPYC 7702P 64-Core Processor  AuthenticAMD  cpu MHz 1996.249
│ │ │ │ +-- beginning computation Thu Jan  1 11:06:24 UTC 2026
│ │ │ │ +-- Linux sbuild 6.12.57+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ │ +6.12.57-1 (2025-11-05) x86_64 GNU/Linux
│ │ │ │ +-- Intel Xeon Processor (Skylake, IBRS)  GenuineIntel  cpu MHz 2099.998
│ │ │ │  -- Macaulay2 1.25.11, compiled with gcc 15.2.0
│ │ │ │ --- res39: res of a generic 3 by 9 matrix over ZZ/101: .153215 seconds
│ │ │ │ +-- res39: res of a generic 3 by 9 matrix over ZZ/101: .192783 seconds
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _r_u_n_B_e_n_c_h_m_a_r_k_s is a _c_o_m_m_a_n_d.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/Benchmark.m2:297:0.
│ │ ├── ./usr/share/doc/Macaulay2/Bertini/dump/rawdocumentation.dump
│ │ │ @@ -515,15 +515,15 @@
│ │ │  Pi4uLikiLCJCZXJ0aW5pIn0sIlJhbmRvbUNvbXBsZXgifSxUVHsiID0+ICJ9LFRUeyIuLi4ifSwi
│ │ │  LCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwie30ifSwiLCAiLFNQQU57fX0sU1BBTntUTzJ7bmV3
│ │ │  IERvY3VtZW50VGFnIGZyb20ge1tiZXJ0aW5pVXNlckhvbW90b3B5LFJhbmRvbVJlYWxdLCJiZXJ0
│ │ │  aW5pVXNlckhvbW90b3B5KC4uLixSYW5kb21SZWFsPT4uLi4pIiwiQmVydGluaSJ9LCJSYW5kb21S
│ │ │  ZWFsIn0sVFR7IiA9PiAifSxUVHsiLi4uIn0sIiwgIixTUEFOeyJkZWZhdWx0IHZhbHVlICIsInt9
│ │ │  In0sIiwgIixTUEFOe319LFNQQU57VE8ye25ldyBEb2N1bWVudFRhZyBmcm9tIHsiVG9wRGlyZWN0
│ │ │  b3J5IiwiVG9wRGlyZWN0b3J5IiwiQmVydGluaSJ9LCJUb3BEaXJlY3RvcnkifSxUVHsiID0+ICJ9
│ │ │ -LFRUeyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwiXCIvdG1wL00yLTI4NzA2LTAv
│ │ │ +LFRUeyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwiXCIvdG1wL00yLTQxMDM3LTAv
│ │ │  MFwiIn0sIiwgIixTUEFOeyJPcHRpb24gdG8gY2hhbmdlIGRpcmVjdG9yeSBmb3IgZmlsZSBzdG9y
│ │ │  YWdlLiJ9fSxTUEFOe1RPMntuZXcgRG9jdW1lbnRUYWcgZnJvbSB7W2JlcnRpbmlVc2VySG9tb3Rv
│ │ │  cHksVmVyYm9zZV0sImJlcnRpbmlVc2VySG9tb3RvcHkoLi4uLFZlcmJvc2U9Pi4uLikiLCJCZXJ0
│ │ │  aW5pIn0sIlZlcmJvc2UifSxUVHsiID0+ICJ9LFRUeyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQg
│ │ │  dmFsdWUgIiwiZmFsc2UifSwiLCAiLFNQQU57Ik9wdGlvbiB0byBzaWxlbmNlIGFkZGl0aW9uYWwg
│ │ │  b3V0cHV0In19fSwgc3ltYm9sIERvY3VtZW50VGFnID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsi
│ │ │  YmVydGluaVVzZXJIb21vdG9weSIsImJlcnRpbmlVc2VySG9tb3RvcHkiLCJCZXJ0aW5pIn0sIEtl
│ │ │ @@ -1100,15 +1100,15 @@
│ │ │  ZXJ0aW5pUGFyYW1ldGVySG9tb3RvcHkoLi4uLFJhbmRvbVJlYWw9Pi4uLikiLCJCZXJ0aW5pIn0s
│ │ │  IlJhbmRvbVJlYWwifSxUVHsiID0+ICJ9LFRUeyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQgdmFs
│ │ │  dWUgIiwie30ifSwiLCAiLFNQQU57ImFuIG9wdGlvbiB3aGljaCBkZXNpZ25hdGVzIHN5bWJvbHMv
│ │ │  c3RyaW5ncy92YXJpYWJsZXMgdGhhdCB3aWxsIGJlIHNldCB0byBiZSBhIHJhbmRvbSByZWFsIG51
│ │ │  bWJlciBvciByYW5kb20gY29tcGxleCBudW1iZXIifX0sU1BBTntUTzJ7bmV3IERvY3VtZW50VGFn
│ │ │  IGZyb20geyJUb3BEaXJlY3RvcnkiLCJUb3BEaXJlY3RvcnkiLCJCZXJ0aW5pIn0sIlRvcERpcmVj
│ │ │  dG9yeSJ9LFRUeyIgPT4gIn0sVFR7Ii4uLiJ9LCIsICIsU1BBTnsiZGVmYXVsdCB2YWx1ZSAiLCJc
│ │ │ -Ii90bXAvTTItMjg3MDYtMC8wXCIifSwiLCAiLFNQQU57Ik9wdGlvbiB0byBjaGFuZ2UgZGlyZWN0
│ │ │ +Ii90bXAvTTItNDEwMzctMC8wXCIifSwiLCAiLFNQQU57Ik9wdGlvbiB0byBjaGFuZ2UgZGlyZWN0
│ │ │  b3J5IGZvciBmaWxlIHN0b3JhZ2UuIn19LFNQQU57VE8ye25ldyBEb2N1bWVudFRhZyBmcm9tIHtb
│ │ │  YmVydGluaVBhcmFtZXRlckhvbW90b3B5LFZlcmJvc2VdLCJiZXJ0aW5pUGFyYW1ldGVySG9tb3Rv
│ │ │  cHkoLi4uLFZlcmJvc2U9Pi4uLikiLCJCZXJ0aW5pIn0sIlZlcmJvc2UifSxUVHsiID0+ICJ9LFRU
│ │ │  eyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwiZmFsc2UifSwiLCAiLFNQQU57Ik9w
│ │ │  dGlvbiB0byBzaWxlbmNlIGFkZGl0aW9uYWwgb3V0cHV0In19fSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsiYmVydGluaVBhcmFtZXRlckhvbW90b3B5IiwiYmVy
│ │ │  dGluaVBhcmFtZXRlckhvbW90b3B5IiwiQmVydGluaSJ9LCBLZXkgPT4gYmVydGluaVBhcmFtZXRl
│ │ │ @@ -2449,15 +2449,15 @@
│ │ │  YWw9Pi4uLikiLCJCZXJ0aW5pIn0sIlJhbmRvbVJlYWwifSxUVHsiID0+ICJ9LFRUeyIuLi4ifSwi
│ │ │  LCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwie30ifSwiLCAiLFNQQU57ImFuIG9wdGlvbiB3aGlj
│ │ │  aCBkZXNpZ25hdGVzIHN5bWJvbHMvc3RyaW5ncy92YXJpYWJsZXMgdGhhdCB3aWxsIGJlIHNldCB0
│ │ │  byBiZSBhIHJhbmRvbSByZWFsIG51bWJlciBvciByYW5kb20gY29tcGxleCBudW1iZXIifX0sU1BB
│ │ │  TntUTzJ7bmV3IERvY3VtZW50VGFnIGZyb20ge1tiZXJ0aW5pWmVyb0RpbVNvbHZlLFRvcERpcmVj
│ │ │  dG9yeV0sImJlcnRpbmlaZXJvRGltU29sdmUoLi4uLFRvcERpcmVjdG9yeT0+Li4uKSIsIkJlcnRp
│ │ │  bmkifSwiVG9wRGlyZWN0b3J5In0sVFR7IiA9PiAifSxUVHsiLi4uIn0sIiwgIixTUEFOeyJkZWZh
│ │ │ -dWx0IHZhbHVlICIsIlwiL3RtcC9NMi0yODcwNi0wLzBcIiJ9LCIsICIsU1BBTnsiT3B0aW9uIHRv
│ │ │ +dWx0IHZhbHVlICIsIlwiL3RtcC9NMi00MTAzNy0wLzBcIiJ9LCIsICIsU1BBTnsiT3B0aW9uIHRv
│ │ │  IGNoYW5nZSBkaXJlY3RvcnkgZm9yIGZpbGUgc3RvcmFnZS4ifX0sU1BBTntUTzJ7bmV3IERvY3Vt
│ │ │  ZW50VGFnIGZyb20ge1tiZXJ0aW5pWmVyb0RpbVNvbHZlLFVzZVJlZ2VuZXJhdGlvbl0sImJlcnRp
│ │ │  bmlaZXJvRGltU29sdmUoLi4uLFVzZVJlZ2VuZXJhdGlvbj0+Li4uKSIsIkJlcnRpbmkifSwiVXNl
│ │ │  UmVnZW5lcmF0aW9uIn0sVFR7IiA9PiAifSxUVHsiLi4uIn0sIiwgIixTUEFOeyJkZWZhdWx0IHZh
│ │ │  bHVlICIsIi0xIn0sIiwgIixTUEFOe319LFNQQU57VE8ye25ldyBEb2N1bWVudFRhZyBmcm9tIHtb
│ │ │  YmVydGluaVplcm9EaW1Tb2x2ZSxWZXJib3NlXSwiYmVydGluaVplcm9EaW1Tb2x2ZSguLi4sVmVy
│ │ │  Ym9zZT0+Li4uKSIsIkJlcnRpbmkifSwiVmVyYm9zZSJ9LFRUeyIgPT4gIn0sVFR7Ii4uLiJ9LCIs
│ │ ├── ./usr/share/doc/Macaulay2/Bertini/html/_bertini__Parameter__Homotopy.html
│ │ │ @@ -72,15 +72,15 @@
│ │ │              
  • HomVariableGroup => ..., default value {}, an option to group variables and use multihomogeneous homotopies
    • 
│ │ │              
  • M2Precision (missing documentation)
│ │ │   => ..., default value 53, 
    • 
│ │ │              
  • OutputStyle (missing documentation)
│ │ │   => ..., default value "OutPoints", 
    • 
│ │ │              
  • RandomComplex => ..., default value {}, an option which designates symbols/strings/variables that will be set to be a random real number or random complex number
    • 
│ │ │              
  • RandomReal => ..., default value {}, an option which designates symbols/strings/variables that will be set to be a random real number or random complex number
    • 
│ │ │ -            
  • TopDirectory => ..., default value "/tmp/M2-28706-0/0", Option to change directory for file storage.
    • 
│ │ │ +            
  • TopDirectory => ..., default value "/tmp/M2-41037-0/0", Option to change directory for file storage.
    • 
│ │ │              
  • Verbose => ..., default value false, Option to silence additional output
    • 
│ │ │            
│ │ │          
│ │ │          
  • Outputs:          
      
│ │ │              
    • S, a list, a list whose entries are lists of solutions for each target system
      • 
│ │ │            
    
│ │ │          
    • 
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │              "OutPoints",
│ │ │ │            o _R_a_n_d_o_m_C_o_m_p_l_e_x => ..., default value {}, an option which designates
│ │ │ │              symbols/strings/variables that will be set to be a random real
│ │ │ │              number or random complex number
│ │ │ │            o _R_a_n_d_o_m_R_e_a_l => ..., default value {}, an option which designates
│ │ │ │              symbols/strings/variables that will be set to be a random real
│ │ │ │              number or random complex number
│ │ │ │ -          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-28706-0/0", Option to
│ │ │ │ +          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-41037-0/0", Option to
│ │ │ │              change directory for file storage.
│ │ │ │            o _V_e_r_b_o_s_e => ..., default value false, Option to silence additional
│ │ │ │              output
│ │ │ │      * Outputs:
│ │ │ │            o S, a _l_i_s_t, a list whose entries are lists of solutions for each
│ │ │ │              target system
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ ├── ./usr/share/doc/Macaulay2/Bertini/html/_bertini__User__Homotopy.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │   => ..., default value 53, 
│ │ │              
  • OutputStyle (missing documentation)
│ │ │   => ..., default value "OutPoints", 
    • 
│ │ │              
  • RandomComplex (missing documentation)
│ │ │   => ..., default value {}, 
    • 
│ │ │              
  • RandomReal (missing documentation)
│ │ │   => ..., default value {}, 
    • 
│ │ │ -            
  • TopDirectory => ..., default value "/tmp/M2-28706-0/0", Option to change directory for file storage.
    • 
│ │ │ +            
  • TopDirectory => ..., default value "/tmp/M2-41037-0/0", Option to change directory for file storage.
    • 
│ │ │              
  • Verbose => ..., default value false, Option to silence additional output
    • 
│ │ │            
│ │ │          
│ │ │          
  • Outputs:          
      
│ │ │              
    • S0, a list, a list of solutions to the target system
      • 
│ │ │            
    
│ │ │          
    • 
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,15 +21,15 @@
│ │ │ │              value {},
│ │ │ │            o HomVariableGroup (missing documentation) => ..., default value {},
│ │ │ │            o M2Precision (missing documentation) => ..., default value 53,
│ │ │ │            o OutputStyle (missing documentation) => ..., default value
│ │ │ │              "OutPoints",
│ │ │ │            o RandomComplex (missing documentation) => ..., default value {},
│ │ │ │            o RandomReal (missing documentation) => ..., default value {},
│ │ │ │ -          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-28706-0/0", Option to
│ │ │ │ +          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-41037-0/0", Option to
│ │ │ │              change directory for file storage.
│ │ │ │            o _V_e_r_b_o_s_e => ..., default value false, Option to silence additional
│ │ │ │              output
│ │ │ │      * Outputs:
│ │ │ │            o S0, a _l_i_s_t, a list of solutions to the target system
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This method calls Bertini to track a user-defined homotopy. The user needs to
│ │ ├── ./usr/share/doc/Macaulay2/Bertini/html/_bertini__Zero__Dim__Solve.html
│ │ │ @@ -79,15 +79,15 @@
│ │ │   => ..., default value "main_data", 
│ │ │              
  • NameSolutionsFile (missing documentation)
│ │ │   => ..., default value "raw_solutions", 
    • 
│ │ │              
  • OutputStyle (missing documentation)
│ │ │   => ..., default value "OutPoints", 
    • 
│ │ │              
  • RandomComplex => ..., default value {}, an option which designates symbols/strings/variables that will be set to be a random real number or random complex number
    • 
│ │ │              
  • RandomReal => ..., default value {}, an option which designates symbols/strings/variables that will be set to be a random real number or random complex number
    • 
│ │ │ -            
  • TopDirectory => ..., default value "/tmp/M2-28706-0/0", Option to change directory for file storage.
    • 
│ │ │ +            
  • TopDirectory => ..., default value "/tmp/M2-41037-0/0", Option to change directory for file storage.
    • 
│ │ │              
  • UseRegeneration (missing documentation)
│ │ │   => ..., default value -1, 
    • 
│ │ │              
  • Verbose => ..., default value false, Option to silence additional output
    • 
│ │ │            
│ │ │          
│ │ │          
  • Outputs:          
      
│ │ │              
    • S, a list, a list of points that are contained in the variety of F
      • 
│ │ │ ├── html2text {}
│ │ │ │ @@ -32,15 +32,15 @@
│ │ │ │              "OutPoints",
│ │ │ │            o _R_a_n_d_o_m_C_o_m_p_l_e_x => ..., default value {}, an option which designates
│ │ │ │              symbols/strings/variables that will be set to be a random real
│ │ │ │              number or random complex number
│ │ │ │            o _R_a_n_d_o_m_R_e_a_l => ..., default value {}, an option which designates
│ │ │ │              symbols/strings/variables that will be set to be a random real
│ │ │ │              number or random complex number
│ │ │ │ -          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-28706-0/0", Option to
│ │ │ │ +          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-41037-0/0", Option to
│ │ │ │              change directory for file storage.
│ │ │ │            o UseRegeneration (missing documentation) => ..., default value -1,
│ │ │ │            o _V_e_r_b_o_s_e => ..., default value false, Option to silence additional
│ │ │ │              output
│ │ │ │      * Outputs:
│ │ │ │            o S, a _l_i_s_t, a list of points that are contained in the variety of F
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp1.out
│ │ │ @@ -76,15 +76,15 @@
│ │ │  i8 : A = action(RI,S7)
│ │ │  
│ │ │  o8 = Complex with 15 actors
│ │ │  
│ │ │  o8 : ActionOnComplex
│ │ │  
│ │ │  i9 : elapsedTime c = character A
│ │ │ - -- .559135s elapsed
│ │ │ + -- .394847s elapsed
│ │ │  
│ │ │  o9 = Character over R
│ │ │        
│ │ │       (0, {0}) => | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
│ │ │       (1, {2}) => | 0 -1 1 -1 0 0 0 -1 2 0 2 2 2 6 14 |
│ │ │       (2, {3}) => | 0 1 0 0 -1 1 -1 -1 -1 -1 -1 1 -1 5 35 |
│ │ │       (3, {4}) => | 0 -1 0 0 1 1 1 -1 -1 1 -1 -1 -1 -5 35 |
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp2.out
│ │ │ @@ -100,15 +100,15 @@
│ │ │  i6 : A=action(RI,S6)
│ │ │  
│ │ │  o6 = Complex with 11 actors
│ │ │  
│ │ │  o6 : ActionOnComplex
│ │ │  
│ │ │  i7 : elapsedTime c=character A
│ │ │ - -- .474916s elapsed
│ │ │ + -- .451823s elapsed
│ │ │  
│ │ │  o7 = Character over R
│ │ │        
│ │ │       (0, {0}) => | 1 1 1 1 1 1 1 1 1 1 1 |
│ │ │       (1, {5}) => | 0 1 0 2 0 1 3 0 2 4 6 |
│ │ │       (1, {7}) => | 0 0 0 0 0 1 3 0 4 16 60 |
│ │ │       (1, {9}) => | 0 0 0 0 2 2 2 0 4 8 20 |
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp3.out
│ │ │ @@ -187,27 +187,27 @@
│ │ │  i19 : A2 = action(RI2,G,Sub=>false)
│ │ │  
│ │ │  o19 = Complex with 6 actors
│ │ │  
│ │ │  o19 : ActionOnComplex
│ │ │  
│ │ │  i20 : elapsedTime a1 = character A1
│ │ │ - -- .852762s elapsed
│ │ │ + -- .692703s elapsed
│ │ │  
│ │ │  o20 = Character over R
│ │ │         
│ │ │        (0, {0}) => | 1 1 1 1 1 1 |
│ │ │        (1, {8}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │        (2, {11}) => | 1 1 1 1 1 1 |
│ │ │        (2, {13}) => | 1 1 1 1 1 1 |
│ │ │  
│ │ │  o20 : Character
│ │ │  
│ │ │  i21 : elapsedTime a2 = character A2
│ │ │ - -- 34.344s elapsed
│ │ │ + -- 25.4401s elapsed
│ │ │  
│ │ │  o21 = Character over R
│ │ │         
│ │ │        (0, {0}) => | 1 1 1 1 1 1 |
│ │ │        (1, {16}) => | 6 2 0 0 -1 -1 |
│ │ │        (2, {19}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │        (2, {21}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │ @@ -297,15 +297,15 @@
│ │ │  i31 : B = action(M,G,Sub=>false)
│ │ │  
│ │ │  o31 = Module with 6 actors
│ │ │  
│ │ │  o31 : ActionOnGradedModule
│ │ │  
│ │ │  i32 : elapsedTime b = character(B,21)
│ │ │ - -- 14.2617s elapsed
│ │ │ + -- 11.4504s elapsed
│ │ │  
│ │ │  o32 = Character over R
│ │ │         
│ │ │        (0, {21}) => | 1 1 1 1 1 1 |
│ │ │  
│ │ │  o32 : Character
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp1.html
│ │ │ @@ -162,15 +162,15 @@
│ │ │  
│ │ │  o8 : ActionOnComplex
    • 
│ │ │              
    
│ │ │                
    i9 : elapsedTime c = character A
│ │ │ - -- .559135s elapsed
│ │ │ + -- .394847s elapsed
│ │ │  
│ │ │  o9 = Character over R
│ │ │        
│ │ │       (0, {0}) => | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
│ │ │       (1, {2}) => | 0 -1 1 -1 0 0 0 -1 2 0 2 2 2 6 14 |
│ │ │       (2, {3}) => | 0 1 0 0 -1 1 -1 -1 -1 -1 -1 1 -1 5 35 |
│ │ │       (3, {4}) => | 0 -1 0 0 1 1 1 -1 -1 1 -1 -1 -1 -5 35 |
│ │ │ ├── html2text {}
│ │ │ │ @@ -91,15 +91,15 @@
│ │ │ │  o7 : List
│ │ │ │  i8 : A = action(RI,S7)
│ │ │ │  
│ │ │ │  o8 = Complex with 15 actors
│ │ │ │  
│ │ │ │  o8 : ActionOnComplex
│ │ │ │  i9 : elapsedTime c = character A
│ │ │ │ - -- .559135s elapsed
│ │ │ │ + -- .394847s elapsed
│ │ │ │  
│ │ │ │  o9 = Character over R
│ │ │ │  
│ │ │ │       (0, {0}) => | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
│ │ │ │       (1, {2}) => | 0 -1 1 -1 0 0 0 -1 2 0 2 2 2 6 14 |
│ │ │ │       (2, {3}) => | 0 1 0 0 -1 1 -1 -1 -1 -1 -1 1 -1 5 35 |
│ │ │ │       (3, {4}) => | 0 -1 0 0 1 1 1 -1 -1 1 -1 -1 -1 -5 35 |
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp2.html
│ │ │ @@ -180,15 +180,15 @@
│ │ │  
│ │ │  o6 : ActionOnComplex
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime c=character A
│ │ │ - -- .474916s elapsed
│ │ │ + -- .451823s elapsed
│ │ │  
│ │ │  o7 = Character over R
│ │ │        
│ │ │       (0, {0}) => | 1 1 1 1 1 1 1 1 1 1 1 |
│ │ │       (1, {5}) => | 0 1 0 2 0 1 3 0 2 4 6 |
│ │ │       (1, {7}) => | 0 0 0 0 0 1 3 0 4 16 60 |
│ │ │       (1, {9}) => | 0 0 0 0 2 2 2 0 4 8 20 |
│ │ │ ├── html2text {}
│ │ │ │ @@ -113,15 +113,15 @@
│ │ │ │  o5 : List
│ │ │ │  i6 : A=action(RI,S6)
│ │ │ │  
│ │ │ │  o6 = Complex with 11 actors
│ │ │ │  
│ │ │ │  o6 : ActionOnComplex
│ │ │ │  i7 : elapsedTime c=character A
│ │ │ │ - -- .474916s elapsed
│ │ │ │ + -- .451823s elapsed
│ │ │ │  
│ │ │ │  o7 = Character over R
│ │ │ │  
│ │ │ │       (0, {0}) => | 1 1 1 1 1 1 1 1 1 1 1 |
│ │ │ │       (1, {5}) => | 0 1 0 2 0 1 3 0 2 4 6 |
│ │ │ │       (1, {7}) => | 0 0 0 0 0 1 3 0 4 16 60 |
│ │ │ │       (1, {9}) => | 0 0 0 0 2 2 2 0 4 8 20 |
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp3.html
│ │ │ @@ -310,30 +310,30 @@
│ │ │  
│ │ │  o19 : ActionOnComplex
    
│ │ │              
    
│ │ │                
    i20 : elapsedTime a1 = character A1
│ │ │ - -- .852762s elapsed
│ │ │ + -- .692703s elapsed
│ │ │  
│ │ │  o20 = Character over R
│ │ │         
│ │ │        (0, {0}) => | 1 1 1 1 1 1 |
│ │ │        (1, {8}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │        (2, {11}) => | 1 1 1 1 1 1 |
│ │ │        (2, {13}) => | 1 1 1 1 1 1 |
│ │ │  
│ │ │  o20 : Character
    
│ │ │              
    
│ │ │                
    i21 : elapsedTime a2 = character A2
│ │ │ - -- 34.344s elapsed
│ │ │ + -- 25.4401s elapsed
│ │ │  
│ │ │  o21 = Character over R
│ │ │         
│ │ │        (0, {0}) => | 1 1 1 1 1 1 |
│ │ │        (1, {16}) => | 6 2 0 0 -1 -1 |
│ │ │        (2, {19}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │        (2, {21}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │ @@ -467,15 +467,15 @@
│ │ │  
│ │ │  o31 : ActionOnGradedModule
    
│ │ │              
    
│ │ │                
    i32 : elapsedTime b = character(B,21)
│ │ │ - -- 14.2617s elapsed
│ │ │ + -- 11.4504s elapsed
│ │ │  
│ │ │  o32 = Character over R
│ │ │         
│ │ │        (0, {21}) => | 1 1 1 1 1 1 |
│ │ │  
│ │ │  o32 : Character
    
│ │ │              
    
│ │ │                
    i23 : time j=bruns F.dd_3;
│ │ │ - -- used 0.354218s (cpu); 0.289007s (thread); 0s (gc)
│ │ │ + -- used 0.265543s (cpu); 0.19889s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of S
    
│ │ │              
    
│ │ │                
    i24 : betti res j
│ │ │ ├── html2text {}
│ │ │ │ @@ -230,15 +230,15 @@
│ │ │ │  o22 = total: 1 5 8 5 1
│ │ │ │            0: 1 . . . .
│ │ │ │            1: . 4 2 . .
│ │ │ │            2: . 1 6 5 1
│ │ │ │  
│ │ │ │  o22 : BettiTally
│ │ │ │  i23 : time j=bruns F.dd_3;
│ │ │ │ - -- used 0.354218s (cpu); 0.289007s (thread); 0s (gc)
│ │ │ │ + -- used 0.265543s (cpu); 0.19889s (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__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 y, Cell of dimension 0 with label x, Cell of dimension 0 with label z}}
│ │ │                 1 => {Cell of dimension 1 with label x*y}
│ │ │  
│ │ │  o7 : HashTable
│ │ │  
│ │ │  i8 : R = QQ;
│ │ │  
│ │ │  i9 : P = convexHull matrix {{1,1,-1,-1},{1,-1,1,-1}};
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_cells_lp__Z__Z_cm__Cell__Complex_rp.out
│ │ │ @@ -10,17 +10,17 @@
│ │ │  
│ │ │  i5 : exy = newSimplexCell {vx,vy};
│ │ │  
│ │ │  i6 : C = cellComplex(R,{exy,vz});
│ │ │  
│ │ │  i7 : cells(0,C)
│ │ │  
│ │ │ -o7 = {Cell of dimension 0 with label y, Cell of dimension 0 with label x,
│ │ │ +o7 = {Cell of dimension 0 with label x, Cell of dimension 0 with label z,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     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
│ │ │ @@ -22,20 +22,20 @@
│ │ │  
│ │ │  i11 : T = new HashTable from {v0 => a^2*b, v1 => b*c^2, v2 => b^2, v3 => a*c};
│ │ │  
│ │ │  i12 : relabeledC = relabelCellComplex(C,T);
│ │ │  
│ │ │  i13 : for c in cells(0,relabeledC) list cellLabel(c)
│ │ │  
│ │ │ -        2      2   2
│ │ │ -o13 = {a b, b*c , b , a*c}
│ │ │ +          2   2        2
│ │ │ +o13 = {b*c , b , a*c, a b}
│ │ │  
│ │ │  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*c , a b }
│ │ │  
│ │ │  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__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 y, Cell of dimension 0 with label x, Cell of dimension 0 with label z}}
│ │ │                 1 => {Cell of dimension 1 with label x*y}
│ │ │  
│ │ │  o7 : HashTable
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │  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}}
│ │ │ │ +label x, Cell of dimension 0 with label z}}
│ │ │ │                 1 => {Cell of dimension 1 with label x*y}
│ │ │ │  
│ │ │ │  o7 : HashTable
│ │ │ │  i8 : R = QQ;
│ │ │ │  i9 : P = convexHull matrix {{1,1,-1,-1},{1,-1,1,-1}};
│ │ │ │  i10 : C = cellComplex(R,P);
│ │ │ │  i11 : cells C
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_cells_lp__Z__Z_cm__Cell__Complex_rp.html
│ │ │ @@ -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 x, Cell of dimension 0 with label z,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     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 x, Cell of dimension 0 with label z,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     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
│ │ │ @@ -136,26 +136,26 @@
│ │ │                
    i12 : relabeledC = relabelCellComplex(C,T);
    
│ │ │              
    
│ │ │                
    i13 : for c in cells(0,relabeledC) list cellLabel(c)
│ │ │  
│ │ │ -        2      2   2
│ │ │ -o13 = {a b, b*c , b , a*c}
│ │ │ +          2   2        2
│ │ │ +o13 = {b*c , b , a*c, a b}
│ │ │  
│ │ │  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*c , a b }
│ │ │  
│ │ │  o14 : List
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -31,22 +31,22 @@
│ │ │ │  i8 : v1 = verts#1;
│ │ │ │  i9 : v2 = verts#2;
│ │ │ │  i10 : v3 = verts#3;
│ │ │ │  i11 : T = new HashTable from {v0 => a^2*b, v1 => b*c^2, v2 => b^2, v3 => a*c};
│ │ │ │  i12 : relabeledC = relabelCellComplex(C,T);
│ │ │ │  i13 : for c in cells(0,relabeledC) list cellLabel(c)
│ │ │ │  
│ │ │ │ -        2      2   2
│ │ │ │ -o13 = {a b, b*c , b , a*c}
│ │ │ │ +          2   2        2
│ │ │ │ +o13 = {b*c , b , a*c, a b}
│ │ │ │  
│ │ │ │  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*c , a b }
│ │ │ │  
│ │ │ │  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 0.297722s (cpu); 0.239584s (thread); 0s (gc)
│ │ │ + -- used 0.3389s (cpu); 0.263959s (thread); 0s (gc)
│ │ │  
│ │ │  i14 : isQuasiIsomorphism m
│ │ │  
│ │ │  o14 = true
│ │ │  
│ │ │  i15 : E[1] == source m
│ │ ├── ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_resolution__Of__Chain__Complex.out
│ │ │ @@ -27,18 +27,18 @@
│ │ │  i5 : C = res(R^1/(ideal vars R))**(R^1/(ideal vars R)^5);
│ │ │  
│ │ │  i6 : mods = for i from 0 to max C list pushForward(f, C_i);
│ │ │  
│ │ │  i7 : C = chainComplex for i from min C+1 to max C list map(mods_(i-1),mods_i,substitute(matrix C.dd_i,S));
│ │ │  
│ │ │  i8 : time m = resolutionOfChainComplex C;
│ │ │ - -- used 0.0968336s (cpu); 0.0968327s (thread); 0s (gc)
│ │ │ + -- used 0.103578s (cpu); 0.103579s (thread); 0s (gc)
│ │ │  
│ │ │  i9 : time n = cartanEilenbergResolution C;
│ │ │ - -- used 0.226002s (cpu); 0.156796s (thread); 0s (gc)
│ │ │ + -- used 0.24087s (cpu); 0.164874s (thread); 0s (gc)
│ │ │  
│ │ │  i10 : betti source m
│ │ │  
│ │ │               0  1  2   3   4   5   6   7
│ │ │  o10 = total: 1 19 80 181 312 484 447 156
│ │ │            0: 1  3  3   1   .   .   .   .
│ │ │            1: .  .  1   3   3   .   .   .
│ │ ├── ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_minimize_lp__Chain__Complex_rp.html
│ │ │ @@ -181,15 +181,15 @@
│ │ │          
    
│ │ │            
    Now we minimize the result. The free summand we added to the end maps to zero, and thus is part of the minimization.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ @@ -301,15 +301,15 @@
│ │ │  
│ │ │  o21 : A
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -160,15 +160,15 @@
│ │ │ │  
│ │ │ │                2              2
│ │ │ │  o14 = ideal (x x  - x x x , x x )
│ │ │ │                0 3    1 2 4   2 5
│ │ │ │  
│ │ │ │  o14 : Ideal of R
│ │ │ │  i15 : time csmK=CSM(A,K)
│ │ │ │ - -- used 0.984576s (cpu); 0.463326s (thread); 0s (gc)
│ │ │ │ + -- used 1.23554s (cpu); 0.414859s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          2 2     2         2    2            2
│ │ │ │  o15 = 7h h  + 5h h  + 4h h  + h  + 3h h  + h
│ │ │ │          1 2     1 2     1 2    1     1 2    2
│ │ │ │  
│ │ │ │  o15 : A
│ │ │ │  i16 : csmKHash= CSM(A,K,Output=>HashForm)
│ │ │ │ @@ -199,15 +199,15 @@
│ │ │ │  
│ │ │ │          2 2     2         2     2             2
│ │ │ │  o21 = 9h h  + 9h h  + 9h h  + 3h  + 7h h  + 3h  + 3h  + 2h
│ │ │ │          1 2     1 2     1 2     1     1 2     2     1     2
│ │ │ │  
│ │ │ │  o21 : A
│ │ │ │  i22 : time CSM(A,K,m)
│ │ │ │ - -- used 0.111403s (cpu); 0.0580925s (thread); 0s (gc)
│ │ │ │ + -- used 0.107807s (cpu); 0.0734502s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          2 2     2         2    2            2
│ │ │ │  o22 = 7h h  + 5h h  + 4h h  + h  + 3h h  + h
│ │ │ │          1 2     1 2     1 2    1     1 2    2
│ │ │ │  
│ │ │ │  o22 : A
│ │ │ │  In the case where the ambient space is a toric variety which is not a product
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Check__Smooth.html
│ │ │ @@ -72,15 +72,15 @@
│ │ │  
│ │ │  o2 : NormalToricVariety
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i13 : time m = minimize (E[1]);
│ │ │ - -- used 0.297722s (cpu); 0.239584s (thread); 0s (gc)
    
│ │ │ + -- used 0.3389s (cpu); 0.263959s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i14 : isQuasiIsomorphism m
│ │ │  
│ │ │  o14 = true
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -81,15 +81,15 @@
│ │ │ │  o11 : ChainComplex
│ │ │ │  i12 : isMinimalChainComplex E
│ │ │ │  
│ │ │ │  o12 = false
│ │ │ │  Now we minimize the result. The free summand we added to the end maps to zero,
│ │ │ │  and thus is part of the minimization.
│ │ │ │  i13 : time m = minimize (E[1]);
│ │ │ │ - -- used 0.297722s (cpu); 0.239584s (thread); 0s (gc)
│ │ │ │ + -- used 0.3389s (cpu); 0.263959s (thread); 0s (gc)
│ │ │ │  i14 : isQuasiIsomorphism m
│ │ │ │  
│ │ │ │  o14 = true
│ │ │ │  i15 : E[1] == source m
│ │ │ │  
│ │ │ │  o15 = true
│ │ │ │  i16 : E' = target m
│ │ ├── ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_resolution__Of__Chain__Complex.html
│ │ │ @@ -129,21 +129,21 @@
│ │ │                  		
│ │ │                
    i7 : C = chainComplex for i from min C+1 to max C list map(mods_(i-1),mods_i,substitute(matrix C.dd_i,S));
    
│ │ │              
    
│ │ │                
    i8 : time m = resolutionOfChainComplex C;
│ │ │ - -- used 0.0968336s (cpu); 0.0968327s (thread); 0s (gc)
    
│ │ │ + -- used 0.103578s (cpu); 0.103579s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i9 : time n = cartanEilenbergResolution C;
│ │ │ - -- used 0.226002s (cpu); 0.156796s (thread); 0s (gc)
    
│ │ │ + -- used 0.24087s (cpu); 0.164874s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i10 : betti source m
│ │ │  
│ │ │               0  1  2   3   4   5   6   7
│ │ │ ├── html2text {}
│ │ │ │ @@ -49,17 +49,17 @@
│ │ │ │  
│ │ │ │  o4 : RingMap R <-- S
│ │ │ │  i5 : C = res(R^1/(ideal vars R))**(R^1/(ideal vars R)^5);
│ │ │ │  i6 : mods = for i from 0 to max C list pushForward(f, C_i);
│ │ │ │  i7 : C = chainComplex for i from min C+1 to max C list map(mods_(i-
│ │ │ │  1),mods_i,substitute(matrix C.dd_i,S));
│ │ │ │  i8 : time m = resolutionOfChainComplex C;
│ │ │ │ - -- used 0.0968336s (cpu); 0.0968327s (thread); 0s (gc)
│ │ │ │ + -- used 0.103578s (cpu); 0.103579s (thread); 0s (gc)
│ │ │ │  i9 : time n = cartanEilenbergResolution C;
│ │ │ │ - -- used 0.226002s (cpu); 0.156796s (thread); 0s (gc)
│ │ │ │ + -- used 0.24087s (cpu); 0.164874s (thread); 0s (gc)
│ │ │ │  i10 : betti source m
│ │ │ │  
│ │ │ │               0  1  2   3   4   5   6   7
│ │ │ │  o10 = total: 1 19 80 181 312 484 447 156
│ │ │ │            0: 1  3  3   1   .   .   .   .
│ │ │ │            1: .  .  1   3   3   .   .   .
│ │ │ │            2: .  1  3   3   2   .   .   .
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___C__S__M.out
│ │ │ @@ -83,15 +83,15 @@
│ │ │                2              2
│ │ │  o14 = ideal (x x  - x x x , x x )
│ │ │                0 3    1 2 4   2 5
│ │ │  
│ │ │  o14 : Ideal of R
│ │ │  
│ │ │  i15 : time csmK=CSM(A,K)
│ │ │ - -- used 0.984576s (cpu); 0.463326s (thread); 0s (gc)
│ │ │ + -- used 1.23554s (cpu); 0.414859s (thread); 0s (gc)
│ │ │  
│ │ │          2 2     2         2    2            2
│ │ │  o15 = 7h h  + 5h h  + 4h h  + h  + 3h h  + h
│ │ │          1 2     1 2     1 2    1     1 2    2
│ │ │  
│ │ │  o15 : A
│ │ │  
│ │ │ @@ -124,15 +124,15 @@
│ │ │          2 2     2         2     2             2
│ │ │  o21 = 9h h  + 9h h  + 9h h  + 3h  + 7h h  + 3h  + 3h  + 2h
│ │ │          1 2     1 2     1 2     1     1 2     2     1     2
│ │ │  
│ │ │  o21 : A
│ │ │  
│ │ │  i22 : time CSM(A,K,m)
│ │ │ - -- used 0.111403s (cpu); 0.0580925s (thread); 0s (gc)
│ │ │ + -- used 0.107807s (cpu); 0.0734502s (thread); 0s (gc)
│ │ │  
│ │ │          2 2     2         2    2            2
│ │ │  o22 = 7h h  + 5h h  + 4h h  + h  + 3h h  + h
│ │ │          1 2     1 2     1 2    1     1 2    2
│ │ │  
│ │ │  o22 : A
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Check__Smooth.out
│ │ │ @@ -9,28 +9,28 @@
│ │ │  i2 : U = toricProjectiveSpace 7
│ │ │  
│ │ │  o2 = U
│ │ │  
│ │ │  o2 : NormalToricVariety
│ │ │  
│ │ │  i3 : time CSM U
│ │ │ - -- used 0.246174s (cpu); 0.164624s (thread); 0s (gc)
│ │ │ + -- used 0.244192s (cpu); 0.163054s (thread); 0s (gc)
│ │ │  
│ │ │         7      6      5      4      3      2
│ │ │  o3 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │         7      7      7      7      7      7     7
│ │ │  
│ │ │                                                  ZZ[x ..x ]
│ │ │                                                      0   7
│ │ │  o3 : -----------------------------------------------------------------------------------------------
│ │ │       (x x x x x x x x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x )
│ │ │         0 1 2 3 4 5 6 7     0    1     0    2     0    3     0    4     0    5     0    6     0    7
│ │ │  
│ │ │  i4 : time CSM(U,CheckSmooth=>false)
│ │ │ - -- used 0.374312s (cpu); 0.295125s (thread); 0s (gc)
│ │ │ + -- used 0.413007s (cpu); 0.313062s (thread); 0s (gc)
│ │ │  
│ │ │         7      6      5      4      3      2
│ │ │  o4 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │         7      7      7      7      7      7     7
│ │ │  
│ │ │                                                  ZZ[x ..x ]
│ │ │                                                      0   7
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Comp__Method.out
│ │ │ @@ -18,29 +18,29 @@
│ │ │  i3 : R=ZZ/32749[v_0..v_5];
│ │ │  
│ │ │  i4 : I=ideal(4*v_3*v_1*v_2-8*v_1*v_3^2,v_5*(v_0*v_1*v_4-v_2^3));
│ │ │  
│ │ │  o4 : Ideal of R
│ │ │  
│ │ │  i5 : time CSM(I,CompMethod=>ProjectiveDegree)
│ │ │ - -- used 0.629302s (cpu); 0.301613s (thread); 0s (gc)
│ │ │ + -- used 0.866703s (cpu); 0.376965s (thread); 0s (gc)
│ │ │  
│ │ │         5      4      3      2
│ │ │  o5 = 6h  + 14h  + 14h  + 10h
│ │ │         1      1      1      1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o5 : ------
│ │ │          6
│ │ │         h
│ │ │          1
│ │ │  
│ │ │  i6 : time CSM(I,CompMethod=>PnResidual)
│ │ │ - -- used 2.13127s (cpu); 1.82455s (thread); 0s (gc)
│ │ │ + -- used 2.4222s (cpu); 2.14452s (thread); 0s (gc)
│ │ │  
│ │ │         5      4      3      2
│ │ │  o6 = 6H  + 14H  + 14H  + 10H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o6 : -----
│ │ │          6
│ │ │ @@ -53,29 +53,29 @@
│ │ │  i8 : S=QQ[s_0..s_3];
│ │ │  
│ │ │  i9 : K=ideal(4*s_3*s_2-s_2^2,(s_0*s_1*s_3-s_2^3));
│ │ │  
│ │ │  o9 : Ideal of S
│ │ │  
│ │ │  i10 : time CSM(K,CompMethod=>ProjectiveDegree)
│ │ │ - -- used 0.279106s (cpu); 0.195272s (thread); 0s (gc)
│ │ │ + -- used 0.342009s (cpu); 0.251525s (thread); 0s (gc)
│ │ │  
│ │ │          3     2
│ │ │  o10 = 3h  + 5h
│ │ │          1     1
│ │ │  
│ │ │        ZZ[h ]
│ │ │            1
│ │ │  o10 : ------
│ │ │           4
│ │ │          h
│ │ │           1
│ │ │  
│ │ │  i11 : time CSM(K,CompMethod=>PnResidual)
│ │ │ - -- used 0.0825143s (cpu); 0.0825209s (thread); 0s (gc)
│ │ │ + -- used 0.102148s (cpu); 0.10217s (thread); 0s (gc)
│ │ │  
│ │ │          3     2
│ │ │  o11 = 3H  + 5H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o11 : -----
│ │ │           4
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler.out
│ │ │ @@ -21,20 +21,20 @@
│ │ │               2                                                        2
│ │ │       - 14254x  - 11226x x  + 2653x x  + 12365x x  - 10226x x  - 12696x )
│ │ │               3         0 4        1 4         2 4         3 4         4
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : time Euler(I,InputIsSmooth=>true)
│ │ │ - -- used 0.0576959s (cpu); 0.0354928s (thread); 0s (gc)
│ │ │ + -- used 0.0677474s (cpu); 0.0448798s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 4
│ │ │  
│ │ │  i5 : time Euler I
│ │ │ - -- used 0.254114s (cpu); 0.146833s (thread); 0s (gc)
│ │ │ + -- used 0.306022s (cpu); 0.184479s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 4
│ │ │  
│ │ │  i6 : EulerIHash=Euler(I,Output=>HashForm);
│ │ │  
│ │ │  i7 : A=ring EulerIHash#"CSM"
│ │ │  
│ │ │ @@ -62,20 +62,20 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       - x x )
│ │ │          0 3
│ │ │  
│ │ │  o9 : Ideal of R
│ │ │  
│ │ │  i10 : time Euler(J,Method=>DirectCompleteInt)
│ │ │ - -- used 0.182967s (cpu); 0.0690199s (thread); 0s (gc)
│ │ │ + -- used 0.145412s (cpu); 0.0950761s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 2
│ │ │  
│ │ │  i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
│ │ │ - -- used 0.190575s (cpu); 0.0845693s (thread); 0s (gc)
│ │ │ + -- used 0.286246s (cpu); 0.13218s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = 2
│ │ │  
│ │ │  i12 : R=MultiProjCoordRing({2,2})
│ │ │  
│ │ │  o12 = R
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler__Affine.out
│ │ │ @@ -13,12 +13,12 @@
│ │ │              2    2    2
│ │ │  o3 = ideal(x  + x  + x  - 1)
│ │ │              1    2    3
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : time EulerAffine I
│ │ │ - -- used 0.0874131s (cpu); 0.0492159s (thread); 0s (gc)
│ │ │ + -- used 0.0757855s (cpu); 0.0618365s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 2
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Inds__Of__Smooth.out
│ │ │ @@ -7,29 +7,29 @@
│ │ │  o1 : PolynomialRing
│ │ │  
│ │ │  i2 : I=ideal(R_0*R_1*R_3-R_0^2*R_3,random({0,1},R),random({1,2},R));
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : time CSM(I,Method=>DirectCompletInt)
│ │ │ - -- used 2.58397s (cpu); 1.12003s (thread); 0s (gc)
│ │ │ + -- used 6.25189s (cpu); 1.40623s (thread); 0s (gc)
│ │ │  
│ │ │         2 2     2         2
│ │ │  o3 = 2h h  + 2h h  + 5h h
│ │ │         1 2     1 2     1 2
│ │ │  
│ │ │       ZZ[h ..h ]
│ │ │           1   2
│ │ │  o3 : ----------
│ │ │          3   3
│ │ │        (h , h )
│ │ │          1   2
│ │ │  
│ │ │  i4 : time CSM(I,Method=>DirectCompletInt,IndsOfSmooth=>{1,2})
│ │ │ - -- used 2.73505s (cpu); 1.2427s (thread); 0s (gc)
│ │ │ + -- used 6.11753s (cpu); 1.41944s (thread); 0s (gc)
│ │ │  
│ │ │         2 2     2         2
│ │ │  o4 = 2h h  + 2h h  + 5h h
│ │ │         1 2     1 2     1 2
│ │ │  
│ │ │       ZZ[h ..h ]
│ │ │           1   2
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Input__Is__Smooth.out
│ │ │ @@ -3,43 +3,43 @@
│ │ │  i1 : R = ZZ/32749[x_0..x_4];
│ │ │  
│ │ │  i2 : I=ideal(random(2,R),random(2,R),random(1,R));
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : time CSM I
│ │ │ - -- used 0.852243s (cpu); 0.438332s (thread); 0s (gc)
│ │ │ + -- used 1.02425s (cpu); 0.50674s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o3 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o3 : ------
│ │ │          5
│ │ │         h
│ │ │          1
│ │ │  
│ │ │  i4 : time CSM(I,InputIsSmooth=>true)
│ │ │ - -- used 0.0911228s (cpu); 0.0316495s (thread); 0s (gc)
│ │ │ + -- used 0.0775952s (cpu); 0.041121s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o4 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o4 : ------
│ │ │          5
│ │ │         h
│ │ │          1
│ │ │  
│ │ │  i5 : time Chern I
│ │ │ - -- used 0.0751141s (cpu); 0.0309694s (thread); 0s (gc)
│ │ │ + -- used 0.0656896s (cpu); 0.0406127s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o5 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Method.out
│ │ │ @@ -7,29 +7,29 @@
│ │ │  o1 : PolynomialRing
│ │ │  
│ │ │  i2 : I=ideal(random(2,R),random(1,R),R_0*R_1*R_6-R_0^3);
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : time CSM I
│ │ │ - -- used 1.71002s (cpu); 0.984712s (thread); 0s (gc)
│ │ │ + -- used 2.74846s (cpu); 1.07222s (thread); 0s (gc)
│ │ │  
│ │ │          5      4     3
│ │ │  o3 = 12h  + 10h  + 6h
│ │ │          1      1     1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o3 : ------
│ │ │          7
│ │ │         h
│ │ │          1
│ │ │  
│ │ │  i4 : time CSM(I,Method=>DirectCompleteInt)
│ │ │ - -- used 0.438602s (cpu); 0.209232s (thread); 0s (gc)
│ │ │ + -- used 0.6721s (cpu); 0.261928s (thread); 0s (gc)
│ │ │  
│ │ │          5      4     3
│ │ │  o4 = 12h  + 10h  + 6h
│ │ │          1      1     1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___C__S__M.html
│ │ │ @@ -234,15 +234,15 @@
│ │ │  
│ │ │  o14 : Ideal of R
    
│ │ │              
    
│ │ │                
    i15 : time csmK=CSM(A,K)
│ │ │ - -- used 0.984576s (cpu); 0.463326s (thread); 0s (gc)
│ │ │ + -- used 1.23554s (cpu); 0.414859s (thread); 0s (gc)
│ │ │  
│ │ │          2 2     2         2    2            2
│ │ │  o15 = 7h h  + 5h h  + 4h h  + h  + 3h h  + h
│ │ │          1 2     1 2     1 2    1     1 2    2
│ │ │  
│ │ │  o15 : A
    
│ │ │              
    
│ │ │                
    i22 : time CSM(A,K,m)
│ │ │ - -- used 0.111403s (cpu); 0.0580925s (thread); 0s (gc)
│ │ │ + -- used 0.107807s (cpu); 0.0734502s (thread); 0s (gc)
│ │ │  
│ │ │          2 2     2         2    2            2
│ │ │  o22 = 7h h  + 5h h  + 4h h  + h  + 3h h  + h
│ │ │          1 2     1 2     1 2    1     1 2    2
│ │ │  
│ │ │  o22 : A
    
│ │ │              
    
│ │ │                
    i3 : time CSM U
│ │ │ - -- used 0.246174s (cpu); 0.164624s (thread); 0s (gc)
│ │ │ + -- used 0.244192s (cpu); 0.163054s (thread); 0s (gc)
│ │ │  
│ │ │         7      6      5      4      3      2
│ │ │  o3 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │         7      7      7      7      7      7     7
│ │ │  
│ │ │                                                  ZZ[x ..x ]
│ │ │                                                      0   7
│ │ │ @@ -88,15 +88,15 @@
│ │ │       (x x x x x x x x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x )
│ │ │         0 1 2 3 4 5 6 7     0    1     0    2     0    3     0    4     0    5     0    6     0    7
    
│ │ │              
    
│ │ │                
    i4 : time CSM(U,CheckSmooth=>false)
│ │ │ - -- used 0.374312s (cpu); 0.295125s (thread); 0s (gc)
│ │ │ + -- used 0.413007s (cpu); 0.313062s (thread); 0s (gc)
│ │ │  
│ │ │         7      6      5      4      3      2
│ │ │  o4 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │         7      7      7      7      7      7     7
│ │ │  
│ │ │                                                  ZZ[x ..x ]
│ │ │                                                      0   7
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,30 +16,30 @@
│ │ │ │  o1 : Package
│ │ │ │  i2 : U = toricProjectiveSpace 7
│ │ │ │  
│ │ │ │  o2 = U
│ │ │ │  
│ │ │ │  o2 : NormalToricVariety
│ │ │ │  i3 : time CSM U
│ │ │ │ - -- used 0.246174s (cpu); 0.164624s (thread); 0s (gc)
│ │ │ │ + -- used 0.244192s (cpu); 0.163054s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         7      6      5      4      3      2
│ │ │ │  o3 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │ │         7      7      7      7      7      7     7
│ │ │ │  
│ │ │ │                                                  ZZ[x ..x ]
│ │ │ │                                                      0   7
│ │ │ │  o3 : --------------------------------------------------------------------------
│ │ │ │  ---------------------
│ │ │ │       (x x x x x x x x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x ,
│ │ │ │  - x  + x , - x  + x )
│ │ │ │         0 1 2 3 4 5 6 7     0    1     0    2     0    3     0    4     0    5
│ │ │ │  0    6     0    7
│ │ │ │  i4 : time CSM(U,CheckSmooth=>false)
│ │ │ │ - -- used 0.374312s (cpu); 0.295125s (thread); 0s (gc)
│ │ │ │ + -- used 0.413007s (cpu); 0.313062s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         7      6      5      4      3      2
│ │ │ │  o4 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │ │         7      7      7      7      7      7     7
│ │ │ │  
│ │ │ │                                                  ZZ[x ..x ]
│ │ │ │                                                      0   7
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Comp__Method.html
│ │ │ @@ -92,15 +92,15 @@
│ │ │  
│ │ │  o4 : Ideal of R
    
│ │ │              
    
│ │ │                
    i5 : time CSM(I,CompMethod=>ProjectiveDegree)
│ │ │ - -- used 0.629302s (cpu); 0.301613s (thread); 0s (gc)
│ │ │ + -- used 0.866703s (cpu); 0.376965s (thread); 0s (gc)
│ │ │  
│ │ │         5      4      3      2
│ │ │  o5 = 6h  + 14h  + 14h  + 10h
│ │ │         1      1      1      1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -109,15 +109,15 @@
│ │ │         h
│ │ │          1
    
│ │ │              
    
│ │ │                
    i6 : time CSM(I,CompMethod=>PnResidual)
│ │ │ - -- used 2.13127s (cpu); 1.82455s (thread); 0s (gc)
│ │ │ + -- used 2.4222s (cpu); 2.14452s (thread); 0s (gc)
│ │ │  
│ │ │         5      4      3      2
│ │ │  o6 = 6H  + 14H  + 14H  + 10H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o6 : -----
│ │ │          6
│ │ │ @@ -142,15 +142,15 @@
│ │ │  
│ │ │  o9 : Ideal of S
    
│ │ │              
    
│ │ │                
    i10 : time CSM(K,CompMethod=>ProjectiveDegree)
│ │ │ - -- used 0.279106s (cpu); 0.195272s (thread); 0s (gc)
│ │ │ + -- used 0.342009s (cpu); 0.251525s (thread); 0s (gc)
│ │ │  
│ │ │          3     2
│ │ │  o10 = 3h  + 5h
│ │ │          1     1
│ │ │  
│ │ │        ZZ[h ]
│ │ │            1
│ │ │ @@ -159,15 +159,15 @@
│ │ │          h
│ │ │           1
    
│ │ │              
    
│ │ │                
    i11 : time CSM(K,CompMethod=>PnResidual)
│ │ │ - -- used 0.0825143s (cpu); 0.0825209s (thread); 0s (gc)
│ │ │ + -- used 0.102148s (cpu); 0.10217s (thread); 0s (gc)
│ │ │  
│ │ │          3     2
│ │ │  o11 = 3H  + 5H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o11 : -----
│ │ │           4
│ │ │ ├── html2text {}
│ │ │ │ @@ -32,28 +32,28 @@
│ │ │ │  using the regenerative cascade implemented in Bertini. This is done by choosing
│ │ │ │  the option bertini, provided Bertini is _i_n_s_t_a_l_l_e_d_ _a_n_d_ _c_o_n_f_i_g_u_r_e_d.
│ │ │ │  i3 : R=ZZ/32749[v_0..v_5];
│ │ │ │  i4 : I=ideal(4*v_3*v_1*v_2-8*v_1*v_3^2,v_5*(v_0*v_1*v_4-v_2^3));
│ │ │ │  
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : time CSM(I,CompMethod=>ProjectiveDegree)
│ │ │ │ - -- used 0.629302s (cpu); 0.301613s (thread); 0s (gc)
│ │ │ │ + -- used 0.866703s (cpu); 0.376965s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         5      4      3      2
│ │ │ │  o5 = 6h  + 14h  + 14h  + 10h
│ │ │ │         1      1      1      1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ │ │  o5 : ------
│ │ │ │          6
│ │ │ │         h
│ │ │ │          1
│ │ │ │  i6 : time CSM(I,CompMethod=>PnResidual)
│ │ │ │ - -- used 2.13127s (cpu); 1.82455s (thread); 0s (gc)
│ │ │ │ + -- used 2.4222s (cpu); 2.14452s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         5      4      3      2
│ │ │ │  o6 = 6H  + 14H  + 14H  + 10H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o6 : -----
│ │ │ │          6
│ │ │ │ @@ -62,28 +62,28 @@
│ │ │ │  
│ │ │ │  o7 = 2
│ │ │ │  i8 : S=QQ[s_0..s_3];
│ │ │ │  i9 : K=ideal(4*s_3*s_2-s_2^2,(s_0*s_1*s_3-s_2^3));
│ │ │ │  
│ │ │ │  o9 : Ideal of S
│ │ │ │  i10 : time CSM(K,CompMethod=>ProjectiveDegree)
│ │ │ │ - -- used 0.279106s (cpu); 0.195272s (thread); 0s (gc)
│ │ │ │ + -- used 0.342009s (cpu); 0.251525s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          3     2
│ │ │ │  o10 = 3h  + 5h
│ │ │ │          1     1
│ │ │ │  
│ │ │ │        ZZ[h ]
│ │ │ │            1
│ │ │ │  o10 : ------
│ │ │ │           4
│ │ │ │          h
│ │ │ │           1
│ │ │ │  i11 : time CSM(K,CompMethod=>PnResidual)
│ │ │ │ - -- used 0.0825143s (cpu); 0.0825209s (thread); 0s (gc)
│ │ │ │ + -- used 0.102148s (cpu); 0.10217s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          3     2
│ │ │ │  o11 = 3H  + 5H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o11 : -----
│ │ │ │           4
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler.html
│ │ │ @@ -125,23 +125,23 @@
│ │ │  
│ │ │  o3 : Ideal of R
    
│ │ │              
    
│ │ │                
    i4 : time Euler(I,InputIsSmooth=>true)
│ │ │ - -- used 0.0576959s (cpu); 0.0354928s (thread); 0s (gc)
│ │ │ + -- used 0.0677474s (cpu); 0.0448798s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 4
    
│ │ │              
    
│ │ │                
    i5 : time Euler I
│ │ │ - -- used 0.254114s (cpu); 0.146833s (thread); 0s (gc)
│ │ │ + -- used 0.306022s (cpu); 0.184479s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 4
    
│ │ │              
    
│ │ │                
    i6 : EulerIHash=Euler(I,Output=>HashForm);
    
│ │ │ @@ -189,23 +189,23 @@
│ │ │          
    
│ │ │            
    Note that the ideal J above is a complete intersection, thus we may change the method option which may speed computation in some cases. We may also note that the ideal generated by the first 2 generators of I defines a smooth scheme and input this information into the method. This may also improve computation speed.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : time Euler(J,Method=>DirectCompleteInt)
│ │ │ - -- used 0.182967s (cpu); 0.0690199s (thread); 0s (gc)
│ │ │ + -- used 0.145412s (cpu); 0.0950761s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 2
    
│ │ │              
    
│ │ │                
    i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
│ │ │ - -- used 0.190575s (cpu); 0.0845693s (thread); 0s (gc)
│ │ │ + -- used 0.286246s (cpu); 0.13218s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = 2
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Now consider an example in \PP^2 \times \PP^2.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -74,19 +74,19 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │               2                                                        2
│ │ │ │       - 14254x  - 11226x x  + 2653x x  + 12365x x  - 10226x x  - 12696x )
│ │ │ │               3         0 4        1 4         2 4         3 4         4
│ │ │ │  
│ │ │ │  o3 : Ideal of R
│ │ │ │  i4 : time Euler(I,InputIsSmooth=>true)
│ │ │ │ - -- used 0.0576959s (cpu); 0.0354928s (thread); 0s (gc)
│ │ │ │ + -- used 0.0677474s (cpu); 0.0448798s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 4
│ │ │ │  i5 : time Euler I
│ │ │ │ - -- used 0.254114s (cpu); 0.146833s (thread); 0s (gc)
│ │ │ │ + -- used 0.306022s (cpu); 0.184479s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 4
│ │ │ │  i6 : EulerIHash=Euler(I,Output=>HashForm);
│ │ │ │  i7 : A=ring EulerIHash#"CSM"
│ │ │ │  
│ │ │ │  o7 = A
│ │ │ │  
│ │ │ │ @@ -114,19 +114,19 @@
│ │ │ │  o9 : Ideal of R
│ │ │ │  Note that the ideal J above is a complete intersection, thus we may change the
│ │ │ │  method option which may speed computation in some cases. We may also note that
│ │ │ │  the ideal generated by the first 2 generators of I defines a smooth scheme and
│ │ │ │  input this information into the method. This may also improve computation
│ │ │ │  speed.
│ │ │ │  i10 : time Euler(J,Method=>DirectCompleteInt)
│ │ │ │ - -- used 0.182967s (cpu); 0.0690199s (thread); 0s (gc)
│ │ │ │ + -- used 0.145412s (cpu); 0.0950761s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = 2
│ │ │ │  i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
│ │ │ │ - -- used 0.190575s (cpu); 0.0845693s (thread); 0s (gc)
│ │ │ │ + -- used 0.286246s (cpu); 0.13218s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = 2
│ │ │ │  Now consider an example in \PP^2 \times \PP^2.
│ │ │ │  i12 : R=MultiProjCoordRing({2,2})
│ │ │ │  
│ │ │ │  o12 = R
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler__Affine.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │              
    
│ │ │                
    i4 : time EulerAffine I
│ │ │ - -- used 0.0874131s (cpu); 0.0492159s (thread); 0s (gc)
│ │ │ + -- used 0.0757855s (cpu); 0.0618365s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 2
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Observe that the algorithm is a probabilistic algorithm and may give a wrong answer with a small but nonzero probability. Read more under probabilistic algorithm.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,15 +23,15 @@
│ │ │ │  
│ │ │ │              2    2    2
│ │ │ │  o3 = ideal(x  + x  + x  - 1)
│ │ │ │              1    2    3
│ │ │ │  
│ │ │ │  o3 : Ideal of R
│ │ │ │  i4 : time EulerAffine I
│ │ │ │ - -- used 0.0874131s (cpu); 0.0492159s (thread); 0s (gc)
│ │ │ │ + -- used 0.0757855s (cpu); 0.0618365s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 2
│ │ │ │  Observe that the algorithm is a probabilistic algorithm and may give a wrong
│ │ │ │  answer with a small but nonzero probability. Read more under _p_r_o_b_a_b_i_l_i_s_t_i_c
│ │ │ │  _a_l_g_o_r_i_t_h_m.
│ │ │ │  ********** WWaayyss ttoo uussee EEuulleerrAAffffiinnee:: **********
│ │ │ │      * EulerAffine(Ideal)
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Inds__Of__Smooth.html
│ │ │ @@ -70,15 +70,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time CSM(I,Method=>DirectCompletInt)
│ │ │ - -- used 2.58397s (cpu); 1.12003s (thread); 0s (gc)
│ │ │ + -- used 6.25189s (cpu); 1.40623s (thread); 0s (gc)
│ │ │  
│ │ │         2 2     2         2
│ │ │  o3 = 2h h  + 2h h  + 5h h
│ │ │         1 2     1 2     1 2
│ │ │  
│ │ │       ZZ[h ..h ]
│ │ │           1   2
│ │ │ @@ -87,15 +87,15 @@
│ │ │        (h , h )
│ │ │          1   2
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time CSM(I,Method=>DirectCompletInt,IndsOfSmooth=>{1,2})
│ │ │ - -- used 2.73505s (cpu); 1.2427s (thread); 0s (gc)
│ │ │ + -- used 6.11753s (cpu); 1.41944s (thread); 0s (gc)
│ │ │  
│ │ │         2 2     2         2
│ │ │  o4 = 2h h  + 2h h  + 5h h
│ │ │         1 2     1 2     1 2
│ │ │  
│ │ │       ZZ[h ..h ]
│ │ │           1   2
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,28 +16,28 @@
│ │ │ │  o1 = R
│ │ │ │  
│ │ │ │  o1 : PolynomialRing
│ │ │ │  i2 : I=ideal(R_0*R_1*R_3-R_0^2*R_3,random({0,1},R),random({1,2},R));
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : time CSM(I,Method=>DirectCompletInt)
│ │ │ │ - -- used 2.58397s (cpu); 1.12003s (thread); 0s (gc)
│ │ │ │ + -- used 6.25189s (cpu); 1.40623s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         2 2     2         2
│ │ │ │  o3 = 2h h  + 2h h  + 5h h
│ │ │ │         1 2     1 2     1 2
│ │ │ │  
│ │ │ │       ZZ[h ..h ]
│ │ │ │           1   2
│ │ │ │  o3 : ----------
│ │ │ │          3   3
│ │ │ │        (h , h )
│ │ │ │          1   2
│ │ │ │  i4 : time CSM(I,Method=>DirectCompletInt,IndsOfSmooth=>{1,2})
│ │ │ │ - -- used 2.73505s (cpu); 1.2427s (thread); 0s (gc)
│ │ │ │ + -- used 6.11753s (cpu); 1.41944s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         2 2     2         2
│ │ │ │  o4 = 2h h  + 2h h  + 5h h
│ │ │ │         1 2     1 2     1 2
│ │ │ │  
│ │ │ │       ZZ[h ..h ]
│ │ │ │           1   2
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Input__Is__Smooth.html
│ │ │ @@ -66,15 +66,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time CSM I
│ │ │ - -- used 0.852243s (cpu); 0.438332s (thread); 0s (gc)
│ │ │ + -- used 1.02425s (cpu); 0.50674s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o3 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -83,15 +83,15 @@
│ │ │         h
│ │ │          1
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time CSM(I,InputIsSmooth=>true)
│ │ │ - -- used 0.0911228s (cpu); 0.0316495s (thread); 0s (gc)
│ │ │ + -- used 0.0775952s (cpu); 0.041121s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o4 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -105,15 +105,15 @@
│ │ │          
    
│ │ │            
    Note that one could, equivalently, use the command Chern instead in this case.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : time Chern I
│ │ │ - -- used 0.0751141s (cpu); 0.0309694s (thread); 0s (gc)
│ │ │ + -- used 0.0656896s (cpu); 0.0406127s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o5 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,42 +9,42 @@
│ │ │ │  input ideal is known to define a smooth subscheme setting this option to true
│ │ │ │  will speed up computations (it is set to false by default).
│ │ │ │  i1 : R = ZZ/32749[x_0..x_4];
│ │ │ │  i2 : I=ideal(random(2,R),random(2,R),random(1,R));
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : time CSM I
│ │ │ │ - -- used 0.852243s (cpu); 0.438332s (thread); 0s (gc)
│ │ │ │ + -- used 1.02425s (cpu); 0.50674s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         3
│ │ │ │  o3 = 4h
│ │ │ │         1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ │ │  o3 : ------
│ │ │ │          5
│ │ │ │         h
│ │ │ │          1
│ │ │ │  i4 : time CSM(I,InputIsSmooth=>true)
│ │ │ │ - -- used 0.0911228s (cpu); 0.0316495s (thread); 0s (gc)
│ │ │ │ + -- used 0.0775952s (cpu); 0.041121s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         3
│ │ │ │  o4 = 4h
│ │ │ │         1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ │ │  o4 : ------
│ │ │ │          5
│ │ │ │         h
│ │ │ │          1
│ │ │ │  Note that one could, equivalently, use the command _C_h_e_r_n instead in this case.
│ │ │ │  i5 : time Chern I
│ │ │ │ - -- used 0.0751141s (cpu); 0.0309694s (thread); 0s (gc)
│ │ │ │ + -- used 0.0656896s (cpu); 0.0406127s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         3
│ │ │ │  o5 = 4h
│ │ │ │         1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Method.html
│ │ │ @@ -70,15 +70,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │              
    
│ │ │                
    i3 : time CSM I
│ │ │ - -- used 1.71002s (cpu); 0.984712s (thread); 0s (gc)
│ │ │ + -- used 2.74846s (cpu); 1.07222s (thread); 0s (gc)
│ │ │  
│ │ │          5      4     3
│ │ │  o3 = 12h  + 10h  + 6h
│ │ │          1      1     1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -87,15 +87,15 @@
│ │ │         h
│ │ │          1
    
│ │ │              
    
│ │ │                
    i4 : time CSM(I,Method=>DirectCompleteInt)
│ │ │ - -- used 0.438602s (cpu); 0.209232s (thread); 0s (gc)
│ │ │ + -- used 0.6721s (cpu); 0.261928s (thread); 0s (gc)
│ │ │  
│ │ │          5      4     3
│ │ │  o4 = 12h  + 10h  + 6h
│ │ │          1      1     1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,28 +18,28 @@
│ │ │ │  o1 = R
│ │ │ │  
│ │ │ │  o1 : PolynomialRing
│ │ │ │  i2 : I=ideal(random(2,R),random(1,R),R_0*R_1*R_6-R_0^3);
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : time CSM I
│ │ │ │ - -- used 1.71002s (cpu); 0.984712s (thread); 0s (gc)
│ │ │ │ + -- used 2.74846s (cpu); 1.07222s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          5      4     3
│ │ │ │  o3 = 12h  + 10h  + 6h
│ │ │ │          1      1     1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ │ │  o3 : ------
│ │ │ │          7
│ │ │ │         h
│ │ │ │          1
│ │ │ │  i4 : time CSM(I,Method=>DirectCompleteInt)
│ │ │ │ - -- used 0.438602s (cpu); 0.209232s (thread); 0s (gc)
│ │ │ │ + -- used 0.6721s (cpu); 0.261928s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          5      4     3
│ │ │ │  o4 = 12h  + 10h  + 6h
│ │ │ │          1      1     1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal__Net_lp__Hash__Table_cm__Hash__Table_cm__Elim__Tree_cm__Digraph_rp.out
│ │ │ @@ -16,32 +16,32 @@
│ │ │  
│ │ │  o2 : Digraph
│ │ │  
│ │ │  i3 : G = chordalGraph digraph hashTable{a=>{b,c},b=>{c},c=>{d},d=>{}};
│ │ │  
│ │ │  i4 : tree = elimTree G
│ │ │  
│ │ │ -o4 = ElimTree{a => b   }
│ │ │ +o4 = ElimTree{a => c}
│ │ │                b => c
│ │ │                c => d
│ │ │ -              d => null
│ │ │ +              d => b
│ │ │  
│ │ │  o4 : ElimTree
│ │ │  
│ │ │  i5 : rnk = hashTable{"a0"=>a, "a1"=>a, "b0"=>b, "b1"=>b, "b2"=>b,
│ │ │                       "c0"=>c, "d0"=>d, "c1"=>c, "d1"=>d};
│ │ │  
│ │ │  i6 : eqs = hashTable{"a0" => ({a},{}), "a1" => ({},{}),
│ │ │                       "b0" => ({b},{}), "b1" => ({},{}), "b2" => ({b},{}),
│ │ │                       "c0" => ({c},{}), "c1" => ({},{}),
│ │ │                       "d0" => ({},{}), "d1" => ({d},{}) };
│ │ │  
│ │ │  i7 : chordalNet(eqs,rnk,tree,DG)
│ │ │  
│ │ │ -o7 = ChordalNet{ d => { , d}    }
│ │ │ -                 b => {b,  , b}
│ │ │ -                 a => {a,  }
│ │ │ +o7 = ChordalNet{ a => {a,  }    }
│ │ │                   c => { , c}
│ │ │ +                 d => { , d}
│ │ │ +                 b => {b,  , b}
│ │ │  
│ │ │  o7 : ChordalNet
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Chordal/html/_chordal__Net_lp__Hash__Table_cm__Hash__Table_cm__Elim__Tree_cm__Digraph_rp.html
│ │ │ @@ -102,18 +102,18 @@
│ │ │                
    i3 : G = chordalGraph digraph hashTable{a=>{b,c},b=>{c},c=>{d},d=>{}};
    
│ │ │              
    
│ │ │                
    i4 : tree = elimTree G
│ │ │  
│ │ │ -o4 = ElimTree{a => b   }
│ │ │ +o4 = ElimTree{a => c}
│ │ │                b => c
│ │ │                c => d
│ │ │ -              d => null
│ │ │ +              d => b
│ │ │  
│ │ │  o4 : ElimTree
    
│ │ │              
    
│ │ │                
    i5 : rnk = hashTable{"a0"=>a, "a1"=>a, "b0"=>b, "b1"=>b, "b2"=>b,
│ │ │ @@ -128,18 +128,18 @@
│ │ │                       "d0" => ({},{}), "d1" => ({d},{}) };
    
│ │ │              
    
│ │ │                
    i7 : chordalNet(eqs,rnk,tree,DG)
│ │ │  
│ │ │ -o7 = ChordalNet{ d => { , d}    }
│ │ │ -                 b => {b,  , b}
│ │ │ -                 a => {a,  }
│ │ │ +o7 = ChordalNet{ a => {a,  }    }
│ │ │                   c => { , c}
│ │ │ +                 d => { , d}
│ │ │ +                 b => {b,  , b}
│ │ │  
│ │ │  o7 : ChordalNet
    
│ │ │              
    
│ │ │          
    
│ │ │        
│ │ │ ├── html2text {}
│ │ │ │ @@ -32,32 +32,32 @@
│ │ │ │               d0 => {}
│ │ │ │               d1 => {}
│ │ │ │  
│ │ │ │  o2 : Digraph
│ │ │ │  i3 : G = chordalGraph digraph hashTable{a=>{b,c},b=>{c},c=>{d},d=>{}};
│ │ │ │  i4 : tree = elimTree G
│ │ │ │  
│ │ │ │ -o4 = ElimTree{a => b   }
│ │ │ │ +o4 = ElimTree{a => c}
│ │ │ │                b => c
│ │ │ │                c => d
│ │ │ │ -              d => null
│ │ │ │ +              d => b
│ │ │ │  
│ │ │ │  o4 : ElimTree
│ │ │ │  i5 : rnk = hashTable{"a0"=>a, "a1"=>a, "b0"=>b, "b1"=>b, "b2"=>b,
│ │ │ │                       "c0"=>c, "d0"=>d, "c1"=>c, "d1"=>d};
│ │ │ │  i6 : eqs = hashTable{"a0" => ({a},{}), "a1" => ({},{}),
│ │ │ │                       "b0" => ({b},{}), "b1" => ({},{}), "b2" => ({b},{}),
│ │ │ │                       "c0" => ({c},{}), "c1" => ({},{}),
│ │ │ │                       "d0" => ({},{}), "d1" => ({d},{}) };
│ │ │ │  i7 : chordalNet(eqs,rnk,tree,DG)
│ │ │ │  
│ │ │ │ -o7 = ChordalNet{ d => { , d}    }
│ │ │ │ -                 b => {b,  , b}
│ │ │ │ -                 a => {a,  }
│ │ │ │ +o7 = ChordalNet{ a => {a,  }    }
│ │ │ │                   c => { , c}
│ │ │ │ +                 d => { , d}
│ │ │ │ +                 b => {b,  , b}
│ │ │ │  
│ │ │ │  o7 : ChordalNet
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_i_s_p_l_a_y_N_e_t -- displays a chordal network using Graphivz
│ │ │ │      * _d_i_g_r_a_p_h_(_C_h_o_r_d_a_l_N_e_t_) -- digraph associated to a chordal network
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _c_h_o_r_d_a_l_N_e_t_(_H_a_s_h_T_a_b_l_e_,_H_a_s_h_T_a_b_l_e_,_E_l_i_m_T_r_e_e_,_D_i_g_r_a_p_h_) -- construct chordal
│ │ ├── ./usr/share/doc/Macaulay2/CohomCalg/example-output/___Cohom__Calg.out
│ │ │ @@ -184,15 +184,15 @@
│ │ │        {0, -1, 0, 0, 0, -1}, {0, 0, -1, 0, 0, -1}, {0, 0, 0, -1, 0, -1}, {0,
│ │ │        -----------------------------------------------------------------------
│ │ │        0, 0, 0, -1, -1}}
│ │ │  
│ │ │  o19 : List
│ │ │  
│ │ │  i20 : elapsedTime hvecs = cohomCalg(X, D2)
│ │ │ - -- 2.67408s elapsed
│ │ │ + -- 3.11184s elapsed
│ │ │  
│ │ │  o20 = {{0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0},
│ │ │        -----------------------------------------------------------------------
│ │ │        {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0}, {0,
│ │ │        -----------------------------------------------------------------------
│ │ │        0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -265,45 +265,45 @@
│ │ │  i22 : degree(X_3 + X_7 + X_8)
│ │ │  
│ │ │  o22 = {0, 0, 1, 2, 0, -1}
│ │ │  
│ │ │  o22 : List
│ │ │  
│ │ │  i23 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 + X_8)
│ │ │ - -- .302915s elapsed
│ │ │ + -- .514668s elapsed
│ │ │  
│ │ │  o23 = {1, 0, 0, 0, 0}
│ │ │  
│ │ │  o23 : List
│ │ │  
│ │ │  i24 : elapsedTime cohomvec2 = for j from 0 to dim X list rank HH^j(X, OO_X(0,0,1,2,0,-1))
│ │ │ - -- 10.9792s elapsed
│ │ │ + -- 9.21423s elapsed
│ │ │  
│ │ │  o24 = {1, 0, 0, 0, 0}
│ │ │  
│ │ │  o24 : List
│ │ │  
│ │ │  i25 : assert(cohomvec1 == cohomvec2)
│ │ │  
│ │ │  i26 : degree(X_3 + X_7 - X_8)
│ │ │  
│ │ │  o26 = {0, 0, 1, 2, -2, -1}
│ │ │  
│ │ │  o26 : List
│ │ │  
│ │ │  i27 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 - X_8)
│ │ │ - -- .342126s elapsed
│ │ │ + -- .530444s elapsed
│ │ │  
│ │ │  o27 = {0, 0, 0, 0, 0}
│ │ │  
│ │ │  o27 : List
│ │ │  
│ │ │  i28 : elapsedTime cohomvec2 = elapsedTime for j from 0 to dim X list rank HH^j(X, OO_X(0,0,1,2,-2,-1))
│ │ │ - -- .517346s elapsed
│ │ │ - -- .517373s elapsed
│ │ │ + -- .492547s elapsed
│ │ │ + -- .492574s elapsed
│ │ │  
│ │ │  o28 = {0, 0, 0, 0, 0}
│ │ │  
│ │ │  o28 : List
│ │ │  
│ │ │  i29 : assert(cohomvec1 == cohomvec2)
│ │ ├── ./usr/share/doc/Macaulay2/CohomCalg/html/index.html
│ │ │ @@ -309,15 +309,15 @@
│ │ │  
│ │ │  o19 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i20 : elapsedTime hvecs = cohomCalg(X, D2)
│ │ │ - -- 2.67408s elapsed
│ │ │ + -- 3.11184s elapsed
│ │ │  
│ │ │  o20 = {{0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0},
│ │ │        -----------------------------------------------------------------------
│ │ │        {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0}, {0,
│ │ │        -----------------------------------------------------------------------
│ │ │        0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -399,25 +399,25 @@
│ │ │  
│ │ │  o22 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i23 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 + X_8)
│ │ │ - -- .302915s elapsed
│ │ │ + -- .514668s elapsed
│ │ │  
│ │ │  o23 = {1, 0, 0, 0, 0}
│ │ │  
│ │ │  o23 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i24 : elapsedTime cohomvec2 = for j from 0 to dim X list rank HH^j(X, OO_X(0,0,1,2,0,-1))
│ │ │ - -- 10.9792s elapsed
│ │ │ + -- 9.21423s elapsed
│ │ │  
│ │ │  o24 = {1, 0, 0, 0, 0}
│ │ │  
│ │ │  o24 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -433,26 +433,26 @@
│ │ │  
│ │ │  o26 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i27 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 - X_8)
│ │ │ - -- .342126s elapsed
│ │ │ + -- .530444s elapsed
│ │ │  
│ │ │  o27 = {0, 0, 0, 0, 0}
│ │ │  
│ │ │  o27 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i28 : elapsedTime cohomvec2 = elapsedTime for j from 0 to dim X list rank HH^j(X, OO_X(0,0,1,2,-2,-1))
│ │ │ - -- .517346s elapsed
│ │ │ - -- .517373s elapsed
│ │ │ + -- .492547s elapsed
│ │ │ + -- .492574s elapsed
│ │ │  
│ │ │  o28 = {0, 0, 0, 0, 0}
│ │ │  
│ │ │  o28 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -182,15 +182,15 @@
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        {0, -1, 0, 0, 0, -1}, {0, 0, -1, 0, 0, -1}, {0, 0, 0, -1, 0, -1}, {0,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        0, 0, 0, -1, -1}}
│ │ │ │  
│ │ │ │  o19 : List
│ │ │ │  i20 : elapsedTime hvecs = cohomCalg(X, D2)
│ │ │ │ - -- 2.67408s elapsed
│ │ │ │ + -- 3.11184s elapsed
│ │ │ │  
│ │ │ │  o20 = {{0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0}, {0,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ @@ -262,42 +262,42 @@
│ │ │ │                         {2, 2, 3, 1, -4, -6} => {{0, 1, 0, 0, 0}, {{1, 1x1*x2}}}
│ │ │ │  i22 : degree(X_3 + X_7 + X_8)
│ │ │ │  
│ │ │ │  o22 = {0, 0, 1, 2, 0, -1}
│ │ │ │  
│ │ │ │  o22 : List
│ │ │ │  i23 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 + X_8)
│ │ │ │ - -- .302915s elapsed
│ │ │ │ + -- .514668s elapsed
│ │ │ │  
│ │ │ │  o23 = {1, 0, 0, 0, 0}
│ │ │ │  
│ │ │ │  o23 : List
│ │ │ │  i24 : elapsedTime cohomvec2 = for j from 0 to dim X list rank HH^j(X, OO_X
│ │ │ │  (0,0,1,2,0,-1))
│ │ │ │ - -- 10.9792s elapsed
│ │ │ │ + -- 9.21423s elapsed
│ │ │ │  
│ │ │ │  o24 = {1, 0, 0, 0, 0}
│ │ │ │  
│ │ │ │  o24 : List
│ │ │ │  i25 : assert(cohomvec1 == cohomvec2)
│ │ │ │  i26 : degree(X_3 + X_7 - X_8)
│ │ │ │  
│ │ │ │  o26 = {0, 0, 1, 2, -2, -1}
│ │ │ │  
│ │ │ │  o26 : List
│ │ │ │  i27 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 - X_8)
│ │ │ │ - -- .342126s elapsed
│ │ │ │ + -- .530444s elapsed
│ │ │ │  
│ │ │ │  o27 = {0, 0, 0, 0, 0}
│ │ │ │  
│ │ │ │  o27 : List
│ │ │ │  i28 : elapsedTime cohomvec2 = elapsedTime for j from 0 to dim X list rank HH^j
│ │ │ │  (X, OO_X(0,0,1,2,-2,-1))
│ │ │ │ - -- .517346s elapsed
│ │ │ │ - -- .517373s elapsed
│ │ │ │ + -- .492547s elapsed
│ │ │ │ + -- .492574s elapsed
│ │ │ │  
│ │ │ │  o28 = {0, 0, 0, 0, 0}
│ │ │ │  
│ │ │ │  o28 : List
│ │ │ │  i29 : assert(cohomvec1 == cohomvec2)
│ │ │ │  _c_o_h_o_m_C_a_l_g computes cohomology vectors by calling CohomCalg. It also stashes
│ │ │ │  it's results in the toric variety's cache table, so computations need not be
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash.out
│ │ │ @@ -35,15 +35,15 @@
│ │ │  o5 : QuotientRing
│ │ │  
│ │ │  i6 : len = 10
│ │ │  
│ │ │  o6 = 10
│ │ │  
│ │ │  i7 : time G = EisenbudShamash(ff,F,len)
│ │ │ - -- used 6.66292s (cpu); 4.89797s (thread); 0s (gc)
│ │ │ + -- used 7.6472s (cpu); 5.77844s (thread); 0s (gc)
│ │ │  
│ │ │       /    S   \1     /    S   \5     /    S   \12     /    S   \20     /    S   \28     /    S   \36     /    S   \44     /    S   \52     /    S   \60     /    S   \68     /    S   \76
│ │ │  o7 = |--------|  <-- |--------|  <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|
│ │ │       |  2   3 |      |  2   3 |      |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |
│ │ │       |(x , x )|      |(x , x )|      |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|
│ │ │       \  0   1 /      \  0   1 /      \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /
│ │ │                                                                                                                                                                                
│ │ │ @@ -140,37 +140,37 @@
│ │ │  i19 : R1 = R/ideal ff
│ │ │  
│ │ │  o19 = R1
│ │ │  
│ │ │  o19 : QuotientRing
│ │ │  
│ │ │  i20 : FF = time Shamash(R1,F,4)
│ │ │ - -- used 0.169356s (cpu); 0.0960445s (thread); 0s (gc)
│ │ │ + -- used 0.205348s (cpu); 0.115054s (thread); 0s (gc)
│ │ │  
│ │ │          1       6       18       38       66
│ │ │  o20 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │                                           
│ │ │        0       1       2        3        4
│ │ │  
│ │ │  o20 : Complex
│ │ │  
│ │ │  i21 : GG = time EisenbudShamash(ff,F,4)
│ │ │ - -- used 0.937459s (cpu); 0.720912s (thread); 0s (gc)
│ │ │ + -- used 1.20834s (cpu); 0.951486s (thread); 0s (gc)
│ │ │  
│ │ │        / R\1     / R\6     / R\18     / R\38     / R\66
│ │ │  o21 = |--|  <-- |--|  <-- |--|   <-- |--|   <-- |--|
│ │ │        | 3|      | 3|      | 3|       | 3|       | 3|
│ │ │        \c /      \c /      \c /       \c /       \c /
│ │ │                                                   
│ │ │        0         1         2          3          4
│ │ │  
│ │ │  o21 : Complex
│ │ │  
│ │ │  i22 : GG = time EisenbudShamash(R1,F[2],4)
│ │ │ - -- used 0.93026s (cpu); 0.702132s (thread); 0s (gc)
│ │ │ + -- used 1.08345s (cpu); 0.842864s (thread); 0s (gc)
│ │ │  
│ │ │          1       6       18       38       66
│ │ │  o22 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │                                           
│ │ │        -2      -1      0        1        2
│ │ │  
│ │ │  o22 : Complex
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_sum__Two__Monomials.out
│ │ │ @@ -2,21 +2,21 @@
│ │ │  
│ │ │  i1 : setRandomSeed 0
│ │ │   -- setting random seed to 0
│ │ │  
│ │ │  o1 = 0
│ │ │  
│ │ │  i2 : sumTwoMonomials(2,3)
│ │ │ - -- used 0.370239s (cpu); 0.319739s (thread); 0s (gc)
│ │ │ + -- used 0.5834s (cpu); 0.411115s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{2, 2}, {1, 2}} => 3}
│ │ │  
│ │ │ - -- used 0.210104s (cpu); 0.138159s (thread); 0s (gc)
│ │ │ + -- used 0.325698s (cpu); 0.189051s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │  
│ │ │ - -- used 3.697e-06s (cpu); 3.326e-06s (thread); 0s (gc)
│ │ │ + -- used 3.417e-06s (cpu); 3.212e-06s (thread); 0s (gc)
│ │ │  4
│ │ │  Tally{}
│ │ │  
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_two__Monomials.out
│ │ │ @@ -2,23 +2,23 @@
│ │ │  
│ │ │  i1 : setRandomSeed 0
│ │ │   -- setting random seed to 0
│ │ │  
│ │ │  o1 = 0
│ │ │  
│ │ │  i2 : twoMonomials(2,3)
│ │ │ - -- used 0.802517s (cpu); 0.585358s (thread); 0s (gc)
│ │ │ + -- used 1.21967s (cpu); 0.739341s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{1, 1}} => 2        }
│ │ │        {{2, 2}, {1, 2}} => 4
│ │ │  
│ │ │ - -- used 0.401303s (cpu); 0.335055s (thread); 0s (gc)
│ │ │ + -- used 0.587918s (cpu); 0.427198s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 2}
│ │ │        {{3, 3}, {2, 3}} => 1
│ │ │  
│ │ │ - -- used 0.202221s (cpu); 0.138045s (thread); 0s (gc)
│ │ │ + -- used 0.21924s (cpu); 0.137787s (thread); 0s (gc)
│ │ │  4
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │  
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Eisenbud__Shamash.html
│ │ │ @@ -131,15 +131,15 @@
│ │ │  
│ │ │  o6 = 10
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time G = EisenbudShamash(ff,F,len)
│ │ │ - -- used 6.66292s (cpu); 4.89797s (thread); 0s (gc)
│ │ │ + -- used 7.6472s (cpu); 5.77844s (thread); 0s (gc)
│ │ │  
│ │ │       /    S   \1     /    S   \5     /    S   \12     /    S   \20     /    S   \28     /    S   \36     /    S   \44     /    S   \52     /    S   \60     /    S   \68     /    S   \76
│ │ │  o7 = |--------|  <-- |--------|  <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|
│ │ │       |  2   3 |      |  2   3 |      |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |
│ │ │       |(x , x )|      |(x , x )|      |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|
│ │ │       \  0   1 /      \  0   1 /      \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /
│ │ │                                                                                                                                                                                
│ │ │ @@ -295,28 +295,28 @@
│ │ │  
│ │ │  o19 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i20 : FF = time Shamash(R1,F,4)
│ │ │ - -- used 0.169356s (cpu); 0.0960445s (thread); 0s (gc)
│ │ │ + -- used 0.205348s (cpu); 0.115054s (thread); 0s (gc)
│ │ │  
│ │ │          1       6       18       38       66
│ │ │  o20 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │                                           
│ │ │        0       1       2        3        4
│ │ │  
│ │ │  o20 : Complex
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i21 : GG = time EisenbudShamash(ff,F,4)
│ │ │ - -- used 0.937459s (cpu); 0.720912s (thread); 0s (gc)
│ │ │ + -- used 1.20834s (cpu); 0.951486s (thread); 0s (gc)
│ │ │  
│ │ │        / R\1     / R\6     / R\18     / R\38     / R\66
│ │ │  o21 = |--|  <-- |--|  <-- |--|   <-- |--|   <-- |--|
│ │ │        | 3|      | 3|      | 3|       | 3|       | 3|
│ │ │        \c /      \c /      \c /       \c /       \c /
│ │ │                                                   
│ │ │        0         1         2          3          4
│ │ │ @@ -328,15 +328,15 @@
│ │ │          
    
│ │ │            
    The function also deals correctly with complexes F where min F is not 0:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i22 : GG = time EisenbudShamash(R1,F[2],4)
│ │ │ - -- used 0.93026s (cpu); 0.702132s (thread); 0s (gc)
│ │ │ + -- used 1.08345s (cpu); 0.842864s (thread); 0s (gc)
│ │ │  
│ │ │          1       6       18       38       66
│ │ │  o22 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │                                           
│ │ │        -2      -1      0        1        2
│ │ │  
│ │ │  o22 : Complex
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -49,15 +49,15 @@
│ │ │ │  o5 = R
│ │ │ │  
│ │ │ │  o5 : QuotientRing
│ │ │ │  i6 : len = 10
│ │ │ │  
│ │ │ │  o6 = 10
│ │ │ │  i7 : time G = EisenbudShamash(ff,F,len)
│ │ │ │ - -- used 6.66292s (cpu); 4.89797s (thread); 0s (gc)
│ │ │ │ + -- used 7.6472s (cpu); 5.77844s (thread); 0s (gc)
│ │ │ │  
│ │ │ │       /    S   \1     /    S   \5     /    S   \12     /    S   \20     /    S
│ │ │ │  \28     /    S   \36     /    S   \44     /    S   \52     /    S   \60     /
│ │ │ │  S   \68     /    S   \76
│ │ │ │  o7 = |--------|  <-- |--------|  <-- |--------|   <-- |--------|   <-- |-------
│ │ │ │  -|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |-
│ │ │ │  -------|   <-- |--------|
│ │ │ │ @@ -165,36 +165,36 @@
│ │ │ │  o18 : Matrix R  <-- R
│ │ │ │  i19 : R1 = R/ideal ff
│ │ │ │  
│ │ │ │  o19 = R1
│ │ │ │  
│ │ │ │  o19 : QuotientRing
│ │ │ │  i20 : FF = time Shamash(R1,F,4)
│ │ │ │ - -- used 0.169356s (cpu); 0.0960445s (thread); 0s (gc)
│ │ │ │ + -- used 0.205348s (cpu); 0.115054s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          1       6       18       38       66
│ │ │ │  o20 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │ │  
│ │ │ │        0       1       2        3        4
│ │ │ │  
│ │ │ │  o20 : Complex
│ │ │ │  i21 : GG = time EisenbudShamash(ff,F,4)
│ │ │ │ - -- used 0.937459s (cpu); 0.720912s (thread); 0s (gc)
│ │ │ │ + -- used 1.20834s (cpu); 0.951486s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        / R\1     / R\6     / R\18     / R\38     / R\66
│ │ │ │  o21 = |--|  <-- |--|  <-- |--|   <-- |--|   <-- |--|
│ │ │ │        | 3|      | 3|      | 3|       | 3|       | 3|
│ │ │ │        \c /      \c /      \c /       \c /       \c /
│ │ │ │  
│ │ │ │        0         1         2          3          4
│ │ │ │  
│ │ │ │  o21 : Complex
│ │ │ │  The function also deals correctly with complexes F where min F is not 0:
│ │ │ │  i22 : GG = time EisenbudShamash(R1,F[2],4)
│ │ │ │ - -- used 0.93026s (cpu); 0.702132s (thread); 0s (gc)
│ │ │ │ + -- used 1.08345s (cpu); 0.842864s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          1       6       18       38       66
│ │ │ │  o22 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │ │  
│ │ │ │        -2      -1      0        1        2
│ │ │ │  
│ │ │ │  o22 : Complex
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_sum__Two__Monomials.html
│ │ │ @@ -79,23 +79,23 @@
│ │ │  
│ │ │  o1 = 0
│ │ │              
    
│ │ │                
    i2 : sumTwoMonomials(2,3)
│ │ │ - -- used 0.370239s (cpu); 0.319739s (thread); 0s (gc)
│ │ │ + -- used 0.5834s (cpu); 0.411115s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{2, 2}, {1, 2}} => 3}
│ │ │  
│ │ │ - -- used 0.210104s (cpu); 0.138159s (thread); 0s (gc)
│ │ │ + -- used 0.325698s (cpu); 0.189051s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │  
│ │ │ - -- used 3.697e-06s (cpu); 3.326e-06s (thread); 0s (gc)
│ │ │ + -- used 3.417e-06s (cpu); 3.212e-06s (thread); 0s (gc)
│ │ │  4
│ │ │  Tally{}
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,23 +18,23 @@
│ │ │ │  appropriate syzygy M of M0 = R/(m1+m2) where m1 and m2 are monomials of the
│ │ │ │  same degree.
│ │ │ │  i1 : setRandomSeed 0
│ │ │ │   -- setting random seed to 0
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  i2 : sumTwoMonomials(2,3)
│ │ │ │ - -- used 0.370239s (cpu); 0.319739s (thread); 0s (gc)
│ │ │ │ + -- used 0.5834s (cpu); 0.411115s (thread); 0s (gc)
│ │ │ │  2
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 3}
│ │ │ │  
│ │ │ │ - -- used 0.210104s (cpu); 0.138159s (thread); 0s (gc)
│ │ │ │ + -- used 0.325698s (cpu); 0.189051s (thread); 0s (gc)
│ │ │ │  3
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │ │  
│ │ │ │ - -- used 3.697e-06s (cpu); 3.326e-06s (thread); 0s (gc)
│ │ │ │ + -- used 3.417e-06s (cpu); 3.212e-06s (thread); 0s (gc)
│ │ │ │  4
│ │ │ │  Tally{}
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_w_o_M_o_n_o_m_i_a_l_s -- tally the sequences of BRanks for certain examples
│ │ │ │  ********** WWaayyss ttoo uussee ssuummTTwwooMMoonnoommiiaallss:: **********
│ │ │ │      * sumTwoMonomials(ZZ,ZZ)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_two__Monomials.html
│ │ │ @@ -83,25 +83,25 @@
│ │ │  
│ │ │  o1 = 0
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : twoMonomials(2,3)
│ │ │ - -- used 0.802517s (cpu); 0.585358s (thread); 0s (gc)
│ │ │ + -- used 1.21967s (cpu); 0.739341s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{1, 1}} => 2        }
│ │ │        {{2, 2}, {1, 2}} => 4
│ │ │  
│ │ │ - -- used 0.401303s (cpu); 0.335055s (thread); 0s (gc)
│ │ │ + -- used 0.587918s (cpu); 0.427198s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 2}
│ │ │        {{3, 3}, {2, 3}} => 1
│ │ │  
│ │ │ - -- used 0.202221s (cpu); 0.138045s (thread); 0s (gc)
│ │ │ + -- used 0.21924s (cpu); 0.137787s (thread); 0s (gc)
│ │ │  4
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,25 +20,25 @@
│ │ │ │  that is, for an appropriate syzygy M of M0 = R/(m1, m2) where m1 and m2 are
│ │ │ │  monomials of the same degree.
│ │ │ │  i1 : setRandomSeed 0
│ │ │ │   -- setting random seed to 0
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  i2 : twoMonomials(2,3)
│ │ │ │ - -- used 0.802517s (cpu); 0.585358s (thread); 0s (gc)
│ │ │ │ + -- used 1.21967s (cpu); 0.739341s (thread); 0s (gc)
│ │ │ │  2
│ │ │ │  Tally{{{1, 1}} => 2        }
│ │ │ │        {{2, 2}, {1, 2}} => 4
│ │ │ │  
│ │ │ │ - -- used 0.401303s (cpu); 0.335055s (thread); 0s (gc)
│ │ │ │ + -- used 0.587918s (cpu); 0.427198s (thread); 0s (gc)
│ │ │ │  3
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 2}
│ │ │ │        {{3, 3}, {2, 3}} => 1
│ │ │ │  
│ │ │ │ - -- used 0.202221s (cpu); 0.138045s (thread); 0s (gc)
│ │ │ │ + -- used 0.21924s (cpu); 0.137787s (thread); 0s (gc)
│ │ │ │  4
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_w_o_M_o_n_o_m_i_a_l_s -- tally the sequences of BRanks for certain examples
│ │ │ │  ********** WWaayyss ttoo uussee ttwwooMMoonnoommiiaallss:: **********
│ │ │ │      * twoMonomials(ZZ,ZZ)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/ConformalBlocks/dump/rawdocumentation.dump
│ │ │ @@ -1,8 +1,8 @@
│ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sun Dec 14 14:09:54 2025
│ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sun Dec 14 14:09:53 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │  #:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=22
│ │ │  Y2Fub25pY2FsRGl2aXNvck0wbmJhcg==
│ │ ├── ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Cosmological_spcorrelator_spfor_spthe_sp2-site_spchain.out
│ │ │ @@ -27,18 +27,18 @@
│ │ │       - ϵ*z*dy + 2ϵ  - ϵ, x*dx + y*dy + z*dz - 2ϵ)
│ │ │  
│ │ │  o7 : Ideal of D
│ │ │  
│ │ │  i8 : assert(holonomicRank I == 4)
│ │ │  
│ │ │  i9 : elapsedTime A = connectionMatrices I;
│ │ │ - -- 2.81078s elapsed
│ │ │ + -- 2.36245s elapsed
│ │ │  
│ │ │  i10 : elapsedTime assert isIntegrable A
│ │ │ - -- 5.94013s elapsed
│ │ │ + -- 3.95334s elapsed
│ │ │  
│ │ │  i11 : netList A
│ │ │  
│ │ │        +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  o11 = || 2ϵ/x                                                                                                                                                                                                                                                                                                                                                                                                                                    -y/x                                                                                                                                                                                                                                                                                                                                                                                                                                                                   -z/x                                                                                                                                                                                                                                                                                                                                                                                                                                                  0                                                                                                                             |                                                                                                                                                                                                                                                                                                                                     |
│ │ │        || (4x2y2ϵ^2+4xy2zϵ^2-2x2z2ϵ^2-2y2z2ϵ^2-4xz3ϵ^2+x3zϵ-3xy2zϵ+2xz3ϵ)/(2x4y2+2x3y3+x4yz+2x3y2z+x2y3z-x4z2-x3yz2-x2y2z2-xy3z2-x3z3-2x2yz3-xy2z3)                                                                                                                                                                                                                                                                                               (2x3y2ϵ-2x2y3ϵ+2x3yzϵ-2xy3zϵ-x3z2ϵ+x2yz2ϵ-xy2z2ϵ+y3z2ϵ-2x3yz+2xy3z)/(2x4y2+2x3y3+x4yz+2x3y2z+x2y3z-x4z2-x3yz2-x2y2z2-xy3z2-x3z3-2x2yz3-xy2z3)                                                                                                                                                                                                                                                                                                                          (-2x2y2zϵ-x3z2ϵ-3xy2z2ϵ+x2z3ϵ+y2z3ϵ+4xz4ϵ+2xy2z2-2xz4)/(2x4y2+2x3y3+x4yz+2x3y2z+x2y3z-x4z2-x3yz2-x2y2z2-xy3z2-x3z3-2x2yz3-xy2z3)                                                                                                                                                                                                                                                                                                                      (-xyz+xz2+yz2-z3)/(2x2y+2xy2-x2z-2xyz-y2z)                                                                                    |                                                                                                                                                                                                                                                                                                                                     |
│ │ │        || (-2xyz2ϵ^2-2y2z2ϵ^2-4yz3ϵ^2+2x2y2ϵ+x2yzϵ+xy2zϵ+2y2z2ϵ+2yz3ϵ)/(2x3y2z+x3yz2+x2y2z2-x3z3-xy2z3-x2z4-xyz4)                                                                                                                                                                                                                                                                                                                                 (x2yz2ϵ+2xy2z2ϵ+y3z2ϵ+2xyz3ϵ+2y2z3ϵ-2x2y3-x2y2z-xy3z-x2yz2-y3z2-xyz3-y2z3)/(2x3y2z+x3yz2+x2y2z2-x3z3-xy2z3-x2z4-xyz4)                                                                                                                                                                                                                                                                                                                                                  (2x2y2ϵ+x2yzϵ+xy2zϵ-2x2z2ϵ+xyz2ϵ+y2z2ϵ-2xz3ϵ+2yz3ϵ-2x2y2-x2yz-xy2z+x2z2-y2z2+xz3-yz3)/(2x3y2+x3yz+x2y2z-x3z2-xy2z2-x2z3-xyz3)                                                                                                                                                                                                                                                                                                                         (-yz+z2)/(2xy-xz-yz)                                                                                                          |                                                                                                                                                                                                                                                                                                                                     |
│ │ │ @@ -56,24 +56,24 @@
│ │ │        +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  
│ │ │  i12 : F = baseFractionField D;
│ │ │  
│ │ │  i13 : B = {1_D,dx,dy,dx*dy};
│ │ │  
│ │ │  i14 : elapsedTime g = gaugeMatrix(I, B);
│ │ │ - -- .73523s elapsed
│ │ │ + -- .491211s elapsed
│ │ │  
│ │ │                4      4
│ │ │  o14 : Matrix F  <-- F
│ │ │  
│ │ │  i15 : elapsedTime A1 = gaugeTransform(g, A);
│ │ │ - -- 1.57042s elapsed
│ │ │ + -- 1.07411s elapsed
│ │ │  
│ │ │  i16 : elapsedTime assert isIntegrable A1
│ │ │ - -- .808615s elapsed
│ │ │ + -- .787232s elapsed
│ │ │  
│ │ │  i17 : netList A1
│ │ │  
│ │ │        +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  o17 = || 0                            1                      0                                         0                                                      |                                                                         |
│ │ │        || (-2ϵ^2+ϵ)/(x2-z2)            (3xϵ+zϵ-2x)/(x2-z2)    (yϵ+zϵ)/(x2-z2)                           (-y-z)/(x-z)                                           |                                                                         |
│ │ │        || 0                            0                      0                                         1                                                      |                                                                         |
│ │ │ @@ -96,18 +96,18 @@
│ │ │                {0, 0, ϵ*(y^2-z^2), ϵ*(x+y)*(y+z)},
│ │ │                {0, 0, 0, -(x+y)*(x+z)*(y+z)}});
│ │ │  
│ │ │                4      4
│ │ │  o18 : Matrix F  <-- F
│ │ │  
│ │ │  i19 : elapsedTime A2 = gaugeTransform(changeEps, A1);
│ │ │ - -- .496173s elapsed
│ │ │ + -- .31763s elapsed
│ │ │  
│ │ │  i20 : elapsedTime assert isIntegrable A2
│ │ │ - -- .830479s elapsed
│ │ │ + -- .609243s elapsed
│ │ │  
│ │ │  i21 : netList A2
│ │ │  
│ │ │        +-------------------------------------------------------------------------------------------+
│ │ │  o21 = || ϵ/(x+z) 2zϵ/(x2-z2) 0       0                      |                                     |
│ │ │        || 0       ϵ/(x-z)     0       ϵ/(x+y)                |                                     |
│ │ │        || 0       0           ϵ/(x+z) (-yϵ+zϵ)/(x2+xy+xz+yz) |                                     |
│ │ ├── ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Massless_spone-loop_sptriangle_sp__Feynman_spdiagram.out
│ │ │ @@ -16,18 +16,18 @@
│ │ │  
│ │ │                     2
│ │ │  o6 = {1, dx, dy, dy }
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : elapsedTime A = connectionMatrices I;
│ │ │ - -- .267226s elapsed
│ │ │ + -- .203972s elapsed
│ │ │  
│ │ │  i8 : elapsedTime assert isIntegrable A
│ │ │ - -- .203883s elapsed
│ │ │ + -- .16896s elapsed
│ │ │  
│ │ │  i9 : netList A
│ │ │  
│ │ │       +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  o9 = || 0                                                       1                                               0                                                                     0                                                                      ||
│ │ │       || 0                                                       -1/x                                            1/x                                                                   y/x                                                                    ||
│ │ │       || -1/2xy                                                  -1/y                                            (-x-3y+1)/2xy                                                         (-x-y+1)/2x                                                            ||
│ │ ├── ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Cosmological_spcorrelator_spfor_spthe_sp2-site_spchain.html
│ │ │ @@ -118,21 +118,21 @@
│ │ │          
    
│ │ │            
    Then, we compute the system in connection form and verify that it meets the integrability conditions.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -82,15 +82,15 @@
│ │ │ │                2                 2
│ │ │ │       - a*c + e         - b*c + f
│ │ │ │       ----------*v, x + ----------*v)
│ │ │ │        d*e - a*f         d*e - a*f
│ │ │ │  
│ │ │ │  o5 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
│ │ │ │  i6 : time phi^** q
│ │ │ │ - -- used 0.157531s (cpu); 0.157527s (thread); 0s (gc)
│ │ │ │ + -- used 0.166215s (cpu); 0.166214s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                  e        d        c        b        a
│ │ │ │  o6 = ideal (u - -*v, t - -*v, z - -*v, y - -*v, x - -*v)
│ │ │ │                  f        f        f        f        f
│ │ │ │  
│ │ │ │  o6 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
│ │ │ │  i7 : oo == p
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Segre__Class.html
│ │ │ @@ -134,59 +134,59 @@
│ │ │                x x  - 2x x x x  + x x  - 2x x x x  - 2x x x x  + 4x x x x  + x x  + 4x x x x  - 2x x x x  - 2x x x x  - 2x x x x  + x x
│ │ │                 3 4     2 3 4 5    2 5     1 3 4 6     1 2 5 6     0 3 5 6    1 6     1 2 4 7     0 3 4 7     0 2 5 7     0 1 6 7    0 7
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
    
│ │ │                
    i9 : elapsedTime A = connectionMatrices I;
│ │ │ - -- 2.81078s elapsed
    
│ │ │ + -- 2.36245s elapsed
│ │ │              
    
│ │ │                
    i10 : elapsedTime assert isIntegrable A
│ │ │ - -- 5.94013s elapsed
    
│ │ │ + -- 3.95334s elapsed
│ │ │              
    
│ │ │                
    i11 : netList A
│ │ │  
│ │ │        +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ @@ -167,30 +167,30 @@
│ │ │              
    		
│ │ │                
    i13 : B = {1_D,dx,dy,dx*dy};
    
│ │ │              
    
│ │ │                
    i14 : elapsedTime g = gaugeMatrix(I, B);
│ │ │ - -- .73523s elapsed
│ │ │ + -- .491211s elapsed
│ │ │  
│ │ │                4      4
│ │ │  o14 : Matrix F  <-- F
    
│ │ │              
    
│ │ │                
    i15 : elapsedTime A1 = gaugeTransform(g, A);
│ │ │ - -- 1.57042s elapsed
    
│ │ │ + -- 1.07411s elapsed
│ │ │              
    
│ │ │                
    i16 : elapsedTime assert isIntegrable A1
│ │ │ - -- .808615s elapsed
    
│ │ │ + -- .787232s elapsed
│ │ │              
    
│ │ │                
    i17 : netList A1
│ │ │  
│ │ │        +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ @@ -227,21 +227,21 @@
│ │ │                4      4
│ │ │  o18 : Matrix F  <-- F
    
│ │ │              
    
│ │ │                
    i19 : elapsedTime A2 = gaugeTransform(changeEps, A1);
│ │ │ - -- .496173s elapsed
    
│ │ │ + -- .31763s elapsed
│ │ │              
    
│ │ │                
    i20 : elapsedTime assert isIntegrable A2
│ │ │ - -- .830479s elapsed
    
│ │ │ + -- .609243s elapsed
│ │ │              
    
│ │ │                
    i21 : netList A2
│ │ │  
│ │ │        +-------------------------------------------------------------------------------------------+
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,17 +41,17 @@
│ │ │ │  
│ │ │ │  o7 : Ideal of D
│ │ │ │  First, we check that the system has finite holonomic rank using _h_o_l_o_n_o_m_i_c_R_a_n_k.
│ │ │ │  i8 : assert(holonomicRank I == 4)
│ │ │ │  Then, we compute the system in connection form and verify that it meets the
│ │ │ │  integrability conditions.
│ │ │ │  i9 : elapsedTime A = connectionMatrices I;
│ │ │ │ - -- 2.81078s elapsed
│ │ │ │ + -- 2.36245s elapsed
│ │ │ │  i10 : elapsedTime assert isIntegrable A
│ │ │ │ - -- 5.94013s elapsed
│ │ │ │ + -- 3.95334s elapsed
│ │ │ │  i11 : netList A
│ │ │ │  
│ │ │ │        +----------------------------------------------------------------------------------------------------------
│ │ │ │  -----------------------------------------------------------------------------------------------------------------
│ │ │ │  -----------------------------------------------------------------------------------------------------------------
│ │ │ │  -----------------------------------------------------------------------------------------------------------------
│ │ │ │  -----------------------------------------------------------------------------------------------------------------
│ │ │ │ @@ -227,22 +227,22 @@
│ │ │ │  -----------------------------------------------------------------------------------+
│ │ │ │  Next, we use _g_a_u_g_e_ _m_a_t_r_i_x for changing base to a base given by suitable set of
│ │ │ │  standard monomials, and compute the _g_a_u_g_e_ _t_r_a_n_s_f_o_r_m with respect to this gauge
│ │ │ │  matrix.
│ │ │ │  i12 : F = baseFractionField D;
│ │ │ │  i13 : B = {1_D,dx,dy,dx*dy};
│ │ │ │  i14 : elapsedTime g = gaugeMatrix(I, B);
│ │ │ │ - -- .73523s elapsed
│ │ │ │ + -- .491211s elapsed
│ │ │ │  
│ │ │ │                4      4
│ │ │ │  o14 : Matrix F  <-- F
│ │ │ │  i15 : elapsedTime A1 = gaugeTransform(g, A);
│ │ │ │ - -- 1.57042s elapsed
│ │ │ │ + -- 1.07411s elapsed
│ │ │ │  i16 : elapsedTime assert isIntegrable A1
│ │ │ │ - -- .808615s elapsed
│ │ │ │ + -- .787232s elapsed
│ │ │ │  i17 : netList A1
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  --------------------------------------------------------------------------+
│ │ │ │  o17 = || 0                            1                      0
│ │ │ │  0                                                      |
│ │ │ │ @@ -300,17 +300,17 @@
│ │ │ │                {0, ϵ*(x^2-z^2), 0, ϵ*(x+y)*(x+z)},
│ │ │ │                {0, 0, ϵ*(y^2-z^2), ϵ*(x+y)*(y+z)},
│ │ │ │                {0, 0, 0, -(x+y)*(x+z)*(y+z)}});
│ │ │ │  
│ │ │ │                4      4
│ │ │ │  o18 : Matrix F  <-- F
│ │ │ │  i19 : elapsedTime A2 = gaugeTransform(changeEps, A1);
│ │ │ │ - -- .496173s elapsed
│ │ │ │ + -- .31763s elapsed
│ │ │ │  i20 : elapsedTime assert isIntegrable A2
│ │ │ │ - -- .830479s elapsed
│ │ │ │ + -- .609243s elapsed
│ │ │ │  i21 : netList A2
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------+
│ │ │ │  o21 = || ϵ/(x+z) 2zϵ/(x2-z2) 0       0                      |
│ │ │ │  |
│ │ │ │        || 0       ϵ/(x-z)     0       ϵ/(x+y)                |
│ │ ├── ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Massless_spone-loop_sptriangle_sp__Feynman_spdiagram.html
│ │ │ @@ -100,21 +100,21 @@
│ │ │          
    
│ │ │            
    Finally, we can compute the connection matrices.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : elapsedTime A = connectionMatrices I;
│ │ │ - -- .267226s elapsed
    
│ │ │ + -- .203972s elapsed
│ │ │              
    
│ │ │                
    i8 : elapsedTime assert isIntegrable A
│ │ │ - -- .203883s elapsed
    
│ │ │ + -- .16896s elapsed
│ │ │              
    
│ │ │                
    i9 : netList A
│ │ │  
│ │ │       +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,17 +20,17 @@
│ │ │ │  
│ │ │ │                     2
│ │ │ │  o6 = {1, dx, dy, dy }
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  Finally, we can compute the connection matrices.
│ │ │ │  i7 : elapsedTime A = connectionMatrices I;
│ │ │ │ - -- .267226s elapsed
│ │ │ │ + -- .203972s elapsed
│ │ │ │  i8 : elapsedTime assert isIntegrable A
│ │ │ │ - -- .203883s elapsed
│ │ │ │ + -- .16896s elapsed
│ │ │ │  i9 : netList A
│ │ │ │  
│ │ │ │       +-------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------+
│ │ │ │  o9 = || 0                                                       1
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Chern__Schwartz__Mac__Pherson.out
│ │ │ @@ -13,27 +13,27 @@
│ │ │  o2 = ideal (- x  + x x , - x x  + x x , - x  + x x )
│ │ │                 1    0 2     1 2    0 3     2    1 3
│ │ │  
│ │ │  o2 : Ideal of GF 78125[x ..x ]
│ │ │                          0   4
│ │ │  
│ │ │  i3 : time ChernSchwartzMacPherson C
│ │ │ - -- used 2.25862s (cpu); 1.18322s (thread); 0s (gc)
│ │ │ + -- used 2.36974s (cpu); 1.25502s (thread); 0s (gc)
│ │ │  
│ │ │         4     3     2
│ │ │  o3 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o3 : -----
│ │ │          5
│ │ │         H
│ │ │  
│ │ │  i4 : time ChernSchwartzMacPherson(C,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 1.61784s (cpu); 1.14031s (thread); 0s (gc)
│ │ │ + -- used 1.43309s (cpu); 0.986049s (thread); 0s (gc)
│ │ │  
│ │ │         4     3     2
│ │ │  o4 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o4 : -----
│ │ │          5
│ │ │ @@ -62,27 +62,27 @@
│ │ │          0,2 1,3    0,1 2,3
│ │ │  
│ │ │                  ZZ
│ │ │  o8 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   ]
│ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
│ │ │  
│ │ │  i9 : time ChernClass G
│ │ │ - -- used 0.439952s (cpu); 0.258932s (thread); 0s (gc)
│ │ │ + -- used 0.350846s (cpu); 0.194791s (thread); 0s (gc)
│ │ │  
│ │ │          9      8      7      6      5      4     3
│ │ │  o9 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o9 : -----
│ │ │         10
│ │ │        H
│ │ │  
│ │ │  i10 : time ChernClass(G,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.132347s (cpu); 0.0449789s (thread); 0s (gc)
│ │ │ + -- used 0.132431s (cpu); 0.0387954s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6      5      4     3
│ │ │  o10 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o10 : -----
│ │ │          10
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Cremona.out
│ │ │ @@ -1,56 +1,56 @@
│ │ │  -- -*- M2-comint -*- hash: 10433409267944421825
│ │ │  
│ │ │  i1 : ZZ/300007[t_0..t_6];
│ │ │  
│ │ │  i2 : time phi = toMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ - -- used 0.00430215s (cpu); 0.00429842s (thread); 0s (gc)
│ │ │ + -- used 0.00524112s (cpu); 0.00523976s (thread); 0s (gc)
│ │ │  
│ │ │              ZZ              ZZ                3                2    2                2        2                      2                  2    2                 2                       3                2    2                2                                 2                           2    2                                  2        2                      2                  2                        2                         2    2                 2                       3                2    2
│ │ │  o2 = map (------[t ..t ], ------[x ..x ], {- t  + 2t t t  - t t  - t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t , - t t  + t t t  + t t t  - t t t  - t t  + t t t , - t t t  + t t  + t t  - t t t  - t t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t t  + t t t  - t t  - t t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t })
│ │ │            300007  0   6   300007  0   9       2     1 2 3    0 3    1 4    0 2 4     2 3    1 3    1 2 4    0 3 4    1 5    0 2 5     2 3    2 4    1 3 4    0 4    1 2 5    0 3 5     3     2 3 4    1 4    2 5    1 3 5     2 4    1 3 4    1 2 5    0 3 5    1 6    0 2 6     2 3 4    1 4    2 5    0 4 5    1 2 6    0 3 6     3 4    2 4    2 3 5    1 4 5    2 6    1 3 6     2 4    2 3 5    1 4 5    0 5    1 3 6    0 4 6     3 4    3 5    2 4 5    1 5    2 3 6    1 4 6     4     3 4 5    2 5    3 6    2 4 6
│ │ │  
│ │ │                 ZZ                 ZZ
│ │ │  o2 : RingMap ------[t ..t ] <-- ------[x ..x ]
│ │ │               300007  0   6      300007  0   9
│ │ │  
│ │ │  i3 : time J = kernel(phi,2)
│ │ │ - -- used 0.137231s (cpu); 0.0699679s (thread); 0s (gc)
│ │ │ + -- used 0.149399s (cpu); 0.0730355s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = ideal (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x 
│ │ │               6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4
│ │ │       ------------------------------------------------------------------------
│ │ │       - x x  + x x , x x  - x x  + x x )
│ │ │          1 6    0 8   2 4    1 5    0 7
│ │ │  
│ │ │                  ZZ
│ │ │  o3 : Ideal of ------[x ..x ]
│ │ │                300007  0   9
│ │ │  
│ │ │  i4 : time degreeMap phi
│ │ │ - -- used 0.02944s (cpu); 0.0294445s (thread); 0s (gc)
│ │ │ + -- used 0.0334176s (cpu); 0.0334224s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
│ │ │  
│ │ │  i5 : time projectiveDegrees phi
│ │ │ - -- used 0.68756s (cpu); 0.487586s (thread); 0s (gc)
│ │ │ + -- used 0.716496s (cpu); 0.549996s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : time projectiveDegrees(phi,NumDegrees=>0)
│ │ │ - -- used 0.0623207s (cpu); 0.0622653s (thread); 0s (gc)
│ │ │ + -- used 0.0704585s (cpu); 0.0704671s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = {5}
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ - -- used 0.0021486s (cpu); 0.00214939s (thread); 0s (gc)
│ │ │ + -- used 0.0025549s (cpu); 0.00255898s (thread); 0s (gc)
│ │ │  
│ │ │                                                                         ZZ
│ │ │                                                                       ------[x ..x ]
│ │ │              ZZ                                                       300007  0   9                                                  3                2    2                2        2                      2                  2    2                 2                       3                2    2                2                                 2                           2    2                                  2        2                      2                  2                        2                         2    2                 2                       3                2    2
│ │ │  o7 = map (------[t ..t ], ----------------------------------------------------------------------------------------------------, {- t  + 2t t t  - t t  - t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t , - t t  + t t t  + t t t  - t t t  - t t  + t t t , - t t t  + t t  + t t  - t t t  - t t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t t  + t t t  - t t  - t t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t })
│ │ │            300007  0   6   (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )      2     1 2 3    0 3    1 4    0 2 4     2 3    1 3    1 2 4    0 3 4    1 5    0 2 5     2 3    2 4    1 3 4    0 4    1 2 5    0 3 5     3     2 3 4    1 4    2 5    1 3 5     2 4    1 3 4    1 2 5    0 3 5    1 6    0 2 6     2 3 4    1 4    2 5    0 4 5    1 2 6    0 3 6     3 4    2 4    2 3 5    1 4 5    2 6    1 3 6     2 4    2 3 5    1 4 5    0 5    1 3 6    0 4 6     3 4    3 5    2 4 5    1 5    2 3 6    1 4 6     4     3 4 5    2 5    3 6    2 4 6
│ │ │                              6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │ @@ -59,15 +59,15 @@
│ │ │                                                                             ------[x ..x ]
│ │ │                 ZZ                                                          300007  0   9
│ │ │  o7 : RingMap ------[t ..t ] <-- ----------------------------------------------------------------------------------------------------
│ │ │               300007  0   6      (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )
│ │ │                                    6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │  
│ │ │  i8 : time psi = inverseMap phi
│ │ │ - -- used 0.474252s (cpu); 0.394534s (thread); 0s (gc)
│ │ │ + -- used 0.422925s (cpu); 0.422931s (thread); 0s (gc)
│ │ │  
│ │ │                                                         ZZ
│ │ │                                                       ------[x ..x ]
│ │ │                                                       300007  0   9                                                ZZ              3                2               2    2                        2                          2     2        2                               2                                   2               2             2                       3                                                 2                 2    2                                  2    2                 2                                                 3                         2      2    2      2                                              2
│ │ │  o8 = map (----------------------------------------------------------------------------------------------------, ------[t ..t ], {x  - 2x x x  + x x  - x x x  + x x  + x x  + x x x  - x x x  + x x  - 2x x x  - x x x  - 2x x , x x  - x x  - x x x  + x x x  + x x x  + x x  - 2x x x  - x x x  + x x x , x x  - x x x  + x x  - x x x  + x x  - x x x  - x x x , x  - x x x  + x x x  + x x x  - 2x x x  - x x x , x x  - x x x  + x x  + x x  - x x x  - x x x  - x x x , x x  - x x  - x x x  + x x  + x x x  + x x x  - 2x x x  - x x x  + x x x , x  - 2x x x  - x x x  + x x  + x x  + x x  + x x  + x x x  - 2x x x  - x x x  - x x x  - 2x x })
│ │ │            (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )  300007  0   6     2     1 2 3    0 3    1 2 5    0 5    1 6    0 2 6    0 4 6    1 7     0 2 7    0 4 7     0 9   2 3    1 3    1 2 6    0 3 6    0 5 6    1 8     0 2 8    0 4 8    0 1 9   2 3    1 3 6    0 6    0 3 8    1 9    0 2 9    0 4 9   3    1 3 8    0 6 8    1 2 9     0 3 9    0 5 9   3 6    2 3 8    0 8    2 9    1 3 9    0 6 9    0 7 9   3 6    3 8    2 6 8    1 8    2 3 9    2 5 9     1 6 9    1 7 9    0 8 9   6     3 6 8    5 6 8    2 8    4 8    3 9    5 9    2 6 9     4 6 9    2 7 9    4 7 9     0 9
│ │ │              6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │ @@ -76,32 +76,32 @@
│ │ │                                                          ------[x ..x ]
│ │ │                                                          300007  0   9                                                   ZZ
│ │ │  o8 : RingMap ---------------------------------------------------------------------------------------------------- <-- ------[t ..t ]
│ │ │               (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )     300007  0   6
│ │ │                 6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │  
│ │ │  i9 : time isInverseMap(phi,psi)
│ │ │ - -- used 0.00931603s (cpu); 0.00931855s (thread); 0s (gc)
│ │ │ + -- used 0.0109647s (cpu); 0.0109661s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : time degreeMap psi
│ │ │ - -- used 0.458493s (cpu); 0.294259s (thread); 0s (gc)
│ │ │ + -- used 0.486731s (cpu); 0.262439s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 1
│ │ │  
│ │ │  i11 : time projectiveDegrees psi
│ │ │ - -- used 5.29004s (cpu); 4.63968s (thread); 0s (gc)
│ │ │ + -- used 5.44371s (cpu); 5.02492s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o11 : List
│ │ │  
│ │ │  i12 : time phi = rationalMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ - -- used 0.00220238s (cpu); 0.00220313s (thread); 0s (gc)
│ │ │ + -- used 0.002611s (cpu); 0.00261541s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │                     300007  0   1   2   3   4   5   6
│ │ │                       ZZ
│ │ │        target: Proj(------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ @@ -147,15 +147,15 @@
│ │ │                         - t  + 2t t t  - t t  - t t  + t t t
│ │ │                            4     3 4 5    2 5    3 6    2 4 6
│ │ │                        }
│ │ │  
│ │ │  o12 : RationalMap (cubic rational map from PP^6 to PP^9)
│ │ │  
│ │ │  i13 : time phi = rationalMap(phi,Dominant=>2)
│ │ │ - -- used 0.15731s (cpu); 0.0849869s (thread); 0s (gc)
│ │ │ + -- used 0.17472s (cpu); 0.090802s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │                     300007  0   1   2   3   4   5   6
│ │ │                                     ZZ
│ │ │        target: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │ @@ -217,15 +217,15 @@
│ │ │                         - t  + 2t t t  - t t  - t t  + t t t
│ │ │                            4     3 4 5    2 5    3 6    2 4 6
│ │ │                        }
│ │ │  
│ │ │  o13 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9)
│ │ │  
│ │ │  i14 : time phi^(-1)
│ │ │ - -- used 0.512059s (cpu); 0.426868s (thread); 0s (gc)
│ │ │ + -- used 0.440018s (cpu); 0.440021s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = -- rational map --
│ │ │                                     ZZ
│ │ │        source: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │                                   300007  0   1   2   3   4   5   6   7   8   9
│ │ │                {
│ │ │                 x x  - x x  + x x ,
│ │ │ @@ -275,71 +275,71 @@
│ │ │                         x  - 2x x x  - x x x  + x x  + x x  + x x  + x x  + x x x  - 2x x x  - x x x  - x x x  - 2x x
│ │ │                          6     3 6 8    5 6 8    2 8    4 8    3 9    5 9    2 6 9     4 6 9    2 7 9    4 7 9     0 9
│ │ │                        }
│ │ │  
│ │ │  o14 : RationalMap (cubic birational map from 6-dimensional subvariety of PP^9 to PP^6)
│ │ │  
│ │ │  i15 : time degrees phi^(-1)
│ │ │ - -- used 0.347749s (cpu); 0.274141s (thread); 0s (gc)
│ │ │ + -- used 0.475373s (cpu); 0.340152s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : time degrees phi
│ │ │ - -- used 0.0180113s (cpu); 0.0176993s (thread); 0s (gc)
│ │ │ + -- used 0.0793952s (cpu); 0.0267941s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o16 : List
│ │ │  
│ │ │  i17 : time describe phi
│ │ │ - -- used 0.00320718s (cpu); 0.00320744s (thread); 0s (gc)
│ │ │ + -- used 0.00372634s (cpu); 0.00373177s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = rational map defined by forms of degree 3
│ │ │        source variety: PP^6
│ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        dominance: true
│ │ │        birationality: true (the inverse map is already calculated)
│ │ │        projective degrees: {1, 3, 9, 17, 21, 15, 5}
│ │ │        coefficient ring: ZZ/300007
│ │ │  
│ │ │  i18 : time describe phi^(-1)
│ │ │ - -- used 0.00997997s (cpu); 0.0099807s (thread); 0s (gc)
│ │ │ + -- used 0.0109202s (cpu); 0.0109263s (thread); 0s (gc)
│ │ │  
│ │ │  o18 = rational map defined by forms of degree 3
│ │ │        source variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        target variety: PP^6
│ │ │        dominance: true
│ │ │        birationality: true (the inverse map is already calculated)
│ │ │        projective degrees: {5, 15, 21, 17, 9, 3, 1}
│ │ │        number of minimal representatives: 1
│ │ │        dimension base locus: 4
│ │ │        degree base locus: 24
│ │ │        coefficient ring: ZZ/300007
│ │ │  
│ │ │  i19 : time (f,g) = graph phi^-1; f;
│ │ │ - -- used 0.00950571s (cpu); 0.00950659s (thread); 0s (gc)
│ │ │ + -- used 0.0115313s (cpu); 0.0115369s (thread); 0s (gc)
│ │ │  
│ │ │  o20 : MultihomogeneousRationalMap (birational map from 6-dimensional subvariety of PP^9 x PP^6 to 6-dimensional subvariety of PP^9)
│ │ │  
│ │ │  i21 : time degrees f
│ │ │ - -- used 1.33327s (cpu); 0.95512s (thread); 0s (gc)
│ │ │ + -- used 1.20731s (cpu); 0.987305s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = {904, 508, 268, 130, 56, 20, 5}
│ │ │  
│ │ │  o21 : List
│ │ │  
│ │ │  i22 : time degree f
│ │ │ - -- used 1.625e-05s (cpu); 1.593e-05s (thread); 0s (gc)
│ │ │ + -- used 2.1074e-05s (cpu); 1.9557e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = 1
│ │ │  
│ │ │  i23 : time describe f
│ │ │ - -- used 0.00161465s (cpu); 0.00161555s (thread); 0s (gc)
│ │ │ + -- used 0.00165199s (cpu); 0.00165748s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = rational map defined by multiforms of degree {1, 0}
│ │ │        source variety: 6-dimensional subvariety of PP^9 x PP^6 cut out by 20 hypersurfaces of degrees ({1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{2, 0},{2, 0},{2, 0},{2, 0},{2, 0})
│ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        dominance: true
│ │ │        birationality: true
│ │ │        projective degrees: {904, 508, 268, 130, 56, 20, 5}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Euler__Characteristic.out
│ │ │ @@ -3,18 +3,18 @@
│ │ │  i1 : I = Grassmannian(1,4,CoefficientRing=>ZZ/190181);
│ │ │  
│ │ │                  ZZ
│ │ │  o1 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   ]
│ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
│ │ │  
│ │ │  i2 : time EulerCharacteristic I
│ │ │ - -- used 0.265746s (cpu); 0.15298s (thread); 0s (gc)
│ │ │ + -- used 0.313648s (cpu); 0.171991s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
│ │ │  
│ │ │  i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0115158s (cpu); 0.0109715s (thread); 0s (gc)
│ │ │ + -- used 0.0746203s (cpu); 0.0206391s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = 10
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Rational__Map_sp!.out
│ │ │ @@ -8,15 +8,15 @@
│ │ │  
│ │ │  o3 = rational map defined by forms of degree 2
│ │ │       source variety: PP^5
│ │ │       target variety: PP^5
│ │ │       coefficient ring: QQ
│ │ │  
│ │ │  i4 : time phi! ;
│ │ │ - -- used 0.0532429s (cpu); 0.0528992s (thread); 0s (gc)
│ │ │ + -- used 0.0726521s (cpu); 0.0609858s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (Cremona transformation of PP^5 of type (2,2))
│ │ │  
│ │ │  i5 : describe phi
│ │ │  
│ │ │  o5 = rational map defined by forms of degree 2
│ │ │       source variety: PP^5
│ │ │ @@ -37,15 +37,15 @@
│ │ │  
│ │ │  o8 = rational map defined by forms of degree 2
│ │ │       source variety: PP^4
│ │ │       target variety: PP^5
│ │ │       coefficient ring: QQ
│ │ │  
│ │ │  i9 : time phi! ;
│ │ │ - -- used 0.0360627s (cpu); 0.0357049s (thread); 0s (gc)
│ │ │ + -- used 0.0662474s (cpu); 0.0459304s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : RationalMap (quadratic rational map from PP^4 to PP^5)
│ │ │  
│ │ │  i10 : describe phi
│ │ │  
│ │ │  o10 = rational map defined by forms of degree 2
│ │ │        source variety: PP^4
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Rational__Map_sp^_st_st_sp__Ideal.out
│ │ │ @@ -67,15 +67,15 @@
│ │ │       - a*c + e         - b*c + f
│ │ │       ----------*v, x + ----------*v)
│ │ │        d*e - a*f         d*e - a*f
│ │ │  
│ │ │  o5 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
│ │ │  
│ │ │  i6 : time phi^** q
│ │ │ - -- used 0.157531s (cpu); 0.157527s (thread); 0s (gc)
│ │ │ + -- used 0.166215s (cpu); 0.166214s (thread); 0s (gc)
│ │ │  
│ │ │                  e        d        c        b        a
│ │ │  o6 = ideal (u - -*v, t - -*v, z - -*v, y - -*v, x - -*v)
│ │ │                  f        f        f        f        f
│ │ │  
│ │ │  o6 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Segre__Class.out
│ │ │ @@ -47,50 +47,50 @@
│ │ │                                                                            P7
│ │ │  o3 : Ideal of -------------------------------------------------------------------------------------------------------------------------
│ │ │                 2 2                2 2                                        2 2                                                    2 2
│ │ │                x x  - 2x x x x  + x x  - 2x x x x  - 2x x x x  + 4x x x x  + x x  + 4x x x x  - 2x x x x  - 2x x x x  - 2x x x x  + x x
│ │ │                 3 4     2 3 4 5    2 5     1 3 4 6     1 2 5 6     0 3 5 6    1 6     1 2 4 7     0 3 4 7     0 2 5 7     0 1 6 7    0 7
│ │ │  
│ │ │  i4 : time SegreClass X
│ │ │ - -- used 0.865825s (cpu); 0.519842s (thread); 0s (gc)
│ │ │ + -- used 0.787156s (cpu); 0.53615s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5       4      3
│ │ │  o4 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o4 : -----
│ │ │          8
│ │ │         H
│ │ │  
│ │ │  i5 : time SegreClass lift(X,P7)
│ │ │ - -- used 0.565093s (cpu); 0.36901s (thread); 0s (gc)
│ │ │ + -- used 0.620994s (cpu); 0.36341s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o5 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o5 : -----
│ │ │          8
│ │ │         H
│ │ │  
│ │ │  i6 : time SegreClass(X,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0212909s (cpu); 0.0208755s (thread); 0s (gc)
│ │ │ + -- used 0.0674279s (cpu); 0.0316446s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5       4      3
│ │ │  o6 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o6 : -----
│ │ │          8
│ │ │         H
│ │ │  
│ │ │  i7 : time SegreClass(lift(X,P7),Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.097715s (cpu); 0.0973659s (thread); 0s (gc)
│ │ │ + -- used 0.127674s (cpu); 0.114673s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o7 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o7 : -----
│ │ │          8
│ │ │ @@ -98,22 +98,22 @@
│ │ │  
│ │ │  i8 : o4 == o6 and o5 == o7
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : use ZZ/100003[x_0..x_6]
│ │ │  
│ │ │ -o9 =   ZZ
│ │ │ - ------[x ..x ]
│ │ │ - 100003  0   6
│ │ │ +       ZZ
│ │ │ +o9 = ------[x ..x ]
│ │ │ +     100003  0   6
│ │ │  
│ │ │  o9 : PolynomialRing
│ │ │  
│ │ │  i10 : time phi = inverseMap toMap(minors(2,matrix{{x_0,x_1,x_3,x_4,x_5},{x_1,x_2,x_4,x_5,x_6}}),Dominant=>2)
│ │ │ - -- used 0.216303s (cpu); 0.10276s (thread); 0s (gc)
│ │ │ + -- used 0.0662291s (cpu); 0.0662326s (thread); 0s (gc)
│ │ │  
│ │ │                                                          ZZ
│ │ │                                                        ------[y ..y ]
│ │ │                                                        100003  0   9                                                ZZ              2                              2
│ │ │  o10 = map (----------------------------------------------------------------------------------------------------, ------[x ..x ], {y  - y y  - y y , y y  - y y , y  - y y  - y y , y y  + y y  - y y , y y  - y y , y y  - y y  - y y , y y  - y y  - y y })
│ │ │             (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y )  100003  0   6     3    0 5    1 6   3 4    1 7   4    2 7    0 9   2 5    3 5    1 8   4 5    1 9   4 8    2 9    3 9   7 8    4 9    6 9
│ │ │               5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6    1 7    0 8   2 3    1 4    0 5
│ │ │ @@ -122,15 +122,15 @@
│ │ │                                                           ------[y ..y ]
│ │ │                                                           100003  0   9                                                   ZZ
│ │ │  o10 : RingMap ---------------------------------------------------------------------------------------------------- <-- ------[x ..x ]
│ │ │                (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y )     100003  0   6
│ │ │                  5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6    1 7    0 8   2 3    1 4    0 5
│ │ │  
│ │ │  i11 : time SegreClass phi
│ │ │ - -- used 0.338604s (cpu); 0.228164s (thread); 0s (gc)
│ │ │ + -- used 0.372775s (cpu); 0.253258s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6     5
│ │ │  o11 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o11 : -----
│ │ │          10
│ │ │ @@ -150,27 +150,27 @@
│ │ │                                                            100003  0   9
│ │ │  o12 : Ideal of ----------------------------------------------------------------------------------------------------
│ │ │                 (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y )
│ │ │                   5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6    1 7    0 8   2 3    1 4    0 5
│ │ │  
│ │ │  i13 : -- Segre class of B in G(1,4)
│ │ │        time SegreClass B
│ │ │ - -- used 0.39646s (cpu); 0.292129s (thread); 0s (gc)
│ │ │ + -- used 0.423745s (cpu); 0.294539s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6     5
│ │ │  o13 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o13 : -----
│ │ │          10
│ │ │         H
│ │ │  
│ │ │  i14 : -- Segre class of B in P^9
│ │ │        time SegreClass lift(B,ambient ring B)
│ │ │ - -- used 1.41797s (cpu); 0.900612s (thread); 0s (gc)
│ │ │ + -- used 1.54782s (cpu); 0.936395s (thread); 0s (gc)
│ │ │  
│ │ │             9       8       7      6     5
│ │ │  o14 = 2764H  - 984H  + 294H  - 67H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o14 : -----
│ │ │          10
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_abstract__Rational__Map.out
│ │ │ @@ -17,32 +17,32 @@
│ │ │  
│ │ │  o3 = QQ[u ..u ]
│ │ │           0   5
│ │ │  
│ │ │  o3 : PolynomialRing
│ │ │  
│ │ │  i4 : time psi = abstractRationalMap(P4,P5,f)
│ │ │ - -- used 0.000410019s (cpu); 0.000406292s (thread); 0s (gc)
│ │ │ + -- used 0.000464119s (cpu); 0.000458369s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = -- rational map --
│ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │                        0   1   2   3   4   5
│ │ │       defining forms: given by a function
│ │ │  
│ │ │  o4 : AbstractRationalMap (rational map from PP^4 to PP^5)
│ │ │  
│ │ │  i5 : time projectiveDegrees(psi,3)
│ │ │ - -- used 0.298853s (cpu); 0.18527s (thread); 0s (gc)
│ │ │ + -- used 0.358626s (cpu); 0.201614s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
│ │ │  
│ │ │  i6 : time rationalMap psi
│ │ │ - -- used 0.504026s (cpu); 0.366499s (thread); 0s (gc)
│ │ │ + -- used 0.468152s (cpu); 0.384639s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = -- rational map --
│ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │                        0   1   2   3   4   5
│ │ │       defining forms: {
│ │ │ @@ -113,48 +113,48 @@
│ │ │                  1    0 2     1 2    0 3     2    1 3
│ │ │  
│ │ │                   ZZ
│ │ │  o13 : Ideal of -----[x ..x ]
│ │ │                 65521  0   3
│ │ │  
│ │ │  i14 : time T = abstractRationalMap(I,"OADP")
│ │ │ - -- used 0.149185s (cpu); 0.0769299s (thread); 0s (gc)
│ │ │ + -- used 0.163849s (cpu); 0.0752081s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │        defining forms: given by a function
│ │ │  
│ │ │  o14 : AbstractRationalMap (rational map from PP^3 to PP^3)
│ │ │  
│ │ │  i15 : time projectiveDegrees(T,2)
│ │ │ - -- used 4.07764s (cpu); 2.12022s (thread); 0s (gc)
│ │ │ + -- used 4.43247s (cpu); 2.24391s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3
│ │ │  
│ │ │  i16 : time T2 = T * T
│ │ │ - -- used 2.8564e-05s (cpu); 2.8273e-05s (thread); 0s (gc)
│ │ │ + -- used 2.6402e-05s (cpu); 2.5293e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │        defining forms: given by a function
│ │ │  
│ │ │  o16 : AbstractRationalMap (rational map from PP^3 to PP^3)
│ │ │  
│ │ │  i17 : time projectiveDegrees(T2,2)
│ │ │ - -- used 6.65901s (cpu); 3.45699s (thread); 0s (gc)
│ │ │ + -- used 7.01159s (cpu); 3.54843s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = 1
│ │ │  
│ │ │  i18 : p = apply(3,i->random(ZZ/65521))|{1}
│ │ │  
│ │ │  o18 = {-6648, -23396, -12311, 1}
│ │ │  
│ │ │ @@ -169,15 +169,15 @@
│ │ │  i20 : T q
│ │ │  
│ │ │  o20 = {-6648, -23396, -12311, 1}
│ │ │  
│ │ │  o20 : List
│ │ │  
│ │ │  i21 : time f = rationalMap T
│ │ │ - -- used 5.38367s (cpu); 2.92119s (thread); 0s (gc)
│ │ │ + -- used 5.82316s (cpu); 2.9113s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_approximate__Inverse__Map.out
│ │ │ @@ -54,15 +54,15 @@
│ │ │  -- approximateInverseMap: step 4 of 10
│ │ │  -- approximateInverseMap: step 5 of 10
│ │ │  -- approximateInverseMap: step 6 of 10
│ │ │  -- approximateInverseMap: step 7 of 10
│ │ │  -- approximateInverseMap: step 8 of 10
│ │ │  -- approximateInverseMap: step 9 of 10
│ │ │  -- approximateInverseMap: step 10 of 10
│ │ │ - -- used 0.272017s (cpu); 0.206969s (thread); 0s (gc)
│ │ │ + -- used 0.287583s (cpu); 0.226178s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                    ZZ
│ │ │       source: Proj(--[t , t , t , t , t , t , t , t , t ])
│ │ │                    97  0   1   2   3   4   5   6   7   8
│ │ │                                  ZZ
│ │ │       target: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │ @@ -109,15 +109,15 @@
│ │ │  
│ │ │  i4 : assert(phi * psi == 1 and psi * phi == 1)
│ │ │  
│ │ │  i5 : time psi' = approximateInverseMap(phi,CodimBsInv=>5);
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ - -- used 0.227935s (cpu); 0.16233s (thread); 0s (gc)
│ │ │ + -- used 0.23606s (cpu); 0.178632s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
│ │ │  
│ │ │  i6 : assert(psi == psi')
│ │ │  
│ │ │  i7 : phi = rationalMap map(P8,ZZ/97[x_0..x_11]/ideal(x_1*x_3-8*x_2*x_3+25*x_3^2-25*x_2*x_4-22*x_3*x_4+x_0*x_5+13*x_2*x_5+41*x_3*x_5-x_0*x_6+12*x_2*x_6+25*x_1*x_7+25*x_3*x_7+23*x_5*x_7-3*x_6*x_7+2*x_0*x_8+11*x_1*x_8-37*x_3*x_8-23*x_4*x_8-33*x_6*x_8+8*x_0*x_9+10*x_1*x_9-25*x_2*x_9-9*x_3*x_9+3*x_4*x_9+24*x_5*x_9-27*x_6*x_9-5*x_0*x_10+28*x_1*x_10+37*x_2*x_10+9*x_4*x_10+27*x_6*x_10-25*x_0*x_11+9*x_2*x_11+27*x_4*x_11-27*x_5*x_11,x_2^2+17*x_2*x_3-14*x_3^2-13*x_2*x_4+34*x_3*x_4+44*x_0*x_5-30*x_2*x_5+27*x_3*x_5+31*x_2*x_6-36*x_3*x_6-x_0*x_7+13*x_1*x_7+8*x_3*x_7+9*x_5*x_7+46*x_6*x_7+41*x_0*x_8-7*x_1*x_8-34*x_3*x_8-9*x_4*x_8-46*x_6*x_8-17*x_0*x_9+32*x_1*x_9-8*x_2*x_9-35*x_3*x_9-46*x_4*x_9+26*x_5*x_9+17*x_6*x_9+15*x_0*x_10+35*x_1*x_10+34*x_2*x_10+20*x_4*x_10+14*x_0*x_11+36*x_1*x_11+35*x_2*x_11-17*x_4*x_11,x_1*x_2-40*x_2*x_3+28*x_3^2-x_0*x_4+5*x_2*x_4-16*x_3*x_4+5*x_0*x_5-36*x_2*x_5+37*x_3*x_5+48*x_2*x_6-5*x_1*x_7-5*x_3*x_7+x_5*x_7+20*x_6*x_7+10*x_0*x_8+34*x_1*x_8+41*x_3*x_8-x_4*x_8+x_6*x_8+40*x_0*x_9-32*x_1*x_9+5*x_2*x_9-11*x_3*x_9-20*x_4*x_9+45*x_5*x_9-14*x_6*x_9-25*x_0*x_10+45*x_1*x_10-41*x_2*x_10-46*x_4*x_10+8*x_6*x_10-28*x_0*x_11+11*x_2*x_11+14*x_4*x_11-8*x_5*x_11),{t_4^2+t_0*t_5+t_1*t_5+35*t_2*t_5+10*t_3*t_5+25*t_4*t_5-5*t_5^2-14*t_0*t_6-14*t_1*t_6-5*t_2*t_6-13*t_4*t_6+37*t_5*t_6+22*t_6^2-31*t_3*t_7+26*t_4*t_7+12*t_5*t_7-45*t_6*t_7-46*t_3*t_8+37*t_4*t_8+28*t_5*t_8+33*t_6*t_8,t_3*t_4+4*t_0*t_5+39*t_1*t_5-40*t_2*t_5+40*t_3*t_5+26*t_4*t_5-20*t_5^2+41*t_0*t_6+36*t_1*t_6-22*t_2*t_6+36*t_4*t_6-30*t_5*t_6-13*t_6^2-25*t_3*t_7+5*t_4*t_7-35*t_5*t_7+10*t_6*t_7+11*t_3*t_8+46*t_4*t_8+29*t_5*t_8+28*t_6*t_8,t_2*t_4-5*t_0*t_5-40*t_1*t_5+12*t_2*t_5+47*t_3*t_5+37*t_4*t_5+25*t_5^2-27*t_0*t_6-22*t_1*t_6+27*t_2*t_6-23*t_4*t_6+5*t_5*t_6-13*t_6^2-39*t_3*t_7-29*t_4*t_7+9*t_5*t_7+39*t_6*t_7+36*t_3*t_8+13*t_4*t_8+26*t_5*t_8+37*t_6*t_8,t_0*t_4-t_0*t_5-8*t_1*t_5-35*t_2*t_5-10*t_3*t_5-33*t_4*t_5+5*t_5^2+15*t_0*t_6+15*t_1*t_6+5*t_2*t_6+15*t_4*t_6-38*t_5*t_6-22*t_6^2+31*t_3*t_7-25*t_4*t_7-19*t_5*t_7+47*t_6*t_7+46*t_3*t_8-36*t_4*t_8-35*t_5*t_8-31*t_6*t_8,t_2*t_3-t_0*t_5-t_1*t_5-35*t_2*t_5-10*t_3*t_5-33*t_4*t_5+5*t_5^2+14*t_0*t_6+14*t_1*t_6+5*t_2*t_6+14*t_4*t_6-31*t_5*t_6-24*t_6^2+32*t_3*t_7-25*t_4*t_7-19*t_5*t_7+47*t_6*t_7+46*t_3*t_8-36*t_4*t_8-35*t_5*t_8-31*t_6*t_8,t_1*t_3-7*t_1*t_5+t_1*t_6+t_4*t_6-7*t_5*t_6+2*t_6^2-t_3*t_7,t_0*t_3-46*t_0*t_5-39*t_1*t_5-43*t_2*t_5-41*t_3*t_5-26*t_4*t_5-28*t_5^2-35*t_0*t_6-36*t_1*t_6+20*t_2*t_6-36*t_4*t_6+9*t_5*t_6+15*t_6^2+26*t_3*t_7-5*t_4*t_7+35*t_5*t_7-10*t_6*t_7-10*t_3*t_8-46*t_4*t_8+47*t_5*t_8-25*t_6*t_8,t_2^2-46*t_1*t_4-33*t_0*t_5-45*t_1*t_5-39*t_2*t_5-39*t_3*t_5-46*t_4*t_5-29*t_5^2-48*t_0*t_6-38*t_1*t_6-30*t_2*t_6+19*t_4*t_6-44*t_5*t_6-47*t_6^2-36*t_0*t_7-46*t_1*t_7+t_2*t_7-44*t_3*t_7+48*t_4*t_7-14*t_5*t_7+4*t_6*t_7-36*t_0*t_8-46*t_1*t_8+47*t_2*t_8-34*t_3*t_8-24*t_4*t_8-12*t_5*t_8-47*t_6*t_8+47*t_7*t_8,t_1*t_2+6*t_1*t_5+5*t_0*t_6-2*t_1*t_6-t_4*t_6-t_5*t_6+5*t_0*t_7+t_1*t_7-2*t_2*t_7-7*t_5*t_7+2*t_6*t_7-2*t_1*t_8+3*t_7*t_8,t_0*t_2+t_1*t_4+5*t_0*t_5+32*t_1*t_5-20*t_2*t_5-47*t_3*t_5-37*t_4*t_5-25*t_5^2+19*t_0*t_6+22*t_1*t_6-25*t_2*t_6+25*t_4*t_6-5*t_5*t_6+13*t_6^2+5*t_0*t_7+t_1*t_7+39*t_3*t_7+28*t_4*t_7-9*t_5*t_7-39*t_6*t_7+4*t_0*t_8+t_1*t_8-36*t_3*t_8-14*t_4*t_8-26*t_5*t_8-37*t_6*t_8,t_0*t_1-39*t_1*t_4+40*t_1*t_5-37*t_0*t_6-39*t_1*t_6+19*t_4*t_6-39*t_5*t_6-38*t_0*t_7+39*t_1*t_7+19*t_2*t_7+18*t_5*t_7-19*t_6*t_7+19*t_1*t_8+20*t_7*t_8,t_0^2+12*t_1*t_4+20*t_0*t_5+27*t_1*t_5-8*t_2*t_5+37*t_3*t_5+28*t_4*t_5+30*t_5^2-46*t_0*t_6+24*t_1*t_6-40*t_2*t_6+25*t_4*t_6+16*t_5*t_6-35*t_6^2+29*t_0*t_7+12*t_1*t_7-35*t_2*t_7-8*t_3*t_7-18*t_4*t_7+42*t_5*t_7-12*t_6*t_7-6*t_0*t_8+12*t_1*t_8-15*t_3*t_8+9*t_4*t_8+20*t_5*t_8-30*t_6*t_8+4*t_7*t_8})
│ │ │  
│ │ │ @@ -192,15 +192,15 @@
│ │ │  o7 : RationalMap (quadratic rational map from PP^8 to 8-dimensional subvariety of PP^11)
│ │ │  
│ │ │  i8 : -- without the option 'CodimBsInv=>4', it takes about triple time 
│ │ │       time psi=approximateInverseMap(phi,CodimBsInv=>4)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ - -- used 2.31467s (cpu); 1.78736s (thread); 0s (gc)
│ │ │ + -- used 2.10674s (cpu); 1.7984s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = -- rational map --
│ │ │                                  ZZ
│ │ │       source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │                                  97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │               {
│ │ │                                  2
│ │ │ @@ -258,15 +258,15 @@
│ │ │  
│ │ │  i10 : -- in this case we can remedy enabling the option Certify
│ │ │        time psi = approximateInverseMap(phi,CodimBsInv=>4,Certify=>true)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │  Certify: output certified!
│ │ │ - -- used 3.88333s (cpu); 3.11658s (thread); 0s (gc)
│ │ │ + -- used 2.90154s (cpu); 2.53s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = -- rational map --
│ │ │                                   ZZ
│ │ │        source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │                                   97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │                {
│ │ │                                   2
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_degree__Map.out
│ │ │ @@ -9,27 +9,27 @@
│ │ │                                   2                  2                             2                                       2                                                2                                                           2                                                                       2                                                                              2                                                                                            2         2                 2                             2                                       2                                              2                                                           2                                                                   2                                                                               2                                                                                          2        2                   2                            2                                      2                                                  2                                                          2                                                                      2                                                                               2                                                                                            2        2                   2                             2                                      2                                                 2                                                          2                                                                    2                                                                               2                                                                                          2         2                2                          2                                      2                                                 2                                                           2                                                                       2                                                                            2                                                                                          2        2                  2                         2                                       2                                                2                                                           2                                                                      2                                                                               2                                                                                        2       2                  2                             2                                    2                                                2                                                          2                                                                    2                                                                               2                                                                                       2       2                 2                           2                                      2                                                 2                                                            2                                                                       2                                                                                2                                                                                           2        2                   2                             2                                       2                                                  2                                                            2                                                                   2                                                                            2                                                                                          2      2                 2                            2                                       2                                                2                                                         2                                                                        2                                                                               2                                                                                          2     2                   2                             2                                      2                                                   2                                                          2                                                                     2                                                                               2                                                                                          2         2                2                            2                                       2                                                 2                                                           2                                                                      2                                                                                  2                                                                                              2      2                  2                            2                                    2                                                2                                                            2                                                                    2                                                                                2                                                                                          2       2                  2                            2                                    2                                                   2                                                        2                                                                         2                                                                               2                                                                                           2       2                  2                             2                                       2                                                 2                                                          2                                                                       2                                                                               2                                                                                       2
│ │ │  o4 = map (ringP8, ringP14, {- 95x  + 181x x  + 1028x  - 1384x x  - 1455x x  + 559x  - 502x x  + 1264x x  - 162x x  + 1209x  - 180x x  - 504x x  - 1168x x  - 676x x  + 501x  + 73x x  + 1263x x  + 1035x x  + 844x x  + 1593x x  + 785x  + 982x x  - 412x x  + 1335x x  + 1136x x  + 826x x  + 1078x x  + 1158x  + 335x x  - 982x x  - 1479x x  - 15x x  + 1363x x  + 1397x x  - 575x x  - 71x  + 1255x x  - 1138x x  - 1590x x  + 604x x  + 1182x x  - 63x x  - 1382x x  - 1255x x  - 613x , - 1444x  + 575x x  + 767x  - 1495x x  + 1631x x  - 217x  - 294x x  - 1511x x  - 504x x  - 1284x  - 1459x x  + 152x x  + 141x x  - 10x x  - 95x  + 1056x x  + 654x x  + 1397x x  - 930x x  + 578x x  - 696x  + 759x x  + 733x x  + 505x x  - 609x x  + 526x x  - 659x x  + 846x  + 1253x x  - 1519x x  + 635x x  + 576x x  + 54x x  - 1261x x  - 822x x  - 257x  - 986x x  + 356x x  - 1488x x  - 1561x x  - 850x x  - 85x x  - 1350x x  - 783x x  - 1335x , - 871x  + 1006x x  - 1399x  - 1636x x  - 699x x  - 769x  - 307x x  - 1645x x  - 502x x  - 719x  + 1405x x  + 870x x  - 1133x x  + 425x x  - 1203x  - 1601x x  + 117x x  - 382x x  + 318x x  - 117x x  - 560x  + 1135x x  + 1468x x  + 869x x  - 943x x  - 335x x  - 1218x x  + 201x  - 11x x  + 540x x  - 710x x  - 489x x  + 1605x x  + 1663x x  - 423x x  + 1246x  + 97x x  - 644x x  + 1655x x  + 1219x x  + 1476x x  + 1355x x  + 1594x x  + 893x x  + 1150x , - 143x  + 1240x x  - 1042x  + 1649x x  + 1024x x  + 794x  + 1442x x  - 1263x x  + 537x x  - 82x  - 734x x  - 1569x x  - 798x x  - 366x x  + 1289x  - 569x x  - 254x x  + 237x x  - 1234x x  - 807x x  + 264x  - 202x x  - 616x x  + 44x x  + 1465x x  + 685x x  + 1630x x  - 406x  - 123x x  - 4x x  + 1583x x  + 1235x x  + 162x x  + 1034x x  - 1035x x  + 737x  + 660x x  + 1459x x  - 359x x  - 1291x x  + 1638x x  - 325x x  - 631x x  + 73x x  - 1471x , - 1340x  + 31x x  - 994x  - 880x x  - 89x x  + 574x  + 760x x  - 1054x x  + 772x x  - 239x  - 443x x  + 1240x x  + 637x x  - 1423x x  + 320x  - 1363x x  - 1139x x  - 158x x  - 325x x  - 1578x x  + 32x  + 695x x  + 305x x  + 1012x x  + 1492x x  + 1290x x  + 1579x x  - 342x  - 83x x  - 104x x  + 998x x  - 92x x  + 1554x x  + 201x x  - 237x x  + 160x  - 228x x  - 543x x  - 1147x x  - 376x x  + 1313x x  + 603x x  + 106x x  - 1361x x  + 699x , - 228x  - 1510x x  + 277x  - 4x x  - 22x x  - 1526x  + 234x x  + 969x x  + 1253x x  - 1426x  - 1474x x  + 947x x  + 194x x  - 316x x  - 988x  - 1211x x  + 1087x x  + 536x x  - 491x x  + 870x x  - 659x  + 1490x x  - 469x x  + 1190x x  + 807x x  + 650x x  + 448x x  - 1353x  - 218x x  + 759x x  - 253x x  + 830x x  - 1080x x  - 143x x  - 1313x x  - 374x  - 180x x  + 741x x  + 742x x  - 1254x x  + 458x x  - 345x x  + 597x x  + 1567x x  - 31x , 1120x  + 709x x  - 1538x  - 1048x x  - 162x x  - 1518x  - 73x x  + 380x x  + 533x x  - 286x  + 1374x x  - 74x x  - 22x x  + 1535x x  - 1071x  - 839x x  - 560x x  + 928x x  + 335x x  - 1008x x  + 810x  - 448x x  - 357x x  - 107x x  + 40x x  + 784x x  - 1423x x  + 1276x  + 147x x  + 443x x  - 598x x  - 1077x x  - 1214x x  + 322x x  - 1408x x  + 72x  - 63x x  - 1513x x  - 791x x  + 11x x  + 77x x  + 836x x  - 1100x x  + 1637x x  - 788x , 1331x  + 318x x  - 704x  + 51x x  + 275x x  + 1149x  + 1526x x  + 768x x  + 414x x  - 782x  - 262x x  + 686x x  - 380x x  + 1377x x  + 1077x  + 1650x x  - 1129x x  - 508x x  + 846x x  + 1513x x  + 460x  - 1626x x  - 1024x x  + 862x x  + 1352x x  - 188x x  - 1382x x  - 650x  + 55x x  - 326x x  + 1037x x  + 705x x  - 667x x  + 1483x x  + 1661x x  - 1652x  - 1052x x  - 692x x  - 542x x  + 162x x  + 582x x  - 1369x x  + 934x x  + 1392x x  + 1227x , - 346x  + 1408x x  - 1225x  - 1536x x  - 1028x x  - 985x  - 210x x  - 1312x x  + 915x x  + 1633x  - 202x x  - 1636x x  - 1653x x  - 480x x  - 1260x  - 813x x  - 1623x x  - 1429x x  + 1094x x  - 747x x  + 955x  + 898x x  - 795x x  - 35x x  - 566x x  + 1631x x  - 324x x  + 926x  - 132x x  - 9x x  - 1290x x  - 543x x  + 902x x  + 735x x  - 342x x  - 400x  + 900x x  - 463x x  + 694x x  - 1262x x  - 1449x x  - 448x x  - 1402x x  - 731x x  - 996x , 301x  + 166x x  - 955x  - 739x x  - 1199x x  - 319x  + 1047x x  - 532x x  + 902x x  + 1195x  - 663x x  + 1215x x  - 534x x  - 332x x  - 973x  + 772x x  - 308x x  + 315x x  - 454x x  - 483x x  - 239x  - 1313x x  - 419x x  - 1340x x  - 1388x x  - 1340x x  - 1665x x  - 333x  - 465x x  - 1084x x  + 676x x  - 1612x x  - 288x x  + 11x x  - 1170x x  - 189x  + 498x x  - 889x x  + 693x x  + 1460x x  - 473x x  - 414x x  - 122x x  - 1659x x  - 1421x , 14x  - 1049x x  + 1506x  + 1235x x  + 642x x  - 1034x  + 460x x  + 150x x  + 760x x  - 1246x  - 1407x x  + 1570x x  + 1403x x  - 1610x x  - 431x  + 574x x  + 893x x  - 657x x  + 417x x  + 1362x x  + 224x  + 268x x  + 1097x x  + 1132x x  + 148x x  + 1331x x  - 77x x  - 756x  + 228x x  + 136x x  - 1484x x  - 1478x x  - 13x x  + 1620x x  - 701x x  - 769x  - 760x x  - 492x x  - 1077x x  - 1249x x  - 834x x  - 395x x  - 1358x x  - 988x x  + 113x , - 1634x  - 13x x  + 805x  - 21x x  - 1655x x  + 1479x  - 1510x x  - 646x x  + 225x x  - 1411x  + 1227x x  - 1108x x  + 1291x x  - 59x x  - 142x  + 586x x  - 676x x  + 655x x  - 1476x x  + 453x x  - 1076x  - 1152x x  + 1373x x  - 1191x x  - 416x x  + 699x x  + 317x x  + 825x  - 1560x x  - 488x x  - 1035x x  - 1561x x  - 644x x  - 1178x x  - 1320x x  + 158x  + 889x x  + 1444x x  - 1486x x  - 1211x x  + 1269x x  - 1228x x  + 568x x  + 1591x x  + 1207x , 105x  - 538x x  - 1222x  - 277x x  + 716x x  - 1067x  - 428x x  + 154x x  - 469x x  + 77x  + 538x x  - 179x x  + 921x x  - 223x x  + 1093x  - 262x x  + 1299x x  + 631x x  + 1486x x  - 1280x x  - 121x  - 50x x  - 978x x  - 694x x  - 531x x  + 505x x  + 1412x x  - 1061x  + 1202x x  + 448x x  - 187x x  + 1276x x  - 121x x  + 1361x x  + 697x x  + 682x  + 1592x x  + 705x x  - 227x x  - 7x x  - 1423x x  - 1446x x  - 1578x x  + 1511x x  + 917x , 1270x  - 391x x  - 1116x  - 287x x  + 653x x  + 1643x  + 1623x x  + 514x x  - 14x x  - 90x  + 1232x x  - 1434x x  + 1296x x  + 1522x x  + 136x  - 623x x  - 607x x  + 18x x  + 896x x  - 29x x  + 1059x  - 1053x x  + 1643x x  + 1652x x  - 1190x x  - 1073x x  + 1470x x  - 944x  - 93x x  - 187x x  - 994x x  - 1415x x  - 229x x  - 796x x  + 1642x x  + 1600x  - 344x x  + 905x x  + 1032x x  - 538x x  - 891x x  + 1243x x  + 1290x x  + 490x x  - 1148x , 1613x  + 175x x  - 1346x  - 1000x x  - 1217x x  - 729x  - 1296x x  + 1456x x  + 745x x  + 539x  + 525x x  - 811x x  + 753x x  + 1362x x  + 1629x  - 840x x  + 513x x  + 429x x  + 842x x  + 1414x x  - 308x  + 1415x x  - 1461x x  - 1135x x  + 701x x  + 766x x  + 785x x  + 1503x  + 147x x  + 929x x  - 1220x x  - 853x x  + 493x x  + 226x x  + 1416x x  + 280x  - 7x x  + 1632x x  + 520x x  + 1259x x  + 157x x  + 1596x x  + 655x x  - 42x x  - 586x })
│ │ │                                   0       0 1        1        0 2        1 2       2       0 3        1 3       2 3        3       0 4       1 4        2 4       3 4       4      0 5        1 5        2 5       3 5        4 5       5       0 6       1 6        2 6        3 6       4 6        5 6        6       0 7       1 7        2 7      3 7        4 7        5 7       6 7      7        0 8        1 8        2 8       3 8        4 8      5 8        6 8        7 8       8         0       0 1       1        0 2        1 2       2       0 3        1 3       2 3        3        0 4       1 4       2 4      3 4      4        0 5       1 5        2 5       3 5       4 5       5       0 6       1 6       2 6       3 6       4 6       5 6       6        0 7        1 7       2 7       3 7      4 7        5 7       6 7       7       0 8       1 8        2 8        3 8       4 8      5 8        6 8       7 8        8        0        0 1        1        0 2       1 2       2       0 3        1 3       2 3       3        0 4       1 4        2 4       3 4        4        0 5       1 5       2 5       3 5       4 5       5        0 6        1 6       2 6       3 6       4 6        5 6       6      0 7       1 7       2 7       3 7        4 7        5 7       6 7        7      0 8       1 8        2 8        3 8        4 8        5 8        6 8       7 8        8        0        0 1        1        0 2        1 2       2        0 3        1 3       2 3      3       0 4        1 4       2 4       3 4        4       0 5       1 5       2 5        3 5       4 5       5       0 6       1 6      2 6        3 6       4 6        5 6       6       0 7     1 7        2 7        3 7       4 7        5 7        6 7       7       0 8        1 8       2 8        3 8        4 8       5 8       6 8      7 8        8         0      0 1       1       0 2      1 2       2       0 3        1 3       2 3       3       0 4        1 4       2 4        3 4       4        0 5        1 5       2 5       3 5        4 5      5       0 6       1 6        2 6        3 6        4 6        5 6       6      0 7       1 7       2 7      3 7        4 7       5 7       6 7       7       0 8       1 8        2 8       3 8        4 8       5 8       6 8        7 8       8        0        0 1       1     0 2      1 2        2       0 3       1 3        2 3        3        0 4       1 4       2 4       3 4       4        0 5        1 5       2 5       3 5       4 5       5        0 6       1 6        2 6       3 6       4 6       5 6        6       0 7       1 7       2 7       3 7        4 7       5 7        6 7       7       0 8       1 8       2 8        3 8       4 8       5 8       6 8        7 8      8       0       0 1        1        0 2       1 2        2      0 3       1 3       2 3       3        0 4      1 4      2 4        3 4        4       0 5       1 5       2 5       3 5        4 5       5       0 6       1 6       2 6      3 6       4 6        5 6        6       0 7       1 7       2 7        3 7        4 7       5 7        6 7      7      0 8        1 8       2 8      3 8      4 8       5 8        6 8        7 8       8       0       0 1       1      0 2       1 2        2        0 3       1 3       2 3       3       0 4       1 4       2 4        3 4        4        0 5        1 5       2 5       3 5        4 5       5        0 6        1 6       2 6        3 6       4 6        5 6       6      0 7       1 7        2 7       3 7       4 7        5 7        6 7        7        0 8       1 8       2 8       3 8       4 8        5 8       6 8        7 8        8        0        0 1        1        0 2        1 2       2       0 3        1 3       2 3        3       0 4        1 4        2 4       3 4        4       0 5        1 5        2 5        3 5       4 5       5       0 6       1 6      2 6       3 6        4 6       5 6       6       0 7     1 7        2 7       3 7       4 7       5 7       6 7       7       0 8       1 8       2 8        3 8        4 8       5 8        6 8       7 8       8      0       0 1       1       0 2        1 2       2        0 3       1 3       2 3        3       0 4        1 4       2 4       3 4       4       0 5       1 5       2 5       3 5       4 5       5        0 6       1 6        2 6        3 6        4 6        5 6       6       0 7        1 7       2 7        3 7       4 7      5 7        6 7       7       0 8       1 8       2 8        3 8       4 8       5 8       6 8        7 8        8     0        0 1        1        0 2       1 2        2       0 3       1 3       2 3        3        0 4        1 4        2 4        3 4       4       0 5       1 5       2 5       3 5        4 5       5       0 6        1 6        2 6       3 6        4 6      5 6       6       0 7       1 7        2 7        3 7      4 7        5 7       6 7       7       0 8       1 8        2 8        3 8       4 8       5 8        6 8       7 8       8         0      0 1       1      0 2        1 2        2        0 3       1 3       2 3        3        0 4        1 4        2 4      3 4       4       0 5       1 5       2 5        3 5       4 5        5        0 6        1 6        2 6       3 6       4 6       5 6       6        0 7       1 7        2 7        3 7       4 7        5 7        6 7       7       0 8        1 8        2 8        3 8        4 8        5 8       6 8        7 8        8      0       0 1        1       0 2       1 2        2       0 3       1 3       2 3      3       0 4       1 4       2 4       3 4        4       0 5        1 5       2 5        3 5        4 5       5      0 6       1 6       2 6       3 6       4 6        5 6        6        0 7       1 7       2 7        3 7       4 7        5 7       6 7       7        0 8       1 8       2 8     3 8        4 8        5 8        6 8        7 8       8       0       0 1        1       0 2       1 2        2        0 3       1 3      2 3      3        0 4        1 4        2 4        3 4       4       0 5       1 5      2 5       3 5      4 5        5        0 6        1 6        2 6        3 6        4 6        5 6       6      0 7       1 7       2 7        3 7       4 7       5 7        6 7        7       0 8       1 8        2 8       3 8       4 8        5 8        6 8       7 8        8       0       0 1        1        0 2        1 2       2        0 3        1 3       2 3       3       0 4       1 4       2 4        3 4        4       0 5       1 5       2 5       3 5        4 5       5        0 6        1 6        2 6       3 6       4 6       5 6        6       0 7       1 7        2 7       3 7       4 7       5 7        6 7       7     0 8        1 8       2 8        3 8       4 8        5 8       6 8      7 8       8
│ │ │  
│ │ │  o4 : RingMap ringP8 <-- ringP14
│ │ │  
│ │ │  i5 : time degreeMap phi
│ │ │ - -- used 0.0453696s (cpu); 0.0453707s (thread); 0s (gc)
│ │ │ + -- used 0.0553537s (cpu); 0.0550886s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 1
│ │ │  
│ │ │  i6 : -- Compose phi:P^8--->P^14 with a linear projection P^14--->P^8 from a general subspace of P^14 
│ │ │       -- of dimension 5 (so that the composition phi':P^8--->P^8 must have degree equal to deg(G(1,5))=14)
│ │ │       phi'=phi*map(ringP14,ringP8,for i to 8 list random(1,ringP14))
│ │ │  
│ │ │                                   2                  2                           2                                      2                                                 2                                                           2                                                                   2                                                                              2                                                                                          2        2                  2                              2                                       2                                                2                                                             2                                                                  2                                                                              2                                                                                            2        2                  2                             2                                       2                                                2                                                           2                                                                      2                                                                              2                                                                                         2         2                 2                            2                                       2                                                  2                                                             2                                                                    2                                                                                2                                                                                             2       2                   2                            2                                     2                                                2                                                          2                                                                  2                                                                                   2                                                                                            2        2                2                           2                                      2                                                  2                                                            2                                                                      2                                                                                 2                                                                                          2   2                   2                           2                                     2                                                  2                                                           2                                                                    2                                                                              2                                                                                         2      2                  2                           2                                      2                                                  2                                                             2                                                                       2                                                                              2                                                                                          2         2                  2                            2                                     2                                                 2                                                              2                                                                    2                                                                               2                                                                                        2
│ │ │  o6 = map (ringP8, ringP8, {- 780x  - 506x x  + 1537x  - 132x x  - 928x x  + 386x  - 102x x  + 422x x  + 725x x  - 1073x  - 905x x  - 830x x  + 1500x x  + 276x x  + 1533x  - 653x x  + 1558x x  + 939x x  - 1432x x  + 462x x  - 329x  - 92x x  + 661x x  - 1298x x  - 684x x  + 70x x  - 715x x  + 1093x  + 581x x  + 329x x  + 454x x  - 911x x  - 84x x  - 1452x x  - 809x x  + 1202x  + 1353x x  + 1503x x  + 482x x  + 893x x  - 643x x  + 598x x  + 110x x  + 1064x x  - 472x , - 522x  - 583x x  + 1339x  + 1535x x  - 1317x x  + 1113x  - 169x x  + 1440x x  - 1657x x  + 721x  + 40x x  - 1576x x  - 367x x  + 257x x  - 1454x  + 1612x x  + 1529x x  - 1068x x  + 560x x  - 1441x x  + 608x  - 92x x  - 1006x x  + 285x x  + 102x x  - 397x x  + 66x x  - 643x  - 38x x  + 1380x x  + 1069x x  - 426x x  + 1147x x  + 982x x  + 10x x  - 662x  + 16x x  + 1561x x  + 1597x x  + 512x x  + 1288x x  - 1253x x  + 1317x x  + 1481x x  - 354x , - 640x  - 1551x x  + 469x  + 1482x x  - 1593x x  - 986x  + 471x x  + 612x x  + 1228x x  + 1156x  - 731x x  + 1503x x  - 628x x  + 674x x  - 799x  + 1137x x  + 844x x  + 589x x  - 666x x  + 829x x  - 1024x  - 170x x  + 450x x  + 1497x x  + 1204x x  - 907x x  + 1621x x  - 417x  + 1297x x  + 1444x x  + 4x x  + 398x x  + 996x x  - 1031x x  + 239x x  + 303x  + 1215x x  - 83x x  + 1571x x  - 1543x x  - 925x x  - 694x x  + 151x x  - 520x x  + 880x , - 1210x  - 222x x  + 185x  + 245x x  + 1059x x  - 322x  + 238x x  + 962x x  + 1260x x  - 1581x  + 50x x  + 1352x x  - 1465x x  + 1555x x  + 1333x  + 1362x x  + 1365x x  + 1168x x  - 1401x x  + 149x x  - 652x  + 1378x x  - 557x x  - 112x x  + 26x x  + 315x x  + 111x x  + 1592x  - 283x x  - 1454x x  + 907x x  + 212x x  + 400x x  + 1049x x  - 882x x  - 1429x  - 183x x  + 1571x x  - 1286x x  - 1179x x  + 1319x x  + 240x x  - 1100x x  + 1500x x  - 348x , 1051x  - 1325x x  + 1354x  - 346x x  - 1532x x  - 466x  + 163x x  - 659x x  - 291x x  + 966x  + 789x x  + 393x x  + 403x x  - 1199x x  - 570x  - 93x x  - 492x x  - 418x x  + 713x x  - 1323x x  - 1384x  - 830x x  - 54x x  - 306x x  + 709x x  + 421x x  - 954x x  - 299x  + 1053x x  - 1080x x  + 686x x  + 170x x  - 1272x x  - 1661x x  + 1235x x  + 1553x  - 1454x x  - 1411x x  - 1195x x  - 962x x  + 737x x  - 390x x  + 957x x  + 1538x x  + 1234x , - 509x  + 9x x  - 1563x  - 710x x  - 642x x  + 541x  + 220x x  - 1214x x  - 16x x  + 1008x  - 1088x x  + 755x x  - 886x x  - 1433x x  + 1154x  + 1627x x  - 1547x x  - 951x x  + 866x x  + 163x x  - 1142x  - 668x x  + 1361x x  + 1324x x  - 490x x  + 282x x  - 1133x x  - 612x  + 805x x  - 126x x  + 1296x x  - 973x x  + 1271x x  - 1646x x  + 844x x  + 1073x  - 1452x x  - 1112x x  - 141x x  + 176x x  - 1579x x  - 78x x  + 848x x  - 1365x x  + 711x , x  + 1543x x  - 1076x  + 493x x  - 526x x  + 868x  - 582x x  - 996x x  + 206x x  - 419x  + 1258x x  - 391x x  + 1002x x  - 1539x x  + 931x  - 1504x x  + 810x x  + 324x x  + 1356x x  + 313x x  + 772x  + 299x x  + 1186x x  + 718x x  + 407x x  - 64x x  - 828x x  - 1393x  + 94x x  - 290x x  - 766x x  + 950x x  - 640x x  + 265x x  - 1640x x  - 1403x  - 126x x  + 891x x  - 1519x x  - 927x x  - 1335x x  - 1448x x  - x x  - 1103x x  - 1152x , 821x  + 558x x  - 1174x  - 168x x  + 986x x  + 790x  + 549x x  + 817x x  + 1396x x  + 695x  + 1211x x  + 878x x  - 1061x x  - 1244x x  - 880x  + 1409x x  - 567x x  + 1240x x  + 1126x x  - 1262x x  + 490x  + 1553x x  + 1276x x  + 805x x  + 576x x  - 1076x x  + 1617x x  - 495x  - 750x x  - 277x x  + 544x x  + 1479x x  - 784x x  - 64x x  - 1203x x  + 405x  + 1013x x  + 604x x  + 1301x x  + 1003x x  + 235x x  + 696x x  + 939x x  - 714x x  - 879x , - 1452x  + 727x x  - 1159x  + 449x x  - 1169x x  + 732x  + 575x x  - 600x x  + 924x x  - 837x  + 1298x x  - 860x x  + 1010x x  + 774x x  + 319x  + 1087x x  - 1120x x  + 1439x x  + 1175x x  - 1648x x  + 985x  - 1317x x  - 878x x  + 399x x  - 1339x x  + 70x x  - 463x x  + 470x  - 628x x  - 907x x  + 748x x  + 98x x  + 1150x x  + 1140x x  + 1308x x  + 621x  + 369x x  - 991x x  - 1186x x  + 61x x  - 907x x  - 681x x  - 1528x x  + 717x x  + 854x })
│ │ │                                   0       0 1        1       0 2       1 2       2       0 3       1 3       2 3        3       0 4       1 4        2 4       3 4        4       0 5        1 5       2 5        3 5       4 5       5      0 6       1 6        2 6       3 6      4 6       5 6        6       0 7       1 7       2 7       3 7      4 7        5 7       6 7        7        0 8        1 8       2 8       3 8       4 8       5 8       6 8        7 8       8        0       0 1        1        0 2        1 2        2       0 3        1 3        2 3       3      0 4        1 4       2 4       3 4        4        0 5        1 5        2 5       3 5        4 5       5      0 6        1 6       2 6       3 6       4 6      5 6       6      0 7        1 7        2 7       3 7        4 7       5 7      6 7       7      0 8        1 8        2 8       3 8        4 8        5 8        6 8        7 8       8        0        0 1       1        0 2        1 2       2       0 3       1 3        2 3        3       0 4        1 4       2 4       3 4       4        0 5       1 5       2 5       3 5       4 5        5       0 6       1 6        2 6        3 6       4 6        5 6       6        0 7        1 7     2 7       3 7       4 7        5 7       6 7       7        0 8      1 8        2 8        3 8       4 8       5 8       6 8       7 8       8         0       0 1       1       0 2        1 2       2       0 3       1 3        2 3        3      0 4        1 4        2 4        3 4        4        0 5        1 5        2 5        3 5       4 5       5        0 6       1 6       2 6      3 6       4 6       5 6        6       0 7        1 7       2 7       3 7       4 7        5 7       6 7        7       0 8        1 8        2 8        3 8        4 8       5 8        6 8        7 8       8       0        0 1        1       0 2        1 2       2       0 3       1 3       2 3       3       0 4       1 4       2 4        3 4       4      0 5       1 5       2 5       3 5        4 5        5       0 6      1 6       2 6       3 6       4 6       5 6       6        0 7        1 7       2 7       3 7        4 7        5 7        6 7        7        0 8        1 8        2 8       3 8       4 8       5 8       6 8        7 8        8        0     0 1        1       0 2       1 2       2       0 3        1 3      2 3        3        0 4       1 4       2 4        3 4        4        0 5        1 5       2 5       3 5       4 5        5       0 6        1 6        2 6       3 6       4 6        5 6       6       0 7       1 7        2 7       3 7        4 7        5 7       6 7        7        0 8        1 8       2 8       3 8        4 8      5 8       6 8        7 8       8   0        0 1        1       0 2       1 2       2       0 3       1 3       2 3       3        0 4       1 4        2 4        3 4       4        0 5       1 5       2 5        3 5       4 5       5       0 6        1 6       2 6       3 6      4 6       5 6        6      0 7       1 7       2 7       3 7       4 7       5 7        6 7        7       0 8       1 8        2 8       3 8        4 8        5 8    6 8        7 8        8      0       0 1        1       0 2       1 2       2       0 3       1 3        2 3       3        0 4       1 4        2 4        3 4       4        0 5       1 5        2 5        3 5        4 5       5        0 6        1 6       2 6       3 6        4 6        5 6       6       0 7       1 7       2 7        3 7       4 7      5 7        6 7       7        0 8       1 8        2 8        3 8       4 8       5 8       6 8       7 8       8         0       0 1        1       0 2        1 2       2       0 3       1 3       2 3       3        0 4       1 4        2 4       3 4       4        0 5        1 5        2 5        3 5        4 5       5        0 6       1 6       2 6        3 6      4 6       5 6       6       0 7       1 7       2 7      3 7        4 7        5 7        6 7       7       0 8       1 8        2 8      3 8       4 8       5 8        6 8       7 8       8
│ │ │  
│ │ │  o6 : RingMap ringP8 <-- ringP8
│ │ │  
│ │ │  i7 : time degreeMap phi'
│ │ │ - -- used 1.24889s (cpu); 0.706401s (thread); 0s (gc)
│ │ │ + -- used 1.22619s (cpu); 0.716078s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = 14
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_force__Image.out
│ │ │ @@ -5,14 +5,14 @@
│ │ │  o2 : Ideal of P6
│ │ │  
│ │ │  i3 : Phi = rationalMap(X,Dominant=>2);
│ │ │  
│ │ │  o3 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9)
│ │ │  
│ │ │  i4 : time forceImage(Phi,ideal 0_(target Phi))
│ │ │ - -- used 0.000607219s (cpu); 0.000601939s (thread); 0s (gc)
│ │ │ + -- used 0.000859741s (cpu); 0.000852916s (thread); 0s (gc)
│ │ │  
│ │ │  i5 : Phi;
│ │ │  
│ │ │  o5 : RationalMap (cubic dominant rational map from PP^6 to 6-dimensional subvariety of PP^9)
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_graph.out
│ │ │ @@ -35,15 +35,15 @@
│ │ │                        - x  + x x
│ │ │                           3    2 4
│ │ │                       }
│ │ │  
│ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │  
│ │ │  i3 : time (p1,p2) = graph phi;
│ │ │ - -- used 0.0188178s (cpu); 0.0184354s (thread); 0s (gc)
│ │ │ + -- used 0.081288s (cpu); 0.0303614s (thread); 0s (gc)
│ │ │  
│ │ │  i4 : p1
│ │ │  
│ │ │  o4 = -- rational map --
│ │ │                                    ZZ                                 ZZ
│ │ │       source: subvariety of Proj(------[x , x , x , x , x ]) x Proj(------[y , y , y , y , y , y ]) defined by
│ │ │                                  190181  0   1   2   3   4          190181  0   1   2   3   4   5
│ │ │ @@ -173,15 +173,15 @@
│ │ │  i8 : projectiveDegrees p2
│ │ │  
│ │ │  o8 = {51, 28, 14, 6, 2}
│ │ │  
│ │ │  o8 : List
│ │ │  
│ │ │  i9 : time g = graph p2;
│ │ │ - -- used 0.0317165s (cpu); 0.0312624s (thread); 0s (gc)
│ │ │ + -- used 0.0536877s (cpu); 0.037804s (thread); 0s (gc)
│ │ │  
│ │ │  i10 : g_0;
│ │ │  
│ │ │  o10 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to PP^4)
│ │ │  
│ │ │  i11 : g_1;
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_ideal_lp__Rational__Map_rp.out
│ │ │ @@ -33,15 +33,15 @@
│ │ │                        x  - x x
│ │ │                         1    0 3
│ │ │                       }
│ │ │  
│ │ │  o2 : RationalMap (quadratic rational map from hypersurface in PP^5 to PP^4)
│ │ │  
│ │ │  i3 : time ideal phi
│ │ │ - -- used 0.0035599s (cpu); 0.00355522s (thread); 0s (gc)
│ │ │ + -- used 0.00385233s (cpu); 0.00384916s (thread); 0s (gc)
│ │ │  
│ │ │               2                                     2                      
│ │ │  o3 = ideal (x  - x x , x x  - x x  + x x , x x  - x  + x x , x x  - x x  +
│ │ │               4    3 5   2 4    3 4    1 5   2 3    3    1 4   1 2    1 3  
│ │ │       ------------------------------------------------------------------------
│ │ │              2
│ │ │       x x , x  - x x )
│ │ │ @@ -108,15 +108,15 @@
│ │ │                        y
│ │ │                         4
│ │ │                       }
│ │ │  
│ │ │  o5 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^5 x PP^4 to PP^4)
│ │ │  
│ │ │  i6 : time ideal phi'
│ │ │ - -- used 0.0930691s (cpu); 0.0930488s (thread); 0s (gc)
│ │ │ + -- used 0.103877s (cpu); 0.103879s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = ideal 1
│ │ │  
│ │ │                                                                                                              QQ[x ..x , y ..y ]
│ │ │                                                                                                                  0   5   0   4
│ │ │  o6 : Ideal of --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
│ │ │                                                                                                                                                                                                       2
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_inverse__Map.out
│ │ │ @@ -72,15 +72,15 @@
│ │ │                        w w  - w w  + w w
│ │ │                         2 4    1 5    0 6
│ │ │                       }
│ │ │  
│ │ │  o1 : RationalMap (quadratic Cremona transformation of PP^20)
│ │ │  
│ │ │  i2 : time psi = inverseMap phi
│ │ │ - -- used 0.184232s (cpu); 0.12115s (thread); 0s (gc)
│ │ │ + -- used 0.19784s (cpu); 0.113276s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = -- rational map --
│ │ │       source: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  ])
│ │ │                        0   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19   20
│ │ │       target: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  ])
│ │ │                        0   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19   20
│ │ │       defining forms: {
│ │ │ @@ -158,15 +158,15 @@
│ │ │  o4 = map (QQ[w ..w  ], QQ[w ..w  ], {w  w   - w  w   - w  w   - w  w   - w w  , w  w   - w  w   - w  w   - w  w   - w w  , w  w   - w  w   - w  w   - w  w   - w w  , w  w   - w  w   + w  w   - w  w   - w w  , w  w   - w  w   + w  w   + w  w   - w w  , w w   - w w   + w w   + w w   + w w  , w  w   - w  w   + w  w   - w  w   - w w  , w  w   - w  w   + w  w   + w  w   - w w  , w w   - w w   + w w   + w w   + w w  , w  w   - w  w   - w  w   + w  w   - w w  , w w   - w w   - w w   + w w   + w w  , w  w   - w  w   - w  w   + w  w   - w w  , w  w   - w  w   - w  w   + w  w   - w w  , w w   - w w   - w w   + w w   + w w  , w w   - w w   - w w   + w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w   - w w   - w w   - w w   + w w  , w w   - w w   - w w   - w w   + w w  , w w   - w w   - w w   - w w   + w w  , w w   - w w   - w w   - w w   + w w  , w w   - w w   - w w   + w w   - w w  , w w   - w w   + w w   + w w   - w w  , w w   - w w   - w w   - w w   + w w  , w w   - w w   - w w   - w w   + w w  , w w   - w w   - w w   + w w   - w w  , w w   - w w   - w w   + w w   - w w  , w w  - w w  - w w  + w w  - w w })
│ │ │                0   26       0   26     21 22    20 23    15 24    10 25    0 26   19 22    18 23    16 24    11 25    1 26   19 20    18 21    17 24    12 25    2 26   15 19    16 21    17 23    13 25    3 26   10 19    11 21    12 23    13 24    4 26   0 19    1 21    2 23    3 24    4 25   15 18    16 20    17 22    14 25    5 26   10 18    11 20    12 22    14 24    6 26   0 18    1 20    2 22    5 24    6 25   12 16    11 17    13 18    14 19    7 26   2 16    1 17    3 18    5 19    7 25   12 15    10 17    13 20    14 21    8 26   11 15    10 16    13 22    14 23    9 26   2 15    0 17    3 20    5 21    8 25   1 15    0 16    3 22    5 23    9 25   5 13    3 14    7 15    8 16    9 17   5 12    2 14    6 17    8 18    7 20   3 12    2 13    4 17    8 19    7 21   5 11    1 14    6 16    9 18    7 22   3 11    1 13    4 16    9 19    7 23   2 11    1 12    4 18    6 19    7 24   7 10    8 11    9 12    6 13    4 14   5 10    0 14    6 15    9 20    8 22   3 10    0 13    4 15    9 21    8 23   2 10    0 12    4 20    6 21    8 24   1 10    0 11    4 22    6 23    9 24   4 5    3 6    0 7    1 8    2 9
│ │ │  
│ │ │  o4 : RingMap QQ[w ..w  ] <-- QQ[w ..w  ]
│ │ │                   0   26          0   26
│ │ │  
│ │ │  i5 : time psi = inverseMap phi
│ │ │ - -- used 0.371428s (cpu); 0.224639s (thread); 0s (gc)
│ │ │ + -- used 0.350088s (cpu); 0.208419s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = map (QQ[w ..w  ], QQ[w ..w  ], {- w w   + w w   + w  w   - w  w   - w w  , - w w   + w w   + w  w   - w  w   - w w  , - w w   + w w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , w  w   - w  w   + w  w   - w  w   - w w  , - w  w   + w  w   - w  w   + w  w   - w  w  , - w  w   + w  w   - w  w   + w  w   - w  w  , w w   - w w   + w w   + w  w   - w  w  , - w w   + w w   + w  w   + w w   - w w  , - w w   + w w   + w  w   + w w   - w w  , - w w   - w  w   + w  w   + w w   - w w  , - w w   - w  w   + w  w   + w w   - w w  , w  w   - w  w   + w w   - w w   + w w  , w  w   - w w   + w w   - w w   + w w  , w  w   - w w   + w w   - w w   + w w  , w w  - w w   + w w   - w w   + w w  , w w  - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w  - w w  - w w   + w w   - w w  , - w w  + w w  + w w   - w w   + w w  , w w  - w w  - w w  + w w   - w w  })
│ │ │                0   26       0   26       5 22    8 23    14 24    13 25    0 26     5 18    8 19    14 20    10 25    1 26     5 16    8 17    13 20    10 24    2 26     5 15    14 17    13 19    10 23    3 26     5 21    20 23    19 24    17 25    4 26     8 15    14 16    13 18    10 22    6 26     8 21    20 22    18 24    16 25    7 26   17 18    16 19    15 20    10 21    9 26     13 21    17 22    16 23    15 24    11 26     14 21    19 22    18 23    15 25    12 26   0 21    4 22    7 23    12 24    11 25     4 18    7 19    12 20    1 21    9 25     4 16    7 17    11 20    2 21    9 24     4 15    12 17    11 19    3 21    9 23     7 15    12 16    11 18    6 21    9 22   12 13    11 14    0 15    3 22    6 23   10 12    9 14    1 15    3 18    6 19   10 11    9 13    2 15    3 16    6 17   8 9    7 10    1 16    2 18    6 20   5 9    4 10    1 17    2 19    3 20   8 11    7 13    0 16    2 22    6 24   5 11    4 13    0 17    2 23    3 24   8 12    7 14    0 18    1 22    6 25   5 12    4 14    0 19    1 23    3 25   5 7    4 8    0 20    1 24    2 25     5 6    3 8    0 10    1 13    2 14   4 6    3 7    0 9    1 11    2 12
│ │ │  
│ │ │  o5 : RingMap QQ[w ..w  ] <-- QQ[w ..w  ]
│ │ │                   0   26          0   26
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_inverse_lp__Rational__Map_rp.out
│ │ │ @@ -28,15 +28,15 @@
│ │ │                        - -------x  + ---------x x  + ------------x x  - ----------x x  - -----x  - -----------x x  + -------------x x x  + -------------x x x  - --------x x  - ----------x x  + -------------x x x  - ----------x x  - -----------x x  + ----------x x  + ------x  + -----------x x  + ----------x x x  - -----------x x x  - -------x x  + -------------x x x  + ------------x x x x  - -----------x x x  + -----------x x x  - ------------x x x  + ----------x x  - -----------x x  - ------------x x x  - ---------x x  - ------------x x x  - -----------x x x  + -----------x x  - ----------x x  + -------x x  + --------x x  + ------x  + ---------x x  - ------------x x x  - -------------x x x  - ----------x x  + --------------x x x  + -------------x x x x  - ------------x x x  + -------------x x x  + ------------x x x  + ----------x x  + -----------x x x  - -------------x x x x  - ----------x x x  + --------------x x x x  - -------------x x x x  + -------------x x x  - ------------x x x  + ---------x x x  - ------------x x x  + ---------x x  - ---------x x  - -----------x x x  - ----------x x  + -----------x x x  + -----------x x x  + ----------x x  - -----------x x x  - -----------x x x  - ------------x x x  - ----------x x  + ---------x x  - ------x x  - --------x x  - ----------x x  - -----x
│ │ │                           290304 0    3888000  0 1    2939328000  0 1    163296000 0 1   20250 1    228614400  0 2    41150592000  0 1 2    41150592000  0 1 2    3888000 1 2     3572100  0 2    10287648000  0 1 2    342921600 1 2    114307200  0 2    63504000  1 2    25200 2     76204800  0 3    42336000  0 1 3    428652000  0 1 3    212625 1 3     5334336000  0 2 3    9601804800  0 1 2 3    489888000  1 2 3    222264000  0 2 3    12002256000 1 2 3    66679200  2 3    666792000  0 3     666792000  0 1 3    47628000 1 3    1333584000  0 2 3    444528000  1 2 3    777924000  2 3    55566000  0 3    105840 1 3    3472875 2 3    11025 3    4665600  0 4    2939328000  0 1 4     4898880000  0 1 4    29160000  1 4     41150592000  0 2 4    20575296000  0 1 2 4    4898880000  1 2 4    20575296000  0 2 4    1371686400  1 2 4    95256000  2 4     40824000  0 3 4     8573040000  0 1 3 4    11664000  1 3 4     24004512000  0 2 3 4    34292160000  1 2 3 4    12002256000  2 3 4     333396000  0 3 4    5292000  1 3 4    1333584000  2 3 4    3969000  3 4    6804000  0 4    272160000  0 1 4    58320000  1 4    190512000  0 2 4    4898880000 1 2 4    190512000 2 4    476280000  0 3 4    204120000  1 3 4    2857680000  2 3 4    23814000  3 4    30618000 0 4    46656 1 4   12757500 2 4    51030000  3 4   30375 4
│ │ │                       }
│ │ │  
│ │ │  o2 : RationalMap (rational map from PP^4 to PP^4)
│ │ │  
│ │ │  i3 : time inverse phi
│ │ │ - -- used 0.056969s (cpu); 0.0569687s (thread); 0s (gc)
│ │ │ + -- used 0.064242s (cpu); 0.0641291s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[x , x , x , x , x ])
│ │ │                        0   1   2   3   4
│ │ │       defining forms: {
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_is__Birational.out
│ │ │ @@ -40,18 +40,18 @@
│ │ │                        - t  + t t
│ │ │                           3    2 4
│ │ │                       }
│ │ │  
│ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │  
│ │ │  i3 : time isBirational phi
│ │ │ - -- used 0.0193201s (cpu); 0.0193206s (thread); 0s (gc)
│ │ │ + -- used 0.0222861s (cpu); 0.0222858s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 : time isBirational(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0136925s (cpu); 0.0132915s (thread); 0s (gc)
│ │ │ + -- used 0.0265349s (cpu); 0.0147412s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_is__Dominant.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  
│ │ │  i2 : phi = rationalMap ideal jacobian ideal det matrix{{x_0..x_4},{x_1..x_5},{x_2..x_6},{x_3..x_7},{x_4..x_8}};
│ │ │  
│ │ │  o2 : RationalMap (rational map from PP^8 to PP^8)
│ │ │  
│ │ │  i3 : time isDominant(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 2.58345s (cpu); 2.01134s (thread); 0s (gc)
│ │ │ + -- used 2.56366s (cpu); 2.23958s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 : P7 = ZZ/101[x_0..x_7];
│ │ │  
│ │ │  i5 : -- hyperelliptic curve of genus 3
│ │ │       C = ideal(x_4*x_5+23*x_5^2-23*x_0*x_6-18*x_1*x_6+6*x_2*x_6+37*x_3*x_6+23*x_4*x_6-26*x_5*x_6+2*x_6^2-25*x_0*x_7+45*x_1*x_7+30*x_2*x_7-49*x_3*x_7-49*x_4*x_7+50*x_5*x_7,x_3*x_5-24*x_5^2+21*x_0*x_6+x_1*x_6+46*x_3*x_6+27*x_4*x_6+5*x_5*x_6+35*x_6^2+20*x_0*x_7-23*x_1*x_7+8*x_2*x_7-22*x_3*x_7+20*x_4*x_7-15*x_5*x_7,x_2*x_5+47*x_5^2-40*x_0*x_6+37*x_1*x_6-25*x_2*x_6-22*x_3*x_6-8*x_4*x_6+27*x_5*x_6+15*x_6^2-23*x_0*x_7-42*x_1*x_7+27*x_2*x_7+35*x_3*x_7+39*x_4*x_7+24*x_5*x_7,x_1*x_5+15*x_5^2+49*x_0*x_6+8*x_1*x_6-31*x_2*x_6+9*x_3*x_6+38*x_4*x_6-36*x_5*x_6-30*x_6^2-33*x_0*x_7+26*x_1*x_7+32*x_2*x_7+27*x_3*x_7+6*x_4*x_7+36*x_5*x_7,x_0*x_5+30*x_5^2-11*x_0*x_6-38*x_1*x_6+13*x_2*x_6-32*x_3*x_6-30*x_4*x_6+4*x_5*x_6-28*x_6^2-30*x_0*x_7-6*x_1*x_7-45*x_2*x_7+34*x_3*x_7+20*x_4*x_7+48*x_5*x_7,x_3*x_4+46*x_5^2-37*x_0*x_6+27*x_1*x_6+33*x_2*x_6+8*x_3*x_6-32*x_4*x_6+42*x_5*x_6-34*x_6^2-37*x_0*x_7-28*x_1*x_7+10*x_2*x_7-27*x_3*x_7-42*x_4*x_7-8*x_5*x_7,x_2*x_4-25*x_5^2-4*x_0*x_6+2*x_1*x_6-31*x_2*x_6-5*x_3*x_6+16*x_4*x_6-24*x_5*x_6+31*x_6^2-30*x_0*x_7+32*x_1*x_7+12*x_2*x_7-40*x_3*x_7+3*x_4*x_7-28*x_5*x_7,x_0*x_4+15*x_5^2+48*x_0*x_6-50*x_1*x_6+46*x_2*x_6-48*x_3*x_6-23*x_4*x_6-28*x_5*x_6+39*x_6^2+38*x_1*x_7-5*x_3*x_7+5*x_4*x_7-34*x_5*x_7,x_3^2-31*x_5^2+41*x_0*x_6-30*x_1*x_6-4*x_2*x_6+43*x_3*x_6+23*x_4*x_6+7*x_5*x_6+31*x_6^2-19*x_0*x_7+25*x_1*x_7-49*x_2*x_7-16*x_3*x_7-45*x_4*x_7+25*x_5*x_7,x_2*x_3+13*x_5^2-45*x_0*x_6-22*x_1*x_6+33*x_2*x_6-26*x_3*x_6-21*x_4*x_6+34*x_5*x_6-21*x_6^2-47*x_0*x_7-10*x_1*x_7+29*x_2*x_7-46*x_3*x_7-x_4*x_7+20*x_5*x_7,x_1*x_3+22*x_5^2+4*x_0*x_6+3*x_1*x_6+45*x_2*x_6+37*x_3*x_6+17*x_4*x_6+36*x_5*x_6-2*x_6^2-31*x_0*x_7+3*x_1*x_7-12*x_2*x_7+19*x_3*x_7+28*x_4*x_7+30*x_5*x_7,x_0*x_3-47*x_5^2-43*x_0*x_6+6*x_1*x_6-40*x_2*x_6+21*x_3*x_6+26*x_4*x_6-5*x_5*x_6-5*x_6^2+4*x_0*x_7-15*x_1*x_7+18*x_2*x_7-31*x_3*x_7+50*x_4*x_7-46*x_5*x_7,x_2^2+4*x_5^2+31*x_0*x_6+41*x_1*x_6+31*x_2*x_6+28*x_3*x_6+42*x_4*x_6-28*x_5*x_6-4*x_6^2-7*x_0*x_7+15*x_1*x_7-9*x_2*x_7+31*x_3*x_7+3*x_4*x_7+7*x_5*x_7,x_1*x_2-46*x_5^2-6*x_0*x_6-50*x_1*x_6+32*x_2*x_6-10*x_3*x_6+42*x_4*x_6+33*x_5*x_6+18*x_6^2-9*x_0*x_7-20*x_1*x_7+45*x_2*x_7-9*x_3*x_7+10*x_4*x_7-8*x_5*x_7,x_0*x_2-9*x_5^2+34*x_0*x_6-45*x_1*x_6+19*x_2*x_6+24*x_3*x_6+23*x_4*x_6-37*x_5*x_6-44*x_6^2+24*x_0*x_7-33*x_2*x_7+41*x_3*x_7-40*x_4*x_7+4*x_5*x_7,x_1^2+x_1*x_4+x_4^2-28*x_5^2-33*x_0*x_6-17*x_1*x_6+11*x_3*x_6+20*x_4*x_6+25*x_5*x_6-21*x_6^2-22*x_0*x_7+24*x_1*x_7-14*x_2*x_7+5*x_3*x_7-39*x_4*x_7-18*x_5*x_7,x_0*x_1-47*x_5^2-5*x_0*x_6-9*x_1*x_6-45*x_2*x_6+48*x_3*x_6+45*x_4*x_6-29*x_5*x_6+3*x_6^2+29*x_0*x_7+40*x_1*x_7+46*x_2*x_7+27*x_3*x_7-36*x_4*x_7-39*x_5*x_7,x_0^2-31*x_5^2+36*x_0*x_6-30*x_1*x_6-10*x_2*x_6+42*x_3*x_6+9*x_4*x_6+34*x_5*x_6-6*x_6^2+48*x_0*x_7-47*x_1*x_7-19*x_2*x_7+25*x_3*x_7+28*x_4*x_7+34*x_5*x_7);
│ │ │ @@ -21,12 +21,12 @@
│ │ │  
│ │ │  i6 : phi = rationalMap(C,3,2);
│ │ │  
│ │ │  o6 : RationalMap (cubic rational map from PP^7 to PP^7)
│ │ │  
│ │ │  i7 : time isDominant(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 3.88844s (cpu); 2.54882s (thread); 0s (gc)
│ │ │ + -- used 3.81151s (cpu); 2.74143s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = false
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_kernel_lp__Ring__Map_cm__Z__Z_rp.out
│ │ │ @@ -6,23 +6,23 @@
│ │ │  o1 = map (QQ[x ..x ], QQ[y ..y  ], {- 5x x  + x x  + x x  + 35x x  - 7x x  + x x  - x x  - 49x  - 5x x  + 2x x  - x x  + 27x x  - 4x  + x x  - 7x x  + 2x x  - 2x x  + 14x x  - 4x x , - x x  - 6x x  - 5x x  + 2x x  + x x  + x x  - 5x x  - x x  + 2x x  + 7x x  - 2x x  + 2x x  - 3x x , - 25x  + 9x x  + 10x x  - 2x x  - x  + 29x x  - x x  - 7x x  - 13x x  + 3x x  + x x  - x x  + 2x x  - x x  + 7x x  - 2x x  - 8x x  + 2x x  - 3x x , x x  + x x  + x  + 7x x  - 9x x  + 12x x  - 4x  + 2x x  + 2x x  - 14x x  + 4x x  + x x  - x x  - 14x x  + x x , - 5x x  + x x  - 7x x  + 8x x  - 5x x  + 2x x  - x x  + x x  - x x  + 7x x  - 2x x  - x x  + 7x x  - 2x x , x x  + x  - 7x x  - 8x x  + x x  + x x  + 2x x  - x x  + x x  - 7x x  + 2x x  + x x  - 7x x  + 2x x , x x  + x  - 8x x  + x x  + 6x x  - 2x  + x x  + x x  - 7x x  + 2x x  + x x  - 7x x  + 2x x , x x  - 7x x  + x x  + x x  - 7x x  + 2x  - x x , - 4x x  + x x  - x  - 7x x  + 8x x  + x x  - x x  - 6x x  + 2x  - x x  - x x  + 7x x  - 2x x  - x x  + 7x x  - 2x x , - 5x x  + 2x  + x x  - x  - x x  + 8x x  - 10x x  + 2x x  + 2x x  - 2x x  + 14x x  - 4x x  + 5x x  - 3x x  - 2x x  + 7x x  - 2x x  - 3x x , - 5x x  + x x  + x x  - 4x x  - x x  + x x  + x x , x x  - x x  + 5x x  + x x  - 14x x  - x x  - 8x x  - 8x x  + 2x x  + 4x x  + 2x x  + 4x x  + 3x x  - 7x x  + 2x x  - 3x x })
│ │ │                0   8       0   11        0 3    2 4    3 4      0 5     2 5    3 5    4 5      5     0 6     2 6    4 6      5 6     6    4 7     5 7     6 7     4 8      5 8     6 8     1 2     1 5     0 6     1 6    4 6    5 6     0 7    1 7     2 7     5 7     6 7     1 8     7 8       0     0 2      0 4     2 4    4      0 5    2 5     4 5      0 6     4 6    5 6    0 7     2 7    4 7     5 7     6 7     0 8     4 8     7 8   2 4    3 4    4     2 5     4 5      5 6     6     3 7     4 7      5 7     6 7    3 8    4 8      5 8    6 8      0 4    2 4     2 5     4 5     0 6     2 6    4 6    5 6    4 7     5 7     6 7    4 8     5 8     6 8   0 4    4     1 5     4 5    0 6    1 6     4 6    5 6    4 7     5 7     6 7    4 8     5 8     6 8   2 3    4     4 5    4 6     5 6     6    3 7    4 7     5 7     6 7    4 8     5 8     6 8   1 3     1 5    1 6    4 6     5 6     6    3 7      0 3    3 4    4     0 5     4 5    0 6    4 6     5 6     6    3 7    4 7     5 7     6 7    4 8     5 8     6 8      0 2     2    2 4    4    2 5     4 5      0 6     5 6     2 7     4 7      5 7     6 7     0 8     2 8     4 8     5 8     6 8     7 8      0 1    1 2    1 4     0 6    1 6    4 6    0 7   0 2    1 2     0 4    1 4      1 5    2 5     4 5     0 6     1 6     4 6     2 7     0 8     1 8     5 8     6 8     7 8
│ │ │  
│ │ │  o1 : RingMap QQ[x ..x ] <-- QQ[y ..y  ]
│ │ │                   0   8          0   11
│ │ │  
│ │ │  i2 : time kernel(phi,1)
│ │ │ - -- used 0.0174446s (cpu); 0.0174411s (thread); 0s (gc)
│ │ │ + -- used 0.021294s (cpu); 0.0212939s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal ()
│ │ │  
│ │ │  o2 : Ideal of QQ[y ..y  ]
│ │ │                    0   11
│ │ │  
│ │ │  i3 : time kernel(phi,2)
│ │ │ - -- used 0.911685s (cpu); 0.449335s (thread); 0s (gc)
│ │ │ + -- used 1.04194s (cpu); 0.476185s (thread); 0s (gc)
│ │ │  
│ │ │                             2                                                
│ │ │  o3 = ideal (y y  + y y  + y  + 5y y  + y y  + 5y y  - y y  - 4y y  - 5y y  -
│ │ │               2 4    3 4    4     2 5    3 5     4 5    1 6     2 6     5 6  
│ │ │       ------------------------------------------------------------------------
│ │ │                                                                             
│ │ │       4y y  - 2y y  - y y  + 4y y  - 5y y  - 4y y  + 3y y  - 4y y  - y y   -
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_parametrize_lp__Ideal_rp.out
│ │ │ @@ -26,15 +26,15 @@
│ │ │                8           9
│ │ │  
│ │ │                   ZZ
│ │ │  o2 : Ideal of --------[x ..x ]
│ │ │                10000019  0   9
│ │ │  
│ │ │  i3 : time parametrize L
│ │ │ - -- used 0.00500618s (cpu); 0.00500168s (thread); 0s (gc)
│ │ │ + -- used 0.00577994s (cpu); 0.0057765s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                       ZZ
│ │ │       source: Proj(--------[t , t , t , t , t , t ])
│ │ │                    10000019  0   1   2   3   4   5
│ │ │                       ZZ
│ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ @@ -116,15 +116,15 @@
│ │ │               5 9           6 9           7 9           8 9           9
│ │ │  
│ │ │                   ZZ
│ │ │  o4 : Ideal of --------[x ..x ]
│ │ │                10000019  0   9
│ │ │  
│ │ │  i5 : time parametrize Q
│ │ │ - -- used 0.5466s (cpu); 0.394392s (thread); 0s (gc)
│ │ │ + -- used 0.530896s (cpu); 0.427076s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = -- rational map --
│ │ │                       ZZ
│ │ │       source: Proj(--------[t , t , t , t , t , t , t ])
│ │ │                    10000019  0   1   2   3   4   5   6
│ │ │                       ZZ
│ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_point_lp__Quotient__Ring_rp.out
│ │ │ @@ -1,15 +1,15 @@
│ │ │  -- -*- M2-comint -*- hash: 3560583829489988690
│ │ │  
│ │ │  i1 : f = inverseMap specialQuadraticTransformation(9,ZZ/33331);
│ │ │  
│ │ │  o1 : RationalMap (cubic rational map from 8-dimensional subvariety of PP^11 to PP^8)
│ │ │  
│ │ │  i2 : time p = point source f
│ │ │ - -- used 0.463068s (cpu); 0.208194s (thread); 0s (gc)
│ │ │ + -- used 0.470352s (cpu); 0.225542s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal (y   - 9235y  , y  + 11075y  , y  - 5847y  , y  + 7396y  , y  +
│ │ │               10        11   9         11   8        11   7        11   6  
│ │ │       ------------------------------------------------------------------------
│ │ │       13530y  , y  + 4359y  , y  - 2924y  , y  + 13040y  , y  + 6904y  , y  -
│ │ │             11   5        11   4        11   3         11   2        11   1  
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -20,12 +20,12 @@
│ │ │                                                             -----[y ..y  ]
│ │ │                                                             33331  0   11
│ │ │  o2 : Ideal of -------------------------------------------------------------------------------------------------------
│ │ │                (y y  - y y  + y y  , y y  - y y  + y y  , y y  - y y  + y y  , y y  - y y  + y y , y y  - y y  + y y )
│ │ │                  6 7    5 8    4 11   3 7    2 8    1 11   3 5    2 6    0 11   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │  
│ │ │  i3 : time p == f^* f p
│ │ │ - -- used 0.212468s (cpu); 0.135812s (thread); 0s (gc)
│ │ │ + -- used 0.219581s (cpu); 0.136401s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_projective__Degrees.out
│ │ │ @@ -8,15 +8,15 @@
│ │ │                       0   4              0   5       1    0 2     1 2    0 3     2    1 3     1 3    0 4     2 3    1 4     3    2 4
│ │ │  
│ │ │  o2 : RingMap GF 109561[t ..t ] <-- GF 109561[x ..x ]
│ │ │                          0   4                 0   5
│ │ │  
│ │ │  i3 : time projectiveDegrees(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0151376s (cpu); 0.0148069s (thread); 0s (gc)
│ │ │ + -- used 0.0641293s (cpu); 0.0231389s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = {1, 2, 4, 4, 2}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : psi=inverseMap(toMap(phi,Dominant=>infinity))
│ │ │  
│ │ │ @@ -30,15 +30,15 @@
│ │ │                           0   5
│ │ │  o4 : RingMap ------------------ <-- GF 109561[t ..t ]
│ │ │               x x  - x x  + x x                 0   4
│ │ │                2 3    1 4    0 5
│ │ │  
│ │ │  i5 : time projectiveDegrees(psi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0116489s (cpu); 0.0113647s (thread); 0s (gc)
│ │ │ + -- used 0.0796457s (cpu); 0.0209875s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {2, 4, 4, 2, 1}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : -- Cremona transformation of P^6 defined by the quadrics through a rational octic surface
│ │ │       phi = map specialCremonaTransformation(7,ZZ/300007)
│ │ │ @@ -48,21 +48,21 @@
│ │ │            300007  0   6   300007  0   6     2 4    1 5          0 4          1 4          4         0 5          1 5         2 5          4 5         5          3 6         4 6         5 6   2 3    0 5          1 3          1 4          4         0 5          1 5         2 5          4 5         5          3 6         4 6         5 6        0 3         1 4         3 4         4          0 5         1 5         2 5          3 5          4 5         5         3 6          4 6         5 6          0 1          1         0 2          1 2         2          1 4          1 5         2 5          0 6         1 6         2 6         0          1         0 2         1 2         2         1 4          4         0 5         1 5          2 5          4 5         5         0 6         1 6          2 6          3 6         4 6         5 6
│ │ │  
│ │ │                 ZZ                 ZZ
│ │ │  o6 : RingMap ------[x ..x ] <-- ------[x ..x ]
│ │ │               300007  0   6      300007  0   6
│ │ │  
│ │ │  i7 : time projectiveDegrees phi
│ │ │ - -- used 5.859e-05s (cpu); 5.361e-05s (thread); 0s (gc)
│ │ │ + -- used 6.1895e-05s (cpu); 5.4573e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {1, 2, 4, 8, 8, 4, 1}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : time projectiveDegrees(phi,NumDegrees=>1)
│ │ │ - -- used 2.6109e-05s (cpu); 2.5939e-05s (thread); 0s (gc)
│ │ │ + -- used 3.6877e-05s (cpu); 3.674e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = {4, 1}
│ │ │  
│ │ │  o8 : List
│ │ │  
│ │ │  i9 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_rational__Map_lp__Ideal_cm__Z__Z_cm__Z__Z_rp.out
│ │ │ @@ -3,15 +3,15 @@
│ │ │  i1 : ZZ/33331[x_0..x_6]; V = ideal(x_4^2-x_3*x_5,x_2*x_4-x_1*x_5,x_2*x_3-x_1*x_4,x_2^2-x_0*x_5,x_1*x_2-x_0*x_4,x_1^2-x_0*x_3,x_6);
│ │ │  
│ │ │                  ZZ
│ │ │  o2 : Ideal of -----[x ..x ]
│ │ │                33331  0   6
│ │ │  
│ │ │  i3 : time phi = rationalMap(V,3,2)
│ │ │ - -- used 0.095321s (cpu); 0.0953215s (thread); 0s (gc)
│ │ │ + -- used 0.109934s (cpu); 0.109934s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                      ZZ
│ │ │       source: Proj(-----[x , x , x , x , x , x , x ])
│ │ │                    33331  0   1   2   3   4   5   6
│ │ │                      ZZ
│ │ │       target: Proj(-----[y , y , y , y , y , y , y , y , y , y , y  , y  , y  , y  ])
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_rational__Map_lp__Ring_cm__Tally_rp.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │                     0         1         2         3        4         5
│ │ │  
│ │ │  o4 : Ideal of X
│ │ │  
│ │ │  i5 : D = new Tally from {H => 2,C => 1};
│ │ │  
│ │ │  i6 : time phi = rationalMap D
│ │ │ - -- used 0.0301481s (cpu); 0.0301432s (thread); 0s (gc)
│ │ │ + -- used 0.0341706s (cpu); 0.0341558s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = -- rational map --
│ │ │                                    ZZ
│ │ │       source: subvariety of Proj(-----[x , x , x , x , x , x ]) defined by
│ │ │                                  65521  0   1   2   3   4   5
│ │ │               {
│ │ │                   2                  2
│ │ │ @@ -123,13 +123,13 @@
│ │ │                        x x x  + x x x  + x x x  + x x  + x x x  - 2x x x  + x x
│ │ │                         0 1 5    0 2 5    1 2 5    2 5    1 4 5     2 4 5    4 5
│ │ │                       }
│ │ │  
│ │ │  o6 : RationalMap (cubic rational map from surface in PP^5 to PP^20)
│ │ │  
│ │ │  i7 : time ? image(phi,"F4")
│ │ │ - -- used 1.24116s (cpu); 0.701199s (thread); 0s (gc)
│ │ │ + -- used 1.42653s (cpu); 0.648079s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = surface of degree 38 and sectional genus 20 in PP^20 cut out by 153
│ │ │       hypersurfaces of degree 2
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_special__Cremona__Transformation.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1330846641081
│ │ │  
│ │ │  i1 : time apply(1..12,i -> describe specialCremonaTransformation(i,ZZ/3331))
│ │ │ - -- used 1.59376s (cpu); 1.16591s (thread); 0s (gc)
│ │ │ + -- used 1.50129s (cpu); 1.16668s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = (rational map defined by forms of degree 3,
│ │ │        source variety: PP^3                      
│ │ │        target variety: PP^3                      
│ │ │        dominance: true                           
│ │ │        birationality: true                       
│ │ │        projective degrees: {1, 3, 3, 1}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_special__Cubic__Transformation.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1730018912715498288
│ │ │  
│ │ │  i1 : time specialCubicTransformation 9
│ │ │ - -- used 0.0954111s (cpu); 0.0954104s (thread); 0s (gc)
│ │ │ + -- used 0.0897767s (cpu); 0.089776s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x , x , x ])
│ │ │                        0   1   2   3   4   5   6
│ │ │       target: subvariety of Proj(QQ[t , t , t , t , t , t , t , t , t , t ]) defined by
│ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │               {
│ │ │ @@ -62,15 +62,15 @@
│ │ │                        8x x  - 12x x  + 24x  - 11x x  + 17x x x  - 24x x  - 10x x  + 11x x  - 3x  - 6x x  + 28x x x  - 70x x  - 21x x x  + 47x x x  - 13x x  - 14x x  + 66x x  - 22x x  - 20x  + 2x x  - 2x x x  - 10x x  - 11x x x  + 8x x x  - 5x x  + 3x x x  + 23x x x  - 11x x x  - 12x x  + 3x x  - 3x x  - 2x x  + 3x x  + x  - 11x x  + 14x x x  + 34x x  - 6x x x  - 16x x x  + 3x x  - 15x x x  - 66x x x  + 12x x x  + 30x x  - 19x x x  + 2x x x  - 5x x x  - 2x x x  - 7x x  + 6x x  + 21x x  - 3x x  - 21x x  + x x  + 5x  - 8x x  + 7x x x  - 32x x  - 13x x x  + 28x x x  - 9x x  + 70x x x  - 27x x x  - 36x x  + x x x  + 4x x x  - 7x x x  - 2x x x  + 3x x  - 25x x x  - 23x x x  + 4x x x  + 27x x x  - 14x x x  - 9x x  - 2x x  + 10x x  - 6x x  - 10x x  + 3x x  - 2x x
│ │ │                          0 1      0 1      1      0 2      0 1 2      1 2      0 2      1 2     2     0 3      0 1 3      1 3      0 2 3      1 2 3      2 3      0 3      1 3      2 3      3     0 4     0 1 4      1 4      0 2 4     1 2 4     2 4     0 3 4      1 3 4      2 3 4      3 4     0 4     1 4     2 4     3 4    4      0 5      0 1 5      1 5     0 2 5      1 2 5     2 5      0 3 5      1 3 5      2 3 5      3 5      0 4 5     1 4 5     2 4 5     3 4 5     4 5     0 5      1 5     2 5      3 5    4 5     5     0 6     0 1 6      1 6      0 2 6      1 2 6     2 6      1 3 6      2 3 6      3 6    0 4 6     1 4 6     2 4 6     3 4 6     4 6      0 5 6      1 5 6     2 5 6      3 5 6      4 5 6     5 6     0 6      1 6     2 6      3 6     4 6     5 6
│ │ │                       }
│ │ │  
│ │ │  o1 : RationalMap (cubic birational map from PP^6 to 6-dimensional subvariety of PP^9)
│ │ │  
│ │ │  i2 : time describe oo
│ │ │ - -- used 0.0182984s (cpu); 0.0182857s (thread); 0s (gc)
│ │ │ + -- used 0.0185673s (cpu); 0.0185692s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = rational map defined by forms of degree 3
│ │ │       source variety: PP^6
│ │ │       target variety: complete intersection of type (2,2,2) in PP^9
│ │ │       dominance: true
│ │ │       birationality: true
│ │ │       projective degrees: {1, 3, 9, 17, 21, 16, 8}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_special__Quadratic__Transformation.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1729200582376678705
│ │ │  
│ │ │  i1 : time specialQuadraticTransformation 4
│ │ │ - -- used 0.0733927s (cpu); 0.0733921s (thread); 0s (gc)
│ │ │ + -- used 0.0755322s (cpu); 0.0755319s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x , x , x , x , x ])
│ │ │                        0   1   2   3   4   5   6   7   8
│ │ │       target: subvariety of Proj(QQ[y , y , y , y , y , y , y , y , y , y ]) defined by
│ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │               {
│ │ │ @@ -50,15 +50,15 @@
│ │ │                        x x  - x x  + x x  - x x  - x  - x x
│ │ │                         0 1    0 4    3 6    4 6    6    5 7
│ │ │                       }
│ │ │  
│ │ │  o1 : RationalMap (quadratic birational map from PP^8 to hypersurface in PP^9)
│ │ │  
│ │ │  i2 : time describe oo
│ │ │ - -- used 0.110232s (cpu); 0.0307456s (thread); 0s (gc)
│ │ │ + -- used 0.11261s (cpu); 0.031117s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = rational map defined by forms of degree 2
│ │ │       source variety: PP^8
│ │ │       target variety: hypersurface of degree 3 in PP^9
│ │ │       dominance: true
│ │ │       birationality: true
│ │ │       projective degrees: {1, 2, 4, 8, 16, 21, 17, 9, 3}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_to__External__String_lp__Rational__Map_rp.out
│ │ │ @@ -7,34 +7,34 @@
│ │ │  i2 : str = toExternalString phi;
│ │ │  
│ │ │  i3 : #str
│ │ │  
│ │ │  o3 = 6927
│ │ │  
│ │ │  i4 : time phi' = value str;
│ │ │ - -- used 0.0234197s (cpu); 0.0234189s (thread); 0s (gc)
│ │ │ + -- used 0.0252531s (cpu); 0.0252537s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
│ │ │  
│ │ │  i5 : time describe phi'
│ │ │ - -- used 0.00540039s (cpu); 0.00540078s (thread); 0s (gc)
│ │ │ + -- used 0.0059402s (cpu); 0.00594602s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = rational map defined by forms of degree 3
│ │ │       source variety: PP^3
│ │ │       target variety: smooth quadric hypersurface in PP^4
│ │ │       dominance: true
│ │ │       birationality: true (the inverse map is already calculated)
│ │ │       projective degrees: {1, 3, 4, 2}
│ │ │       number of minimal representatives: 1
│ │ │       dimension base locus: 1
│ │ │       degree base locus: 5
│ │ │       coefficient ring: ZZ/33331
│ │ │  
│ │ │  i6 : time describe inverse phi'
│ │ │ - -- used 0.0044104s (cpu); 0.00441117s (thread); 0s (gc)
│ │ │ + -- used 0.00506706s (cpu); 0.00507227s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = rational map defined by forms of degree 2
│ │ │       source variety: smooth quadric hypersurface in PP^4
│ │ │       target variety: PP^3
│ │ │       dominance: true
│ │ │       birationality: true (the inverse map is already calculated)
│ │ │       projective degrees: {2, 4, 3, 1}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Chern__Schwartz__Mac__Pherson.html
│ │ │ @@ -97,30 +97,30 @@
│ │ │  o2 : Ideal of GF 78125[x ..x ]
│ │ │                          0   4
    
│ │ │              
    
│ │ │                
    i3 : time ChernSchwartzMacPherson C
│ │ │ - -- used 2.25862s (cpu); 1.18322s (thread); 0s (gc)
│ │ │ + -- used 2.36974s (cpu); 1.25502s (thread); 0s (gc)
│ │ │  
│ │ │         4     3     2
│ │ │  o3 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o3 : -----
│ │ │          5
│ │ │         H
    
│ │ │              
    
│ │ │                
    i4 : time ChernSchwartzMacPherson(C,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 1.61784s (cpu); 1.14031s (thread); 0s (gc)
│ │ │ + -- used 1.43309s (cpu); 0.986049s (thread); 0s (gc)
│ │ │  
│ │ │         4     3     2
│ │ │  o4 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o4 : -----
│ │ │          5
│ │ │ @@ -167,30 +167,30 @@
│ │ │  o8 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   ]
│ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
    
│ │ │              
    
│ │ │                
    i9 : time ChernClass G
│ │ │ - -- used 0.439952s (cpu); 0.258932s (thread); 0s (gc)
│ │ │ + -- used 0.350846s (cpu); 0.194791s (thread); 0s (gc)
│ │ │  
│ │ │          9      8      7      6      5      4     3
│ │ │  o9 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o9 : -----
│ │ │         10
│ │ │        H
    
│ │ │              
    
│ │ │                
    i10 : time ChernClass(G,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.132347s (cpu); 0.0449789s (thread); 0s (gc)
│ │ │ + -- used 0.132431s (cpu); 0.0387954s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6      5      4     3
│ │ │  o10 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o10 : -----
│ │ │          10
│ │ │ ├── html2text {}
│ │ │ │ @@ -39,26 +39,26 @@
│ │ │ │                 2                           2
│ │ │ │  o2 = ideal (- x  + x x , - x x  + x x , - x  + x x )
│ │ │ │                 1    0 2     1 2    0 3     2    1 3
│ │ │ │  
│ │ │ │  o2 : Ideal of GF 78125[x ..x ]
│ │ │ │                          0   4
│ │ │ │  i3 : time ChernSchwartzMacPherson C
│ │ │ │ - -- used 2.25862s (cpu); 1.18322s (thread); 0s (gc)
│ │ │ │ + -- used 2.36974s (cpu); 1.25502s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         4     3     2
│ │ │ │  o3 = 3H  + 5H  + 3H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o3 : -----
│ │ │ │          5
│ │ │ │         H
│ │ │ │  i4 : time ChernSchwartzMacPherson(C,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 1.61784s (cpu); 1.14031s (thread); 0s (gc)
│ │ │ │ + -- used 1.43309s (cpu); 0.986049s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         4     3     2
│ │ │ │  o4 = 3H  + 5H  + 3H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o4 : -----
│ │ │ │          5
│ │ │ │ @@ -88,26 +88,26 @@
│ │ │ │          0,2 1,3    0,1 2,3
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o8 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p
│ │ │ │  ]
│ │ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
│ │ │ │  i9 : time ChernClass G
│ │ │ │ - -- used 0.439952s (cpu); 0.258932s (thread); 0s (gc)
│ │ │ │ + -- used 0.350846s (cpu); 0.194791s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          9      8      7      6      5      4     3
│ │ │ │  o9 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o9 : -----
│ │ │ │         10
│ │ │ │        H
│ │ │ │  i10 : time ChernClass(G,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.132347s (cpu); 0.0449789s (thread); 0s (gc)
│ │ │ │ + -- used 0.132431s (cpu); 0.0387954s (thread); 0s (gc)
│ │ │ │  
│ │ │ │           9      8      7      6      5      4     3
│ │ │ │  o10 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o10 : -----
│ │ │ │          10
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Euler__Characteristic.html
│ │ │ @@ -85,24 +85,24 @@
│ │ │  o1 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   ]
│ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
    
│ │ │              
    
│ │ │                
    i2 : time EulerCharacteristic I
│ │ │ - -- used 0.265746s (cpu); 0.15298s (thread); 0s (gc)
│ │ │ + -- used 0.313648s (cpu); 0.171991s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
    
│ │ │              
    
│ │ │                
    i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0115158s (cpu); 0.0109715s (thread); 0s (gc)
│ │ │ + -- used 0.0746203s (cpu); 0.0206391s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = 10
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -31,20 +31,20 @@
│ │ │ │  i1 : I = Grassmannian(1,4,CoefficientRing=>ZZ/190181);
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o1 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p
│ │ │ │  ]
│ │ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
│ │ │ │  i2 : time EulerCharacteristic I
│ │ │ │ - -- used 0.265746s (cpu); 0.15298s (thread); 0s (gc)
│ │ │ │ + -- used 0.313648s (cpu); 0.171991s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = 10
│ │ │ │  i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0115158s (cpu); 0.0109715s (thread); 0s (gc)
│ │ │ │ + -- used 0.0746203s (cpu); 0.0206391s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = 10
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  No test is made to see if the projective variety is smooth.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _e_u_l_e_r_(_P_r_o_j_e_c_t_i_v_e_V_a_r_i_e_t_y_) -- topological Euler characteristic of a
│ │ │ │        (smooth) projective variety
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Rational__Map_sp!.html
│ │ │ @@ -86,15 +86,15 @@
│ │ │       target variety: PP^5
│ │ │       coefficient ring: QQ
    
│ │ │              
    
│ │ │                
    i4 : time phi! ;
│ │ │ - -- used 0.0532429s (cpu); 0.0528992s (thread); 0s (gc)
│ │ │ + -- used 0.0726521s (cpu); 0.0609858s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (Cremona transformation of PP^5 of type (2,2))
    
│ │ │              
    
│ │ │                
    i5 : describe phi
│ │ │ @@ -127,15 +127,15 @@
│ │ │       target variety: PP^5
│ │ │       coefficient ring: QQ
    
│ │ │              
    
│ │ │                
    i9 : time phi! ;
│ │ │ - -- used 0.0360627s (cpu); 0.0357049s (thread); 0s (gc)
│ │ │ + -- used 0.0662474s (cpu); 0.0459304s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : RationalMap (quadratic rational map from PP^4 to PP^5)
    
│ │ │              
    
│ │ │                
    i10 : describe phi
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,15 +21,15 @@
│ │ │ │  i3 : describe phi
│ │ │ │  
│ │ │ │  o3 = rational map defined by forms of degree 2
│ │ │ │       source variety: PP^5
│ │ │ │       target variety: PP^5
│ │ │ │       coefficient ring: QQ
│ │ │ │  i4 : time phi! ;
│ │ │ │ - -- used 0.0532429s (cpu); 0.0528992s (thread); 0s (gc)
│ │ │ │ + -- used 0.0726521s (cpu); 0.0609858s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : RationalMap (Cremona transformation of PP^5 of type (2,2))
│ │ │ │  i5 : describe phi
│ │ │ │  
│ │ │ │  o5 = rational map defined by forms of degree 2
│ │ │ │       source variety: PP^5
│ │ │ │       target variety: PP^5
│ │ │ │ @@ -47,15 +47,15 @@
│ │ │ │  i8 : describe phi
│ │ │ │  
│ │ │ │  o8 = rational map defined by forms of degree 2
│ │ │ │       source variety: PP^4
│ │ │ │       target variety: PP^5
│ │ │ │       coefficient ring: QQ
│ │ │ │  i9 : time phi! ;
│ │ │ │ - -- used 0.0360627s (cpu); 0.0357049s (thread); 0s (gc)
│ │ │ │ + -- used 0.0662474s (cpu); 0.0459304s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 : RationalMap (quadratic rational map from PP^4 to PP^5)
│ │ │ │  i10 : describe phi
│ │ │ │  
│ │ │ │  o10 = rational map defined by forms of degree 2
│ │ │ │        source variety: PP^4
│ │ │ │        target variety: PP^5
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Rational__Map_sp^_st_st_sp__Ideal.html
│ │ │ @@ -153,15 +153,15 @@
│ │ │  
│ │ │  o5 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
    
│ │ │              
    
│ │ │                
    i6 : time phi^** q
│ │ │ - -- used 0.157531s (cpu); 0.157527s (thread); 0s (gc)
│ │ │ + -- used 0.166215s (cpu); 0.166214s (thread); 0s (gc)
│ │ │  
│ │ │                  e        d        c        b        a
│ │ │  o6 = ideal (u - -*v, t - -*v, z - -*v, y - -*v, x - -*v)
│ │ │                  f        f        f        f        f
│ │ │  
│ │ │  o6 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
    
│ │ │              
    
│ │ │                
    i4 : time SegreClass X
│ │ │ - -- used 0.865825s (cpu); 0.519842s (thread); 0s (gc)
│ │ │ + -- used 0.787156s (cpu); 0.53615s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5       4      3
│ │ │  o4 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o4 : -----
│ │ │          8
│ │ │         H
    
│ │ │              
    
│ │ │                
    i5 : time SegreClass lift(X,P7)
│ │ │ - -- used 0.565093s (cpu); 0.36901s (thread); 0s (gc)
│ │ │ + -- used 0.620994s (cpu); 0.36341s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o5 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o5 : -----
│ │ │          8
│ │ │         H
    
│ │ │              
    
│ │ │                
    i6 : time SegreClass(X,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0212909s (cpu); 0.0208755s (thread); 0s (gc)
│ │ │ + -- used 0.0674279s (cpu); 0.0316446s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5       4      3
│ │ │  o6 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o6 : -----
│ │ │          8
│ │ │         H
    
│ │ │              
    
│ │ │                
    i7 : time SegreClass(lift(X,P7),Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.097715s (cpu); 0.0973659s (thread); 0s (gc)
│ │ │ + -- used 0.127674s (cpu); 0.114673s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o7 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o7 : -----
│ │ │          8
│ │ │ @@ -203,25 +203,25 @@
│ │ │          
    
│ │ │          
    The method also accepts as input a ring map phi representing a rational map $\Phi:X\dashrightarrow Y$ between projective varieties. In this case, the method returns the push-forward to the Chow ring of the ambient projective space of $X$ of the Segre class of the base locus of $\Phi$ in $X$, i.e., it basically computes SegreClass ideal matrix phi. In the next example, we compute the Segre class of the base locus of a birational map $\mathbb{G}(1,4)\subset\mathbb{P}^9 \dashrightarrow \mathbb{P}^6$.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │          
    
│ │ │                
    i9 : use ZZ/100003[x_0..x_6]
│ │ │  
│ │ │ -o9 =   ZZ
│ │ │ - ------[x ..x ]
│ │ │ - 100003  0   6
│ │ │ +       ZZ
│ │ │ +o9 = ------[x ..x ]
│ │ │ +     100003  0   6
│ │ │  
│ │ │  o9 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i10 : time phi = inverseMap toMap(minors(2,matrix{{x_0,x_1,x_3,x_4,x_5},{x_1,x_2,x_4,x_5,x_6}}),Dominant=>2)
│ │ │ - -- used 0.216303s (cpu); 0.10276s (thread); 0s (gc)
│ │ │ + -- used 0.0662291s (cpu); 0.0662326s (thread); 0s (gc)
│ │ │  
│ │ │                                                          ZZ
│ │ │                                                        ------[y ..y ]
│ │ │                                                        100003  0   9                                                ZZ              2                              2
│ │ │  o10 = map (----------------------------------------------------------------------------------------------------, ------[x ..x ], {y  - y y  - y y , y y  - y y , y  - y y  - y y , y y  + y y  - y y , y y  - y y , y y  - y y  - y y , y y  - y y  - y y })
│ │ │             (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y )  100003  0   6     3    0 5    1 6   3 4    1 7   4    2 7    0 9   2 5    3 5    1 8   4 5    1 9   4 8    2 9    3 9   7 8    4 9    6 9
│ │ │               5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6    1 7    0 8   2 3    1 4    0 5
│ │ │ @@ -233,15 +233,15 @@
│ │ │                (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y )     100003  0   6
│ │ │                  5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6    1 7    0 8   2 3    1 4    0 5
    
│ │ │              
    
│ │ │                
    i11 : time SegreClass phi
│ │ │ - -- used 0.338604s (cpu); 0.228164s (thread); 0s (gc)
│ │ │ + -- used 0.372775s (cpu); 0.253258s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6     5
│ │ │  o11 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o11 : -----
│ │ │          10
│ │ │ @@ -267,30 +267,30 @@
│ │ │                   5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6    1 7    0 8   2 3    1 4    0 5
    
│ │ │              
    
│ │ │                
    i13 : -- Segre class of B in G(1,4)
│ │ │        time SegreClass B
│ │ │ - -- used 0.39646s (cpu); 0.292129s (thread); 0s (gc)
│ │ │ + -- used 0.423745s (cpu); 0.294539s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6     5
│ │ │  o13 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o13 : -----
│ │ │          10
│ │ │         H
    
│ │ │              
    
│ │ │                
    i14 : -- Segre class of B in P^9
│ │ │        time SegreClass lift(B,ambient ring B)
│ │ │ - -- used 1.41797s (cpu); 0.900612s (thread); 0s (gc)
│ │ │ + -- used 1.54782s (cpu); 0.936395s (thread); 0s (gc)
│ │ │  
│ │ │             9       8       7      6     5
│ │ │  o14 = 2764H  - 984H  + 294H  - 67H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o14 : -----
│ │ │          10
│ │ │ ├── html2text {}
│ │ │ │ @@ -81,47 +81,47 @@
│ │ │ │                 2 2                2 2                                        2
│ │ │ │  2                                                    2 2
│ │ │ │                x x  - 2x x x x  + x x  - 2x x x x  - 2x x x x  + 4x x x x  + x x
│ │ │ │  + 4x x x x  - 2x x x x  - 2x x x x  - 2x x x x  + x x
│ │ │ │                 3 4     2 3 4 5    2 5     1 3 4 6     1 2 5 6     0 3 5 6    1
│ │ │ │  6     1 2 4 7     0 3 4 7     0 2 5 7     0 1 6 7    0 7
│ │ │ │  i4 : time SegreClass X
│ │ │ │ - -- used 0.865825s (cpu); 0.519842s (thread); 0s (gc)
│ │ │ │ + -- used 0.787156s (cpu); 0.53615s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            7        6       5       4      3
│ │ │ │  o4 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o4 : -----
│ │ │ │          8
│ │ │ │         H
│ │ │ │  i5 : time SegreClass lift(X,P7)
│ │ │ │ - -- used 0.565093s (cpu); 0.36901s (thread); 0s (gc)
│ │ │ │ + -- used 0.620994s (cpu); 0.36341s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            7        6       5      4      3
│ │ │ │  o5 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o5 : -----
│ │ │ │          8
│ │ │ │         H
│ │ │ │  i6 : time SegreClass(X,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0212909s (cpu); 0.0208755s (thread); 0s (gc)
│ │ │ │ + -- used 0.0674279s (cpu); 0.0316446s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            7        6       5       4      3
│ │ │ │  o6 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o6 : -----
│ │ │ │          8
│ │ │ │         H
│ │ │ │  i7 : time SegreClass(lift(X,P7),Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.097715s (cpu); 0.0973659s (thread); 0s (gc)
│ │ │ │ + -- used 0.127674s (cpu); 0.114673s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            7        6       5      4      3
│ │ │ │  o7 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o7 : -----
│ │ │ │          8
│ │ │ │ @@ -134,22 +134,22 @@
│ │ │ │  method returns the push-forward to the Chow ring of the ambient projective
│ │ │ │  space of $X$ of the Segre class of the base locus of $\Phi$ in $X$, i.e., it
│ │ │ │  basically computes SegreClass ideal matrix phi. In the next example, we compute
│ │ │ │  the Segre class of the base locus of a birational map $\mathbb{G}
│ │ │ │  (1,4)\subset\mathbb{P}^9 \dashrightarrow \mathbb{P}^6$.
│ │ │ │  i9 : use ZZ/100003[x_0..x_6]
│ │ │ │  
│ │ │ │ -o9 =   ZZ
│ │ │ │ - ------[x ..x ]
│ │ │ │ - 100003  0   6
│ │ │ │ +       ZZ
│ │ │ │ +o9 = ------[x ..x ]
│ │ │ │ +     100003  0   6
│ │ │ │  
│ │ │ │  o9 : PolynomialRing
│ │ │ │  i10 : time phi = inverseMap toMap(minors(2,matrix{{x_0,x_1,x_3,x_4,x_5},
│ │ │ │  {x_1,x_2,x_4,x_5,x_6}}),Dominant=>2)
│ │ │ │ - -- used 0.216303s (cpu); 0.10276s (thread); 0s (gc)
│ │ │ │ + -- used 0.0662291s (cpu); 0.0662326s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                                                          ZZ
│ │ │ │                                                        ------[y ..y ]
│ │ │ │                                                        100003  0   9
│ │ │ │  ZZ              2                              2
│ │ │ │  o10 = map (--------------------------------------------------------------------
│ │ │ │  --------------------------------, ------[x ..x ], {y  - y y  - y y , y y  - y y
│ │ │ │ @@ -169,15 +169,15 @@
│ │ │ │  o10 : RingMap -----------------------------------------------------------------
│ │ │ │  ----------------------------------- <-- ------[x ..x ]
│ │ │ │                (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y
│ │ │ │  - y y  + y y , y y  - y y  + y y )     100003  0   6
│ │ │ │                  5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6
│ │ │ │  1 7    0 8   2 3    1 4    0 5
│ │ │ │  i11 : time SegreClass phi
│ │ │ │ - -- used 0.338604s (cpu); 0.228164s (thread); 0s (gc)
│ │ │ │ + -- used 0.372775s (cpu); 0.253258s (thread); 0s (gc)
│ │ │ │  
│ │ │ │           9      8      7      6     5
│ │ │ │  o11 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o11 : -----
│ │ │ │          10
│ │ │ │ @@ -198,26 +198,26 @@
│ │ │ │  ------------------------------------
│ │ │ │                 (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y
│ │ │ │  - y y  + y y , y y  - y y  + y y )
│ │ │ │                   5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2
│ │ │ │  6    1 7    0 8   2 3    1 4    0 5
│ │ │ │  i13 : -- Segre class of B in G(1,4)
│ │ │ │        time SegreClass B
│ │ │ │ - -- used 0.39646s (cpu); 0.292129s (thread); 0s (gc)
│ │ │ │ + -- used 0.423745s (cpu); 0.294539s (thread); 0s (gc)
│ │ │ │  
│ │ │ │           9      8      7      6     5
│ │ │ │  o13 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o13 : -----
│ │ │ │          10
│ │ │ │         H
│ │ │ │  i14 : -- Segre class of B in P^9
│ │ │ │        time SegreClass lift(B,ambient ring B)
│ │ │ │ - -- used 1.41797s (cpu); 0.900612s (thread); 0s (gc)
│ │ │ │ + -- used 1.54782s (cpu); 0.936395s (thread); 0s (gc)
│ │ │ │  
│ │ │ │             9       8       7      6     5
│ │ │ │  o14 = 2764H  - 984H  + 294H  - 67H  + 9H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o14 : -----
│ │ │ │          10
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_abstract__Rational__Map.html
│ │ │ @@ -101,15 +101,15 @@
│ │ │  
│ │ │  o3 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i4 : time psi = abstractRationalMap(P4,P5,f)
│ │ │ - -- used 0.000410019s (cpu); 0.000406292s (thread); 0s (gc)
│ │ │ + -- used 0.000464119s (cpu); 0.000458369s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = -- rational map --
│ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │                        0   1   2   3   4   5
│ │ │       defining forms: given by a function
│ │ │ @@ -119,23 +119,23 @@
│ │ │            
    
│ │ │          
    Now we compute first the degree of the forms defining the abstract map psi and then the corresponding concrete rational map.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │          
    
│ │ │                
    i5 : time projectiveDegrees(psi,3)
│ │ │ - -- used 0.298853s (cpu); 0.18527s (thread); 0s (gc)
│ │ │ + -- used 0.358626s (cpu); 0.201614s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
    
│ │ │              
    
│ │ │                
    i6 : time rationalMap psi
│ │ │ - -- used 0.504026s (cpu); 0.366499s (thread); 0s (gc)
│ │ │ + -- used 0.468152s (cpu); 0.384639s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = -- rational map --
│ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │                        0   1   2   3   4   5
│ │ │       defining forms: {
│ │ │ @@ -233,15 +233,15 @@
│ │ │  o13 : Ideal of -----[x ..x ]
│ │ │                 65521  0   3
    
│ │ │              
    
│ │ │                
    i14 : time T = abstractRationalMap(I,"OADP")
│ │ │ - -- used 0.149185s (cpu); 0.0769299s (thread); 0s (gc)
│ │ │ + -- used 0.163849s (cpu); 0.0752081s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ @@ -253,26 +253,26 @@
│ │ │            
    
│ │ │          
    The degree of the forms defining the abstract map T can be obtained by the following command:
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i15 : time projectiveDegrees(T,2)
│ │ │ - -- used 4.07764s (cpu); 2.12022s (thread); 0s (gc)
│ │ │ + -- used 4.43247s (cpu); 2.24391s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3
    
│ │ │              
    
│ │ │          
    We verify that the composition of T with itself is defined by linear forms:
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i16 : time T2 = T * T
│ │ │ - -- used 2.8564e-05s (cpu); 2.8273e-05s (thread); 0s (gc)
│ │ │ + -- used 2.6402e-05s (cpu); 2.5293e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ @@ -281,15 +281,15 @@
│ │ │  
│ │ │  o16 : AbstractRationalMap (rational map from PP^3 to PP^3)
    
│ │ │              
    
│ │ │                
    i17 : time projectiveDegrees(T2,2)
│ │ │ - -- used 6.65901s (cpu); 3.45699s (thread); 0s (gc)
│ │ │ + -- used 7.01159s (cpu); 3.54843s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = 1
    
│ │ │              
    
│ │ │          
    We verify that the composition of T with itself leaves a random point fixed:
    
│ │ │          
│ │ │ @@ -322,15 +322,15 @@
│ │ │            
│ │ │          
    
│ │ │          
    We now compute the concrete rational map corresponding to T:
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i21 : time f = rationalMap T
│ │ │ - -- used 5.38367s (cpu); 2.92119s (thread); 0s (gc)
│ │ │ + -- used 5.82316s (cpu); 2.9113s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,32 +35,32 @@
│ │ │ │  i3 : P5 := QQ[u_0..u_5]
│ │ │ │  
│ │ │ │  o3 = QQ[u ..u ]
│ │ │ │           0   5
│ │ │ │  
│ │ │ │  o3 : PolynomialRing
│ │ │ │  i4 : time psi = abstractRationalMap(P4,P5,f)
│ │ │ │ - -- used 0.000410019s (cpu); 0.000406292s (thread); 0s (gc)
│ │ │ │ + -- used 0.000464119s (cpu); 0.000458369s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = -- rational map --
│ │ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │ │                        0   1   2   3   4
│ │ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │ │                        0   1   2   3   4   5
│ │ │ │       defining forms: given by a function
│ │ │ │  
│ │ │ │  o4 : AbstractRationalMap (rational map from PP^4 to PP^5)
│ │ │ │  Now we compute first the degree of the forms defining the abstract map psi and
│ │ │ │  then the corresponding concrete rational map.
│ │ │ │  i5 : time projectiveDegrees(psi,3)
│ │ │ │ - -- used 0.298853s (cpu); 0.18527s (thread); 0s (gc)
│ │ │ │ + -- used 0.358626s (cpu); 0.201614s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 2
│ │ │ │  i6 : time rationalMap psi
│ │ │ │ - -- used 0.504026s (cpu); 0.366499s (thread); 0s (gc)
│ │ │ │ + -- used 0.468152s (cpu); 0.384639s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = -- rational map --
│ │ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │ │                        0   1   2   3   4
│ │ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │ │                        0   1   2   3   4   5
│ │ │ │       defining forms: {
│ │ │ │ @@ -139,48 +139,48 @@
│ │ │ │  o13 = ideal (- x  + x x , - x x  + x x , - x  + x x )
│ │ │ │                  1    0 2     1 2    0 3     2    1 3
│ │ │ │  
│ │ │ │                   ZZ
│ │ │ │  o13 : Ideal of -----[x ..x ]
│ │ │ │                 65521  0   3
│ │ │ │  i14 : time T = abstractRationalMap(I,"OADP")
│ │ │ │ - -- used 0.149185s (cpu); 0.0769299s (thread); 0s (gc)
│ │ │ │ + -- used 0.163849s (cpu); 0.0752081s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │                       ZZ
│ │ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │        defining forms: given by a function
│ │ │ │  
│ │ │ │  o14 : AbstractRationalMap (rational map from PP^3 to PP^3)
│ │ │ │  The degree of the forms defining the abstract map T can be obtained by the
│ │ │ │  following command:
│ │ │ │  i15 : time projectiveDegrees(T,2)
│ │ │ │ - -- used 4.07764s (cpu); 2.12022s (thread); 0s (gc)
│ │ │ │ + -- used 4.43247s (cpu); 2.24391s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 3
│ │ │ │  We verify that the composition of T with itself is defined by linear forms:
│ │ │ │  i16 : time T2 = T * T
│ │ │ │ - -- used 2.8564e-05s (cpu); 2.8273e-05s (thread); 0s (gc)
│ │ │ │ + -- used 2.6402e-05s (cpu); 2.5293e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │                       ZZ
│ │ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │        defining forms: given by a function
│ │ │ │  
│ │ │ │  o16 : AbstractRationalMap (rational map from PP^3 to PP^3)
│ │ │ │  i17 : time projectiveDegrees(T2,2)
│ │ │ │ - -- used 6.65901s (cpu); 3.45699s (thread); 0s (gc)
│ │ │ │ + -- used 7.01159s (cpu); 3.54843s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 = 1
│ │ │ │  We verify that the composition of T with itself leaves a random point fixed:
│ │ │ │  i18 : p = apply(3,i->random(ZZ/65521))|{1}
│ │ │ │  
│ │ │ │  o18 = {-6648, -23396, -12311, 1}
│ │ │ │  
│ │ │ │ @@ -193,15 +193,15 @@
│ │ │ │  i20 : T q
│ │ │ │  
│ │ │ │  o20 = {-6648, -23396, -12311, 1}
│ │ │ │  
│ │ │ │  o20 : List
│ │ │ │  We now compute the concrete rational map corresponding to T:
│ │ │ │  i21 : time f = rationalMap T
│ │ │ │ - -- used 5.38367s (cpu); 2.92119s (thread); 0s (gc)
│ │ │ │ + -- used 5.82316s (cpu); 2.9113s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │                       ZZ
│ │ │ │        target: Proj(-----[x , x , x , x ])
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_approximate__Inverse__Map.html
│ │ │ @@ -139,15 +139,15 @@
│ │ │  -- approximateInverseMap: step 4 of 10
│ │ │  -- approximateInverseMap: step 5 of 10
│ │ │  -- approximateInverseMap: step 6 of 10
│ │ │  -- approximateInverseMap: step 7 of 10
│ │ │  -- approximateInverseMap: step 8 of 10
│ │ │  -- approximateInverseMap: step 9 of 10
│ │ │  -- approximateInverseMap: step 10 of 10
│ │ │ - -- used 0.272017s (cpu); 0.206969s (thread); 0s (gc)
│ │ │ + -- used 0.287583s (cpu); 0.226178s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                    ZZ
│ │ │       source: Proj(--[t , t , t , t , t , t , t , t , t ])
│ │ │                    97  0   1   2   3   4   5   6   7   8
│ │ │                                  ZZ
│ │ │       target: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │ @@ -200,15 +200,15 @@
│ │ │            
    
│ │ │                
    i5 : time psi' = approximateInverseMap(phi,CodimBsInv=>5);
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ - -- used 0.227935s (cpu); 0.16233s (thread); 0s (gc)
│ │ │ + -- used 0.23606s (cpu); 0.178632s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
    
│ │ │              
    
│ │ │                
    i6 : assert(psi == psi')
    
│ │ │ @@ -295,15 +295,15 @@
│ │ │            
    
│ │ │                
    i8 : -- without the option 'CodimBsInv=>4', it takes about triple time 
│ │ │       time psi=approximateInverseMap(phi,CodimBsInv=>4)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ - -- used 2.31467s (cpu); 1.78736s (thread); 0s (gc)
│ │ │ + -- used 2.10674s (cpu); 1.7984s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = -- rational map --
│ │ │                                  ZZ
│ │ │       source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │                                  97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │               {
│ │ │                                  2
│ │ │ @@ -367,15 +367,15 @@
│ │ │              
    		
│ │ │                
    i10 : -- in this case we can remedy enabling the option Certify
│ │ │        time psi = approximateInverseMap(phi,CodimBsInv=>4,Certify=>true)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │  Certify: output certified!
│ │ │ - -- used 3.88333s (cpu); 3.11658s (thread); 0s (gc)
│ │ │ + -- used 2.90154s (cpu); 2.53s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = -- rational map --
│ │ │                                   ZZ
│ │ │        source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │                                   97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │                {
│ │ │                                   2
│ │ │ ├── html2text {}
│ │ │ │ @@ -135,15 +135,15 @@
│ │ │ │  -- approximateInverseMap: step 4 of 10
│ │ │ │  -- approximateInverseMap: step 5 of 10
│ │ │ │  -- approximateInverseMap: step 6 of 10
│ │ │ │  -- approximateInverseMap: step 7 of 10
│ │ │ │  -- approximateInverseMap: step 8 of 10
│ │ │ │  -- approximateInverseMap: step 9 of 10
│ │ │ │  -- approximateInverseMap: step 10 of 10
│ │ │ │ - -- used 0.272017s (cpu); 0.206969s (thread); 0s (gc)
│ │ │ │ + -- used 0.287583s (cpu); 0.226178s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = -- rational map --
│ │ │ │                    ZZ
│ │ │ │       source: Proj(--[t , t , t , t , t , t , t , t , t ])
│ │ │ │                    97  0   1   2   3   4   5   6   7   8
│ │ │ │                                  ZZ
│ │ │ │       target: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x ])
│ │ │ │ @@ -252,15 +252,15 @@
│ │ │ │  
│ │ │ │  o3 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
│ │ │ │  i4 : assert(phi * psi == 1 and psi * phi == 1)
│ │ │ │  i5 : time psi' = approximateInverseMap(phi,CodimBsInv=>5);
│ │ │ │  -- approximateInverseMap: step 1 of 3
│ │ │ │  -- approximateInverseMap: step 2 of 3
│ │ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ │ - -- used 0.227935s (cpu); 0.16233s (thread); 0s (gc)
│ │ │ │ + -- used 0.23606s (cpu); 0.178632s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
│ │ │ │  i6 : assert(psi == psi')
│ │ │ │  A more complicated example is the following (here _i_n_v_e_r_s_e_M_a_p takes a lot of
│ │ │ │  time!).
│ │ │ │  i7 : phi = rationalMap map(P8,ZZ/97[x_0..x_11]/ideal(x_1*x_3-8*x_2*x_3+25*x_3^2-25*x_2*x_4-
│ │ │ │  22*x_3*x_4+x_0*x_5+13*x_2*x_5+41*x_3*x_5-x_0*x_6+12*x_2*x_6+25*x_1*x_7+25*x_3*x_7+23*x_5*x_7-
│ │ │ │ @@ -418,15 +418,15 @@
│ │ │ │  
│ │ │ │  o7 : RationalMap (quadratic rational map from PP^8 to 8-dimensional subvariety of PP^11)
│ │ │ │  i8 : -- without the option 'CodimBsInv=>4', it takes about triple time
│ │ │ │       time psi=approximateInverseMap(phi,CodimBsInv=>4)
│ │ │ │  -- approximateInverseMap: step 1 of 3
│ │ │ │  -- approximateInverseMap: step 2 of 3
│ │ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ │ - -- used 2.31467s (cpu); 1.78736s (thread); 0s (gc)
│ │ │ │ + -- used 2.10674s (cpu); 1.7984s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = -- rational map --
│ │ │ │                                  ZZ
│ │ │ │       source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │ │                                  97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │ │               {
│ │ │ │                                  2
│ │ │ │ @@ -526,15 +526,15 @@
│ │ │ │  o9 = false
│ │ │ │  i10 : -- in this case we can remedy enabling the option Certify
│ │ │ │        time psi = approximateInverseMap(phi,CodimBsInv=>4,Certify=>true)
│ │ │ │  -- approximateInverseMap: step 1 of 3
│ │ │ │  -- approximateInverseMap: step 2 of 3
│ │ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 3.88333s (cpu); 3.11658s (thread); 0s (gc)
│ │ │ │ + -- used 2.90154s (cpu); 2.53s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = -- rational map --
│ │ │ │                                   ZZ
│ │ │ │        source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │ │                                   97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │ │                {
│ │ │ │                                   2
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_degree__Map.html
│ │ │ @@ -92,15 +92,15 @@
│ │ │  
│ │ │  o4 : RingMap ringP8 <-- ringP14
    
│ │ │              
    
│ │ │                
    i5 : time degreeMap phi
│ │ │ - -- used 0.0453696s (cpu); 0.0453707s (thread); 0s (gc)
│ │ │ + -- used 0.0553537s (cpu); 0.0550886s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 1
    
│ │ │              
    
│ │ │                
    i6 : -- Compose phi:P^8--->P^14 with a linear projection P^14--->P^8 from a general subspace of P^14 
│ │ │ @@ -113,15 +113,15 @@
│ │ │  
│ │ │  o6 : RingMap ringP8 <-- ringP8
    
│ │ │              
    
│ │ │                
    i7 : time degreeMap phi'
│ │ │ - -- used 1.24889s (cpu); 0.706401s (thread); 0s (gc)
│ │ │ + -- used 1.22619s (cpu); 0.716078s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = 14
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -266,15 +266,15 @@
│ │ │ │  4       0 5       1 5       2 5       3 5        4 5       5        0 6
│ │ │ │  1 6        2 6       3 6       4 6       5 6        6       0 7       1 7
│ │ │ │  2 7       3 7       4 7       5 7        6 7       7     0 8        1 8       2
│ │ │ │  8        3 8       4 8        5 8       6 8      7 8       8
│ │ │ │  
│ │ │ │  o4 : RingMap ringP8 <-- ringP14
│ │ │ │  i5 : time degreeMap phi
│ │ │ │ - -- used 0.0453696s (cpu); 0.0453707s (thread); 0s (gc)
│ │ │ │ + -- used 0.0553537s (cpu); 0.0550886s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 1
│ │ │ │  i6 : -- Compose phi:P^8--->P^14 with a linear projection P^14--->P^8 from a
│ │ │ │  general subspace of P^14
│ │ │ │       -- of dimension 5 (so that the composition phi':P^8--->P^8 must have
│ │ │ │  degree equal to deg(G(1,5))=14)
│ │ │ │       phi'=phi*map(ringP14,ringP8,for i to 8 list random(1,ringP14))
│ │ │ │ @@ -418,15 +418,15 @@
│ │ │ │  0 5        1 5        2 5        3 5        4 5       5        0 6       1 6
│ │ │ │  2 6        3 6      4 6       5 6       6       0 7       1 7       2 7      3
│ │ │ │  7        4 7        5 7        6 7       7       0 8       1 8        2 8
│ │ │ │  3 8       4 8       5 8        6 8       7 8       8
│ │ │ │  
│ │ │ │  o6 : RingMap ringP8 <-- ringP8
│ │ │ │  i7 : time degreeMap phi'
│ │ │ │ - -- used 1.24889s (cpu); 0.706401s (thread); 0s (gc)
│ │ │ │ + -- used 1.22619s (cpu); 0.716078s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = 14
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_e_g_r_e_e_(_R_a_t_i_o_n_a_l_M_a_p_) -- degree of a rational map
│ │ │ │      * _p_r_o_j_e_c_t_i_v_e_D_e_g_r_e_e_s -- projective degrees of a rational map between
│ │ │ │        projective varieties
│ │ │ │  ********** WWaayyss ttoo uussee ddeeggrreeeeMMaapp:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_force__Image.html
│ │ │ @@ -83,15 +83,15 @@
│ │ │  
│ │ │  o3 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time forceImage(Phi,ideal 0_(target Phi))
│ │ │ - -- used 0.000607219s (cpu); 0.000601939s (thread); 0s (gc)
    
│ │ │ + -- used 0.000859741s (cpu); 0.000852916s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : Phi;
│ │ │  
│ │ │  o5 : RationalMap (cubic dominant rational map from PP^6 to 6-dimensional subvariety of PP^9)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,15 +19,15 @@
│ │ │ │  
│ │ │ │  o2 : Ideal of P6
│ │ │ │  i3 : Phi = rationalMap(X,Dominant=>2);
│ │ │ │  
│ │ │ │  o3 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of
│ │ │ │  PP^9)
│ │ │ │  i4 : time forceImage(Phi,ideal 0_(target Phi))
│ │ │ │ - -- used 0.000607219s (cpu); 0.000601939s (thread); 0s (gc)
│ │ │ │ + -- used 0.000859741s (cpu); 0.000852916s (thread); 0s (gc)
│ │ │ │  i5 : Phi;
│ │ │ │  
│ │ │ │  o5 : RationalMap (cubic dominant rational map from PP^6 to 6-dimensional
│ │ │ │  subvariety of PP^9)
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  If the declaration is false, nonsensical answers may result.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_graph.html
│ │ │ @@ -113,15 +113,15 @@
│ │ │  
│ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time (p1,p2) = graph phi;
│ │ │ - -- used 0.0188178s (cpu); 0.0184354s (thread); 0s (gc)
    
│ │ │ + -- used 0.081288s (cpu); 0.0303614s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : p1
│ │ │  
│ │ │  o4 = -- rational map --
│ │ │ @@ -272,15 +272,15 @@
│ │ │            
│ │ │          
│ │ │          
    When the source of the rational map is a multi-projective variety, the method returns all the projections.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -98,15 +98,15 @@
│ │ │ │  
│ │ │ │                        w w  - w w  + w w
│ │ │ │                         2 4    1 5    0 6
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o1 : RationalMap (quadratic Cremona transformation of PP^20)
│ │ │ │  i2 : time psi = inverseMap phi
│ │ │ │ - -- used 0.184232s (cpu); 0.12115s (thread); 0s (gc)
│ │ │ │ + -- used 0.19784s (cpu); 0.113276s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = -- rational map --
│ │ │ │       source: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w
│ │ │ │  , w  , w  , w  , w  , w  , w  , w  ])
│ │ │ │                        0   1   2   3   4   5   6   7   8   9   10   11   12   13
│ │ │ │  14   15   16   17   18   19   20
│ │ │ │       target: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w
│ │ │ │ @@ -216,15 +216,15 @@
│ │ │ │  15    9 20    8 22   3 10    0 13    4 15    9 21    8 23   2 10    0 12    4
│ │ │ │  20    6 21    8 24   1 10    0 11    4 22    6 23    9 24   4 5    3 6    0 7
│ │ │ │  1 8    2 9
│ │ │ │  
│ │ │ │  o4 : RingMap QQ[w ..w  ] <-- QQ[w ..w  ]
│ │ │ │                   0   26          0   26
│ │ │ │  i5 : time psi = inverseMap phi
│ │ │ │ - -- used 0.371428s (cpu); 0.224639s (thread); 0s (gc)
│ │ │ │ + -- used 0.350088s (cpu); 0.208419s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = map (QQ[w ..w  ], QQ[w ..w  ], {- w w   + w w   + w  w   - w  w   - w w  ,
│ │ │ │  - w w   + w w   + w  w   - w  w   - w w  , - w w   + w w   + w  w   - w  w   -
│ │ │ │  w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   -
│ │ │ │  w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   +
│ │ │ │  w  w   - w  w   - w w  , w  w   - w  w   + w  w   - w  w   - w w  , - w  w   +
│ │ │ │  w  w   - w  w   + w  w   - w  w  , - w  w   + w  w   - w  w   + w  w   - w  w
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_inverse_lp__Rational__Map_rp.html
│ │ │ @@ -104,15 +104,15 @@
│ │ │  
│ │ │  o2 : RationalMap (rational map from PP^4 to PP^4)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i9 : time g = graph p2;
│ │ │ - -- used 0.0317165s (cpu); 0.0312624s (thread); 0s (gc)
    
│ │ │ + -- used 0.0536877s (cpu); 0.037804s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i10 : g_0;
│ │ │  
│ │ │  o10 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to PP^4)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -50,15 +50,15 @@
│ │ │ │                        - x  + x x
│ │ │ │                           3    2 4
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in
│ │ │ │  PP^5)
│ │ │ │  i3 : time (p1,p2) = graph phi;
│ │ │ │ - -- used 0.0188178s (cpu); 0.0184354s (thread); 0s (gc)
│ │ │ │ + -- used 0.081288s (cpu); 0.0303614s (thread); 0s (gc)
│ │ │ │  i4 : p1
│ │ │ │  
│ │ │ │  o4 = -- rational map --
│ │ │ │                                    ZZ                                 ZZ
│ │ │ │       source: subvariety of Proj(------[x , x , x , x , x ]) x Proj(------[y , y
│ │ │ │  , y , y , y , y ]) defined by
│ │ │ │                                  190181  0   1   2   3   4          190181  0
│ │ │ │ @@ -192,15 +192,15 @@
│ │ │ │  
│ │ │ │  o8 = {51, 28, 14, 6, 2}
│ │ │ │  
│ │ │ │  o8 : List
│ │ │ │  When the source of the rational map is a multi-projective variety, the method
│ │ │ │  returns all the projections.
│ │ │ │  i9 : time g = graph p2;
│ │ │ │ - -- used 0.0317165s (cpu); 0.0312624s (thread); 0s (gc)
│ │ │ │ + -- used 0.0536877s (cpu); 0.037804s (thread); 0s (gc)
│ │ │ │  i10 : g_0;
│ │ │ │  
│ │ │ │  o10 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety
│ │ │ │  of PP^4 x PP^5 x PP^5 to PP^4)
│ │ │ │  i11 : g_1;
│ │ │ │  
│ │ │ │  o11 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_ideal_lp__Rational__Map_rp.html
│ │ │ @@ -111,15 +111,15 @@
│ │ │  
│ │ │  o2 : RationalMap (quadratic rational map from hypersurface in PP^5 to PP^4)
│ │ │              
    
│ │ │                
    i3 : time ideal phi
│ │ │ - -- used 0.0035599s (cpu); 0.00355522s (thread); 0s (gc)
│ │ │ + -- used 0.00385233s (cpu); 0.00384916s (thread); 0s (gc)
│ │ │  
│ │ │               2                                     2                      
│ │ │  o3 = ideal (x  - x x , x x  - x x  + x x , x x  - x  + x x , x x  - x x  +
│ │ │               4    3 5   2 4    3 4    1 5   2 3    3    1 4   1 2    1 3  
│ │ │       ------------------------------------------------------------------------
│ │ │              2
│ │ │       x x , x  - x x )
│ │ │ @@ -195,15 +195,15 @@
│ │ │  
│ │ │  o5 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^5 x PP^4 to PP^4)
    
│ │ │              
    
│ │ │                
    i6 : time ideal phi'
│ │ │ - -- used 0.0930691s (cpu); 0.0930488s (thread); 0s (gc)
│ │ │ + -- used 0.103877s (cpu); 0.103879s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = ideal 1
│ │ │  
│ │ │                                                                                                              QQ[x ..x , y ..y ]
│ │ │                                                                                                                  0   5   0   4
│ │ │  o6 : Ideal of --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
│ │ │                                                                                                                                                                                                       2
│ │ │ ├── html2text {}
│ │ │ │ @@ -46,15 +46,15 @@
│ │ │ │                         2
│ │ │ │                        x  - x x
│ │ │ │                         1    0 3
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o2 : RationalMap (quadratic rational map from hypersurface in PP^5 to PP^4)
│ │ │ │  i3 : time ideal phi
│ │ │ │ - -- used 0.0035599s (cpu); 0.00355522s (thread); 0s (gc)
│ │ │ │ + -- used 0.00385233s (cpu); 0.00384916s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               2                                     2
│ │ │ │  o3 = ideal (x  - x x , x x  - x x  + x x , x x  - x  + x x , x x  - x x  +
│ │ │ │               4    3 5   2 4    3 4    1 5   2 3    3    1 4   1 2    1 3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │              2
│ │ │ │       x x , x  - x x )
│ │ │ │ @@ -121,15 +121,15 @@
│ │ │ │                        y
│ │ │ │                         4
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o5 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of
│ │ │ │  PP^5 x PP^4 to PP^4)
│ │ │ │  i6 : time ideal phi'
│ │ │ │ - -- used 0.0930691s (cpu); 0.0930488s (thread); 0s (gc)
│ │ │ │ + -- used 0.103877s (cpu); 0.103879s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = ideal 1
│ │ │ │  
│ │ │ │  
│ │ │ │  QQ[x ..x , y ..y ]
│ │ │ │  
│ │ │ │  0   5   0   4
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_inverse__Map.html
│ │ │ @@ -153,15 +153,15 @@
│ │ │  
│ │ │  o1 : RationalMap (quadratic Cremona transformation of PP^20)
    
│ │ │              
    
│ │ │                
    i2 : time psi = inverseMap phi
│ │ │ - -- used 0.184232s (cpu); 0.12115s (thread); 0s (gc)
│ │ │ + -- used 0.19784s (cpu); 0.113276s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = -- rational map --
│ │ │       source: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  ])
│ │ │                        0   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19   20
│ │ │       target: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  ])
│ │ │                        0   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19   20
│ │ │       defining forms: {
│ │ │ @@ -251,15 +251,15 @@
│ │ │  o4 : RingMap QQ[w ..w  ] <-- QQ[w ..w  ]
│ │ │                   0   26          0   26
    
│ │ │              
    
│ │ │                
    i5 : time psi = inverseMap phi
│ │ │ - -- used 0.371428s (cpu); 0.224639s (thread); 0s (gc)
│ │ │ + -- used 0.350088s (cpu); 0.208419s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = map (QQ[w ..w  ], QQ[w ..w  ], {- w w   + w w   + w  w   - w  w   - w w  , - w w   + w w   + w  w   - w  w   - w w  , - w w   + w w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , w  w   - w  w   + w  w   - w  w   - w w  , - w  w   + w  w   - w  w   + w  w   - w  w  , - w  w   + w  w   - w  w   + w  w   - w  w  , w w   - w w   + w w   + w  w   - w  w  , - w w   + w w   + w  w   + w w   - w w  , - w w   + w w   + w  w   + w w   - w w  , - w w   - w  w   + w  w   + w w   - w w  , - w w   - w  w   + w  w   + w w   - w w  , w  w   - w  w   + w w   - w w   + w w  , w  w   - w w   + w w   - w w   + w w  , w  w   - w w   + w w   - w w   + w w  , w w  - w w   + w w   - w w   + w w  , w w  - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w  - w w  - w w   + w w   - w w  , - w w  + w w  + w w   - w w   + w w  , w w  - w w  - w w  + w w   - w w  })
│ │ │                0   26       0   26       5 22    8 23    14 24    13 25    0 26     5 18    8 19    14 20    10 25    1 26     5 16    8 17    13 20    10 24    2 26     5 15    14 17    13 19    10 23    3 26     5 21    20 23    19 24    17 25    4 26     8 15    14 16    13 18    10 22    6 26     8 21    20 22    18 24    16 25    7 26   17 18    16 19    15 20    10 21    9 26     13 21    17 22    16 23    15 24    11 26     14 21    19 22    18 23    15 25    12 26   0 21    4 22    7 23    12 24    11 25     4 18    7 19    12 20    1 21    9 25     4 16    7 17    11 20    2 21    9 24     4 15    12 17    11 19    3 21    9 23     7 15    12 16    11 18    6 21    9 22   12 13    11 14    0 15    3 22    6 23   10 12    9 14    1 15    3 18    6 19   10 11    9 13    2 15    3 16    6 17   8 9    7 10    1 16    2 18    6 20   5 9    4 10    1 17    2 19    3 20   8 11    7 13    0 16    2 22    6 24   5 11    4 13    0 17    2 23    3 24   8 12    7 14    0 18    1 22    6 25   5 12    4 14    0 19    1 23    3 25   5 7    4 8    0 20    1 24    2 25     5 6    3 8    0 10    1 13    2 14   4 6    3 7    0 9    1 11    2 12
│ │ │  
│ │ │  o5 : RingMap QQ[w ..w  ] <-- QQ[w ..w  ]
│ │ │                   0   26          0   26
    
│ │ │              
    
│ │ │                
    i3 : time inverse phi
│ │ │ - -- used 0.056969s (cpu); 0.0569687s (thread); 0s (gc)
│ │ │ + -- used 0.064242s (cpu); 0.0641291s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[x , x , x , x , x ])
│ │ │                        0   1   2   3   4
│ │ │       defining forms: {
│ │ │ ├── html2text {}
│ │ │ │ @@ -290,15 +290,15 @@
│ │ │ │  58320000  1 4    190512000  0 2 4    4898880000 1 2 4    190512000 2 4
│ │ │ │  476280000  0 3 4    204120000  1 3 4    2857680000  2 3 4    23814000  3 4
│ │ │ │  30618000 0 4    46656 1 4   12757500 2 4    51030000  3 4   30375 4
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o2 : RationalMap (rational map from PP^4 to PP^4)
│ │ │ │  i3 : time inverse phi
│ │ │ │ - -- used 0.056969s (cpu); 0.0569687s (thread); 0s (gc)
│ │ │ │ + -- used 0.064242s (cpu); 0.0641291s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = -- rational map --
│ │ │ │       source: Proj(QQ[x , x , x , x , x ])
│ │ │ │                        0   1   2   3   4
│ │ │ │       target: Proj(QQ[x , x , x , x , x ])
│ │ │ │                        0   1   2   3   4
│ │ │ │       defining forms: {
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_is__Birational.html
│ │ │ @@ -123,24 +123,24 @@
│ │ │  
│ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5)
    
│ │ │              
    
│ │ │                
    i3 : time isBirational phi
│ │ │ - -- used 0.0193201s (cpu); 0.0193206s (thread); 0s (gc)
│ │ │ + -- used 0.0222861s (cpu); 0.0222858s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    
│ │ │              
    
│ │ │                
    i4 : time isBirational(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0136925s (cpu); 0.0132915s (thread); 0s (gc)
│ │ │ + -- used 0.0265349s (cpu); 0.0147412s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -58,20 +58,20 @@
│ │ │ │                        - t  + t t
│ │ │ │                           3    2 4
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in
│ │ │ │  PP^5)
│ │ │ │  i3 : time isBirational phi
│ │ │ │ - -- used 0.0193201s (cpu); 0.0193206s (thread); 0s (gc)
│ │ │ │ + -- used 0.0222861s (cpu); 0.0222858s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : time isBirational(phi,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0136925s (cpu); 0.0132915s (thread); 0s (gc)
│ │ │ │ + -- used 0.0265349s (cpu); 0.0147412s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_D_o_m_i_n_a_n_t -- whether a rational map is dominant
│ │ │ │  ********** WWaayyss ttoo uussee iissBBiirraattiioonnaall:: **********
│ │ │ │      * isBirational(RationalMap)
│ │ │ │      * isBirational(RingMap)
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_is__Dominant.html
│ │ │ @@ -86,15 +86,15 @@
│ │ │  o2 : RationalMap (rational map from PP^8 to PP^8)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time isDominant(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 2.58345s (cpu); 2.01134s (thread); 0s (gc)
│ │ │ + -- used 2.56366s (cpu); 2.23958s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : P7 = ZZ/101[x_0..x_7];
    
│ │ │ @@ -115,15 +115,15 @@
│ │ │  o6 : RationalMap (cubic rational map from PP^7 to PP^7)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time isDominant(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 3.88844s (cpu); 2.54882s (thread); 0s (gc)
│ │ │ + -- used 3.81151s (cpu); 2.74143s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = false
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │  i1 : P8 = ZZ/101[x_0..x_8];
│ │ │ │  i2 : phi = rationalMap ideal jacobian ideal det matrix{{x_0..x_4},{x_1..x_5},{x_2..x_6},{x_3..x_7},
│ │ │ │  {x_4..x_8}};
│ │ │ │  
│ │ │ │  o2 : RationalMap (rational map from PP^8 to PP^8)
│ │ │ │  i3 : time isDominant(phi,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 2.58345s (cpu); 2.01134s (thread); 0s (gc)
│ │ │ │ + -- used 2.56366s (cpu); 2.23958s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : P7 = ZZ/101[x_0..x_7];
│ │ │ │  i5 : -- hyperelliptic curve of genus 3
│ │ │ │       C = ideal(x_4*x_5+23*x_5^2-23*x_0*x_6-18*x_1*x_6+6*x_2*x_6+37*x_3*x_6+23*x_4*x_6-
│ │ │ │  26*x_5*x_6+2*x_6^2-25*x_0*x_7+45*x_1*x_7+30*x_2*x_7-49*x_3*x_7-49*x_4*x_7+50*x_5*x_7,x_3*x_5-
│ │ │ │  24*x_5^2+21*x_0*x_6+x_1*x_6+46*x_3*x_6+27*x_4*x_6+5*x_5*x_6+35*x_6^2+20*x_0*x_7-
│ │ │ │ @@ -65,15 +65,15 @@
│ │ │ │  
│ │ │ │  o5 : Ideal of P7
│ │ │ │  i6 : phi = rationalMap(C,3,2);
│ │ │ │  
│ │ │ │  o6 : RationalMap (cubic rational map from PP^7 to PP^7)
│ │ │ │  i7 : time isDominant(phi,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 3.88844s (cpu); 2.54882s (thread); 0s (gc)
│ │ │ │ + -- used 3.81151s (cpu); 2.74143s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = false
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_B_i_r_a_t_i_o_n_a_l -- whether a rational map is birational
│ │ │ │  ********** WWaayyss ttoo uussee iissDDoommiinnaanntt:: **********
│ │ │ │      * isDominant(RationalMap)
│ │ │ │      * isDominant(RingMap)
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_kernel_lp__Ring__Map_cm__Z__Z_rp.html
│ │ │ @@ -90,26 +90,26 @@
│ │ │  o1 : RingMap QQ[x ..x ] <-- QQ[y ..y  ]
│ │ │                   0   8          0   11
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time kernel(phi,1)
│ │ │ - -- used 0.0174446s (cpu); 0.0174411s (thread); 0s (gc)
│ │ │ + -- used 0.021294s (cpu); 0.0212939s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal ()
│ │ │  
│ │ │  o2 : Ideal of QQ[y ..y  ]
│ │ │                    0   11
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time kernel(phi,2)
│ │ │ - -- used 0.911685s (cpu); 0.449335s (thread); 0s (gc)
│ │ │ + -- used 1.04194s (cpu); 0.476185s (thread); 0s (gc)
│ │ │  
│ │ │                             2                                                
│ │ │  o3 = ideal (y y  + y y  + y  + 5y y  + y y  + 5y y  - y y  - 4y y  - 5y y  -
│ │ │               2 4    3 4    4     2 5    3 5     4 5    1 6     2 6     5 6  
│ │ │       ------------------------------------------------------------------------
│ │ │                                                                             
│ │ │       4y y  - 2y y  - y y  + 4y y  - 5y y  - 4y y  + 3y y  - 4y y  - y y   -
│ │ │ ├── html2text {}
│ │ │ │ @@ -69,22 +69,22 @@
│ │ │ │  4 8     5 8     6 8     7 8      0 1    1 2    1 4     0 6    1 6    4 6    0 7
│ │ │ │  0 2    1 2     0 4    1 4      1 5    2 5     4 5     0 6     1 6     4 6     2
│ │ │ │  7     0 8     1 8     5 8     6 8     7 8
│ │ │ │  
│ │ │ │  o1 : RingMap QQ[x ..x ] <-- QQ[y ..y  ]
│ │ │ │                   0   8          0   11
│ │ │ │  i2 : time kernel(phi,1)
│ │ │ │ - -- used 0.0174446s (cpu); 0.0174411s (thread); 0s (gc)
│ │ │ │ + -- used 0.021294s (cpu); 0.0212939s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = ideal ()
│ │ │ │  
│ │ │ │  o2 : Ideal of QQ[y ..y  ]
│ │ │ │                    0   11
│ │ │ │  i3 : time kernel(phi,2)
│ │ │ │ - -- used 0.911685s (cpu); 0.449335s (thread); 0s (gc)
│ │ │ │ + -- used 1.04194s (cpu); 0.476185s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                             2
│ │ │ │  o3 = ideal (y y  + y y  + y  + 5y y  + y y  + 5y y  - y y  - 4y y  - 5y y  -
│ │ │ │               2 4    3 4    4     2 5    3 5     4 5    1 6     2 6     5 6
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │  
│ │ │ │       4y y  - 2y y  - y y  + 4y y  - 5y y  - 4y y  + 3y y  - 4y y  - y y   -
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_parametrize_lp__Ideal_rp.html
│ │ │ @@ -105,15 +105,15 @@
│ │ │  o2 : Ideal of --------[x ..x ]
│ │ │                10000019  0   9
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time parametrize L
│ │ │ - -- used 0.00500618s (cpu); 0.00500168s (thread); 0s (gc)
│ │ │ + -- used 0.00577994s (cpu); 0.0057765s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                       ZZ
│ │ │       source: Proj(--------[t , t , t , t , t , t ])
│ │ │                    10000019  0   1   2   3   4   5
│ │ │                       ZZ
│ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ @@ -201,15 +201,15 @@
│ │ │  o4 : Ideal of --------[x ..x ]
│ │ │                10000019  0   9
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time parametrize Q
│ │ │ - -- used 0.5466s (cpu); 0.394392s (thread); 0s (gc)
│ │ │ + -- used 0.530896s (cpu); 0.427076s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = -- rational map --
│ │ │                       ZZ
│ │ │       source: Proj(--------[t , t , t , t , t , t , t ])
│ │ │                    10000019  0   1   2   3   4   5   6
│ │ │                       ZZ
│ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ ├── html2text {}
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │       - 849671x  + 3034137x )
│ │ │ │                8           9
│ │ │ │  
│ │ │ │                   ZZ
│ │ │ │  o2 : Ideal of --------[x ..x ]
│ │ │ │                10000019  0   9
│ │ │ │  i3 : time parametrize L
│ │ │ │ - -- used 0.00500618s (cpu); 0.00500168s (thread); 0s (gc)
│ │ │ │ + -- used 0.00577994s (cpu); 0.0057765s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │       source: Proj(--------[t , t , t , t , t , t ])
│ │ │ │                    10000019  0   1   2   3   4   5
│ │ │ │                       ZZ
│ │ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ │ @@ -136,15 +136,15 @@
│ │ │ │       1211601x x  - 2168594x x  - 1801762x x  + 3022242x x  + 3618789x )
│ │ │ │               5 9           6 9           7 9           8 9           9
│ │ │ │  
│ │ │ │                   ZZ
│ │ │ │  o4 : Ideal of --------[x ..x ]
│ │ │ │                10000019  0   9
│ │ │ │  i5 : time parametrize Q
│ │ │ │ - -- used 0.5466s (cpu); 0.394392s (thread); 0s (gc)
│ │ │ │ + -- used 0.530896s (cpu); 0.427076s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │       source: Proj(--------[t , t , t , t , t , t , t ])
│ │ │ │                    10000019  0   1   2   3   4   5   6
│ │ │ │                       ZZ
│ │ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_point_lp__Quotient__Ring_rp.html
│ │ │ @@ -78,15 +78,15 @@
│ │ │  
│ │ │  o1 : RationalMap (cubic rational map from 8-dimensional subvariety of PP^11 to PP^8)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time p = point source f
│ │ │ - -- used 0.463068s (cpu); 0.208194s (thread); 0s (gc)
│ │ │ + -- used 0.470352s (cpu); 0.225542s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal (y   - 9235y  , y  + 11075y  , y  - 5847y  , y  + 7396y  , y  +
│ │ │               10        11   9         11   8        11   7        11   6  
│ │ │       ------------------------------------------------------------------------
│ │ │       13530y  , y  + 4359y  , y  - 2924y  , y  + 13040y  , y  + 6904y  , y  -
│ │ │             11   5        11   4        11   3         11   2        11   1  
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -100,15 +100,15 @@
│ │ │                (y y  - y y  + y y  , y y  - y y  + y y  , y y  - y y  + y y  , y y  - y y  + y y , y y  - y y  + y y )
│ │ │                  6 7    5 8    4 11   3 7    2 8    1 11   3 5    2 6    0 11   3 4    1 6    0 8   2 4    1 5    0 7
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time p == f^* f p
│ │ │ - -- used 0.212468s (cpu); 0.135812s (thread); 0s (gc)
│ │ │ + -- used 0.219581s (cpu); 0.136401s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,15 +19,15 @@
│ │ │ │  documentation) , see _p_o_i_n_t_(_M_u_l_t_i_p_r_o_j_e_c_t_i_v_e_V_a_r_i_e_t_y_).
│ │ │ │  Below we verify the birationality of a rational map.
│ │ │ │  i1 : f = inverseMap specialQuadraticTransformation(9,ZZ/33331);
│ │ │ │  
│ │ │ │  o1 : RationalMap (cubic rational map from 8-dimensional subvariety of PP^11 to
│ │ │ │  PP^8)
│ │ │ │  i2 : time p = point source f
│ │ │ │ - -- used 0.463068s (cpu); 0.208194s (thread); 0s (gc)
│ │ │ │ + -- used 0.470352s (cpu); 0.225542s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = ideal (y   - 9235y  , y  + 11075y  , y  - 5847y  , y  + 7396y  , y  +
│ │ │ │               10        11   9         11   8        11   7        11   6
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       13530y  , y  + 4359y  , y  - 2924y  , y  + 13040y  , y  + 6904y  , y  -
│ │ │ │             11   5        11   4        11   3         11   2        11   1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │  o2 : Ideal of -----------------------------------------------------------------
│ │ │ │  --------------------------------------
│ │ │ │                (y y  - y y  + y y  , y y  - y y  + y y  , y y  - y y  + y y  , y
│ │ │ │  y  - y y  + y y , y y  - y y  + y y )
│ │ │ │                  6 7    5 8    4 11   3 7    2 8    1 11   3 5    2 6    0 11
│ │ │ │  3 4    1 6    0 8   2 4    1 5    0 7
│ │ │ │  i3 : time p == f^* f p
│ │ │ │ - -- used 0.212468s (cpu); 0.135812s (thread); 0s (gc)
│ │ │ │ + -- used 0.219581s (cpu); 0.136401s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_a_n_d_o_m_K_R_a_t_i_o_n_a_l_P_o_i_n_t -- pick a random K rational point on the scheme X
│ │ │ │        defined by I
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * point(PolynomialRing)
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_projective__Degrees.html
│ │ │ @@ -89,15 +89,15 @@
│ │ │                          0   4                 0   5
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time projectiveDegrees(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0151376s (cpu); 0.0148069s (thread); 0s (gc)
│ │ │ + -- used 0.0641293s (cpu); 0.0231389s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = {1, 2, 4, 4, 2}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -117,15 +117,15 @@
│ │ │                2 3    1 4    0 5
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time projectiveDegrees(psi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0116489s (cpu); 0.0113647s (thread); 0s (gc)
│ │ │ + -- used 0.0796457s (cpu); 0.0209875s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {2, 4, 4, 2, 1}
│ │ │  
│ │ │  o5 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ @@ -143,25 +143,25 @@
│ │ │  o6 : RingMap ------[x ..x ] <-- ------[x ..x ]
│ │ │               300007  0   6      300007  0   6
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time projectiveDegrees phi
│ │ │ - -- used 5.859e-05s (cpu); 5.361e-05s (thread); 0s (gc)
│ │ │ + -- used 6.1895e-05s (cpu); 5.4573e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {1, 2, 4, 8, 8, 4, 1}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : time projectiveDegrees(phi,NumDegrees=>1)
│ │ │ - -- used 2.6109e-05s (cpu); 2.5939e-05s (thread); 0s (gc)
│ │ │ + -- used 3.6877e-05s (cpu); 3.674e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = {4, 1}
│ │ │  
│ │ │  o8 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -53,15 +53,15 @@
│ │ │ │                       0   4              0   5       1    0 2     1 2    0 3
│ │ │ │  2    1 3     1 3    0 4     2 3    1 4     3    2 4
│ │ │ │  
│ │ │ │  o2 : RingMap GF 109561[t ..t ] <-- GF 109561[x ..x ]
│ │ │ │                          0   4                 0   5
│ │ │ │  i3 : time projectiveDegrees(phi,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0151376s (cpu); 0.0148069s (thread); 0s (gc)
│ │ │ │ + -- used 0.0641293s (cpu); 0.0231389s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = {1, 2, 4, 4, 2}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : psi=inverseMap(toMap(phi,Dominant=>infinity))
│ │ │ │  
│ │ │ │             GF 109561[x ..x ]
│ │ │ │ @@ -76,15 +76,15 @@
│ │ │ │                GF 109561[x ..x ]
│ │ │ │                           0   5
│ │ │ │  o4 : RingMap ------------------ <-- GF 109561[t ..t ]
│ │ │ │               x x  - x x  + x x                 0   4
│ │ │ │                2 3    1 4    0 5
│ │ │ │  i5 : time projectiveDegrees(psi,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0116489s (cpu); 0.0113647s (thread); 0s (gc)
│ │ │ │ + -- used 0.0796457s (cpu); 0.0209875s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = {2, 4, 4, 2, 1}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : -- Cremona transformation of P^6 defined by the quadrics through a
│ │ │ │  rational octic surface
│ │ │ │       phi = map specialCremonaTransformation(7,ZZ/300007)
│ │ │ │ @@ -119,21 +119,21 @@
│ │ │ │  4 5         5         0 6         1 6          2 6          3 6         4 6
│ │ │ │  5 6
│ │ │ │  
│ │ │ │                 ZZ                 ZZ
│ │ │ │  o6 : RingMap ------[x ..x ] <-- ------[x ..x ]
│ │ │ │               300007  0   6      300007  0   6
│ │ │ │  i7 : time projectiveDegrees phi
│ │ │ │ - -- used 5.859e-05s (cpu); 5.361e-05s (thread); 0s (gc)
│ │ │ │ + -- used 6.1895e-05s (cpu); 5.4573e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = {1, 2, 4, 8, 8, 4, 1}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  i8 : time projectiveDegrees(phi,NumDegrees=>1)
│ │ │ │ - -- used 2.6109e-05s (cpu); 2.5939e-05s (thread); 0s (gc)
│ │ │ │ + -- used 3.6877e-05s (cpu); 3.674e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = {4, 1}
│ │ │ │  
│ │ │ │  o8 : List
│ │ │ │  Another way to use this method is by passing an integer i as second argument.
│ │ │ │  However, this is equivalent to first projectiveDegrees(phi,NumDegrees=>i) and
│ │ │ │  generally it is not faster.
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_rational__Map_lp__Ideal_cm__Z__Z_cm__Z__Z_rp.html
│ │ │ @@ -88,15 +88,15 @@
│ │ │  o2 : Ideal of -----[x ..x ]
│ │ │                33331  0   6
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time phi = rationalMap(V,3,2)
│ │ │ - -- used 0.095321s (cpu); 0.0953215s (thread); 0s (gc)
│ │ │ + -- used 0.109934s (cpu); 0.109934s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                      ZZ
│ │ │       source: Proj(-----[x , x , x , x , x , x , x ])
│ │ │                    33331  0   1   2   3   4   5   6
│ │ │                      ZZ
│ │ │       target: Proj(-----[y , y , y , y , y , y , y , y , y , y , y  , y  , y  , y  ])
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │  i1 : ZZ/33331[x_0..x_6]; V = ideal(x_4^2-x_3*x_5,x_2*x_4-x_1*x_5,x_2*x_3-
│ │ │ │  x_1*x_4,x_2^2-x_0*x_5,x_1*x_2-x_0*x_4,x_1^2-x_0*x_3,x_6);
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o2 : Ideal of -----[x ..x ]
│ │ │ │                33331  0   6
│ │ │ │  i3 : time phi = rationalMap(V,3,2)
│ │ │ │ - -- used 0.095321s (cpu); 0.0953215s (thread); 0s (gc)
│ │ │ │ + -- used 0.109934s (cpu); 0.109934s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = -- rational map --
│ │ │ │                      ZZ
│ │ │ │       source: Proj(-----[x , x , x , x , x , x , x ])
│ │ │ │                    33331  0   1   2   3   4   5   6
│ │ │ │                      ZZ
│ │ │ │       target: Proj(-----[y , y , y , y , y , y , y , y , y , y , y  , y  , y  ,
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_rational__Map_lp__Ring_cm__Tally_rp.html
│ │ │ @@ -111,15 +111,15 @@
│ │ │              
│ │ │                
    i5 : D = new Tally from {H => 2,C => 1};
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time phi = rationalMap D
│ │ │ - -- used 0.0301481s (cpu); 0.0301432s (thread); 0s (gc)
│ │ │ + -- used 0.0341706s (cpu); 0.0341558s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = -- rational map --
│ │ │                                    ZZ
│ │ │       source: subvariety of Proj(-----[x , x , x , x , x , x ]) defined by
│ │ │                                  65521  0   1   2   3   4   5
│ │ │               {
│ │ │                   2                  2
│ │ │ @@ -219,15 +219,15 @@
│ │ │  
│ │ │  o6 : RationalMap (cubic rational map from surface in PP^5 to PP^20)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time ? image(phi,"F4")
│ │ │ - -- used 1.24116s (cpu); 0.701199s (thread); 0s (gc)
│ │ │ + -- used 1.42653s (cpu); 0.648079s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = surface of degree 38 and sectional genus 20 in PP^20 cut out by 153
│ │ │       hypersurfaces of degree 2
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    See also the package WeilDivisors, which provides general tools for working with divisors.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │  
│ │ │ │  o4 = ideal(- 32646x  - 28377x  + 26433x  - 29566x  + 3783x  + 26696x )
│ │ │ │                     0         1         2         3        4         5
│ │ │ │  
│ │ │ │  o4 : Ideal of X
│ │ │ │  i5 : D = new Tally from {H => 2,C => 1};
│ │ │ │  i6 : time phi = rationalMap D
│ │ │ │ - -- used 0.0301481s (cpu); 0.0301432s (thread); 0s (gc)
│ │ │ │ + -- used 0.0341706s (cpu); 0.0341558s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = -- rational map --
│ │ │ │                                    ZZ
│ │ │ │       source: subvariety of Proj(-----[x , x , x , x , x , x ]) defined by
│ │ │ │                                  65521  0   1   2   3   4   5
│ │ │ │               {
│ │ │ │                   2                  2
│ │ │ │ @@ -169,15 +169,15 @@
│ │ │ │                                                    2                         2
│ │ │ │                        x x x  + x x x  + x x x  + x x  + x x x  - 2x x x  + x x
│ │ │ │                         0 1 5    0 2 5    1 2 5    2 5    1 4 5     2 4 5    4 5
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o6 : RationalMap (cubic rational map from surface in PP^5 to PP^20)
│ │ │ │  i7 : time ? image(phi,"F4")
│ │ │ │ - -- used 1.24116s (cpu); 0.701199s (thread); 0s (gc)
│ │ │ │ + -- used 1.42653s (cpu); 0.648079s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = surface of degree 38 and sectional genus 20 in PP^20 cut out by 153
│ │ │ │       hypersurfaces of degree 2
│ │ │ │  See also the package _W_e_i_l_D_i_v_i_s_o_r_s, which provides general tools for working
│ │ │ │  with divisors.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_a_t_i_o_n_a_l_M_a_p -- makes a rational map
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_special__Cremona__Transformation.html
│ │ │ @@ -70,15 +70,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    A Cremona transformation is said to be special if the base locus scheme is smooth and irreducible. To ensure this condition, the field K must be large enough but no check is made.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : time apply(1..12,i -> describe specialCremonaTransformation(i,ZZ/3331))
│ │ │ - -- used 1.59376s (cpu); 1.16591s (thread); 0s (gc)
│ │ │ + -- used 1.50129s (cpu); 1.16668s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = (rational map defined by forms of degree 3,
│ │ │        source variety: PP^3                      
│ │ │        target variety: PP^3                      
│ │ │        dominance: true                           
│ │ │        birationality: true                       
│ │ │        projective degrees: {1, 3, 3, 1}
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,15 +16,15 @@
│ │ │ │              K, according to the classification given in Table 1 of _S_p_e_c_i_a_l
│ │ │ │              _c_u_b_i_c_ _C_r_e_m_o_n_a_ _t_r_a_n_s_f_o_r_m_a_t_i_o_n_s_ _o_f_ _P_6_ _a_n_d_ _P_7.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  A Cremona transformation is said to be special if the base locus scheme is
│ │ │ │  smooth and irreducible. To ensure this condition, the field K must be large
│ │ │ │  enough but no check is made.
│ │ │ │  i1 : time apply(1..12,i -> describe specialCremonaTransformation(i,ZZ/3331))
│ │ │ │ - -- used 1.59376s (cpu); 1.16591s (thread); 0s (gc)
│ │ │ │ + -- used 1.50129s (cpu); 1.16668s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = (rational map defined by forms of degree 3,
│ │ │ │        source variety: PP^3
│ │ │ │        target variety: PP^3
│ │ │ │        dominance: true
│ │ │ │        birationality: true
│ │ │ │        projective degrees: {1, 3, 3, 1}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_special__Cubic__Transformation.html
│ │ │ @@ -70,15 +70,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    The field K is required to be large enough.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : time specialCubicTransformation 9
│ │ │ - -- used 0.0954111s (cpu); 0.0954104s (thread); 0s (gc)
│ │ │ + -- used 0.0897767s (cpu); 0.089776s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x , x , x ])
│ │ │                        0   1   2   3   4   5   6
│ │ │       target: subvariety of Proj(QQ[t , t , t , t , t , t , t , t , t , t ]) defined by
│ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │               {
│ │ │ @@ -138,15 +138,15 @@
│ │ │  
│ │ │  o1 : RationalMap (cubic birational map from PP^6 to 6-dimensional subvariety of PP^9)
    
│ │ │              
    
│ │ │                
    i2 : time describe oo
│ │ │ - -- used 0.0182984s (cpu); 0.0182857s (thread); 0s (gc)
│ │ │ + -- used 0.0185673s (cpu); 0.0185692s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = rational map defined by forms of degree 3
│ │ │       source variety: PP^6
│ │ │       target variety: complete intersection of type (2,2,2) in PP^9
│ │ │       dominance: true
│ │ │       birationality: true
│ │ │       projective degrees: {1, 3, 9, 17, 21, 16, 8}
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │            o a _r_a_t_i_o_n_a_l_ _m_a_p, an example of special cubic birational
│ │ │ │              transformation over K, according to the classification given in
│ │ │ │              Table 2 of _S_p_e_c_i_a_l_ _c_u_b_i_c_ _b_i_r_a_t_i_o_n_a_l_ _t_r_a_n_s_f_o_r_m_a_t_i_o_n_s_ _o_f_ _p_r_o_j_e_c_t_i_v_e
│ │ │ │              _s_p_a_c_e_s.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  The field K is required to be large enough.
│ │ │ │  i1 : time specialCubicTransformation 9
│ │ │ │ - -- used 0.0954111s (cpu); 0.0954104s (thread); 0s (gc)
│ │ │ │ + -- used 0.0897767s (cpu); 0.089776s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = -- rational map --
│ │ │ │       source: Proj(QQ[x , x , x , x , x , x , x ])
│ │ │ │                        0   1   2   3   4   5   6
│ │ │ │       target: subvariety of Proj(QQ[t , t , t , t , t , t , t , t , t , t ])
│ │ │ │  defined by
│ │ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │ │ @@ -323,15 +323,15 @@
│ │ │ │  6     4 6      0 5 6      1 5 6     2 5 6      3 5 6      4 5 6     5 6     0 6
│ │ │ │  1 6     2 6      3 6     4 6     5 6
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o1 : RationalMap (cubic birational map from PP^6 to 6-dimensional subvariety of
│ │ │ │  PP^9)
│ │ │ │  i2 : time describe oo
│ │ │ │ - -- used 0.0182984s (cpu); 0.0182857s (thread); 0s (gc)
│ │ │ │ + -- used 0.0185673s (cpu); 0.0185692s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = rational map defined by forms of degree 3
│ │ │ │       source variety: PP^6
│ │ │ │       target variety: complete intersection of type (2,2,2) in PP^9
│ │ │ │       dominance: true
│ │ │ │       birationality: true
│ │ │ │       projective degrees: {1, 3, 9, 17, 21, 16, 8}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_special__Quadratic__Transformation.html
│ │ │ @@ -70,15 +70,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    The field K is required to be large enough.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : time specialQuadraticTransformation 4
│ │ │ - -- used 0.0733927s (cpu); 0.0733921s (thread); 0s (gc)
│ │ │ + -- used 0.0755322s (cpu); 0.0755319s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x , x , x , x , x ])
│ │ │                        0   1   2   3   4   5   6   7   8
│ │ │       target: subvariety of Proj(QQ[y , y , y , y , y , y , y , y , y , y ]) defined by
│ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │               {
│ │ │ @@ -126,15 +126,15 @@
│ │ │  
│ │ │  o1 : RationalMap (quadratic birational map from PP^8 to hypersurface in PP^9)
    
│ │ │              
    
│ │ │                
    i2 : time describe oo
│ │ │ - -- used 0.110232s (cpu); 0.0307456s (thread); 0s (gc)
│ │ │ + -- used 0.11261s (cpu); 0.031117s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = rational map defined by forms of degree 2
│ │ │       source variety: PP^8
│ │ │       target variety: hypersurface of degree 3 in PP^9
│ │ │       dominance: true
│ │ │       birationality: true
│ │ │       projective degrees: {1, 2, 4, 8, 16, 21, 17, 9, 3}
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │            o a _r_a_t_i_o_n_a_l_ _m_a_p, an example of special quadratic birational
│ │ │ │              transformation over K, according to the classification given in
│ │ │ │              Table 1 of _E_x_a_m_p_l_e_s_ _o_f_ _s_p_e_c_i_a_l_ _q_u_a_d_r_a_t_i_c_ _b_i_r_a_t_i_o_n_a_l_ _t_r_a_n_s_f_o_r_m_a_t_i_o_n_s
│ │ │ │              _i_n_t_o_ _c_o_m_p_l_e_t_e_ _i_n_t_e_r_s_e_c_t_i_o_n_s_ _o_f_ _q_u_a_d_r_i_c_s.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  The field K is required to be large enough.
│ │ │ │  i1 : time specialQuadraticTransformation 4
│ │ │ │ - -- used 0.0733927s (cpu); 0.0733921s (thread); 0s (gc)
│ │ │ │ + -- used 0.0755322s (cpu); 0.0755319s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = -- rational map --
│ │ │ │       source: Proj(QQ[x , x , x , x , x , x , x , x , x ])
│ │ │ │                        0   1   2   3   4   5   6   7   8
│ │ │ │       target: subvariety of Proj(QQ[y , y , y , y , y , y , y , y , y , y ])
│ │ │ │  defined by
│ │ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │ │ @@ -78,15 +78,15 @@
│ │ │ │                                                     2
│ │ │ │                        x x  - x x  + x x  - x x  - x  - x x
│ │ │ │                         0 1    0 4    3 6    4 6    6    5 7
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o1 : RationalMap (quadratic birational map from PP^8 to hypersurface in PP^9)
│ │ │ │  i2 : time describe oo
│ │ │ │ - -- used 0.110232s (cpu); 0.0307456s (thread); 0s (gc)
│ │ │ │ + -- used 0.11261s (cpu); 0.031117s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = rational map defined by forms of degree 2
│ │ │ │       source variety: PP^8
│ │ │ │       target variety: hypersurface of degree 3 in PP^9
│ │ │ │       dominance: true
│ │ │ │       birationality: true
│ │ │ │       projective degrees: {1, 2, 4, 8, 16, 21, 17, 9, 3}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_to__External__String_lp__Rational__Map_rp.html
│ │ │ @@ -88,23 +88,23 @@
│ │ │  
│ │ │  o3 = 6927
    
│ │ │              
    
│ │ │                
    i4 : time phi' = value str;
│ │ │ - -- used 0.0234197s (cpu); 0.0234189s (thread); 0s (gc)
│ │ │ + -- used 0.0252531s (cpu); 0.0252537s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
    
│ │ │              
    
│ │ │                
    i5 : time describe phi'
│ │ │ - -- used 0.00540039s (cpu); 0.00540078s (thread); 0s (gc)
│ │ │ + -- used 0.0059402s (cpu); 0.00594602s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = rational map defined by forms of degree 3
│ │ │       source variety: PP^3
│ │ │       target variety: smooth quadric hypersurface in PP^4
│ │ │       dominance: true
│ │ │       birationality: true (the inverse map is already calculated)
│ │ │       projective degrees: {1, 3, 4, 2}
│ │ │ @@ -113,15 +113,15 @@
│ │ │       degree base locus: 5
│ │ │       coefficient ring: ZZ/33331
    
│ │ │              
    
│ │ │                
    i6 : time describe inverse phi'
│ │ │ - -- used 0.0044104s (cpu); 0.00441117s (thread); 0s (gc)
│ │ │ + -- used 0.00506706s (cpu); 0.00507227s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = rational map defined by forms of degree 2
│ │ │       source variety: smooth quadric hypersurface in PP^4
│ │ │       target variety: PP^3
│ │ │       dominance: true
│ │ │       birationality: true (the inverse map is already calculated)
│ │ │       projective degrees: {2, 4, 3, 1}
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,32 +19,32 @@
│ │ │ │  
│ │ │ │  o1 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
│ │ │ │  i2 : str = toExternalString phi;
│ │ │ │  i3 : #str
│ │ │ │  
│ │ │ │  o3 = 6927
│ │ │ │  i4 : time phi' = value str;
│ │ │ │ - -- used 0.0234197s (cpu); 0.0234189s (thread); 0s (gc)
│ │ │ │ + -- used 0.0252531s (cpu); 0.0252537s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
│ │ │ │  i5 : time describe phi'
│ │ │ │ - -- used 0.00540039s (cpu); 0.00540078s (thread); 0s (gc)
│ │ │ │ + -- used 0.0059402s (cpu); 0.00594602s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = rational map defined by forms of degree 3
│ │ │ │       source variety: PP^3
│ │ │ │       target variety: smooth quadric hypersurface in PP^4
│ │ │ │       dominance: true
│ │ │ │       birationality: true (the inverse map is already calculated)
│ │ │ │       projective degrees: {1, 3, 4, 2}
│ │ │ │       number of minimal representatives: 1
│ │ │ │       dimension base locus: 1
│ │ │ │       degree base locus: 5
│ │ │ │       coefficient ring: ZZ/33331
│ │ │ │  i6 : time describe inverse phi'
│ │ │ │ - -- used 0.0044104s (cpu); 0.00441117s (thread); 0s (gc)
│ │ │ │ + -- used 0.00506706s (cpu); 0.00507227s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = rational map defined by forms of degree 2
│ │ │ │       source variety: smooth quadric hypersurface in PP^4
│ │ │ │       target variety: PP^3
│ │ │ │       dominance: true
│ │ │ │       birationality: true (the inverse map is already calculated)
│ │ │ │       projective degrees: {2, 4, 3, 1}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/index.html
│ │ │ @@ -58,29 +58,29 @@
│ │ │              
    		
│ │ │                
    i1 : ZZ/300007[t_0..t_6];
    
│ │ │              
    
│ │ │                
    i2 : time phi = toMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ - -- used 0.00430215s (cpu); 0.00429842s (thread); 0s (gc)
│ │ │ + -- used 0.00524112s (cpu); 0.00523976s (thread); 0s (gc)
│ │ │  
│ │ │              ZZ              ZZ                3                2    2                2        2                      2                  2    2                 2                       3                2    2                2                                 2                           2    2                                  2        2                      2                  2                        2                         2    2                 2                       3                2    2
│ │ │  o2 = map (------[t ..t ], ------[x ..x ], {- t  + 2t t t  - t t  - t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t , - t t  + t t t  + t t t  - t t t  - t t  + t t t , - t t t  + t t  + t t  - t t t  - t t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t t  + t t t  - t t  - t t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t })
│ │ │            300007  0   6   300007  0   9       2     1 2 3    0 3    1 4    0 2 4     2 3    1 3    1 2 4    0 3 4    1 5    0 2 5     2 3    2 4    1 3 4    0 4    1 2 5    0 3 5     3     2 3 4    1 4    2 5    1 3 5     2 4    1 3 4    1 2 5    0 3 5    1 6    0 2 6     2 3 4    1 4    2 5    0 4 5    1 2 6    0 3 6     3 4    2 4    2 3 5    1 4 5    2 6    1 3 6     2 4    2 3 5    1 4 5    0 5    1 3 6    0 4 6     3 4    3 5    2 4 5    1 5    2 3 6    1 4 6     4     3 4 5    2 5    3 6    2 4 6
│ │ │  
│ │ │                 ZZ                 ZZ
│ │ │  o2 : RingMap ------[t ..t ] <-- ------[x ..x ]
│ │ │               300007  0   6      300007  0   9
    
│ │ │              
    
│ │ │                
    i3 : time J = kernel(phi,2)
│ │ │ - -- used 0.137231s (cpu); 0.0699679s (thread); 0s (gc)
│ │ │ + -- used 0.149399s (cpu); 0.0730355s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = ideal (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x 
│ │ │               6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4
│ │ │       ------------------------------------------------------------------------
│ │ │       - x x  + x x , x x  - x x  + x x )
│ │ │          1 6    0 8   2 4    1 5    0 7
│ │ │  
│ │ │ @@ -88,43 +88,43 @@
│ │ │  o3 : Ideal of ------[x ..x ]
│ │ │                300007  0   9
    
│ │ │              
    
│ │ │                
    i4 : time degreeMap phi
│ │ │ - -- used 0.02944s (cpu); 0.0294445s (thread); 0s (gc)
│ │ │ + -- used 0.0334176s (cpu); 0.0334224s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
    
│ │ │              
    
│ │ │                
    i5 : time projectiveDegrees phi
│ │ │ - -- used 0.68756s (cpu); 0.487586s (thread); 0s (gc)
│ │ │ + -- used 0.716496s (cpu); 0.549996s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o5 : List
    
│ │ │              
    
│ │ │                
    i6 : time projectiveDegrees(phi,NumDegrees=>0)
│ │ │ - -- used 0.0623207s (cpu); 0.0622653s (thread); 0s (gc)
│ │ │ + -- used 0.0704585s (cpu); 0.0704671s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = {5}
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ - -- used 0.0021486s (cpu); 0.00214939s (thread); 0s (gc)
│ │ │ + -- used 0.0025549s (cpu); 0.00255898s (thread); 0s (gc)
│ │ │  
│ │ │                                                                         ZZ
│ │ │                                                                       ------[x ..x ]
│ │ │              ZZ                                                       300007  0   9                                                  3                2    2                2        2                      2                  2    2                 2                       3                2    2                2                                 2                           2    2                                  2        2                      2                  2                        2                         2    2                 2                       3                2    2
│ │ │  o7 = map (------[t ..t ], ----------------------------------------------------------------------------------------------------, {- t  + 2t t t  - t t  - t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t , - t t  + t t t  + t t t  - t t t  - t t  + t t t , - t t t  + t t  + t t  - t t t  - t t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t t  + t t t  - t t  - t t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t })
│ │ │            300007  0   6   (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )      2     1 2 3    0 3    1 4    0 2 4     2 3    1 3    1 2 4    0 3 4    1 5    0 2 5     2 3    2 4    1 3 4    0 4    1 2 5    0 3 5     3     2 3 4    1 4    2 5    1 3 5     2 4    1 3 4    1 2 5    0 3 5    1 6    0 2 6     2 3 4    1 4    2 5    0 4 5    1 2 6    0 3 6     3 4    2 4    2 3 5    1 4 5    2 6    1 3 6     2 4    2 3 5    1 4 5    0 5    1 3 6    0 4 6     3 4    3 5    2 4 5    1 5    2 3 6    1 4 6     4     3 4 5    2 5    3 6    2 4 6
│ │ │                              6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │ @@ -136,15 +136,15 @@
│ │ │               300007  0   6      (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )
│ │ │                                    6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
    
│ │ │              
    
│ │ │                
    i8 : time psi = inverseMap phi
│ │ │ - -- used 0.474252s (cpu); 0.394534s (thread); 0s (gc)
│ │ │ + -- used 0.422925s (cpu); 0.422931s (thread); 0s (gc)
│ │ │  
│ │ │                                                         ZZ
│ │ │                                                       ------[x ..x ]
│ │ │                                                       300007  0   9                                                ZZ              3                2               2    2                        2                          2     2        2                               2                                   2               2             2                       3                                                 2                 2    2                                  2    2                 2                                                 3                         2      2    2      2                                              2
│ │ │  o8 = map (----------------------------------------------------------------------------------------------------, ------[t ..t ], {x  - 2x x x  + x x  - x x x  + x x  + x x  + x x x  - x x x  + x x  - 2x x x  - x x x  - 2x x , x x  - x x  - x x x  + x x x  + x x x  + x x  - 2x x x  - x x x  + x x x , x x  - x x x  + x x  - x x x  + x x  - x x x  - x x x , x  - x x x  + x x x  + x x x  - 2x x x  - x x x , x x  - x x x  + x x  + x x  - x x x  - x x x  - x x x , x x  - x x  - x x x  + x x  + x x x  + x x x  - 2x x x  - x x x  + x x x , x  - 2x x x  - x x x  + x x  + x x  + x x  + x x  + x x x  - 2x x x  - x x x  - x x x  - 2x x })
│ │ │            (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )  300007  0   6     2     1 2 3    0 3    1 2 5    0 5    1 6    0 2 6    0 4 6    1 7     0 2 7    0 4 7     0 9   2 3    1 3    1 2 6    0 3 6    0 5 6    1 8     0 2 8    0 4 8    0 1 9   2 3    1 3 6    0 6    0 3 8    1 9    0 2 9    0 4 9   3    1 3 8    0 6 8    1 2 9     0 3 9    0 5 9   3 6    2 3 8    0 8    2 9    1 3 9    0 6 9    0 7 9   3 6    3 8    2 6 8    1 8    2 3 9    2 5 9     1 6 9    1 7 9    0 8 9   6     3 6 8    5 6 8    2 8    4 8    3 9    5 9    2 6 9     4 6 9    2 7 9    4 7 9     0 9
│ │ │              6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │ @@ -156,44 +156,44 @@
│ │ │               (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )     300007  0   6
│ │ │                 6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
    
│ │ │              
    
│ │ │                
    i9 : time isInverseMap(phi,psi)
│ │ │ - -- used 0.00931603s (cpu); 0.00931855s (thread); 0s (gc)
│ │ │ + -- used 0.0109647s (cpu); 0.0109661s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = true
    
│ │ │              
    
│ │ │                
    i10 : time degreeMap psi
│ │ │ - -- used 0.458493s (cpu); 0.294259s (thread); 0s (gc)
│ │ │ + -- used 0.486731s (cpu); 0.262439s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 1
    
│ │ │              
    
│ │ │                
    i11 : time projectiveDegrees psi
│ │ │ - -- used 5.29004s (cpu); 4.63968s (thread); 0s (gc)
│ │ │ + -- used 5.44371s (cpu); 5.02492s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o11 : List
    
│ │ │              
    
│ │ │          
    We repeat the example using the type RationalMap and using deterministic methods.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -53,15 +53,15 @@
│ │ │ │        2
│ │ │ │  o6 = y T
│ │ │ │          2
│ │ │ │  
│ │ │ │  o6 : R[T ..T ]
│ │ │ │          1   3
│ │ │ │  i7 : H = HH(KR)
│ │ │ │ -Finding easy relations           :  -- used 0.0137422s (cpu); 0.0130223s
│ │ │ │ +Finding easy relations           :  -- used 0.0601616s (cpu); 0.023681s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = H
│ │ │ │  
│ │ │ │  o7 : PolynomialRing, 3 skew commutative variable(s)
│ │ │ │  i8 : homologyClass(KR,z1*z2)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Module.html
│ │ │ @@ -129,15 +129,15 @@
│ │ │  
│ │ │  o5 : Complex
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i12 : time phi = rationalMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ - -- used 0.00220238s (cpu); 0.00220313s (thread); 0s (gc)
│ │ │ + -- used 0.002611s (cpu); 0.00261541s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │                     300007  0   1   2   3   4   5   6
│ │ │                       ZZ
│ │ │        target: Proj(------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ @@ -242,15 +242,15 @@
│ │ │  
│ │ │  o12 : RationalMap (cubic rational map from PP^6 to PP^9)
    
│ │ │              
    
│ │ │                
    i13 : time phi = rationalMap(phi,Dominant=>2)
│ │ │ - -- used 0.15731s (cpu); 0.0849869s (thread); 0s (gc)
│ │ │ + -- used 0.17472s (cpu); 0.090802s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │                     300007  0   1   2   3   4   5   6
│ │ │                                     ZZ
│ │ │        target: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │ @@ -315,15 +315,15 @@
│ │ │  
│ │ │  o13 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9)
    
│ │ │              
    
│ │ │                
    i14 : time phi^(-1)
│ │ │ - -- used 0.512059s (cpu); 0.426868s (thread); 0s (gc)
│ │ │ + -- used 0.440018s (cpu); 0.440021s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = -- rational map --
│ │ │                                     ZZ
│ │ │        source: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │                                   300007  0   1   2   3   4   5   6   7   8   9
│ │ │                {
│ │ │                 x x  - x x  + x x ,
│ │ │ @@ -376,49 +376,49 @@
│ │ │  
│ │ │  o14 : RationalMap (cubic birational map from 6-dimensional subvariety of PP^9 to PP^6)
    
│ │ │              
    
│ │ │                
    i15 : time degrees phi^(-1)
│ │ │ - -- used 0.347749s (cpu); 0.274141s (thread); 0s (gc)
│ │ │ + -- used 0.475373s (cpu); 0.340152s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o15 : List
    
│ │ │              
    
│ │ │                
    i16 : time degrees phi
│ │ │ - -- used 0.0180113s (cpu); 0.0176993s (thread); 0s (gc)
│ │ │ + -- used 0.0793952s (cpu); 0.0267941s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o16 : List
    
│ │ │              
    
│ │ │                
    i17 : time describe phi
│ │ │ - -- used 0.00320718s (cpu); 0.00320744s (thread); 0s (gc)
│ │ │ + -- used 0.00372634s (cpu); 0.00373177s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = rational map defined by forms of degree 3
│ │ │        source variety: PP^6
│ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        dominance: true
│ │ │        birationality: true (the inverse map is already calculated)
│ │ │        projective degrees: {1, 3, 9, 17, 21, 15, 5}
│ │ │        coefficient ring: ZZ/300007
    
│ │ │              
    
│ │ │                
    i18 : time describe phi^(-1)
│ │ │ - -- used 0.00997997s (cpu); 0.0099807s (thread); 0s (gc)
│ │ │ + -- used 0.0109202s (cpu); 0.0109263s (thread); 0s (gc)
│ │ │  
│ │ │  o18 = rational map defined by forms of degree 3
│ │ │        source variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        target variety: PP^6
│ │ │        dominance: true
│ │ │        birationality: true (the inverse map is already calculated)
│ │ │        projective degrees: {5, 15, 21, 17, 9, 3, 1}
│ │ │ @@ -427,41 +427,41 @@
│ │ │        degree base locus: 24
│ │ │        coefficient ring: ZZ/300007
    
│ │ │              
    
│ │ │                
    i19 : time (f,g) = graph phi^-1; f;
│ │ │ - -- used 0.00950571s (cpu); 0.00950659s (thread); 0s (gc)
│ │ │ + -- used 0.0115313s (cpu); 0.0115369s (thread); 0s (gc)
│ │ │  
│ │ │  o20 : MultihomogeneousRationalMap (birational map from 6-dimensional subvariety of PP^9 x PP^6 to 6-dimensional subvariety of PP^9)
    
│ │ │              
    
│ │ │                
    i21 : time degrees f
│ │ │ - -- used 1.33327s (cpu); 0.95512s (thread); 0s (gc)
│ │ │ + -- used 1.20731s (cpu); 0.987305s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = {904, 508, 268, 130, 56, 20, 5}
│ │ │  
│ │ │  o21 : List
    
│ │ │              
    
│ │ │                
    i22 : time degree f
│ │ │ - -- used 1.625e-05s (cpu); 1.593e-05s (thread); 0s (gc)
│ │ │ + -- used 2.1074e-05s (cpu); 1.9557e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = 1
    
│ │ │              
    
│ │ │                
    i23 : time describe f
│ │ │ - -- used 0.00161465s (cpu); 0.00161555s (thread); 0s (gc)
│ │ │ + -- used 0.00165199s (cpu); 0.00165748s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = rational map defined by multiforms of degree {1, 0}
│ │ │        source variety: 6-dimensional subvariety of PP^9 x PP^6 cut out by 20 hypersurfaces of degrees ({1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{2, 0},{2, 0},{2, 0},{2, 0},{2, 0})
│ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        dominance: true
│ │ │        birationality: true
│ │ │        projective degrees: {904, 508, 268, 130, 56, 20, 5}
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,15 +25,15 @@
│ │ │ │  map) from a list of $m+1$ homogeneous elements of the same degree in $K
│ │ │ │  [x_0,...,x_n]/I$.
│ │ │ │  Below is an example using the methods provided by this package, dealing with a
│ │ │ │  birational transformation $\Phi:\mathbb{P}^6 \dashrightarrow \mathbb{G}
│ │ │ │  (2,4)\subset\mathbb{P}^9$ of bidegree $(3,3)$.
│ │ │ │  i1 : ZZ/300007[t_0..t_6];
│ │ │ │  i2 : time phi = toMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ │ - -- used 0.00430215s (cpu); 0.00429842s (thread); 0s (gc)
│ │ │ │ + -- used 0.00524112s (cpu); 0.00523976s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              ZZ              ZZ                3                2    2
│ │ │ │  2        2                      2                  2    2                 2
│ │ │ │  3                2    2                2                                 2
│ │ │ │  2    2                                  2        2                      2
│ │ │ │  2                        2                         2    2                 2
│ │ │ │  3                2    2
│ │ │ │ @@ -52,43 +52,43 @@
│ │ │ │  0 5    1 3 6    0 4 6     3 4    3 5    2 4 5    1 5    2 3 6    1 4 6     4
│ │ │ │  3 4 5    2 5    3 6    2 4 6
│ │ │ │  
│ │ │ │                 ZZ                 ZZ
│ │ │ │  o2 : RingMap ------[t ..t ] <-- ------[x ..x ]
│ │ │ │               300007  0   6      300007  0   9
│ │ │ │  i3 : time J = kernel(phi,2)
│ │ │ │ - -- used 0.137231s (cpu); 0.0699679s (thread); 0s (gc)
│ │ │ │ + -- used 0.149399s (cpu); 0.0730355s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = ideal (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x
│ │ │ │               6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       - x x  + x x , x x  - x x  + x x )
│ │ │ │          1 6    0 8   2 4    1 5    0 7
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o3 : Ideal of ------[x ..x ]
│ │ │ │                300007  0   9
│ │ │ │  i4 : time degreeMap phi
│ │ │ │ - -- used 0.02944s (cpu); 0.0294445s (thread); 0s (gc)
│ │ │ │ + -- used 0.0334176s (cpu); 0.0334224s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 1
│ │ │ │  i5 : time projectiveDegrees phi
│ │ │ │ - -- used 0.68756s (cpu); 0.487586s (thread); 0s (gc)
│ │ │ │ + -- used 0.716496s (cpu); 0.549996s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = {1, 3, 9, 17, 21, 15, 5}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : time projectiveDegrees(phi,NumDegrees=>0)
│ │ │ │ - -- used 0.0623207s (cpu); 0.0622653s (thread); 0s (gc)
│ │ │ │ + -- used 0.0704585s (cpu); 0.0704671s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = {5}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ │ - -- used 0.0021486s (cpu); 0.00214939s (thread); 0s (gc)
│ │ │ │ + -- used 0.0025549s (cpu); 0.00255898s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                                                                         ZZ
│ │ │ │                                                                       ------[x
│ │ │ │  ..x ]
│ │ │ │              ZZ                                                       300007  0
│ │ │ │  9                                                  3                2    2
│ │ │ │  2        2                      2                  2    2                 2
│ │ │ │ @@ -123,15 +123,15 @@
│ │ │ │  o7 : RingMap ------[t ..t ] <-- -----------------------------------------------
│ │ │ │  -----------------------------------------------------
│ │ │ │               300007  0   6      (x x  - x x  + x x , x x  - x x  + x x , x x  -
│ │ │ │  x x  + x x , x x  - x x  + x x , x x  - x x  + x x )
│ │ │ │                                    6 7    5 8    4 9   3 7    2 8    1 9   3 5
│ │ │ │  2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │ │  i8 : time psi = inverseMap phi
│ │ │ │ - -- used 0.474252s (cpu); 0.394534s (thread); 0s (gc)
│ │ │ │ + -- used 0.422925s (cpu); 0.422931s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                                                         ZZ
│ │ │ │                                                       ------[x ..x ]
│ │ │ │                                                       300007  0   9
│ │ │ │  ZZ              3                2               2    2
│ │ │ │  2                          2     2        2                               2
│ │ │ │  2               2             2                       3
│ │ │ │ @@ -164,31 +164,31 @@
│ │ │ │  o8 : RingMap ------------------------------------------------------------------
│ │ │ │  ---------------------------------- <-- ------[t ..t ]
│ │ │ │               (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x
│ │ │ │  - x x  + x x , x x  - x x  + x x )     300007  0   6
│ │ │ │                 6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4
│ │ │ │  1 6    0 8   2 4    1 5    0 7
│ │ │ │  i9 : time isInverseMap(phi,psi)
│ │ │ │ - -- used 0.00931603s (cpu); 0.00931855s (thread); 0s (gc)
│ │ │ │ + -- used 0.0109647s (cpu); 0.0109661s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  i10 : time degreeMap psi
│ │ │ │ - -- used 0.458493s (cpu); 0.294259s (thread); 0s (gc)
│ │ │ │ + -- used 0.486731s (cpu); 0.262439s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = 1
│ │ │ │  i11 : time projectiveDegrees psi
│ │ │ │ - -- used 5.29004s (cpu); 4.63968s (thread); 0s (gc)
│ │ │ │ + -- used 5.44371s (cpu); 5.02492s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = {5, 15, 21, 17, 9, 3, 1}
│ │ │ │  
│ │ │ │  o11 : List
│ │ │ │  We repeat the example using the type _R_a_t_i_o_n_a_l_M_a_p and using deterministic
│ │ │ │  methods.
│ │ │ │  i12 : time phi = rationalMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ │ - -- used 0.00220238s (cpu); 0.00220313s (thread); 0s (gc)
│ │ │ │ + -- used 0.002611s (cpu); 0.00261541s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o12 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │ │                     300007  0   1   2   3   4   5   6
│ │ │ │                       ZZ
│ │ │ │        target: Proj(------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ │ @@ -233,15 +233,15 @@
│ │ │ │                            3                2    2
│ │ │ │                         - t  + 2t t t  - t t  - t t  + t t t
│ │ │ │                            4     3 4 5    2 5    3 6    2 4 6
│ │ │ │                        }
│ │ │ │  
│ │ │ │  o12 : RationalMap (cubic rational map from PP^6 to PP^9)
│ │ │ │  i13 : time phi = rationalMap(phi,Dominant=>2)
│ │ │ │ - -- used 0.15731s (cpu); 0.0849869s (thread); 0s (gc)
│ │ │ │ + -- used 0.17472s (cpu); 0.090802s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │ │                     300007  0   1   2   3   4   5   6
│ │ │ │                                     ZZ
│ │ │ │        target: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x
│ │ │ │ @@ -304,15 +304,15 @@
│ │ │ │                         - t  + 2t t t  - t t  - t t  + t t t
│ │ │ │                            4     3 4 5    2 5    3 6    2 4 6
│ │ │ │                        }
│ │ │ │  
│ │ │ │  o13 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of
│ │ │ │  PP^9)
│ │ │ │  i14 : time phi^(-1)
│ │ │ │ - -- used 0.512059s (cpu); 0.426868s (thread); 0s (gc)
│ │ │ │ + -- used 0.440018s (cpu); 0.440021s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 = -- rational map --
│ │ │ │                                     ZZ
│ │ │ │        source: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x
│ │ │ │  ]) defined by
│ │ │ │                                   300007  0   1   2   3   4   5   6   7   8   9
│ │ │ │                {
│ │ │ │ @@ -373,67 +373,67 @@
│ │ │ │                          6     3 6 8    5 6 8    2 8    4 8    3 9    5 9    2 6
│ │ │ │  9     4 6 9    2 7 9    4 7 9     0 9
│ │ │ │                        }
│ │ │ │  
│ │ │ │  o14 : RationalMap (cubic birational map from 6-dimensional subvariety of PP^9
│ │ │ │  to PP^6)
│ │ │ │  i15 : time degrees phi^(-1)
│ │ │ │ - -- used 0.347749s (cpu); 0.274141s (thread); 0s (gc)
│ │ │ │ + -- used 0.475373s (cpu); 0.340152s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = {5, 15, 21, 17, 9, 3, 1}
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  i16 : time degrees phi
│ │ │ │ - -- used 0.0180113s (cpu); 0.0176993s (thread); 0s (gc)
│ │ │ │ + -- used 0.0793952s (cpu); 0.0267941s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = {1, 3, 9, 17, 21, 15, 5}
│ │ │ │  
│ │ │ │  o16 : List
│ │ │ │  i17 : time describe phi
│ │ │ │ - -- used 0.00320718s (cpu); 0.00320744s (thread); 0s (gc)
│ │ │ │ + -- used 0.00372634s (cpu); 0.00373177s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 = rational map defined by forms of degree 3
│ │ │ │        source variety: PP^6
│ │ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5
│ │ │ │  hypersurfaces of degree 2
│ │ │ │        dominance: true
│ │ │ │        birationality: true (the inverse map is already calculated)
│ │ │ │        projective degrees: {1, 3, 9, 17, 21, 15, 5}
│ │ │ │        coefficient ring: ZZ/300007
│ │ │ │  i18 : time describe phi^(-1)
│ │ │ │ - -- used 0.00997997s (cpu); 0.0099807s (thread); 0s (gc)
│ │ │ │ + -- used 0.0109202s (cpu); 0.0109263s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o18 = rational map defined by forms of degree 3
│ │ │ │        source variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5
│ │ │ │  hypersurfaces of degree 2
│ │ │ │        target variety: PP^6
│ │ │ │        dominance: true
│ │ │ │        birationality: true (the inverse map is already calculated)
│ │ │ │        projective degrees: {5, 15, 21, 17, 9, 3, 1}
│ │ │ │        number of minimal representatives: 1
│ │ │ │        dimension base locus: 4
│ │ │ │        degree base locus: 24
│ │ │ │        coefficient ring: ZZ/300007
│ │ │ │  i19 : time (f,g) = graph phi^-1; f;
│ │ │ │ - -- used 0.00950571s (cpu); 0.00950659s (thread); 0s (gc)
│ │ │ │ + -- used 0.0115313s (cpu); 0.0115369s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o20 : MultihomogeneousRationalMap (birational map from 6-dimensional subvariety
│ │ │ │  of PP^9 x PP^6 to 6-dimensional subvariety of PP^9)
│ │ │ │  i21 : time degrees f
│ │ │ │ - -- used 1.33327s (cpu); 0.95512s (thread); 0s (gc)
│ │ │ │ + -- used 1.20731s (cpu); 0.987305s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = {904, 508, 268, 130, 56, 20, 5}
│ │ │ │  
│ │ │ │  o21 : List
│ │ │ │  i22 : time degree f
│ │ │ │ - -- used 1.625e-05s (cpu); 1.593e-05s (thread); 0s (gc)
│ │ │ │ + -- used 2.1074e-05s (cpu); 1.9557e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o22 = 1
│ │ │ │  i23 : time describe f
│ │ │ │ - -- used 0.00161465s (cpu); 0.00161555s (thread); 0s (gc)
│ │ │ │ + -- used 0.00165199s (cpu); 0.00165748s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = rational map defined by multiforms of degree {1, 0}
│ │ │ │        source variety: 6-dimensional subvariety of PP^9 x PP^6 cut out by 20
│ │ │ │  hypersurfaces of degrees ({1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1,
│ │ │ │  1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{2, 0},{2, 0},{2, 0},{2,
│ │ │ │  0},{2, 0})
│ │ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Basic_spoperations_spon_sp__D__G_sp__Algebra_sp__Maps.out
│ │ │ @@ -155,15 +155,15 @@
│ │ │                                    2     2     2       2 2     2 2      2 2      2 2     2 2        2 2       2 2        2       2       2
│ │ │         Differential => {a, b, c, a T , b T , c T , a*b c T , b c T , -a b T , -a c T , b c T T , -a c T T , b c T T , -a T T , c T T , b T T }
│ │ │                                      1     2     3         1       4        6        5       3 4        3 5       2 4      1 7     3 7     2 7
│ │ │  
│ │ │  o16 : DGAlgebra
│ │ │  
│ │ │  i17 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.0138129s (cpu); 0.0130958s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0332124s (cpu); 0.0205266s (thread); 0s (gc)
│ │ │  
│ │ │                            ZZ
│ │ │                           ---[a..c]
│ │ │              ZZ           101
│ │ │  o17 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │             101  1   2           3   1     1
│ │ │                          (c, b, a )
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Basic_spoperations_spon_sp__D__G_sp__Algebras.out
│ │ │ @@ -30,15 +30,15 @@
│ │ │        Underlying algebra => R[S ..S ]
│ │ │                                 1   4
│ │ │        Differential => {a, b, c, d}
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │  
│ │ │  i5 : HB = HH B
│ │ │ -Finding easy relations           :  -- used 0.016973s (cpu); 0.0160828s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0424716s (cpu); 0.0242212s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HB
│ │ │  
│ │ │  o5 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i6 : describe HB
│ │ │  
│ │ │ @@ -68,15 +68,15 @@
│ │ │                                      2
│ │ │        Differential => {a, b, c, d, a T }
│ │ │                                        1
│ │ │  
│ │ │  o9 : DGAlgebra
│ │ │  
│ │ │  i10 : homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
│ │ │ -Finding easy relations           :  -- used 0.0188901s (cpu); 0.0173443s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.254629s (cpu); 0.0538928s (thread); 0s (gc)
│ │ │  
│ │ │         ZZ
│ │ │  o10 = ---[X ..X ]
│ │ │        101  1   3
│ │ │  
│ │ │  o10 : PolynomialRing, 3 skew commutative variable(s)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Algebra__Map.out
│ │ │ @@ -55,15 +55,15 @@
│ │ │                 {2} | 0 |
│ │ │                 {2} | 0 |
│ │ │                 {2} | 1 |
│ │ │  
│ │ │  o6 : ComplexMap
│ │ │  
│ │ │  i7 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.0145697s (cpu); 0.0138207s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.099449s (cpu); 0.027786s (thread); 0s (gc)
│ │ │  
│ │ │                           ZZ
│ │ │                          ---[a..c]
│ │ │             ZZ           101
│ │ │  o7 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │            101  1   2           3   1     1
│ │ │                         (c, b, a )
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___The_sp__Koszul_spcomplex_spas_spa_sp__D__G_sp__Algebra.out
│ │ │ @@ -49,15 +49,15 @@
│ │ │                                1                                                             {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a |     3
│ │ │                                                                                      
│ │ │                                                                                     2
│ │ │  
│ │ │  o6 : Complex
│ │ │  
│ │ │  i7 : HKR = HH KR
│ │ │ -Finding easy relations           :  -- used 0.126962s (cpu); 0.0524544s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.266471s (cpu); 0.0684123s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = HKR
│ │ │  
│ │ │  o7 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i8 : ideal HKR
│ │ │  
│ │ │ @@ -68,15 +68,15 @@
│ │ │  i9 : R' = ZZ/101[a,b,c,d]/ideal{a^3,b^3,c^3,d^3,a*c,a*d,b*c,b*d,a^2*b^2-c^2*d^2}
│ │ │  
│ │ │  o9 = R'
│ │ │  
│ │ │  o9 : QuotientRing
│ │ │  
│ │ │  i10 : HKR' = HH koszulComplexDGA R'
│ │ │ -Finding easy relations           :  -- used 0.54717s (cpu); 0.472623s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.645556s (cpu); 0.609624s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = HKR'
│ │ │  
│ │ │  o10 : QuotientRing
│ │ │  
│ │ │  i11 : numgens HKR'
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_cycles.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │  i3 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │  
│ │ │  o3 = {1, 4, 6, 4, 1}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.019887s (cpu); 0.0171466s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.135124s (cpu); 0.0336625s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i5 : numgens HA
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Algebra.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │  i3 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │  
│ │ │  o3 = {1, 4, 6, 4, 1}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0169473s (cpu); 0.0162102s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0563319s (cpu); 0.0307142s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i5 : R = ZZ/101[a,b,c,d]/ideal{a^4,b^4,c^4,d^4,a^3*b^3*c^3*d^3}
│ │ │  
│ │ │ @@ -46,15 +46,15 @@
│ │ │  i7 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │  
│ │ │  o7 = {1, 5, 10, 10, 4}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0853296s (cpu); 0.0826839s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.109501s (cpu); 0.0958652s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = HA
│ │ │  
│ │ │  o8 : QuotientRing
│ │ │  
│ │ │  i9 : numgens HA
│ │ │  
│ │ │ @@ -114,15 +114,15 @@
│ │ │  i15 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │  
│ │ │  o15 = {1, 7, 7, 1}
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0511448s (cpu); 0.0499304s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0738294s (cpu); 0.0611957s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = HA
│ │ │  
│ │ │  o16 : QuotientRing
│ │ │  
│ │ │  i17 : R = ZZ/101[a,b,c,d]
│ │ │  
│ │ │ @@ -151,14 +151,14 @@
│ │ │         Underlying algebra => S[T ..T ]
│ │ │                                  1   4
│ │ │         Differential => {a, b, c, d}
│ │ │  
│ │ │  o20 : DGAlgebra
│ │ │  
│ │ │  i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
│ │ │ -Finding easy relations           :  -- used 0.0169588s (cpu); 0.016221s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0519669s (cpu); 0.0277195s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = HB
│ │ │  
│ │ │  o21 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i22 :
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Class.out
│ │ │ @@ -43,15 +43,15 @@
│ │ │  o6 = y T
│ │ │          2
│ │ │  
│ │ │  o6 : R[T ..T ]
│ │ │          1   3
│ │ │  
│ │ │  i7 : H = HH(KR)
│ │ │ -Finding easy relations           :  -- used 0.0137422s (cpu); 0.0130223s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0601616s (cpu); 0.023681s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = H
│ │ │  
│ │ │  o7 : PolynomialRing, 3 skew commutative variable(s)
│ │ │  
│ │ │  i8 : homologyClass(KR,z1*z2)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Module.out
│ │ │ @@ -34,15 +34,15 @@
│ │ │  o5 = R  <-- R  <-- R  <-- R  <-- R
│ │ │                                    
│ │ │       0      1      2      3      4
│ │ │  
│ │ │  o5 : Complex
│ │ │  
│ │ │  i6 : HKR = HH(KR)
│ │ │ - -- used 0.27818s (cpu); 0.202362s (thread); 0s (gc)
│ │ │ + -- used 0.564706s (cpu); 0.174931s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o6 = HKR
│ │ │  
│ │ │  o6 : QuotientRing
│ │ │  
│ │ │  i7 : degList = first entries vars Q / degree / first
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_massey__Triple__Product.out
│ │ │ @@ -68,15 +68,15 @@
│ │ │                 2
│ │ │  o9 = (true, x y T T T  - x x y T T T )
│ │ │               2 2 1 2 3    1 2 2 2 3 4
│ │ │  
│ │ │  o9 : Sequence
│ │ │  
│ │ │  i10 : z123 = masseyTripleProduct(KR,z1,z2,z3)
│ │ │ -Finding easy relations           :  -- used 0.517042s (cpu); 0.445082s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.683328s (cpu); 0.581384s (thread); 0s (gc)
│ │ │  
│ │ │               2
│ │ │  o10 = x x y z T T T T
│ │ │         1 2 2   2 3 4 5
│ │ │  
│ │ │  o10 : R[T ..T ]
│ │ │           1   5
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_massey__Triple__Product_lp__D__G__Algebra_cm__Z__Z_cm__Z__Z_cm__Z__Z_rp.out
│ │ │ @@ -27,15 +27,15 @@
│ │ │                                 1   4
│ │ │        Differential => {t , t , t , t }
│ │ │                          1   2   3   4
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │  
│ │ │  i5 : H = HH(KR)
│ │ │ -Finding easy relations           :  -- used 0.138884s (cpu); 0.136232s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.181575s (cpu); 0.167873s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = H
│ │ │  
│ │ │  o5 : QuotientRing
│ │ │  
│ │ │  i6 : masseys = masseyTripleProduct(KR,1,1,1);
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_tor__Algebra_lp__Ring_cm__Ring_rp.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │  i3 : S = R/ideal{a^3*b^3*c^3*d^3}
│ │ │  
│ │ │  o3 = S
│ │ │  
│ │ │  o3 : QuotientRing
│ │ │  
│ │ │  i4 : HB = torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)
│ │ │ - -- used 0.601766s (cpu); 0.506804s (thread); 0s (gc)
│ │ │ + -- used 0.611233s (cpu); 0.502307s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o4 = HB
│ │ │  
│ │ │  o4 : QuotientRing
│ │ │  
│ │ │  i5 : numgens HB
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___Basic_spoperations_spon_sp__D__G_sp__Algebra_sp__Maps.html
│ │ │ @@ -289,15 +289,15 @@
│ │ │          
    
│ │ │            
    One can also obtain the map on homology induced by a DGAlgebra map.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i17 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.0138129s (cpu); 0.0130958s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0332124s (cpu); 0.0205266s (thread); 0s (gc)
│ │ │  
│ │ │                            ZZ
│ │ │                           ---[a..c]
│ │ │              ZZ           101
│ │ │  o17 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │             101  1   2           3   1     1
│ │ │                          (c, b, a )
│ │ │ ├── html2text {}
│ │ │ │ @@ -210,15 +210,15 @@
│ │ │ │  a c T , b c T T , -a c T T , b c T T , -a T T , c T T , b T T }
│ │ │ │                                      1     2     3         1       4        6
│ │ │ │  5       3 4        3 5       2 4      1 7     3 7     2 7
│ │ │ │  
│ │ │ │  o16 : DGAlgebra
│ │ │ │  One can also obtain the map on homology induced by a DGAlgebra map.
│ │ │ │  i17 : HHg = HH g
│ │ │ │ -Finding easy relations           :  -- used 0.0138129s (cpu); 0.0130958s
│ │ │ │ +Finding easy relations           :  -- used 0.0332124s (cpu); 0.0205266s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │                            ZZ
│ │ │ │                           ---[a..c]
│ │ │ │              ZZ           101
│ │ │ │  o17 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │ │             101  1   2           3   1     1
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___Basic_spoperations_spon_sp__D__G_sp__Algebras.html
│ │ │ @@ -113,15 +113,15 @@
│ │ │          
    
│ │ │            
    One can compute the homology algebra of a DGAlgebra using the homology (or HH) command.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -174,15 +174,15 @@
│ │ │  
│ │ │  o9 : DGAlgebra
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -42,15 +42,15 @@
│ │ │ │                                 1   4
│ │ │ │        Differential => {a, b, c, d}
│ │ │ │  
│ │ │ │  o4 : DGAlgebra
│ │ │ │  One can compute the homology algebra of a DGAlgebra using the homology (or HH)
│ │ │ │  command.
│ │ │ │  i5 : HB = HH B
│ │ │ │ -Finding easy relations           :  -- used 0.016973s (cpu); 0.0160828s
│ │ │ │ +Finding easy relations           :  -- used 0.0424716s (cpu); 0.0242212s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = HB
│ │ │ │  
│ │ │ │  o5 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  i6 : describe HB
│ │ │ │  
│ │ │ │ @@ -87,15 +87,15 @@
│ │ │ │                                 1   5
│ │ │ │                                      2
│ │ │ │        Differential => {a, b, c, d, a T }
│ │ │ │                                        1
│ │ │ │  
│ │ │ │  o9 : DGAlgebra
│ │ │ │  i10 : homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
│ │ │ │ -Finding easy relations           :  -- used 0.0188901s (cpu); 0.0173443s
│ │ │ │ +Finding easy relations           :  -- used 0.254629s (cpu); 0.0538928s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │         ZZ
│ │ │ │  o10 = ---[X ..X ]
│ │ │ │        101  1   3
│ │ │ │  
│ │ │ │  o10 : PolynomialRing, 3 skew commutative variable(s)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_sp__D__G__Algebra__Map.html
│ │ │ @@ -144,15 +144,15 @@
│ │ │  
│ │ │  o6 : ComplexMap
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -242,15 +242,15 @@
│ │ │  
│ │ │  o15 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : HB = HH B
│ │ │ -Finding easy relations           :  -- used 0.016973s (cpu); 0.0160828s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0424716s (cpu); 0.0242212s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HB
│ │ │  
│ │ │  o5 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i10 : homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
│ │ │ -Finding easy relations           :  -- used 0.0188901s (cpu); 0.0173443s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.254629s (cpu); 0.0538928s (thread); 0s (gc)
│ │ │  
│ │ │         ZZ
│ │ │  o10 = ---[X ..X ]
│ │ │        101  1   3
│ │ │  
│ │ │  o10 : PolynomialRing, 3 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i7 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.0145697s (cpu); 0.0138207s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.099449s (cpu); 0.027786s (thread); 0s (gc)
│ │ │  
│ │ │                           ZZ
│ │ │                          ---[a..c]
│ │ │             ZZ           101
│ │ │  o7 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │            101  1   2           3   1     1
│ │ │                         (c, b, a )
│ │ │ ├── html2text {}
│ │ │ │ @@ -62,15 +62,15 @@
│ │ │ │       2 : R  <------------- R  : 2
│ │ │ │                 {2} | 0 |
│ │ │ │                 {2} | 0 |
│ │ │ │                 {2} | 1 |
│ │ │ │  
│ │ │ │  o6 : ComplexMap
│ │ │ │  i7 : HHg = HH g
│ │ │ │ -Finding easy relations           :  -- used 0.0145697s (cpu); 0.0138207s
│ │ │ │ +Finding easy relations           :  -- used 0.099449s (cpu); 0.027786s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │                           ZZ
│ │ │ │                          ---[a..c]
│ │ │ │             ZZ           101
│ │ │ │  o7 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │ │            101  1   2           3   1     1
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___The_sp__Koszul_spcomplex_spas_spa_sp__D__G_sp__Algebra.html
│ │ │ @@ -138,15 +138,15 @@
│ │ │          
    
│ │ │            
    Since the Koszul complex is a DG algebra, its homology is itself an algebra.  One can obtain this algebra using the command homology, homologyAlgebra, or HH (all commands work).  This algebra structure can detect whether or not the ring is a complete intersection or Gorenstein.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -166,15 +166,15 @@
│ │ │  
│ │ │  o9 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -76,15 +76,15 @@
│ │ │ │  
│ │ │ │  o6 : Complex
│ │ │ │  Since the Koszul complex is a DG algebra, its homology is itself an algebra.
│ │ │ │  One can obtain this algebra using the command homology, homologyAlgebra, or HH
│ │ │ │  (all commands work). This algebra structure can detect whether or not the ring
│ │ │ │  is a complete intersection or Gorenstein.
│ │ │ │  i7 : HKR = HH KR
│ │ │ │ -Finding easy relations           :  -- used 0.126962s (cpu); 0.0524544s
│ │ │ │ +Finding easy relations           :  -- used 0.266471s (cpu); 0.0684123s
│ │ │ │  (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 0.54717s (cpu); 0.472623s (thread);
│ │ │ │ -0s (gc)
│ │ │ │ +Finding easy relations           :  -- used 0.645556s (cpu); 0.609624s
│ │ │ │ +(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 0.019887s (cpu); 0.0171466s
│ │ │ │ +Finding easy relations           :  -- used 0.135124s (cpu); 0.0336625s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = HA
│ │ │ │  
│ │ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  i5 : numgens HA
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Algebra.html
│ │ │ @@ -103,15 +103,15 @@
│ │ │  
│ │ │  o3 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : HKR = HH KR
│ │ │ -Finding easy relations           :  -- used 0.126962s (cpu); 0.0524544s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.266471s (cpu); 0.0684123s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = HKR
│ │ │  
│ │ │  o7 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i10 : HKR' = HH koszulComplexDGA R'
│ │ │ -Finding easy relations           :  -- used 0.54717s (cpu); 0.472623s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.645556s (cpu); 0.609624s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = HKR'
│ │ │  
│ │ │  o10 : QuotientRing
    
│ │ │              
    
│ │ │                
    i4 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.019887s (cpu); 0.0171466s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.135124s (cpu); 0.0336625s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i4 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0169473s (cpu); 0.0162102s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0563319s (cpu); 0.0307142s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │ @@ -148,15 +148,15 @@
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │                
    i8 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0853296s (cpu); 0.0826839s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.109501s (cpu); 0.0958652s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = HA
│ │ │  
│ │ │  o8 : QuotientRing
    
│ │ │              
    
│ │ │                
    i16 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0511448s (cpu); 0.0499304s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0738294s (cpu); 0.0611957s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = HA
│ │ │  
│ │ │  o16 : QuotientRing
    
│ │ │              
    
│ │ │ @@ -302,15 +302,15 @@
│ │ │  
│ │ │  o20 : DGAlgebra
    
│ │ │              
    
│ │ │                
    i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
│ │ │ -Finding easy relations           :  -- used 0.0169588s (cpu); 0.016221s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0519669s (cpu); 0.0277195s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = HB
│ │ │  
│ │ │  o21 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,15 +33,15 @@
│ │ │ │  o2 : DGAlgebra
│ │ │ │  i3 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │ │  
│ │ │ │  o3 = {1, 4, 6, 4, 1}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : HA = homologyAlgebra(A)
│ │ │ │ -Finding easy relations           :  -- used 0.0169473s (cpu); 0.0162102s
│ │ │ │ +Finding easy relations           :  -- used 0.0563319s (cpu); 0.0307142s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = HA
│ │ │ │  
│ │ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  Note that HA is a graded commutative polynomial ring (i.e. an exterior algebra)
│ │ │ │  since R is a complete intersection.
│ │ │ │ @@ -60,15 +60,15 @@
│ │ │ │  o6 : DGAlgebra
│ │ │ │  i7 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │ │  
│ │ │ │  o7 = {1, 5, 10, 10, 4}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  i8 : HA = homologyAlgebra(A)
│ │ │ │ -Finding easy relations           :  -- used 0.0853296s (cpu); 0.0826839s
│ │ │ │ +Finding easy relations           :  -- used 0.109501s (cpu); 0.0958652s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = HA
│ │ │ │  
│ │ │ │  o8 : QuotientRing
│ │ │ │  i9 : numgens HA
│ │ │ │  
│ │ │ │ @@ -122,15 +122,15 @@
│ │ │ │  o14 : DGAlgebra
│ │ │ │  i15 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │ │  
│ │ │ │  o15 = {1, 7, 7, 1}
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  i16 : HA = homologyAlgebra(A)
│ │ │ │ -Finding easy relations           :  -- used 0.0511448s (cpu); 0.0499304s
│ │ │ │ +Finding easy relations           :  -- used 0.0738294s (cpu); 0.0611957s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = HA
│ │ │ │  
│ │ │ │  o16 : QuotientRing
│ │ │ │  One can check that HA has Poincare duality since R is Gorenstein.
│ │ │ │  If your DGAlgebra has generators in even degrees, then one must specify the
│ │ │ │ @@ -158,15 +158,15 @@
│ │ │ │  o20 = {Ring => S                      }
│ │ │ │         Underlying algebra => S[T ..T ]
│ │ │ │                                  1   4
│ │ │ │         Differential => {a, b, c, d}
│ │ │ │  
│ │ │ │  o20 : DGAlgebra
│ │ │ │  i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
│ │ │ │ -Finding easy relations           :  -- used 0.0169588s (cpu); 0.016221s
│ │ │ │ +Finding easy relations           :  -- used 0.0519669s (cpu); 0.0277195s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = HB
│ │ │ │  
│ │ │ │  o21 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  ********** WWaayyss ttoo uussee hhoommoollooggyyAAllggeebbrraa:: **********
│ │ │ │      * homologyAlgebra(DGAlgebra)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Class.html
│ │ │ @@ -135,15 +135,15 @@
│ │ │  o6 : R[T ..T ]
│ │ │          1   3
    
│ │ │              
    
│ │ │                
    i7 : H = HH(KR)
│ │ │ -Finding easy relations           :  -- used 0.0137422s (cpu); 0.0130223s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0601616s (cpu); 0.023681s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = H
│ │ │  
│ │ │  o7 : PolynomialRing, 3 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i6 : HKR = HH(KR)
│ │ │ - -- used 0.27818s (cpu); 0.202362s (thread); 0s (gc)
│ │ │ + -- used 0.564706s (cpu); 0.174931s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o6 = HKR
│ │ │  
│ │ │  o6 : QuotientRing
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -54,15 +54,15 @@
│ │ │ │        1      4      6      4      1
│ │ │ │  o5 = R  <-- R  <-- R  <-- R  <-- R
│ │ │ │  
│ │ │ │       0      1      2      3      4
│ │ │ │  
│ │ │ │  o5 : Complex
│ │ │ │  i6 : HKR = HH(KR)
│ │ │ │ - -- used 0.27818s (cpu); 0.202362s (thread); 0s (gc)
│ │ │ │ + -- used 0.564706s (cpu); 0.174931s (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 0.138884s (cpu); 0.136232s
│ │ │ │ +Finding easy relations           :  -- used 0.181575s (cpu); 0.167873s
│ │ │ │  (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 0.601766s (cpu); 0.506804s (thread); 0s (gc)
│ │ │ │ + -- used 0.611233s (cpu); 0.502307s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o4 = HB
│ │ │ │  
│ │ │ │  o4 : QuotientRing
│ │ │ │  i5 : numgens HB
│ │ │ │  
│ │ │ │  o5 = 35
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_random__Hyper__Graph.out
│ │ │ @@ -3,25 +3,25 @@
│ │ │  i1 : R = QQ[x_1..x_5];
│ │ │  
│ │ │  i2 : randomHyperGraph(R,{3,2,4})
│ │ │  
│ │ │  i3 : randomHyperGraph(R,{3,2,4})
│ │ │  
│ │ │  o3 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ -                              1   3   4     1   5     2   3   4   5
│ │ │ +                              1   2   3     3   5     1   2   4   5
│ │ │                  "ring" => R
│ │ │                  "vertices" => {x , x , x , x , x }
│ │ │                                  1   2   3   4   5
│ │ │  
│ │ │  o3 : HyperGraph
│ │ │  
│ │ │  i4 : randomHyperGraph(R,{3,2,4})
│ │ │  
│ │ │  o4 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ -                              2   3   4     1   2     1   3   4   5
│ │ │ +                              1   2   4     4   5     1   2   3   5
│ │ │                  "ring" => R
│ │ │                  "vertices" => {x , x , x , x , x }
│ │ │                                  1   2   3   4   5
│ │ │  
│ │ │  o4 : HyperGraph
│ │ │  
│ │ │  i5 : randomHyperGraph(R,{4,4,2,2}) -- impossible, returns null when time/branch limit reached
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/html/_random__Hyper__Graph.html
│ │ │ @@ -88,28 +88,28 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,24 +25,24 @@
│ │ │ │  _T_i_m_e_L_i_m_i_t). The method will return null if it cannot find a hypergraph within
│ │ │ │  the branch and time limits.
│ │ │ │  i1 : R = QQ[x_1..x_5];
│ │ │ │  i2 : randomHyperGraph(R,{3,2,4})
│ │ │ │  i3 : randomHyperGraph(R,{3,2,4})
│ │ │ │  
│ │ │ │  o3 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ │ -                              1   3   4     1   5     2   3   4   5
│ │ │ │ +                              1   2   3     3   5     1   2   4   5
│ │ │ │                  "ring" => R
│ │ │ │                  "vertices" => {x , x , x , x , x }
│ │ │ │                                  1   2   3   4   5
│ │ │ │  
│ │ │ │  o3 : HyperGraph
│ │ │ │  i4 : randomHyperGraph(R,{3,2,4})
│ │ │ │  
│ │ │ │  o4 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ │ -                              2   3   4     1   2     1   3   4   5
│ │ │ │ +                              1   2   4     4   5     1   2   3   5
│ │ │ │                  "ring" => R
│ │ │ │                  "vertices" => {x , x , x , x , x }
│ │ │ │                                  1   2   3   4   5
│ │ │ │  
│ │ │ │  o4 : HyperGraph
│ │ │ │  i5 : randomHyperGraph(R,{4,4,2,2}) -- impossible, returns null when time/branch
│ │ │ │  limit reached
│ │ ├── ./usr/share/doc/Macaulay2/EigenSolver/example-output/___Eigen__Solver.out
│ │ │ @@ -15,14 +15,14 @@
│ │ │       a*b*e*f + a*d*e*f + c*d*e*f, a*b*c*d*e + a*b*c*d*f + a*b*c*e*f +
│ │ │       ------------------------------------------------------------------------
│ │ │       a*b*d*e*f + a*c*d*e*f + b*c*d*e*f, a*b*c*d*e*f - 1)
│ │ │  
│ │ │  o2 : Ideal of QQ[a..f]
│ │ │  
│ │ │  i3 : elapsedTime sols = zeroDimSolve I;
│ │ │ - -- .34144s elapsed
│ │ │ + -- .219083s elapsed
│ │ │  
│ │ │  i4 : #sols -- 156 solutions
│ │ │  
│ │ │  o4 = 156
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/EigenSolver/html/index.html
│ │ │ @@ -80,15 +80,15 @@
│ │ │  
│ │ │  o2 : Ideal of QQ[a..f]
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,22 +29,22 @@
│ │ │ │  i3 : g = x^2+c*x+d
│ │ │ │  
│ │ │ │        2
│ │ │ │  o3 = x  + x*c + d
│ │ │ │  
│ │ │ │  o3 : R
│ │ │ │  i4 : time eliminate(x,ideal(f,g))
│ │ │ │ - -- used 0.00276448s (cpu); 0.00276172s (thread); 0s (gc)
│ │ │ │ + -- used 0.00315082s (cpu); 0.00314671s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                        2    2             2           2
│ │ │ │  o4 = ideal(a*b*c - b*c  - a d + a*c*d - b  + 2b*d - d )
│ │ │ │  
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : time ideal resultant(f,g,x)
│ │ │ │ - -- used 0.00161961s (cpu); 0.00162013s (thread); 0s (gc)
│ │ │ │ + -- used 0.00181431s (cpu); 0.00181563s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                          2    2             2           2
│ │ │ │  o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )
│ │ │ │  
│ │ │ │  o5 : Ideal of R
│ │ │ │  i6 : sylvesterMatrix(f,g,x)
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/html/_eliminate.html
│ │ │ @@ -97,26 +97,26 @@
│ │ │  
│ │ │  o3 : R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,22 +30,22 @@
│ │ │ │  i3 : g = x^2+c*x+d
│ │ │ │  
│ │ │ │        2
│ │ │ │  o3 = x  + x*c + d
│ │ │ │  
│ │ │ │  o3 : R
│ │ │ │  i4 : time eliminate(x,ideal(f,g))
│ │ │ │ - -- used 0.00357906s (cpu); 0.00357709s (thread); 0s (gc)
│ │ │ │ + -- used 0.00314217s (cpu); 0.003138s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                        2    2             2           2
│ │ │ │  o4 = ideal(a*b*c - b*c  - a d + a*c*d - b  + 2b*d - d )
│ │ │ │  
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : time ideal resultant(f,g,x)
│ │ │ │ - -- used 0.00211786s (cpu); 0.0021216s (thread); 0s (gc)
│ │ │ │ + -- used 0.00167105s (cpu); 0.00167243s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                          2    2             2           2
│ │ │ │  o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )
│ │ │ │  
│ │ │ │  o5 : Ideal of R
│ │ │ │  i6 : sylvesterMatrix(f,g,x)
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/html/_resultant_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.html
│ │ │ @@ -105,15 +105,15 @@
│ │ │  
│ │ │  o3 : R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : z123 = masseyTripleProduct(KR,z1,z2,z3)
│ │ │ -Finding easy relations           :  -- used 0.517042s (cpu); 0.445082s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.683328s (cpu); 0.581384s (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 0.517042s (cpu); 0.445082s
│ │ │ │ +Finding easy relations           :  -- used 0.683328s (cpu); 0.581384s
│ │ │ │  (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 0.138884s (cpu); 0.136232s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.181575s (cpu); 0.167873s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = H
│ │ │  
│ │ │  o5 : QuotientRing
    
│ │ │              
    
│ │ │                
    i4 : HB = torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)
│ │ │ - -- used 0.601766s (cpu); 0.506804s (thread); 0s (gc)
│ │ │ + -- used 0.611233s (cpu); 0.502307s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o4 = HB
│ │ │  
│ │ │  o4 : QuotientRing
    
│ │ │              
    
│ │ │                
    i3 : randomHyperGraph(R,{3,2,4})
│ │ │  
│ │ │  o3 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ -                              1   3   4     1   5     2   3   4   5
│ │ │ +                              1   2   3     3   5     1   2   4   5
│ │ │                  "ring" => R
│ │ │                  "vertices" => {x , x , x , x , x }
│ │ │                                  1   2   3   4   5
│ │ │  
│ │ │  o3 : HyperGraph
    
│ │ │              
    
│ │ │                
    i4 : randomHyperGraph(R,{3,2,4})
│ │ │  
│ │ │  o4 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ -                              2   3   4     1   2     1   3   4   5
│ │ │ +                              1   2   4     4   5     1   2   3   5
│ │ │                  "ring" => R
│ │ │                  "vertices" => {x , x , x , x , x }
│ │ │                                  1   2   3   4   5
│ │ │  
│ │ │  o4 : HyperGraph
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime sols = zeroDimSolve I;
│ │ │ - -- .34144s elapsed
    
│ │ │ + -- .219083s elapsed
│ │ │              
    
│ │ │                
    i4 : #sols -- 156 solutions
│ │ │  
│ │ │  o4 = 156
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       a*b*e*f + a*d*e*f + c*d*e*f, a*b*c*d*e + a*b*c*d*f + a*b*c*e*f +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       a*b*d*e*f + a*c*d*e*f + b*c*d*e*f, a*b*c*d*e*f - 1)
│ │ │ │  
│ │ │ │  o2 : Ideal of QQ[a..f]
│ │ │ │  i3 : elapsedTime sols = zeroDimSolve I;
│ │ │ │ - -- .34144s elapsed
│ │ │ │ + -- .219083s elapsed
│ │ │ │  i4 : #sols -- 156 solutions
│ │ │ │  
│ │ │ │  o4 = 156
│ │ │ │  The authors would like to acknowledge the June 2020 Macaulay2 workshop held
│ │ │ │  virtually at Warwick, where this package was first developed.
│ │ │ │  RReeffeerreenncceess:
│ │ │ │      * [1] Sturmfels, Bernd. Solving systems of polynomial equations. No. 97.
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/example-output/_discriminant_lp__Ring__Element_cm__Ring__Element_rp.out
│ │ │ @@ -17,23 +17,23 @@
│ │ │  
│ │ │        2
│ │ │  o3 = x  + x*c + d
│ │ │  
│ │ │  o3 : R
│ │ │  
│ │ │  i4 : time eliminate(x,ideal(f,g))
│ │ │ - -- used 0.00276448s (cpu); 0.00276172s (thread); 0s (gc)
│ │ │ + -- used 0.00315082s (cpu); 0.00314671s (thread); 0s (gc)
│ │ │  
│ │ │                        2    2             2           2
│ │ │  o4 = ideal(a*b*c - b*c  - a d + a*c*d - b  + 2b*d - d )
│ │ │  
│ │ │  o4 : Ideal of R
│ │ │  
│ │ │  i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.00161961s (cpu); 0.00162013s (thread); 0s (gc)
│ │ │ + -- used 0.00181431s (cpu); 0.00181563s (thread); 0s (gc)
│ │ │  
│ │ │                          2    2             2           2
│ │ │  o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )
│ │ │  
│ │ │  o5 : Ideal of R
│ │ │  
│ │ │  i6 : sylvesterMatrix(f,g,x)
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/example-output/_eliminate.out
│ │ │ @@ -17,23 +17,23 @@
│ │ │  
│ │ │        2
│ │ │  o3 = x  + x*c + d
│ │ │  
│ │ │  o3 : R
│ │ │  
│ │ │  i4 : time eliminate(x,ideal(f,g))
│ │ │ - -- used 0.00357906s (cpu); 0.00357709s (thread); 0s (gc)
│ │ │ + -- used 0.00314217s (cpu); 0.003138s (thread); 0s (gc)
│ │ │  
│ │ │                        2    2             2           2
│ │ │  o4 = ideal(a*b*c - b*c  - a d + a*c*d - b  + 2b*d - d )
│ │ │  
│ │ │  o4 : Ideal of R
│ │ │  
│ │ │  i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.00211786s (cpu); 0.0021216s (thread); 0s (gc)
│ │ │ + -- used 0.00167105s (cpu); 0.00167243s (thread); 0s (gc)
│ │ │  
│ │ │                          2    2             2           2
│ │ │  o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )
│ │ │  
│ │ │  o5 : Ideal of R
│ │ │  
│ │ │  i6 : sylvesterMatrix(f,g,x)
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/example-output/_resultant_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.out
│ │ │ @@ -17,15 +17,15 @@
│ │ │  
│ │ │        8    5
│ │ │  o3 = x  + x  + x*c + d
│ │ │  
│ │ │  o3 : R
│ │ │  
│ │ │  i4 : time eliminate(ideal(f,g),x)
│ │ │ - -- used 1.89068s (cpu); 1.67085s (thread); 0s (gc)
│ │ │ + -- used 1.38743s (cpu); 1.25876s (thread); 0s (gc)
│ │ │  
│ │ │              7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o4 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -73,15 +73,15 @@
│ │ │       ------------------------------------------------------------------------
│ │ │                 3          4        4
│ │ │       - 216b*c*d  + 2052a*d  - 1944d )
│ │ │  
│ │ │  o4 : Ideal of R
│ │ │  
│ │ │  i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.0240253s (cpu); 0.0240283s (thread); 0s (gc)
│ │ │ + -- used 0.015875s (cpu); 0.0158768s (thread); 0s (gc)
│ │ │  
│ │ │                7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o5 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/example-output/_sylvester__Matrix_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.out
│ │ │ @@ -19,15 +19,15 @@
│ │ │  
│ │ │        8    5
│ │ │  o4 = x  + x  + x*c + d
│ │ │  
│ │ │  o4 : R
│ │ │  
│ │ │  i5 : time eliminate(ideal(f,g),x)
│ │ │ - -- used 1.67634s (cpu); 1.45099s (thread); 0s (gc)
│ │ │ + -- used 1.46589s (cpu); 1.35028s (thread); 0s (gc)
│ │ │  
│ │ │              7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o5 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -75,15 +75,15 @@
│ │ │       ------------------------------------------------------------------------
│ │ │                 3          4        4
│ │ │       - 216b*c*d  + 2052a*d  - 1944d )
│ │ │  
│ │ │  o5 : Ideal of R
│ │ │  
│ │ │  i6 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.01611s (cpu); 0.016112s (thread); 0s (gc)
│ │ │ + -- used 0.0153981s (cpu); 0.0154021s (thread); 0s (gc)
│ │ │  
│ │ │                7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o6 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/html/_discriminant_lp__Ring__Element_cm__Ring__Element_rp.html
│ │ │ @@ -103,26 +103,26 @@
│ │ │  
│ │ │  o3 : R
│ │ │              
    
│ │ │                
    i4 : time eliminate(x,ideal(f,g))
│ │ │ - -- used 0.00276448s (cpu); 0.00276172s (thread); 0s (gc)
│ │ │ + -- used 0.00315082s (cpu); 0.00314671s (thread); 0s (gc)
│ │ │  
│ │ │                        2    2             2           2
│ │ │  o4 = ideal(a*b*c - b*c  - a d + a*c*d - b  + 2b*d - d )
│ │ │  
│ │ │  o4 : Ideal of R
    
│ │ │              
    
│ │ │                
    i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.00161961s (cpu); 0.00162013s (thread); 0s (gc)
│ │ │ + -- used 0.00181431s (cpu); 0.00181563s (thread); 0s (gc)
│ │ │  
│ │ │                          2    2             2           2
│ │ │  o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )
│ │ │  
│ │ │  o5 : Ideal of R
    
│ │ │              
    
│ │ │                
    i4 : time eliminate(x,ideal(f,g))
│ │ │ - -- used 0.00357906s (cpu); 0.00357709s (thread); 0s (gc)
│ │ │ + -- used 0.00314217s (cpu); 0.003138s (thread); 0s (gc)
│ │ │  
│ │ │                        2    2             2           2
│ │ │  o4 = ideal(a*b*c - b*c  - a d + a*c*d - b  + 2b*d - d )
│ │ │  
│ │ │  o4 : Ideal of R
    
│ │ │              
    
│ │ │                
    i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.00211786s (cpu); 0.0021216s (thread); 0s (gc)
│ │ │ + -- used 0.00167105s (cpu); 0.00167243s (thread); 0s (gc)
│ │ │  
│ │ │                          2    2             2           2
│ │ │  o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )
│ │ │  
│ │ │  o5 : Ideal of R
    
│ │ │              
    
│ │ │                
    i4 : time eliminate(ideal(f,g),x)
│ │ │ - -- used 1.89068s (cpu); 1.67085s (thread); 0s (gc)
│ │ │ + -- used 1.38743s (cpu); 1.25876s (thread); 0s (gc)
│ │ │  
│ │ │              7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o4 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -164,15 +164,15 @@
│ │ │  
│ │ │  o4 : Ideal of R
    
│ │ │              
    
│ │ │                
    i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.0240253s (cpu); 0.0240283s (thread); 0s (gc)
│ │ │ + -- used 0.015875s (cpu); 0.0158768s (thread); 0s (gc)
│ │ │  
│ │ │                7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o5 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │       ------------------------------------------------------------------------
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,15 +35,15 @@
│ │ │ │  i3 : g = x^8+x^5+c*x+d
│ │ │ │  
│ │ │ │        8    5
│ │ │ │  o3 = x  + x  + x*c + d
│ │ │ │  
│ │ │ │  o3 : R
│ │ │ │  i4 : time eliminate(ideal(f,g),x)
│ │ │ │ - -- used 1.89068s (cpu); 1.67085s (thread); 0s (gc)
│ │ │ │ + -- used 1.38743s (cpu); 1.25876s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              7       8     3 5    8     6         3 4      7       3 3 2
│ │ │ │  o4 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7
│ │ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -90,15 +90,15 @@
│ │ │ │       + 792a*b c*d - 1512a*b*c d + 648a*c d - 360a b*d  + 648a c*d  - 504b d
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                 3          4        4
│ │ │ │       - 216b*c*d  + 2052a*d  - 1944d )
│ │ │ │  
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : time ideal resultant(f,g,x)
│ │ │ │ - -- used 0.0240253s (cpu); 0.0240283s (thread); 0s (gc)
│ │ │ │ + -- used 0.015875s (cpu); 0.0158768s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                7       8     3 5    8     6         3 4      7       3 3 2
│ │ │ │  o5 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7
│ │ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/html/_sylvester__Matrix_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.html
│ │ │ @@ -104,15 +104,15 @@
│ │ │  
│ │ │  o4 : R
    
│ │ │              
    
│ │ │                
    i5 : time eliminate(ideal(f,g),x)
│ │ │ - -- used 1.67634s (cpu); 1.45099s (thread); 0s (gc)
│ │ │ + -- used 1.46589s (cpu); 1.35028s (thread); 0s (gc)
│ │ │  
│ │ │              7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o5 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -163,15 +163,15 @@
│ │ │  
│ │ │  o5 : Ideal of R
    
│ │ │              
    
│ │ │                
    i6 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.01611s (cpu); 0.016112s (thread); 0s (gc)
│ │ │ + -- used 0.0153981s (cpu); 0.0154021s (thread); 0s (gc)
│ │ │  
│ │ │                7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o6 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │       ------------------------------------------------------------------------
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,15 +30,15 @@
│ │ │ │  i4 : g = x^8+x^5+c*x+d
│ │ │ │  
│ │ │ │        8    5
│ │ │ │  o4 = x  + x  + x*c + d
│ │ │ │  
│ │ │ │  o4 : R
│ │ │ │  i5 : time eliminate(ideal(f,g),x)
│ │ │ │ - -- used 1.67634s (cpu); 1.45099s (thread); 0s (gc)
│ │ │ │ + -- used 1.46589s (cpu); 1.35028s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              7       8     3 5    8     6         3 4      7       3 3 2
│ │ │ │  o5 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7
│ │ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -85,15 +85,15 @@
│ │ │ │       + 792a*b c*d - 1512a*b*c d + 648a*c d - 360a b*d  + 648a c*d  - 504b d
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                 3          4        4
│ │ │ │       - 216b*c*d  + 2052a*d  - 1944d )
│ │ │ │  
│ │ │ │  o5 : Ideal of R
│ │ │ │  i6 : time ideal resultant(f,g,x)
│ │ │ │ - -- used 0.01611s (cpu); 0.016112s (thread); 0s (gc)
│ │ │ │ + -- used 0.0153981s (cpu); 0.0154021s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                7       8     3 5    8     6         3 4      7       3 3 2
│ │ │ │  o6 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7
│ │ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_lines__Hypersurface.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1331975673177
│ │ │  
│ │ │  i1 : time for n from 2 to 10 list linesHypersurface(n)
│ │ │ - -- used 0.0280835s (cpu); 0.0280868s (thread); 0s (gc)
│ │ │ + -- used 0.0289803s (cpu); 0.0289806s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775,
│ │ │       ------------------------------------------------------------------------
│ │ │       289139638632755625, 520764738758073845321}
│ │ │  
│ │ │  o1 : List
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_rational__Curve.out
│ │ │ @@ -37,83 +37,83 @@
│ │ │  i6 : rationalCurve(2) - rationalCurve(1)/8
│ │ │  
│ │ │  o6 = 609250
│ │ │  
│ │ │  o6 : QQ
│ │ │  
│ │ │  i7 : time for D in T list rationalCurve(2,D) - rationalCurve(1,D)/8
│ │ │ - -- used 0.328797s (cpu); 0.276936s (thread); 0s (gc)
│ │ │ + -- used 0.345874s (cpu); 0.2915s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {609250, 92288, 52812, 22428, 9728}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : time rationalCurve(3)
│ │ │ - -- used 0.228142s (cpu); 0.168847s (thread); 0s (gc)
│ │ │ + -- used 0.130073s (cpu); 0.13008s (thread); 0s (gc)
│ │ │  
│ │ │       8564575000
│ │ │  o8 = ----------
│ │ │           27
│ │ │  
│ │ │  o8 : QQ
│ │ │  
│ │ │  i9 : time for D in T list rationalCurve(3,D)
│ │ │ - -- used 5.32989s (cpu); 4.61697s (thread); 0s (gc)
│ │ │ + -- used 4.91979s (cpu); 4.39581s (thread); 0s (gc)
│ │ │  
│ │ │        8564575000  422690816           4834592  11239424
│ │ │  o9 = {----------, ---------, 6424365, -------, --------}
│ │ │            27          27                 3        27
│ │ │  
│ │ │  o9 : List
│ │ │  
│ │ │  i10 : time rationalCurve(3) - rationalCurve(1)/27
│ │ │ - -- used 0.217214s (cpu); 0.167544s (thread); 0s (gc)
│ │ │ + -- used 0.1301s (cpu); 0.130107s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 317206375
│ │ │  
│ │ │  o10 : QQ
│ │ │  
│ │ │  i11 : time for D in T list rationalCurve(3,D) - rationalCurve(1,D)/27
│ │ │ - -- used 5.31342s (cpu); 4.64208s (thread); 0s (gc)
│ │ │ + -- used 4.96808s (cpu); 4.40223s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {317206375, 15655168, 6424326, 1611504, 416256}
│ │ │  
│ │ │  o11 : List
│ │ │  
│ │ │  i12 : time rationalCurve(4)
│ │ │ - -- used 1.64976s (cpu); 1.4421s (thread); 0s (gc)
│ │ │ + -- used 1.42387s (cpu); 1.30604s (thread); 0s (gc)
│ │ │  
│ │ │        15517926796875
│ │ │  o12 = --------------
│ │ │              64
│ │ │  
│ │ │  o12 : QQ
│ │ │  
│ │ │  i13 : time rationalCurve(4,{4,2})
│ │ │ - -- used 7.46787s (cpu); 5.79404s (thread); 0s (gc)
│ │ │ + -- used 6.76391s (cpu); 5.67641s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = 3883914084
│ │ │  
│ │ │  o13 : QQ
│ │ │  
│ │ │  i14 : time rationalCurve(4) - rationalCurve(2)/8
│ │ │ - -- used 1.65931s (cpu); 1.42972s (thread); 0s (gc)
│ │ │ + -- used 1.54252s (cpu); 1.36125s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 242467530000
│ │ │  
│ │ │  o14 : QQ
│ │ │  
│ │ │  i15 : time rationalCurve(4,{4,2}) - rationalCurve(2,{4,2})/8
│ │ │ - -- used 7.68075s (cpu); 6.13623s (thread); 0s (gc)
│ │ │ + -- used 6.58956s (cpu); 5.47923s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3883902528
│ │ │  
│ │ │  o15 : QQ
│ │ │  
│ │ │  i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8
│ │ │ - -- used 7.9262s (cpu); 6.01692s (thread); 0s (gc)
│ │ │ + -- used 6.80324s (cpu); 5.67876s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 1139448384
│ │ │  
│ │ │  o16 : QQ
│ │ │  
│ │ │  i17 :
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/html/_lines__Hypersurface.html
│ │ │ @@ -71,15 +71,15 @@
│ │ │            
    Computes the number of lines on a general hypersurface of degree 2n - 3 in \mathbb P^n.
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,15 +11,15 @@
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the number of lines on a general hypersurface of degree
│ │ │ │              2n - 3 in \mathbb P^n
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Computes the number of lines on a general hypersurface of degree 2n - 3 in
│ │ │ │  \mathbb P^n.
│ │ │ │  i1 : time for n from 2 to 10 list linesHypersurface(n)
│ │ │ │ - -- used 0.0280835s (cpu); 0.0280868s (thread); 0s (gc)
│ │ │ │ + -- used 0.0289803s (cpu); 0.0289806s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       289139638632755625, 520764738758073845321}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  ********** WWaayyss ttoo uussee lliinneessHHyyppeerrssuurrffaaccee:: **********
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/html/_rational__Curve.html
│ │ │ @@ -152,15 +152,15 @@
│ │ │            
    The numbers of conics on general complete intersection Calabi-Yau threefolds can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
    
│ │ │                
    i1 : time for n from 2 to 10 list linesHypersurface(n)
│ │ │ - -- used 0.0280835s (cpu); 0.0280868s (thread); 0s (gc)
│ │ │ + -- used 0.0289803s (cpu); 0.0289806s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775,
│ │ │       ------------------------------------------------------------------------
│ │ │       289139638632755625, 520764738758073845321}
│ │ │  
│ │ │  o1 : List
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : time for D in T list rationalCurve(2,D) - rationalCurve(1,D)/8
│ │ │ - -- used 0.328797s (cpu); 0.276936s (thread); 0s (gc)
│ │ │ + -- used 0.345874s (cpu); 0.2915s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {609250, 92288, 52812, 22428, 9728}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │ @@ -168,27 +168,27 @@
│ │ │            
    For rational curves of degree 3:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ @@ -198,15 +198,15 @@
│ │ │            
    The number of rational curves of degree 3 on a general quintic threefold can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
    
│ │ │                
    i8 : time rationalCurve(3)
│ │ │ - -- used 0.228142s (cpu); 0.168847s (thread); 0s (gc)
│ │ │ + -- used 0.130073s (cpu); 0.13008s (thread); 0s (gc)
│ │ │  
│ │ │       8564575000
│ │ │  o8 = ----------
│ │ │           27
│ │ │  
│ │ │  o8 : QQ
    
│ │ │              
    
│ │ │                
    i9 : time for D in T list rationalCurve(3,D)
│ │ │ - -- used 5.32989s (cpu); 4.61697s (thread); 0s (gc)
│ │ │ + -- used 4.91979s (cpu); 4.39581s (thread); 0s (gc)
│ │ │  
│ │ │        8564575000  422690816           4834592  11239424
│ │ │  o9 = {----------, ---------, 6424365, -------, --------}
│ │ │            27          27                 3        27
│ │ │  
│ │ │  o9 : List
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : time rationalCurve(3) - rationalCurve(1)/27
│ │ │ - -- used 0.217214s (cpu); 0.167544s (thread); 0s (gc)
│ │ │ + -- used 0.1301s (cpu); 0.130107s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 317206375
│ │ │  
│ │ │  o10 : QQ
    
│ │ │              
    
│ │ │ @@ -214,15 +214,15 @@
│ │ │            
    The numbers of rational curves of degree 3 on general complete intersection Calabi-Yau threefolds can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i11 : time for D in T list rationalCurve(3,D) - rationalCurve(1,D)/27
│ │ │ - -- used 5.31342s (cpu); 4.64208s (thread); 0s (gc)
│ │ │ + -- used 4.96808s (cpu); 4.40223s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {317206375, 15655168, 6424326, 1611504, 416256}
│ │ │  
│ │ │  o11 : List
    
│ │ │              
    
│ │ │ @@ -230,27 +230,27 @@
│ │ │            
    For rational curves of degree 4:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i12 : time rationalCurve(4)
│ │ │ - -- used 1.64976s (cpu); 1.4421s (thread); 0s (gc)
│ │ │ + -- used 1.42387s (cpu); 1.30604s (thread); 0s (gc)
│ │ │  
│ │ │        15517926796875
│ │ │  o12 = --------------
│ │ │              64
│ │ │  
│ │ │  o12 : QQ
    
│ │ │              
    
│ │ │                
    i13 : time rationalCurve(4,{4,2})
│ │ │ - -- used 7.46787s (cpu); 5.79404s (thread); 0s (gc)
│ │ │ + -- used 6.76391s (cpu); 5.67641s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = 3883914084
│ │ │  
│ │ │  o13 : QQ
    
│ │ │              
    
│ │ │ @@ -258,15 +258,15 @@
│ │ │            
    The number of rational curves of degree 4 on a general quintic threefold can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i14 : time rationalCurve(4) - rationalCurve(2)/8
│ │ │ - -- used 1.65931s (cpu); 1.42972s (thread); 0s (gc)
│ │ │ + -- used 1.54252s (cpu); 1.36125s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 242467530000
│ │ │  
│ │ │  o14 : QQ
    
│ │ │              
    
│ │ │ @@ -274,25 +274,25 @@
│ │ │            
    The numbers of rational curves of degree 4 on general complete intersections of types (4,2) and (3,3) in \mathbb P^5 can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i15 : time rationalCurve(4,{4,2}) - rationalCurve(2,{4,2})/8
│ │ │ - -- used 7.68075s (cpu); 6.13623s (thread); 0s (gc)
│ │ │ + -- used 6.58956s (cpu); 5.47923s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3883902528
│ │ │  
│ │ │  o15 : QQ
    
│ │ │              
    
│ │ │                
    i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8
│ │ │ - -- used 7.9262s (cpu); 6.01692s (thread); 0s (gc)
│ │ │ + -- used 6.80324s (cpu); 5.67876s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 1139448384
│ │ │  
│ │ │  o16 : QQ
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -59,85 +59,85 @@
│ │ │ │  
│ │ │ │  o6 = 609250
│ │ │ │  
│ │ │ │  o6 : QQ
│ │ │ │  The numbers of conics on general complete intersection Calabi-Yau threefolds
│ │ │ │  can be computed as follows:
│ │ │ │  i7 : time for D in T list rationalCurve(2,D) - rationalCurve(1,D)/8
│ │ │ │ - -- used 0.328797s (cpu); 0.276936s (thread); 0s (gc)
│ │ │ │ + -- used 0.345874s (cpu); 0.2915s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = {609250, 92288, 52812, 22428, 9728}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  For rational curves of degree 3:
│ │ │ │  i8 : time rationalCurve(3)
│ │ │ │ - -- used 0.228142s (cpu); 0.168847s (thread); 0s (gc)
│ │ │ │ + -- used 0.130073s (cpu); 0.13008s (thread); 0s (gc)
│ │ │ │  
│ │ │ │       8564575000
│ │ │ │  o8 = ----------
│ │ │ │           27
│ │ │ │  
│ │ │ │  o8 : QQ
│ │ │ │  i9 : time for D in T list rationalCurve(3,D)
│ │ │ │ - -- used 5.32989s (cpu); 4.61697s (thread); 0s (gc)
│ │ │ │ + -- used 4.91979s (cpu); 4.39581s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        8564575000  422690816           4834592  11239424
│ │ │ │  o9 = {----------, ---------, 6424365, -------, --------}
│ │ │ │            27          27                 3        27
│ │ │ │  
│ │ │ │  o9 : List
│ │ │ │  The number of rational curves of degree 3 on a general quintic threefold can be
│ │ │ │  computed as follows:
│ │ │ │  i10 : time rationalCurve(3) - rationalCurve(1)/27
│ │ │ │ - -- used 0.217214s (cpu); 0.167544s (thread); 0s (gc)
│ │ │ │ + -- used 0.1301s (cpu); 0.130107s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = 317206375
│ │ │ │  
│ │ │ │  o10 : QQ
│ │ │ │  The numbers of rational curves of degree 3 on general complete intersection
│ │ │ │  Calabi-Yau threefolds can be computed as follows:
│ │ │ │  i11 : time for D in T list rationalCurve(3,D) - rationalCurve(1,D)/27
│ │ │ │ - -- used 5.31342s (cpu); 4.64208s (thread); 0s (gc)
│ │ │ │ + -- used 4.96808s (cpu); 4.40223s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = {317206375, 15655168, 6424326, 1611504, 416256}
│ │ │ │  
│ │ │ │  o11 : List
│ │ │ │  For rational curves of degree 4:
│ │ │ │  i12 : time rationalCurve(4)
│ │ │ │ - -- used 1.64976s (cpu); 1.4421s (thread); 0s (gc)
│ │ │ │ + -- used 1.42387s (cpu); 1.30604s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        15517926796875
│ │ │ │  o12 = --------------
│ │ │ │              64
│ │ │ │  
│ │ │ │  o12 : QQ
│ │ │ │  i13 : time rationalCurve(4,{4,2})
│ │ │ │ - -- used 7.46787s (cpu); 5.79404s (thread); 0s (gc)
│ │ │ │ + -- used 6.76391s (cpu); 5.67641s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 = 3883914084
│ │ │ │  
│ │ │ │  o13 : QQ
│ │ │ │  The number of rational curves of degree 4 on a general quintic threefold can be
│ │ │ │  computed as follows:
│ │ │ │  i14 : time rationalCurve(4) - rationalCurve(2)/8
│ │ │ │ - -- used 1.65931s (cpu); 1.42972s (thread); 0s (gc)
│ │ │ │ + -- used 1.54252s (cpu); 1.36125s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 = 242467530000
│ │ │ │  
│ │ │ │  o14 : QQ
│ │ │ │  The numbers of rational curves of degree 4 on general complete intersections of
│ │ │ │  types (4,2) and (3,3) in \mathbb P^5 can be computed as follows:
│ │ │ │  i15 : time rationalCurve(4,{4,2}) - rationalCurve(2,{4,2})/8
│ │ │ │ - -- used 7.68075s (cpu); 6.13623s (thread); 0s (gc)
│ │ │ │ + -- used 6.58956s (cpu); 5.47923s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 3883902528
│ │ │ │  
│ │ │ │  o15 : QQ
│ │ │ │  i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8
│ │ │ │ - -- used 7.9262s (cpu); 6.01692s (thread); 0s (gc)
│ │ │ │ + -- used 6.80324s (cpu); 5.67876s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = 1139448384
│ │ │ │  
│ │ │ │  o16 : QQ
│ │ │ │  ********** WWaayyss ttoo uussee rraattiioonnaallCCuurrvvee:: **********
│ │ │ │      * rationalCurve(ZZ)
│ │ │ │      * rationalCurve(ZZ,List)
│ │ ├── ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_egb__Toric.out
│ │ │ @@ -10,34 +10,34 @@
│ │ │  o3 = map (R, S, {x , x x , x x , x })
│ │ │                    1   1 0   1 0   0
│ │ │  
│ │ │  o3 : RingMap R <-- S
│ │ │  
│ │ │  i4 : G = egbToric(m, OutFile=>stdio)
│ │ │  3
│ │ │ -     -- used .00198437 seconds
│ │ │ -     -- used .000540734 seconds
│ │ │ +     -- used .0020064 seconds
│ │ │ +     -- used .000584388 seconds
│ │ │  (9, 9)
│ │ │  new stuff found
│ │ │  4
│ │ │ -     -- used .00328788 seconds
│ │ │ -     -- used .00432848 seconds
│ │ │ +     -- used .00407531 seconds
│ │ │ +     -- used .00488458 seconds
│ │ │  (16, 26)
│ │ │  new stuff found
│ │ │  5
│ │ │ -     -- used .00788476 seconds
│ │ │ -     -- used .0260362 seconds
│ │ │ +     -- used .0084479 seconds
│ │ │ +     -- used .026256 seconds
│ │ │  (25, 60)
│ │ │  6
│ │ │ -     -- used .0175752 seconds
│ │ │ -     -- used .212106 seconds
│ │ │ +     -- used .0187941 seconds
│ │ │ +     -- used .201187 seconds
│ │ │  (36, 120)
│ │ │  7
│ │ │ -     -- used .0370974 seconds
│ │ │ -     -- used .791376 seconds
│ │ │ +     -- used .0407566 seconds
│ │ │ +     -- used .809507 seconds
│ │ │  (49, 217)
│ │ │  
│ │ │                                     2
│ │ │  o4 = {- y    + y   , - y   y    + y   , - y   y    + y   y   , - y   y    +
│ │ │           1,0    0,1     1,1 0,0    1,0     2,1 0,0    2,0 1,0     2,1 1,0  
│ │ │       ------------------------------------------------------------------------
│ │ │       y   y   , - y   y    + y   y   , - y   y    + y   y   , - y   y    +
│ │ ├── ./usr/share/doc/Macaulay2/EquivariantGB/html/_egb__Toric.html
│ │ │ @@ -101,34 +101,34 @@
│ │ │  o3 : RingMap R <-- S
    
│ │ │              
    
│ │ │                
    i4 : G = egbToric(m, OutFile=>stdio)
│ │ │  3
│ │ │ -     -- used .00198437 seconds
│ │ │ -     -- used .000540734 seconds
│ │ │ +     -- used .0020064 seconds
│ │ │ +     -- used .000584388 seconds
│ │ │  (9, 9)
│ │ │  new stuff found
│ │ │  4
│ │ │ -     -- used .00328788 seconds
│ │ │ -     -- used .00432848 seconds
│ │ │ +     -- used .00407531 seconds
│ │ │ +     -- used .00488458 seconds
│ │ │  (16, 26)
│ │ │  new stuff found
│ │ │  5
│ │ │ -     -- used .00788476 seconds
│ │ │ -     -- used .0260362 seconds
│ │ │ +     -- used .0084479 seconds
│ │ │ +     -- used .026256 seconds
│ │ │  (25, 60)
│ │ │  6
│ │ │ -     -- used .0175752 seconds
│ │ │ -     -- used .212106 seconds
│ │ │ +     -- used .0187941 seconds
│ │ │ +     -- used .201187 seconds
│ │ │  (36, 120)
│ │ │  7
│ │ │ -     -- used .0370974 seconds
│ │ │ -     -- used .791376 seconds
│ │ │ +     -- used .0407566 seconds
│ │ │ +     -- used .809507 seconds
│ │ │  (49, 217)
│ │ │  
│ │ │                                     2
│ │ │  o4 = {- y    + y   , - y   y    + y   , - y   y    + y   y   , - y   y    +
│ │ │           1,0    0,1     1,1 0,0    1,0     2,1 0,0    2,0 1,0     2,1 1,0  
│ │ │       ------------------------------------------------------------------------
│ │ │       y   y   , - y   y    + y   y   , - y   y    + y   y   , - y   y    +
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,34 +33,34 @@
│ │ │ │                    2               2
│ │ │ │  o3 = map (R, S, {x , x x , x x , x })
│ │ │ │                    1   1 0   1 0   0
│ │ │ │  
│ │ │ │  o3 : RingMap R <-- S
│ │ │ │  i4 : G = egbToric(m, OutFile=>stdio)
│ │ │ │  3
│ │ │ │ -     -- used .00198437 seconds
│ │ │ │ -     -- used .000540734 seconds
│ │ │ │ +     -- used .0020064 seconds
│ │ │ │ +     -- used .000584388 seconds
│ │ │ │  (9, 9)
│ │ │ │  new stuff found
│ │ │ │  4
│ │ │ │ -     -- used .00328788 seconds
│ │ │ │ -     -- used .00432848 seconds
│ │ │ │ +     -- used .00407531 seconds
│ │ │ │ +     -- used .00488458 seconds
│ │ │ │  (16, 26)
│ │ │ │  new stuff found
│ │ │ │  5
│ │ │ │ -     -- used .00788476 seconds
│ │ │ │ -     -- used .0260362 seconds
│ │ │ │ +     -- used .0084479 seconds
│ │ │ │ +     -- used .026256 seconds
│ │ │ │  (25, 60)
│ │ │ │  6
│ │ │ │ -     -- used .0175752 seconds
│ │ │ │ -     -- used .212106 seconds
│ │ │ │ +     -- used .0187941 seconds
│ │ │ │ +     -- used .201187 seconds
│ │ │ │  (36, 120)
│ │ │ │  7
│ │ │ │ -     -- used .0370974 seconds
│ │ │ │ -     -- used .791376 seconds
│ │ │ │ +     -- used .0407566 seconds
│ │ │ │ +     -- used .809507 seconds
│ │ │ │  (49, 217)
│ │ │ │  
│ │ │ │                                     2
│ │ │ │  o4 = {- y    + y   , - y   y    + y   , - y   y    + y   y   , - y   y    +
│ │ │ │           1,0    0,1     1,1 0,0    1,0     2,1 0,0    2,0 1,0     2,1 1,0
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       y   y   , - y   y    + y   y   , - y   y    + y   y   , - y   y    +
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/___Fast__Minors__Strategy__Tutorial.out
│ │ │ @@ -462,50 +462,50 @@
│ │ │                 3 2 4     3 6
│ │ │  o27 = ideal(12x x x  - 4x x )
│ │ │                 3 7 9     3 9
│ │ │  
│ │ │  o27 : Ideal of S
│ │ │  
│ │ │  i28 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Random))
│ │ │ - -- used 0.177969s (cpu); 0.122004s (thread); 0s (gc)
│ │ │ + -- used 0.219958s (cpu); 0.15362s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 2
│ │ │  
│ │ │  i29 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallest))
│ │ │ - -- used 0.324762s (cpu); 0.212652s (thread); 0s (gc)
│ │ │ + -- used 0.355877s (cpu); 0.224017s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = 3
│ │ │  
│ │ │  i30 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallestTerm))
│ │ │ - -- used 0.505075s (cpu); 0.32834s (thread); 0s (gc)
│ │ │ + -- used 0.563589s (cpu); 0.360913s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 1
│ │ │  
│ │ │  i31 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexLargest))
│ │ │ - -- used 0.226825s (cpu); 0.185587s (thread); 0s (gc)
│ │ │ + -- used 0.288154s (cpu); 0.225162s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 2
│ │ │  
│ │ │  i32 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallest))
│ │ │ - -- used 0.386378s (cpu); 0.21271s (thread); 0s (gc)
│ │ │ + -- used 0.44032s (cpu); 0.233399s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = 3
│ │ │  
│ │ │  i33 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallestTerm))
│ │ │ - -- used 0.356033s (cpu); 0.24901s (thread); 0s (gc)
│ │ │ + -- used 0.371217s (cpu); 0.239568s (thread); 0s (gc)
│ │ │  
│ │ │  o33 = 3
│ │ │  
│ │ │  i34 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexLargest))
│ │ │ - -- used 0.299885s (cpu); 0.188636s (thread); 0s (gc)
│ │ │ + -- used 0.345982s (cpu); 0.1988s (thread); 0s (gc)
│ │ │  
│ │ │  o34 = 3
│ │ │  
│ │ │  i35 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Points))
│ │ │ - -- used 15.4766s (cpu); 10.5375s (thread); 0s (gc)
│ │ │ + -- used 17.5223s (cpu); 11.4921s (thread); 0s (gc)
│ │ │  
│ │ │  o35 = 1
│ │ │  
│ │ │  i36 : peek StrategyDefault
│ │ │  
│ │ │  o36 = OptionTable{GRevLexLargest => 0      }
│ │ │                    GRevLexSmallest => 16
│ │ │ @@ -514,15 +514,15 @@
│ │ │                    LexSmallest => 16
│ │ │                    LexSmallestTerm => 16
│ │ │                    Points => 0
│ │ │                    Random => 16
│ │ │                    RandomNonzero => 16
│ │ │  
│ │ │  i37 : time chooseGoodMinors(20, 6, M, J, Strategy=>StrategyDefault, Verbose=>true);
│ │ │ - -- used 0.383136s (cpu); 0.324174s (thread); 0s (gc)
│ │ │ + -- used 0.450381s (cpu); 0.378324s (thread); 0s (gc)
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ @@ -582,15 +582,15 @@
│ │ │  i41 : ptsStratGeometric = new OptionTable from (options chooseGoodMinors)#PointOptions;
│ │ │  
│ │ │  i42 : ptsStratGeometric#ExtendField --look at the default value
│ │ │  
│ │ │  o42 = true
│ │ │  
│ │ │  i43 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratGeometric))
│ │ │ - -- used 0.487214s (cpu); 0.426333s (thread); 0s (gc)
│ │ │ + -- used 0.745732s (cpu); 0.600893s (thread); 0s (gc)
│ │ │  
│ │ │  o43 = 2
│ │ │  
│ │ │  i44 : ptsStratRational = ptsStratGeometric++{ExtendField=>false} --change that value
│ │ │  
│ │ │  o44 = OptionTable{DecompositionStrategy => Decompose}
│ │ │                    DimensionFunction => dim
│ │ │ @@ -605,47 +605,47 @@
│ │ │  o44 : OptionTable
│ │ │  
│ │ │  i45 : ptsStratRational.ExtendField --look at our changed value
│ │ │  
│ │ │  o45 = false
│ │ │  
│ │ │  i46 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratRational))
│ │ │ - -- used 0.506683s (cpu); 0.379698s (thread); 0s (gc)
│ │ │ + -- used 0.4884s (cpu); 0.419871s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = 2
│ │ │  
│ │ │  i47 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefault)
│ │ │ - -- used 3.28944s (cpu); 3.0128s (thread); 0s (gc)
│ │ │ + -- used 3.98407s (cpu); 3.53959s (thread); 0s (gc)
│ │ │  
│ │ │  i48 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultNonRandom)
│ │ │ - -- used 0.905703s (cpu); 0.789473s (thread); 0s (gc)
│ │ │ + -- used 0.864416s (cpu); 0.729779s (thread); 0s (gc)
│ │ │  
│ │ │  o48 = true
│ │ │  
│ │ │  i49 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Random)
│ │ │ - -- used 3.58264s (cpu); 3.36454s (thread); 0s (gc)
│ │ │ + -- used 3.24659s (cpu); 3.02979s (thread); 0s (gc)
│ │ │  
│ │ │  i50 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallest)
│ │ │ - -- used 2.38279s (cpu); 1.9883s (thread); 0s (gc)
│ │ │ + -- used 2.75203s (cpu); 2.24497s (thread); 0s (gc)
│ │ │  
│ │ │  i51 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallestTerm)
│ │ │ - -- used 0.831174s (cpu); 0.71578s (thread); 0s (gc)
│ │ │ + -- used 0.87716s (cpu); 0.815091s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = true
│ │ │  
│ │ │  i52 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallest)
│ │ │ - -- used 2.6018s (cpu); 2.15961s (thread); 0s (gc)
│ │ │ + -- used 3.06605s (cpu); 2.51926s (thread); 0s (gc)
│ │ │  
│ │ │  i53 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallestTerm)
│ │ │ - -- used 3.03003s (cpu); 2.62369s (thread); 0s (gc)
│ │ │ + -- used 3.56947s (cpu); 3.07165s (thread); 0s (gc)
│ │ │  
│ │ │  i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ - -- used 9.15531s (cpu); 7.58445s (thread); 0s (gc)
│ │ │ + -- used 10.7366s (cpu); 8.87559s (thread); 0s (gc)
│ │ │  
│ │ │  o54 = true
│ │ │  
│ │ │  i55 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultWithPoints)
│ │ │ - -- used 7.17456s (cpu); 5.85532s (thread); 0s (gc)
│ │ │ + -- used 7.98092s (cpu); 6.59218s (thread); 0s (gc)
│ │ │  
│ │ │  o55 = true
│ │ │  
│ │ │  i56 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/___Regular__In__Codimension__Tutorial.out
│ │ │ @@ -7,20 +7,20 @@
│ │ │  o2 : Ideal of S
│ │ │  
│ │ │  i3 : dim (S/J)
│ │ │  
│ │ │  o3 = 4
│ │ │  
│ │ │  i4 : time regularInCodimension(1, S/J)
│ │ │ - -- used 0.954389s (cpu); 0.635206s (thread); 0s (gc)
│ │ │ + -- used 1.12832s (cpu); 0.759775s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 : time regularInCodimension(2, S/J)
│ │ │ - -- used 11.1794s (cpu); 8.10432s (thread); 0s (gc)
│ │ │ + -- used 11.8543s (cpu); 8.52938s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : time regularInCodimension(1, S/J, Verbose=>true)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 452.908 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ @@ -87,21 +87,21 @@
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 49, and computed = 39
│ │ │  regularInCodimension:  singularLocus dimension verified by isCodimAtLeast
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 2
│ │ │ -regularInCodimension:  Loop completed, submatrices considered = 49, and compute -- used 1.43458s (cpu); 1.01606s (thread); 0s (gc)
│ │ │ +regularInCodimension:  Loop completed, submatrices considered = 49, and compute -- used 1.60919s (cpu); 1.15948s (thread); 0s (gc)
│ │ │  d = 39.  singular locus dimension appears to be = 2
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : time regularInCodimension(1, S/J, MaxMinors=>10, Verbose=>true)
│ │ │ - -- used 0.175899s (cpu); 0.12452s (thread); 0s (gc)
│ │ │ + -- used 0.230667s (cpu); 0.176996s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │ @@ -115,15 +115,15 @@
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 10, and computed = 10
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop completed, submatrices considered = 10, and computed = 10.  singular locus dimension appears to be = 3
│ │ │  
│ │ │  i8 : time regularInCodimension(1, S/J, MaxMinors=>10, Strategy=>StrategyRandom, Verbose=>true)
│ │ │ - -- used 0.15707s (cpu); 0.106723s (thread); 0s (gc)
│ │ │ + -- used 0.171312s (cpu); 0.117678s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │ @@ -137,15 +137,15 @@
│ │ │  internalChooseMinor: Choosing Random
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 10, and computed = 10
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop completed, submatrices considered = 10, and computed = 10.  singular locus dimension appears to be = 3
│ │ │  
│ │ │  i9 : time regularInCodimension(1, S/J, MaxMinors=>10, MinMinorsFunction => t->3, Verbose=>true)
│ │ │ - -- used 0.627779s (cpu); 0.442058s (thread); 0s (gc)
│ │ │ + -- used 0.661642s (cpu); 0.487505s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 3, and computed = 3
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -165,15 +165,15 @@
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 10, and computed = 10
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop completed, submatrices considered = 10, and computed = 10.  singular locus dimension appears to be = 3
│ │ │  
│ │ │  i10 : time regularInCodimension(1, S/J, MaxMinors=>25, CodimCheckFunction => t->t/5, MinMinorsFunction => t->2, Verbose=>true)
│ │ │ - -- used 0.739844s (cpu); 0.504085s (thread); 0s (gc)
│ │ │ + -- used 0.754931s (cpu); 0.540222s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 25 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 2, and computed = 2
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 4
│ │ │ @@ -214,15 +214,15 @@
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 25, and computed = 23
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop completed, submatrices considered = 25, and computed = 23.  singular locus dimension appears to be = 3
│ │ │  
│ │ │  i11 : time regularInCodimension(1, S/J, MaxMinors=>25, UseOnlyFastCodim => true, Verbose=>true)
│ │ │ - -- used 0.482077s (cpu); 0.314107s (thread); 0s (gc)
│ │ │ + -- used 0.493336s (cpu); 0.327536s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 25 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/___Strategy__Default.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 5509279875405941999
│ │ │  
│ │ │  i1 : T=ZZ/7[a..i]/ideal(f*h-e*i,c*h-b*i,f*g-d*i,e*g-d*h,c*g-a*i,b*g-a*h,c*e-b*f,c*d-a*f,b*d-a*e,g^3-h^2*i-g*i^2,d*g^2-e*h*i-d*i^2,a*g^2-b*h*i-a*i^2,d^2*g-e^2*i-d*f*i,a*d*g-b*e*i-a*f*i,a^2*g-b^2*i-a*c*i,d^3-e^2*f-d*f^2,a*d^2-b*e*f-a*f^2,a^2*d-b^2*f-a*c*f,c^3+f^3-i^3,b*c^2+e*f^2-h*i^2,a*c^2+d*f^2-g*i^2,b^2*c+e^2*f-h^2*i,a*b*c+d*e*f-g*h*i,a^2*c+d^2*f-g^2*i,b^3+e^3-h^3,a*b^2+d*e^2-g*h^2,a^2*b+d^2*e-g^2*h,a^3+e^2*f+d*f^2-h^2*i-g*i^2);
│ │ │  
│ │ │  i2 : elapsedTime regularInCodimension(1, T, Strategy=>StrategyDefault)
│ │ │ - -- 1.73127s elapsed
│ │ │ + -- 1.44643s elapsed
│ │ │  
│ │ │  o2 = true
│ │ │  
│ │ │  i3 : peek StrategyDefault
│ │ │  
│ │ │  o3 = OptionTable{GRevLexLargest => 0      }
│ │ │                   GRevLexSmallest => 16
│ │ │ @@ -16,12 +16,12 @@
│ │ │                   LexSmallest => 16
│ │ │                   LexSmallestTerm => 16
│ │ │                   Points => 0
│ │ │                   Random => 16
│ │ │                   RandomNonzero => 16
│ │ │  
│ │ │  i4 : elapsedTime regularInCodimension(1, T, Strategy=>LexSmallestTerm)
│ │ │ - -- 1.19064s elapsed
│ │ │ + -- .860053s elapsed
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/_is__Codim__At__Least.out
│ │ │ @@ -16,29 +16,29 @@
│ │ │  i5 : r = rank myDiff;
│ │ │  
│ │ │  i6 : J = chooseGoodMinors(15, r, myDiff, Strategy=>StrategyDefaultNonRandom);
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : time isCodimAtLeast(3, J)
│ │ │ - -- used 0.00394593s (cpu); 0.00280101s (thread); 0s (gc)
│ │ │ + -- used 0.00405033s (cpu); 0.00532349s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = true
│ │ │  
│ │ │  i8 : I = ideal(x_2^8*x_10^3-3*x_1*x_2^7*x_10^2*x_11+3*x_1^2*x_2^6*x_10*x_11^2-x_1^3*x_2^5*x_11^3,x_5^5*x_6^3*x_11^3-3*x_5^6*x_6^2*x_11^2*x_12+3*x_5^7*x_6*x_11*x_12^2-x_5^8*x_12^3,x_1^5*x_2^3*x_4^3-3*x_1^6*x_2^2*x_4^2*x_5+3*x_1^7*x_2*x_4*x_5^2-x_1^8*x_5^3,x_6^8*x_11^3-3*x_5*x_6^7*x_11^2*x_12+3*x_5^2*x_6^6*x_11*x_12^2-x_5^3*x_6^5*x_12^3,x_8^3*x_10^8-3*x_7*x_8^2*x_10^7*x_11+3*x_7^2*x_8*x_10^6*x_11^2-x_7^3*x_10^5*x_11^3,x_2^8*x_4^3-3*x_1*x_2^7*x_4^2*x_5+3*x_1^2*x_2^6*x_4*x_5^2-x_1^3*x_2^5*x_5^3,-x_6^3*x_11^8+3*x_5*x_6^2*x_11^7*x_12-3*x_5^2*x_6*x_11^6*x_12^2+x_5^3*x_11^5*x_12^3,-x_6^3*x_7^3*x_9^5+3*x_4*x_6^2*x_7^2*x_9^6-3*x_4^2*x_6*x_7*x_9^7+x_4^3*x_9^8,x_8^8*x_10^3-3*x_7*x_8^7*x_10^2*x_11+3*x_7^2*x_8^6*x_10*x_11^2-x_7^3*x_8^5*x_11^3,x_2^5*x_3^3*x_11^3-3*x_2^6*x_3^2*x_11^2*x_12+3*x_2^7*x_3*x_11*x_12^2-x_2^8*x_12^3);
│ │ │  
│ │ │                 ZZ
│ │ │  o8 : Ideal of ---[x  , x , x , x , x  , x , x , x  , x , x , x , x ]
│ │ │                127  11   8   1   9   12   6   5   10   2   4   3   7
│ │ │  
│ │ │  i9 : time isCodimAtLeast(5, I, PairLimit => 5, Verbose=>true)
│ │ │ - -- used 0.00231368s (cpu); 0.00253938s (thread); 0s (gc)
│ │ │ + -- used 0.00396225s (cpu); 0.00510562s (thread); 0s (gc)
│ │ │  isCodimAtLeast: Computing codim of monomials based on ideal generators.
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : time isCodimAtLeast(5, I, PairLimit => 200, Verbose=>false)
│ │ │ - -- used 6.0604e-05s (cpu); 0.00243851s (thread); 0s (gc)
│ │ │ + -- used 0.00489081s (cpu); 0.00483486s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/_proj__Dim.out
│ │ │ @@ -7,17 +7,17 @@
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : pdim(module I)
│ │ │  
│ │ │  o3 = 2
│ │ │  
│ │ │  i4 : time projDim(module I, Strategy=>StrategyRandom)
│ │ │ - -- used 0.267277s (cpu); 0.158683s (thread); 0s (gc)
│ │ │ + -- used 0.27821s (cpu); 0.167497s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
│ │ │  
│ │ │  i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ - -- used 0.0107026s (cpu); 0.0126789s (thread); 0s (gc)
│ │ │ + -- used 0.0125711s (cpu); 0.0150857s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 1
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/_recursive__Minors.out
│ │ │ @@ -4,20 +4,20 @@
│ │ │  
│ │ │  i2 : M = random(R^{5,5,5,5,5,5}, R^7);
│ │ │  
│ │ │               6      7
│ │ │  o2 : Matrix R  <-- R
│ │ │  
│ │ │  i3 : time I2 = recursiveMinors(4, M, Threads=>0);
│ │ │ - -- used 0.514952s (cpu); 0.461544s (thread); 0s (gc)
│ │ │ + -- used 0.568918s (cpu); 0.507531s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : time I1 = minors(4, M, Strategy=>Cofactor);
│ │ │ - -- used 1.47881s (cpu); 1.27687s (thread); 0s (gc)
│ │ │ + -- used 1.37125s (cpu); 1.25481s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of R
│ │ │  
│ │ │  i5 : I1 == I2
│ │ │  
│ │ │  o5 = true
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/_regular__In__Codimension.out
│ │ │ @@ -17,44 +17,44 @@
│ │ │  i6 : S = T/I;
│ │ │  
│ │ │  i7 : dim S
│ │ │  
│ │ │  o7 = 3
│ │ │  
│ │ │  i8 : time regularInCodimension(1, S)
│ │ │ - -- used 0.654392s (cpu); 0.499543s (thread); 0s (gc)
│ │ │ + -- used 0.734378s (cpu); 0.588417s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : time regularInCodimension(2, S)
│ │ │ - -- used 7.00197s (cpu); 5.27802s (thread); 0s (gc)
│ │ │ + -- used 7.07038s (cpu); 5.35834s (thread); 0s (gc)
│ │ │  
│ │ │  i10 : R = QQ[c, f, g, h]/ideal(g^3+h^3+1,f*g^3+f*h^3+f,c*g^3+c*h^3+c,f^2*g^3+f^2*h^3+f^2,c*f*g^3+c*f*h^3+c*f,c^2*g^3+c^2*h^3+c^2,f^3*g^3+f^3*h^3+f^3,c*f^2*g^3+c*f^2*h^3+c*f^2,c^2*f*g^3+c^2*f*h^3+c^2*f,c^3-f^2-c,c^3*h-f^2*h-c*h,c^3*g-f^2*g-c*g,c^3*h^2-f^2*h^2-c*h^2,c^3*g*h-f^2*g*h-c*g*h,c^3*g^2-f^2*g^2-c*g^2,c^3*h^3-f^2*h^3-c*h^3,c^3*g*h^2-f^2*g*h^2-c*g*h^2,c^3*g^2*h-f^2*g^2*h-c*g^2*h,c^3*g^3+f^2*h^3+c*h^3+f^2+c);
│ │ │  
│ │ │  i11 : dim(R)
│ │ │  
│ │ │  o11 = 2
│ │ │  
│ │ │  i12 : time (dim singularLocus (R))
│ │ │ - -- used 0.0199988s (cpu); 0.0198096s (thread); 0s (gc)
│ │ │ + -- used 0.0239444s (cpu); 0.0211436s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -1
│ │ │  
│ │ │  i13 : time regularInCodimension(2, R)
│ │ │ - -- used 0.182885s (cpu); 0.135147s (thread); 0s (gc)
│ │ │ + -- used 0.198059s (cpu); 0.150608s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = true
│ │ │  
│ │ │  i14 : time regularInCodimension(2, R)
│ │ │ - -- used 0.91949s (cpu); 0.572332s (thread); 0s (gc)
│ │ │ + -- used 1.05492s (cpu); 0.672474s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = true
│ │ │  
│ │ │  i15 : time regularInCodimension(2, R)
│ │ │ - -- used 1.24103s (cpu); 0.866303s (thread); 0s (gc)
│ │ │ + -- used 1.46998s (cpu); 1.00868s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = true
│ │ │  
│ │ │  i16 : time regularInCodimension(2, S, Verbose=>true)
│ │ │  regularInCodimension: ring dimension =3, there are 17325 possible 4 by 4 minors, we will compute up to 327.599 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ @@ -386,15 +386,15 @@
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ -internalChooseMinor: Ch -- used 6.56865s (cpu); 4.94928s (thread); 0s (gc)
│ │ │ +internalChooseMinor: Ch -- used 7.64161s (cpu); 5.64553s (thread); 0s (gc)
│ │ │  oosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ @@ -430,15 +430,15 @@
│ │ │  internalChooseMinor: Choosing Random
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 328, and computed = 180
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 1
│ │ │  regularInCodimension:  Loop completed, submatrices considered = 328, and computed = 180.  singular locus dimension appears to be = 1
│ │ │  
│ │ │  i17 : time regularInCodimension(2, S, Verbose=>true, MaxMinors=>30)
│ │ │ - -- used 1.27203s (cpu); 0.95852s (thread); 0s (gc)
│ │ │ + -- used 1.61781s (cpu); 1.2258s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =3, there are 17325 possible 4 by 4 minors, we will compute up to 30 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ @@ -490,59 +490,59 @@
│ │ │  i18 : StrategyCurrent#Random = 0;
│ │ │  
│ │ │  i19 : StrategyCurrent#LexSmallest = 100;
│ │ │  
│ │ │  i20 : StrategyCurrent#LexSmallestTerm = 0;
│ │ │  
│ │ │  i21 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.30306s (cpu); 0.217176s (thread); 0s (gc)
│ │ │ + -- used 0.377725s (cpu); 0.244506s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = true
│ │ │  
│ │ │  i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.113658s (cpu); 0.0792881s (thread); 0s (gc)
│ │ │ + -- used 0.146987s (cpu); 0.083492s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = true
│ │ │  
│ │ │  i23 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.366044s (cpu); 0.272723s (thread); 0s (gc)
│ │ │ + -- used 0.452641s (cpu); 0.318563s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = true
│ │ │  
│ │ │  i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 1.73381s (cpu); 1.2512s (thread); 0s (gc)
│ │ │ + -- used 1.94543s (cpu); 1.40704s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
│ │ │  
│ │ │  i25 : StrategyCurrent#LexSmallest = 0;
│ │ │  
│ │ │  i26 : StrategyCurrent#LexSmallestTerm = 100;
│ │ │  
│ │ │  i27 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 2.32962s (cpu); 1.64458s (thread); 0s (gc)
│ │ │ + -- used 2.58283s (cpu); 1.8019s (thread); 0s (gc)
│ │ │  
│ │ │  i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 2.38741s (cpu); 1.61396s (thread); 0s (gc)
│ │ │ + -- used 2.72717s (cpu); 1.8169s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = true
│ │ │  
│ │ │  i29 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.462956s (cpu); 0.370163s (thread); 0s (gc)
│ │ │ + -- used 0.477227s (cpu); 0.356693s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = true
│ │ │  
│ │ │  i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.770202s (cpu); 0.604272s (thread); 0s (gc)
│ │ │ + -- used 0.988081s (cpu); 0.792663s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
│ │ │  
│ │ │  i31 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.07872s (cpu); 0.871235s (thread); 0s (gc)
│ │ │ + -- used 1.22464s (cpu); 0.967794s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = true
│ │ │  
│ │ │  i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.85936s (cpu); 1.48073s (thread); 0s (gc)
│ │ │ + -- used 1.98063s (cpu); 1.49924s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = true
│ │ │  
│ │ │  i33 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/___Fast__Minors__Strategy__Tutorial.html
│ │ │ @@ -620,71 +620,71 @@
│ │ │          
    
│ │ │            
    Here the $1$ passed to the function says how many minors to compute.  For instance, let's compute 8 minors for each of these strategies and see if that was enough to verify that the ring is regular in codimension 1.  In other words, if the dimension of $J$ plus the ideal of partial minors is $\leq 1$ (since $S/J$ has dimension 3).
    
│ │ │          
    
│ │ │          
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│ │ │              
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│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i28 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Random))
│ │ │ - -- used 0.177969s (cpu); 0.122004s (thread); 0s (gc)
│ │ │ + -- used 0.219958s (cpu); 0.15362s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 2
    
│ │ │              
    
│ │ │                
    i29 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallest))
│ │ │ - -- used 0.324762s (cpu); 0.212652s (thread); 0s (gc)
│ │ │ + -- used 0.355877s (cpu); 0.224017s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = 3
    
│ │ │              
    
│ │ │                
    i30 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallestTerm))
│ │ │ - -- used 0.505075s (cpu); 0.32834s (thread); 0s (gc)
│ │ │ + -- used 0.563589s (cpu); 0.360913s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 1
    
│ │ │              
    
│ │ │                
    i31 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexLargest))
│ │ │ - -- used 0.226825s (cpu); 0.185587s (thread); 0s (gc)
│ │ │ + -- used 0.288154s (cpu); 0.225162s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 2
    
│ │ │              
    
│ │ │                
    i32 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallest))
│ │ │ - -- used 0.386378s (cpu); 0.21271s (thread); 0s (gc)
│ │ │ + -- used 0.44032s (cpu); 0.233399s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = 3
    
│ │ │              
    
│ │ │                
    i33 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallestTerm))
│ │ │ - -- used 0.356033s (cpu); 0.24901s (thread); 0s (gc)
│ │ │ + -- used 0.371217s (cpu); 0.239568s (thread); 0s (gc)
│ │ │  
│ │ │  o33 = 3
    
│ │ │              
    
│ │ │                
    i34 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexLargest))
│ │ │ - -- used 0.299885s (cpu); 0.188636s (thread); 0s (gc)
│ │ │ + -- used 0.345982s (cpu); 0.1988s (thread); 0s (gc)
│ │ │  
│ │ │  o34 = 3
    
│ │ │              
    
│ │ │                
    i35 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Points))
│ │ │ - -- used 15.4766s (cpu); 10.5375s (thread); 0s (gc)
│ │ │ + -- used 17.5223s (cpu); 11.4921s (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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│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i37 : time chooseGoodMinors(20, 6, M, J, Strategy=>StrategyDefault, Verbose=>true);
│ │ │ - -- used 0.383136s (cpu); 0.324174s (thread); 0s (gc)
│ │ │ + -- used 0.450381s (cpu); 0.378324s (thread); 0s (gc)
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ @@ -820,15 +820,15 @@
│ │ │  
│ │ │  o42 = true
    
│ │ │              
    
│ │ │                
    i43 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratGeometric))
│ │ │ - -- used 0.487214s (cpu); 0.426333s (thread); 0s (gc)
│ │ │ + -- used 0.745732s (cpu); 0.600893s (thread); 0s (gc)
│ │ │  
│ │ │  o43 = 2
    
│ │ │              
    
│ │ │                
    i44 : ptsStratRational = ptsStratGeometric++{ExtendField=>false} --change that value
│ │ │ @@ -852,15 +852,15 @@
│ │ │  
│ │ │  o45 = false
    
│ │ │              
    
│ │ │                
    i46 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratRational))
│ │ │ - -- used 0.506683s (cpu); 0.379698s (thread); 0s (gc)
│ │ │ + -- used 0.4884s (cpu); 0.419871s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = 2
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Other options may also be passed to the RandomPoints package via the PointOptions option.
    
│ │ │ @@ -868,69 +868,69 @@
│ │ │          
    
│ │ │            
    regularInCodimension:  It is reasonable to think that you should find a few minors (with one strategy or another), and see if perhaps the minors you have computed so far are enough to verify our ring is regular in codimension 1.  This is exactly what regularInCodimension does.  One can control at a fine level how frequently new minors are computed, and how frequently the dimension of what we have computed so far is checked, by the option codimCheckFunction.  For more on that, see RegularInCodimensionTutorial and regularInCodimension.  Let us finish running regularInCodimension on our example with several different strategies.
    
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│ │ │          
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│ │ │              
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│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i47 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefault)
│ │ │ - -- used 3.28944s (cpu); 3.0128s (thread); 0s (gc)
    
│ │ │ + -- used 3.98407s (cpu); 3.53959s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i48 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultNonRandom)
│ │ │ - -- used 0.905703s (cpu); 0.789473s (thread); 0s (gc)
│ │ │ + -- used 0.864416s (cpu); 0.729779s (thread); 0s (gc)
│ │ │  
│ │ │  o48 = true
    
│ │ │              
    
│ │ │                
    i49 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Random)
│ │ │ - -- used 3.58264s (cpu); 3.36454s (thread); 0s (gc)
    
│ │ │ + -- used 3.24659s (cpu); 3.02979s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i50 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallest)
│ │ │ - -- used 2.38279s (cpu); 1.9883s (thread); 0s (gc)
    
│ │ │ + -- used 2.75203s (cpu); 2.24497s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i51 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallestTerm)
│ │ │ - -- used 0.831174s (cpu); 0.71578s (thread); 0s (gc)
│ │ │ + -- used 0.87716s (cpu); 0.815091s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = true
    
│ │ │              
    
│ │ │                
    i52 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallest)
│ │ │ - -- used 2.6018s (cpu); 2.15961s (thread); 0s (gc)
    
│ │ │ + -- used 3.06605s (cpu); 2.51926s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i53 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallestTerm)
│ │ │ - -- used 3.03003s (cpu); 2.62369s (thread); 0s (gc)
    
│ │ │ + -- used 3.56947s (cpu); 3.07165s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ - -- used 9.15531s (cpu); 7.58445s (thread); 0s (gc)
│ │ │ + -- used 10.7366s (cpu); 8.87559s (thread); 0s (gc)
│ │ │  
│ │ │  o54 = true
    
│ │ │              
    
│ │ │                
    i55 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultWithPoints)
│ │ │ - -- used 7.17456s (cpu); 5.85532s (thread); 0s (gc)
│ │ │ + -- used 7.98092s (cpu); 6.59218s (thread); 0s (gc)
│ │ │  
│ │ │  o55 = true
    
│ │ │              
    
│ │ │          
    
│ │ │            
    If regularInCodimension outputs nothing, then it couldn't verify that the ring was regular in that codimension.  We set MaxMinors => 100 to keep it from running too long with an ineffective strategy.  Again, even though GRevLexSmallest and GRevLexSmallestTerm are not effective in this particular example, in others they perform better than other strategies.  Note similar considerations also apply to projDim.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -486,44 +486,44 @@
│ │ │ │  o27 : Ideal of S
│ │ │ │  Here the $1$ passed to the function says how many minors to compute. For
│ │ │ │  instance, let's compute 8 minors for each of these strategies and see if that
│ │ │ │  was enough to verify that the ring is regular in codimension 1. In other words,
│ │ │ │  if the dimension of $J$ plus the ideal of partial minors is $\leq 1$ (since $S/
│ │ │ │  J$ has dimension 3).
│ │ │ │  i28 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Random))
│ │ │ │ - -- used 0.177969s (cpu); 0.122004s (thread); 0s (gc)
│ │ │ │ + -- used 0.219958s (cpu); 0.15362s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o28 = 2
│ │ │ │  i29 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallest))
│ │ │ │ - -- used 0.324762s (cpu); 0.212652s (thread); 0s (gc)
│ │ │ │ + -- used 0.355877s (cpu); 0.224017s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o29 = 3
│ │ │ │  i30 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallestTerm))
│ │ │ │ - -- used 0.505075s (cpu); 0.32834s (thread); 0s (gc)
│ │ │ │ + -- used 0.563589s (cpu); 0.360913s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o30 = 1
│ │ │ │  i31 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexLargest))
│ │ │ │ - -- used 0.226825s (cpu); 0.185587s (thread); 0s (gc)
│ │ │ │ + -- used 0.288154s (cpu); 0.225162s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o31 = 2
│ │ │ │  i32 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallest))
│ │ │ │ - -- used 0.386378s (cpu); 0.21271s (thread); 0s (gc)
│ │ │ │ + -- used 0.44032s (cpu); 0.233399s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o32 = 3
│ │ │ │  i33 : time dim (J + chooseGoodMinors(8, 6, M, J,
│ │ │ │  Strategy=>GRevLexSmallestTerm))
│ │ │ │ - -- used 0.356033s (cpu); 0.24901s (thread); 0s (gc)
│ │ │ │ + -- used 0.371217s (cpu); 0.239568s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o33 = 3
│ │ │ │  i34 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexLargest))
│ │ │ │ - -- used 0.299885s (cpu); 0.188636s (thread); 0s (gc)
│ │ │ │ + -- used 0.345982s (cpu); 0.1988s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o34 = 3
│ │ │ │  i35 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Points))
│ │ │ │ - -- used 15.4766s (cpu); 10.5375s (thread); 0s (gc)
│ │ │ │ + -- used 17.5223s (cpu); 11.4921s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o35 = 1
│ │ │ │  Indeed, in this example, even computing determinants of 1,000 random
│ │ │ │  submatrices is not typically enough to verify that $V(J)$ is regular in
│ │ │ │  codimension 1. On the other hand, Points is almost always quite effective at
│ │ │ │  finding valuable submatrices, but can be quite slow. In this particular
│ │ │ │  example, we can see that LexSmallestTerm also performs very well (and does it
│ │ │ │ @@ -544,15 +544,15 @@
│ │ │ │  says that we should use GRevLexSmallest, GRevLexSmallestTerm, LexSmallest,
│ │ │ │  LexSmallestTerm, Random, RandomNonzero all with equal probability (note
│ │ │ │  RandomNonzero, which we have not yet discussed chooses random submatrices where
│ │ │ │  no row or column is zero, which is good for working in sparse matrices). For
│ │ │ │  instance, if we run:
│ │ │ │  i37 : time chooseGoodMinors(20, 6, M, J, Strategy=>StrategyDefault,
│ │ │ │  Verbose=>true);
│ │ │ │ - -- used 0.383136s (cpu); 0.324174s (thread); 0s (gc)
│ │ │ │ + -- used 0.450381s (cpu); 0.378324s (thread); 0s (gc)
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │ @@ -633,15 +633,15 @@
│ │ │ │  i41 : ptsStratGeometric = new OptionTable from (options
│ │ │ │  chooseGoodMinors)#PointOptions;
│ │ │ │  i42 : ptsStratGeometric#ExtendField --look at the default value
│ │ │ │  
│ │ │ │  o42 = true
│ │ │ │  i43 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points,
│ │ │ │  PointOptions=>ptsStratGeometric))
│ │ │ │ - -- used 0.487214s (cpu); 0.426333s (thread); 0s (gc)
│ │ │ │ + -- used 0.745732s (cpu); 0.600893s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o43 = 2
│ │ │ │  i44 : ptsStratRational = ptsStratGeometric++{ExtendField=>false} --change that
│ │ │ │  value
│ │ │ │  
│ │ │ │  o44 = OptionTable{DecompositionStrategy => Decompose}
│ │ │ │                    DimensionFunction => dim
│ │ │ │ @@ -655,58 +655,58 @@
│ │ │ │  
│ │ │ │  o44 : OptionTable
│ │ │ │  i45 : ptsStratRational.ExtendField --look at our changed value
│ │ │ │  
│ │ │ │  o45 = false
│ │ │ │  i46 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points,
│ │ │ │  PointOptions=>ptsStratRational))
│ │ │ │ - -- used 0.506683s (cpu); 0.379698s (thread); 0s (gc)
│ │ │ │ + -- used 0.4884s (cpu); 0.419871s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o46 = 2
│ │ │ │  Other options may also be passed to the _R_a_n_d_o_m_P_o_i_n_t_s package via the
│ │ │ │  _P_o_i_n_t_O_p_t_i_o_n_s option.
│ │ │ │  rreegguullaarrIInnCCooddiimmeennssiioonn:: It is reasonable to think that you should find a few
│ │ │ │  minors (with one strategy or another), and see if perhaps the minors you have
│ │ │ │  computed so far are enough to verify our ring is regular in codimension 1. This
│ │ │ │  is exactly what regularInCodimension does. One can control at a fine level how
│ │ │ │  frequently new minors are computed, and how frequently the dimension of what we
│ │ │ │  have computed so far is checked, by the option codimCheckFunction. For more on
│ │ │ │  that, see _R_e_g_u_l_a_r_I_n_C_o_d_i_m_e_n_s_i_o_n_T_u_t_o_r_i_a_l and _r_e_g_u_l_a_r_I_n_C_o_d_i_m_e_n_s_i_o_n. Let us finish
│ │ │ │  running regularInCodimension on our example with several different strategies.
│ │ │ │  i47 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>StrategyDefault)
│ │ │ │ - -- used 3.28944s (cpu); 3.0128s (thread); 0s (gc)
│ │ │ │ + -- used 3.98407s (cpu); 3.53959s (thread); 0s (gc)
│ │ │ │  i48 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>StrategyDefaultNonRandom)
│ │ │ │ - -- used 0.905703s (cpu); 0.789473s (thread); 0s (gc)
│ │ │ │ + -- used 0.864416s (cpu); 0.729779s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o48 = true
│ │ │ │  i49 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Random)
│ │ │ │ - -- used 3.58264s (cpu); 3.36454s (thread); 0s (gc)
│ │ │ │ + -- used 3.24659s (cpu); 3.02979s (thread); 0s (gc)
│ │ │ │  i50 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>LexSmallest)
│ │ │ │ - -- used 2.38279s (cpu); 1.9883s (thread); 0s (gc)
│ │ │ │ + -- used 2.75203s (cpu); 2.24497s (thread); 0s (gc)
│ │ │ │  i51 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>LexSmallestTerm)
│ │ │ │ - -- used 0.831174s (cpu); 0.71578s (thread); 0s (gc)
│ │ │ │ + -- used 0.87716s (cpu); 0.815091s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o51 = true
│ │ │ │  i52 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>GRevLexSmallest)
│ │ │ │ - -- used 2.6018s (cpu); 2.15961s (thread); 0s (gc)
│ │ │ │ + -- used 3.06605s (cpu); 2.51926s (thread); 0s (gc)
│ │ │ │  i53 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>GRevLexSmallestTerm)
│ │ │ │ - -- used 3.03003s (cpu); 2.62369s (thread); 0s (gc)
│ │ │ │ + -- used 3.56947s (cpu); 3.07165s (thread); 0s (gc)
│ │ │ │  i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ │ - -- used 9.15531s (cpu); 7.58445s (thread); 0s (gc)
│ │ │ │ + -- used 10.7366s (cpu); 8.87559s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o54 = true
│ │ │ │  i55 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>StrategyDefaultWithPoints)
│ │ │ │ - -- used 7.17456s (cpu); 5.85532s (thread); 0s (gc)
│ │ │ │ + -- used 7.98092s (cpu); 6.59218s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o55 = true
│ │ │ │  If regularInCodimension outputs nothing, then it couldn't verify that the ring
│ │ │ │  was regular in that codimension. We set MaxMinors => 100 to keep it from
│ │ │ │  running too long with an ineffective strategy. Again, even though
│ │ │ │  GRevLexSmallest and GRevLexSmallestTerm are not effective in this particular
│ │ │ │  example, in others they perform better than other strategies. Note similar
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/___Regular__In__Codimension__Tutorial.html
│ │ │ @@ -81,23 +81,23 @@
│ │ │          
    
│ │ │            
    It is the cone over $P^2 \times E$ where $E$ is an elliptic curve.  We have embedded it with a Segre embedding inside $P^8$.  In particular, this example is even regular in codimension 3.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : time regularInCodimension(1, S/J)
│ │ │ - -- used 0.954389s (cpu); 0.635206s (thread); 0s (gc)
│ │ │ + -- used 1.12832s (cpu); 0.759775s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
    
│ │ │              
    
│ │ │                
    i5 : time regularInCodimension(2, S/J)
│ │ │ - -- used 11.1794s (cpu); 8.10432s (thread); 0s (gc)
    
│ │ │ + -- used 11.8543s (cpu); 8.52938s (thread); 0s (gc)
│ │ │              
    
│ │ │          
    
│ │ │            
    We try to verify that $S/J$ is regular in codimension 1 or 2 by computing the ideal made up of a small number of minors of the Jacobian matrix. In this example, instead of computing all relevant 1465128 minors to compute the singular locus, and then trying to compute the dimension of the ideal they generate, we instead compute a few of them.  regularInCodimension returns true if it verified that the ring is regular in codim 1 or 2 (respectively) and null if not.  Because of the randomness that exists in terms of selecting minors, the execution time can actually vary quite a bit.   Let's take a look at what is occurring by using the Verbose option.  We go through the output and explain what each line is telling us.
    
│ │ │          
    
│ │ │          
│ │ │ @@ -172,29 +172,29 @@
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 49, and computed = 39
│ │ │  regularInCodimension:  singularLocus dimension verified by isCodimAtLeast
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 2
│ │ │ -regularInCodimension:  Loop completed, submatrices considered = 49, and compute -- used 1.43458s (cpu); 1.01606s (thread); 0s (gc)
│ │ │ +regularInCodimension:  Loop completed, submatrices considered = 49, and compute -- used 1.60919s (cpu); 1.15948s (thread); 0s (gc)
│ │ │  d = 39.  singular locus dimension appears to be = 2
│ │ │  
│ │ │  o6 = true
│ │ │              
│ │ │            
│ │ │          
    
│ │ │          
    
│ │ │            
    MaxMinors.  The first output says that we will compute up to 452.9 minors before giving up.  We can control that by setting the option MaxMinors.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : time regularInCodimension(1, S/J, MaxMinors=>10, Verbose=>true)
│ │ │ - -- used 0.175899s (cpu); 0.12452s (thread); 0s (gc)
│ │ │ + -- used 0.230667s (cpu); 0.176996s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │ @@ -219,15 +219,15 @@
│ │ │          
    
│ │ │            
    Selecting submatrices of the Jacobian.  We also see output like: ``Choosing LexSmallest'' or ``Choosing Random''.  This is saying how we are selecting a given submatrix.  For instance, we can run:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i8 : time regularInCodimension(1, S/J, MaxMinors=>10, Strategy=>StrategyRandom, Verbose=>true)
│ │ │ - -- used 0.15707s (cpu); 0.106723s (thread); 0s (gc)
│ │ │ + -- used 0.171312s (cpu); 0.117678s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │ @@ -252,15 +252,15 @@
│ │ │          
    
│ │ │            
    Computing minors vs considering the dimension of what has been computed.  Periodically we compute the codimension of the partial ideal of minors we have computed so far.  There are two options to control this.  First, we can tell the function when to first compute the dimension of the working partial ideal of minors.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i9 : time regularInCodimension(1, S/J, MaxMinors=>10, MinMinorsFunction => t->3, Verbose=>true)
│ │ │ - -- used 0.627779s (cpu); 0.442058s (thread); 0s (gc)
│ │ │ + -- used 0.661642s (cpu); 0.487505s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 3, and computed = 3
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -291,15 +291,15 @@
│ │ │          
    
│ │ │            
    CodimCheckFunction. The option CodimCheckFunction controls how frequently the dimension of the partial ideal of minors is computed.  For instance, setting CodimCheckFunction => t -> t/5 will say it should compute dimension after every 5 minors are examined.  In general, after the output of the CodimCheckFunction increases by an integer we compute the codimension again.  The default function has the space between computations grow exponentially.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : time regularInCodimension(1, S/J, MaxMinors=>25, CodimCheckFunction => t->t/5, MinMinorsFunction => t->2, Verbose=>true)
│ │ │ - -- used 0.739844s (cpu); 0.504085s (thread); 0s (gc)
│ │ │ + -- used 0.754931s (cpu); 0.540222s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 25 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 2, and computed = 2
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 4
│ │ │ @@ -348,15 +348,15 @@
│ │ │          
    
│ │ │            
    isCodimAtLeast and dim.  We see the lines about the ``isCodimAtLeast failed''.  This means that isCodimAtLeast was not enough on its own to verify that our ring is regular in codimension 1.  After this, ``partial singular locus dimension computed'' indicates we did a complete dimension computation of the partial ideal defining the singular locus.  How isCodimAtLeast is called can be controlled via the options SPairsFunction and PairLimit, which are simply passed to isCodimAtLeast.  You can force the function to only use isCodimAtLeast and not call dimension by setting UseOnlyFastCodim => true.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i11 : time regularInCodimension(1, S/J, MaxMinors=>25, UseOnlyFastCodim => true, Verbose=>true)
│ │ │ - -- used 0.482077s (cpu); 0.314107s (thread); 0s (gc)
│ │ │ + -- used 0.493336s (cpu); 0.327536s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 25 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,19 +24,19 @@
│ │ │ │  i3 : dim (S/J)
│ │ │ │  
│ │ │ │  o3 = 4
│ │ │ │  It is the cone over $P^2 \times E$ where $E$ is an elliptic curve. We have
│ │ │ │  embedded it with a Segre embedding inside $P^8$. In particular, this example is
│ │ │ │  even regular in codimension 3.
│ │ │ │  i4 : time regularInCodimension(1, S/J)
│ │ │ │ - -- used 0.954389s (cpu); 0.635206s (thread); 0s (gc)
│ │ │ │ + -- used 1.12832s (cpu); 0.759775s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  i5 : time regularInCodimension(2, S/J)
│ │ │ │ - -- used 11.1794s (cpu); 8.10432s (thread); 0s (gc)
│ │ │ │ + -- used 11.8543s (cpu); 8.52938s (thread); 0s (gc)
│ │ │ │  We try to verify that $S/J$ is regular in codimension 1 or 2 by computing the
│ │ │ │  ideal made up of a small number of minors of the Jacobian matrix. In this
│ │ │ │  example, instead of computing all relevant 1465128 minors to compute the
│ │ │ │  singular locus, and then trying to compute the dimension of the ideal they
│ │ │ │  generate, we instead compute a few of them. regularInCodimension returns true
│ │ │ │  if it verified that the ring is regular in codim 1 or 2 (respectively) and null
│ │ │ │  if not. Because of the randomness that exists in terms of selecting minors, the
│ │ │ │ @@ -121,22 +121,22 @@
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │  considered: 49, and computed = 39
│ │ │ │  regularInCodimension:  singularLocus dimension verified by isCodimAtLeast
│ │ │ │  regularInCodimension:  partial singular locus dimension computed, = 2
│ │ │ │  regularInCodimension:  Loop completed, submatrices considered = 49, and compute
│ │ │ │ --- used 1.43458s (cpu); 1.01606s (thread); 0s (gc)
│ │ │ │ +-- used 1.60919s (cpu); 1.15948s (thread); 0s (gc)
│ │ │ │  d = 39.  singular locus dimension appears to be = 2
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  MMaaxxMMiinnoorrss.. The first output says that we will compute up to 452.9 minors before
│ │ │ │  giving up. We can control that by setting the option MaxMinors.
│ │ │ │  i7 : time regularInCodimension(1, S/J, MaxMinors=>10, Verbose=>true)
│ │ │ │ - -- used 0.175899s (cpu); 0.12452s (thread); 0s (gc)
│ │ │ │ + -- used 0.230667s (cpu); 0.176996s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 10 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │ @@ -159,15 +159,15 @@
│ │ │ │  There are other finer ways to control the MaxMinors option, but they will not
│ │ │ │  be discussed in this tutorial. See _r_e_g_u_l_a_r_I_n_C_o_d_i_m_e_n_s_i_o_n.
│ │ │ │  SSeelleeccttiinngg ssuubbmmaattrriicceess ooff tthhee JJaaccoobbiiaann.. We also see output like: ``Choosing
│ │ │ │  LexSmallest'' or ``Choosing Random''. This is saying how we are selecting a
│ │ │ │  given submatrix. For instance, we can run:
│ │ │ │  i8 : time regularInCodimension(1, S/J, MaxMinors=>10, Strategy=>StrategyRandom,
│ │ │ │  Verbose=>true)
│ │ │ │ - -- used 0.15707s (cpu); 0.106723s (thread); 0s (gc)
│ │ │ │ + -- used 0.171312s (cpu); 0.117678s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 10 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │ @@ -197,15 +197,15 @@
│ │ │ │  CCoommppuuttiinngg mmiinnoorrss vvss ccoonnssiiddeerriinngg tthhee ddiimmeennssiioonn ooff wwhhaatt hhaass bbeeeenn ccoommppuutteedd..
│ │ │ │  Periodically we compute the codimension of the partial ideal of minors we have
│ │ │ │  computed so far. There are two options to control this. First, we can tell the
│ │ │ │  function when to first compute the dimension of the working partial ideal of
│ │ │ │  minors.
│ │ │ │  i9 : time regularInCodimension(1, S/J, MaxMinors=>10, MinMinorsFunction => t-
│ │ │ │  >3, Verbose=>true)
│ │ │ │ - -- used 0.627779s (cpu); 0.442058s (thread); 0s (gc)
│ │ │ │ + -- used 0.661642s (cpu); 0.487505s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 10 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │ @@ -243,15 +243,15 @@
│ │ │ │  dimension of the partial ideal of minors is computed. For instance, setting
│ │ │ │  CodimCheckFunction => t -> t/5 will say it should compute dimension after every
│ │ │ │  5 minors are examined. In general, after the output of the CodimCheckFunction
│ │ │ │  increases by an integer we compute the codimension again. The default function
│ │ │ │  has the space between computations grow exponentially.
│ │ │ │  i10 : time regularInCodimension(1, S/J, MaxMinors=>25, CodimCheckFunction => t-
│ │ │ │  >t/5, MinMinorsFunction => t->2, Verbose=>true)
│ │ │ │ - -- used 0.739844s (cpu); 0.504085s (thread); 0s (gc)
│ │ │ │ + -- used 0.754931s (cpu); 0.540222s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 25 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │  considered: 2, and computed = 2
│ │ │ │ @@ -308,15 +308,15 @@
│ │ │ │  dimension computed'' indicates we did a complete dimension computation of the
│ │ │ │  partial ideal defining the singular locus. How isCodimAtLeast is called can be
│ │ │ │  controlled via the options SPairsFunction and PairLimit, which are simply
│ │ │ │  passed to _i_s_C_o_d_i_m_A_t_L_e_a_s_t. You can force the function to only use isCodimAtLeast
│ │ │ │  and not call dimension by setting UseOnlyFastCodim => true.
│ │ │ │  i11 : time regularInCodimension(1, S/J, MaxMinors=>25, UseOnlyFastCodim =>
│ │ │ │  true, Verbose=>true)
│ │ │ │ - -- used 0.482077s (cpu); 0.314107s (thread); 0s (gc)
│ │ │ │ + -- used 0.493336s (cpu); 0.327536s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 25 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/___Strategy__Default.html
│ │ │ @@ -68,15 +68,15 @@
│ │ │              
    		
│ │ │                
    i1 : T=ZZ/7[a..i]/ideal(f*h-e*i,c*h-b*i,f*g-d*i,e*g-d*h,c*g-a*i,b*g-a*h,c*e-b*f,c*d-a*f,b*d-a*e,g^3-h^2*i-g*i^2,d*g^2-e*h*i-d*i^2,a*g^2-b*h*i-a*i^2,d^2*g-e^2*i-d*f*i,a*d*g-b*e*i-a*f*i,a^2*g-b^2*i-a*c*i,d^3-e^2*f-d*f^2,a*d^2-b*e*f-a*f^2,a^2*d-b^2*f-a*c*f,c^3+f^3-i^3,b*c^2+e*f^2-h*i^2,a*c^2+d*f^2-g*i^2,b^2*c+e^2*f-h^2*i,a*b*c+d*e*f-g*h*i,a^2*c+d^2*f-g^2*i,b^3+e^3-h^3,a*b^2+d*e^2-g*h^2,a^2*b+d^2*e-g^2*h,a^3+e^2*f+d*f^2-h^2*i-g*i^2);
    
│ │ │              
    
│ │ │                
    i2 : elapsedTime regularInCodimension(1, T, Strategy=>StrategyDefault)
│ │ │ - -- 1.73127s elapsed
│ │ │ + -- 1.44643s elapsed
│ │ │  
│ │ │  o2 = true
    
│ │ │              
    
│ │ │  In this particular example, on one machine, we list average time to completion of each of the above strategies after 100 runs.        
      
│ │ │            
    • StrategyDefault: 1.65 seconds
      • 
│ │ │ @@ -122,15 +122,15 @@
│ │ │            
    • StrategyPoints: choose all submatrices via Points.
      • 
│ │ │            
    • StrategyDefaultWithPoints: like StrategyDefault but replaces the Random and RandomNonZero submatrices as with matrices chosen as in Points.
      • 
│ │ │          
    
│ │ │  Additionally, a MutableHashTable named StrategyCurrent is also exported.  It begins as the default strategy, but the user can modify it.

    Using a single heuristic  Alternatively, if the user only wants to use say LexSmallestTerm they can set, Strategy to point to that symbol, instead of a creating a custom strategy HashTable.  For example:         
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : elapsedTime regularInCodimension(1, T, Strategy=>LexSmallestTerm)
│ │ │ - -- 1.19064s elapsed
│ │ │ + -- .860053s elapsed
│ │ │  
│ │ │  o4 = true
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,15 +41,15 @@
│ │ │ │  i1 : T=ZZ/7[a..i]/ideal(f*h-e*i,c*h-b*i,f*g-d*i,e*g-d*h,c*g-a*i,b*g-a*h,c*e-
│ │ │ │  b*f,c*d-a*f,b*d-a*e,g^3-h^2*i-g*i^2,d*g^2-e*h*i-d*i^2,a*g^2-b*h*i-a*i^2,d^2*g-
│ │ │ │  e^2*i-d*f*i,a*d*g-b*e*i-a*f*i,a^2*g-b^2*i-a*c*i,d^3-e^2*f-d*f^2,a*d^2-b*e*f-
│ │ │ │  a*f^2,a^2*d-b^2*f-a*c*f,c^3+f^3-i^3,b*c^2+e*f^2-h*i^2,a*c^2+d*f^2-
│ │ │ │  g*i^2,b^2*c+e^2*f-h^2*i,a*b*c+d*e*f-g*h*i,a^2*c+d^2*f-g^2*i,b^3+e^3-
│ │ │ │  h^3,a*b^2+d*e^2-g*h^2,a^2*b+d^2*e-g^2*h,a^3+e^2*f+d*f^2-h^2*i-g*i^2);
│ │ │ │  i2 : elapsedTime regularInCodimension(1, T, Strategy=>StrategyDefault)
│ │ │ │ - -- 1.73127s elapsed
│ │ │ │ + -- 1.44643s elapsed
│ │ │ │  
│ │ │ │  o2 = true
│ │ │ │  In this particular example, on one machine, we list average time to completion
│ │ │ │  of each of the above strategies after 100 runs.
│ │ │ │      * StrategyDefault: 1.65 seconds
│ │ │ │      * StrategyRandom: 8.32 seconds
│ │ │ │      * StrategyDefaultNonRandom: 0.99 seconds
│ │ │ │ @@ -135,15 +135,15 @@
│ │ │ │  Additionally, a MutableHashTable named StrategyCurrent is also exported. It
│ │ │ │  begins as the default strategy, but the user can modify it.
│ │ │ │  
│ │ │ │  UUssiinngg aa ssiinnggllee hheeuurriissttiicc Alternatively, if the user only wants to use say
│ │ │ │  LexSmallestTerm they can set, Strategy to point to that symbol, instead of a
│ │ │ │  creating a custom strategy HashTable. For example:
│ │ │ │  i4 : elapsedTime regularInCodimension(1, T, Strategy=>LexSmallestTerm)
│ │ │ │ - -- 1.19064s elapsed
│ │ │ │ + -- .860053s elapsed
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _S_t_r_a_t_e_g_y_D_e_f_a_u_l_t is an _o_p_t_i_o_n_ _t_a_b_l_e.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/FastMinors.m2:1993:0.
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/_is__Codim__At__Least.html
│ │ │ @@ -114,15 +114,15 @@
│ │ │  
│ │ │  o6 : Ideal of R
    
│ │ │              
    
│ │ │                
    i7 : time isCodimAtLeast(3, J)
│ │ │ - -- used 0.00394593s (cpu); 0.00280101s (thread); 0s (gc)
│ │ │ + -- used 0.00405033s (cpu); 0.00532349s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = true
    
│ │ │              
    
│ │ │          
    
│ │ │            
    The function works by computing gb(I, PairLimit=>f(i)) for successive values of i.  Here f(i) is a function that takes t, some approximation of the base degree value of the polynomial ring (for example, in a standard graded polynomial ring, this is probably expected to be \{1\}).  And i is a counting variable. You can provide your own function by calling isCodimAtLeast(n, I, SPairsFunction=>( (i) -> f(i) ), the default function is SPairsFunction=>i->ceiling(1.5^i)   Perhaps more commonly however, the user may want to instead tell the function to compute for larger values of i.  This is done via the option PairLimit.  This is the max value of i to be plugged into SPairsFunction before the function gives up.  In other words, PairLimit=>5 will tell the function to check codimension 5 times.
    
│ │ │ @@ -136,24 +136,24 @@
│ │ │  o8 : Ideal of ---[x  , x , x , x , x  , x , x , x  , x , x , x , x ]
│ │ │                127  11   8   1   9   12   6   5   10   2   4   3   7
    
│ │ │              
    
│ │ │                
    i9 : time isCodimAtLeast(5, I, PairLimit => 5, Verbose=>true)
│ │ │ - -- used 0.00231368s (cpu); 0.00253938s (thread); 0s (gc)
│ │ │ + -- used 0.00396225s (cpu); 0.00510562s (thread); 0s (gc)
│ │ │  isCodimAtLeast: Computing codim of monomials based on ideal generators.
│ │ │  
│ │ │  o9 = true
    
│ │ │              
    
│ │ │                
    i10 : time isCodimAtLeast(5, I, PairLimit => 200, Verbose=>false)
│ │ │ - -- used 6.0604e-05s (cpu); 0.00243851s (thread); 0s (gc)
│ │ │ + -- used 0.00489081s (cpu); 0.00483486s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Notice in the first case the function returned null, because the depth of search was not high enough.  It only computed codim 5 times.  The second returned true, but it did so as soon as the answer was found (and before we hit the PairLimit limit).
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -38,15 +38,15 @@
│ │ │ │               30      12
│ │ │ │  o4 : Matrix R   <-- R
│ │ │ │  i5 : r = rank myDiff;
│ │ │ │  i6 : J = chooseGoodMinors(15, r, myDiff, Strategy=>StrategyDefaultNonRandom);
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : time isCodimAtLeast(3, J)
│ │ │ │ - -- used 0.00394593s (cpu); 0.00280101s (thread); 0s (gc)
│ │ │ │ + -- used 0.00405033s (cpu); 0.00532349s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = true
│ │ │ │  The function works by computing gb(I, PairLimit=>f(i)) for successive values of
│ │ │ │  i. Here f(i) is a function that takes t, some approximation of the base degree
│ │ │ │  value of the polynomial ring (for example, in a standard graded polynomial
│ │ │ │  ring, this is probably expected to be \{1\}). And i is a counting variable. You
│ │ │ │  can provide your own function by calling isCodimAtLeast(n, I, SPairsFunction=>
│ │ │ │ @@ -72,20 +72,20 @@
│ │ │ │  x_7^3*x_8^5*x_11^3,x_2^5*x_3^3*x_11^3-
│ │ │ │  3*x_2^6*x_3^2*x_11^2*x_12+3*x_2^7*x_3*x_11*x_12^2-x_2^8*x_12^3);
│ │ │ │  
│ │ │ │                 ZZ
│ │ │ │  o8 : Ideal of ---[x  , x , x , x , x  , x , x , x  , x , x , x , x ]
│ │ │ │                127  11   8   1   9   12   6   5   10   2   4   3   7
│ │ │ │  i9 : time isCodimAtLeast(5, I, PairLimit => 5, Verbose=>true)
│ │ │ │ - -- used 0.00231368s (cpu); 0.00253938s (thread); 0s (gc)
│ │ │ │ + -- used 0.00396225s (cpu); 0.00510562s (thread); 0s (gc)
│ │ │ │  isCodimAtLeast: Computing codim of monomials based on ideal generators.
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  i10 : time isCodimAtLeast(5, I, PairLimit => 200, Verbose=>false)
│ │ │ │ - -- used 6.0604e-05s (cpu); 0.00243851s (thread); 0s (gc)
│ │ │ │ + -- used 0.00489081s (cpu); 0.00483486s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = true
│ │ │ │  Notice in the first case the function returned null, because the depth of
│ │ │ │  search was not high enough. It only computed codim 5 times. The second returned
│ │ │ │  true, but it did so as soon as the answer was found (and before we hit the
│ │ │ │  PairLimit limit).
│ │ │ │  ********** WWaayyss ttoo uussee iissCCooddiimmAAttLLeeaasstt:: **********
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/_proj__Dim.html
│ │ │ @@ -99,23 +99,23 @@
│ │ │  
│ │ │  o3 = 2
    
│ │ │              
    
│ │ │                
    i4 : time projDim(module I, Strategy=>StrategyRandom)
│ │ │ - -- used 0.267277s (cpu); 0.158683s (thread); 0s (gc)
│ │ │ + -- used 0.27821s (cpu); 0.167497s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
    
│ │ │              
    
│ │ │                
    i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ - -- used 0.0107026s (cpu); 0.0126789s (thread); 0s (gc)
│ │ │ + -- used 0.0125711s (cpu); 0.0150857s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 1
    
│ │ │              
    
│ │ │          
    
│ │ │            
    The option MaxMinors can be used to control how many minors are computed at each step. If this is not specified, the number of minors is a function of the dimension $d$ of the polynomial ring and the possible minors $c$. Specifically it is 10 * d + 2 * log_1.3(c). Otherwise the user can set the option MaxMinors => ZZ to specify that a fixed integer is used for each step.  Alternatively, the user can control the number of minors computed at each step by setting the option MaxMinors => List.  In this case, the list specifies how many minors to be computed at each step, (working backwards). Finally, you can also set MaxMinors to be a custom function of the dimension $d$ of the polynomial ring and the maximum number of minors.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -44,19 +44,19 @@
│ │ │ │  i2 : I = ideal((x^3+y)^2, (x^2+y^2)^2, (x+y^3)^2, (x*y)^2);
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : pdim(module I)
│ │ │ │  
│ │ │ │  o3 = 2
│ │ │ │  i4 : time projDim(module I, Strategy=>StrategyRandom)
│ │ │ │ - -- used 0.267277s (cpu); 0.158683s (thread); 0s (gc)
│ │ │ │ + -- used 0.27821s (cpu); 0.167497s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 1
│ │ │ │  i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ │ - -- used 0.0107026s (cpu); 0.0126789s (thread); 0s (gc)
│ │ │ │ + -- used 0.0125711s (cpu); 0.0150857s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 1
│ │ │ │  The option MaxMinors can be used to control how many minors are computed at
│ │ │ │  each step. If this is not specified, the number of minors is a function of the
│ │ │ │  dimension $d$ of the polynomial ring and the possible minors $c$. Specifically
│ │ │ │  it is 10 * d + 2 * log_1.3(c). Otherwise the user can set the option MaxMinors
│ │ │ │  => ZZ to specify that a fixed integer is used for each step. Alternatively, the
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/_recursive__Minors.html
│ │ │ @@ -92,23 +92,23 @@
│ │ │               6      7
│ │ │  o2 : Matrix R  <-- R
    
│ │ │              
    
│ │ │                
    i3 : time I2 = recursiveMinors(4, M, Threads=>0);
│ │ │ - -- used 0.514952s (cpu); 0.461544s (thread); 0s (gc)
│ │ │ + -- used 0.568918s (cpu); 0.507531s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : Ideal of R
    
│ │ │              
    
│ │ │                
    i4 : time I1 = minors(4, M, Strategy=>Cofactor);
│ │ │ - -- used 1.47881s (cpu); 1.27687s (thread); 0s (gc)
│ │ │ + -- used 1.37125s (cpu); 1.25481s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of R
    
│ │ │              
    
│ │ │                
    i5 : I1 == I2
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,19 +27,19 @@
│ │ │ │  strategy for minors
│ │ │ │  i1 : R = QQ[x,y];
│ │ │ │  i2 : M = random(R^{5,5,5,5,5,5}, R^7);
│ │ │ │  
│ │ │ │               6      7
│ │ │ │  o2 : Matrix R  <-- R
│ │ │ │  i3 : time I2 = recursiveMinors(4, M, Threads=>0);
│ │ │ │ - -- used 0.514952s (cpu); 0.461544s (thread); 0s (gc)
│ │ │ │ + -- used 0.568918s (cpu); 0.507531s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : Ideal of R
│ │ │ │  i4 : time I1 = minors(4, M, Strategy=>Cofactor);
│ │ │ │ - -- used 1.47881s (cpu); 1.27687s (thread); 0s (gc)
│ │ │ │ + -- used 1.37125s (cpu); 1.25481s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : I1 == I2
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_i_n_o_r_s -- ideal generated by minors
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/_regular__In__Codimension.html
│ │ │ @@ -131,23 +131,23 @@
│ │ │  
│ │ │  o7 = 3
    
│ │ │              
    
│ │ │                
    i8 : time regularInCodimension(1, S)
│ │ │ - -- used 0.654392s (cpu); 0.499543s (thread); 0s (gc)
│ │ │ + -- used 0.734378s (cpu); 0.588417s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
    
│ │ │              
    
│ │ │                
    i9 : time regularInCodimension(2, S)
│ │ │ - -- used 7.00197s (cpu); 5.27802s (thread); 0s (gc)
    
│ │ │ + -- used 7.07038s (cpu); 5.35834s (thread); 0s (gc)
│ │ │              
    
│ │ │          
    
│ │ │            
    There are numerous examples where regularInCodimension is several orders of magnitude faster that calls of dim singularLocus.
    
│ │ │          
    
│ │ │          
    
│ │ │ @@ -165,39 +165,39 @@
│ │ │  
│ │ │  o11 = 2
    
│ │ │              
    
│ │ │                
    i12 : time (dim singularLocus (R))
│ │ │ - -- used 0.0199988s (cpu); 0.0198096s (thread); 0s (gc)
│ │ │ + -- used 0.0239444s (cpu); 0.0211436s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -1
    
│ │ │              
    
│ │ │                
    i13 : time regularInCodimension(2, R)
│ │ │ - -- used 0.182885s (cpu); 0.135147s (thread); 0s (gc)
│ │ │ + -- used 0.198059s (cpu); 0.150608s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = true
    
│ │ │              
    
│ │ │                
    i14 : time regularInCodimension(2, R)
│ │ │ - -- used 0.91949s (cpu); 0.572332s (thread); 0s (gc)
│ │ │ + -- used 1.05492s (cpu); 0.672474s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = true
    
│ │ │              
    
│ │ │                
    i15 : time regularInCodimension(2, R)
│ │ │ - -- used 1.24103s (cpu); 0.866303s (thread); 0s (gc)
│ │ │ + -- used 1.46998s (cpu); 1.00868s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = true
    
│ │ │              
    
│ │ │          
    
│ │ │            
    The function works by choosing interesting looking submatrices, computing their determinants, and periodically (based on a logarithmic growth setting), computing the dimension of a subideal of the Jacobian. The option Verbose can be used to see this in action.
    
│ │ │ @@ -537,15 +537,15 @@
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ -internalChooseMinor: Ch -- used 6.56865s (cpu); 4.94928s (thread); 0s (gc)
│ │ │ +internalChooseMinor: Ch -- used 7.64161s (cpu); 5.64553s (thread); 0s (gc)
│ │ │  oosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ @@ -589,15 +589,15 @@
│ │ │          
    
│ │ │            
    The maximum number of minors considered can be controlled by the option MaxMinors.  Alternatively, it can be controlled in a more precise way by passing a function to the option MaxMinors.  This function should have two inputs; the first is minimum number of minors needed to determine whether the ring is regular in codimension n, and the second is the total number of minors available in the Jacobian. The function regularInCodimension does not recompute determinants, so MaxMinors or is only an upper bound on the number of minors computed.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i17 : time regularInCodimension(2, S, Verbose=>true, MaxMinors=>30)
│ │ │ - -- used 1.27203s (cpu); 0.95852s (thread); 0s (gc)
│ │ │ + -- used 1.61781s (cpu); 1.2258s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =3, there are 17325 possible 4 by 4 minors, we will compute up to 30 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ @@ -666,39 +666,39 @@
│ │ │              
    		
│ │ │                
    i20 : StrategyCurrent#LexSmallestTerm = 0;
    
│ │ │              
    
│ │ │                
    i21 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.30306s (cpu); 0.217176s (thread); 0s (gc)
│ │ │ + -- used 0.377725s (cpu); 0.244506s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = true
    
│ │ │              
    
│ │ │                
    i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.113658s (cpu); 0.0792881s (thread); 0s (gc)
│ │ │ + -- used 0.146987s (cpu); 0.083492s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = true
    
│ │ │              
    
│ │ │                
    i23 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.366044s (cpu); 0.272723s (thread); 0s (gc)
│ │ │ + -- used 0.452641s (cpu); 0.318563s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = true
    
│ │ │              
    
│ │ │                
    i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 1.73381s (cpu); 1.2512s (thread); 0s (gc)
│ │ │ + -- used 1.94543s (cpu); 1.40704s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
    
│ │ │              
    
│ │ │                
    i25 : StrategyCurrent#LexSmallest = 0;
    
│ │ │ @@ -708,53 +708,53 @@
│ │ │                  		
│ │ │                
    i26 : StrategyCurrent#LexSmallestTerm = 100;
    
│ │ │              
    
│ │ │                
    i27 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 2.32962s (cpu); 1.64458s (thread); 0s (gc)
    
│ │ │ + -- used 2.58283s (cpu); 1.8019s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 2.38741s (cpu); 1.61396s (thread); 0s (gc)
│ │ │ + -- used 2.72717s (cpu); 1.8169s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = true
    
│ │ │              
    
│ │ │                
    i29 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.462956s (cpu); 0.370163s (thread); 0s (gc)
│ │ │ + -- used 0.477227s (cpu); 0.356693s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = true
    
│ │ │              
    
│ │ │                
    i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.770202s (cpu); 0.604272s (thread); 0s (gc)
│ │ │ + -- used 0.988081s (cpu); 0.792663s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
    
│ │ │              
    
│ │ │                
    i31 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.07872s (cpu); 0.871235s (thread); 0s (gc)
│ │ │ + -- used 1.22464s (cpu); 0.967794s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = true
    
│ │ │              
    
│ │ │                
    i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.85936s (cpu); 1.48073s (thread); 0s (gc)
│ │ │ + -- used 1.98063s (cpu); 1.49924s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = true
    
│ │ │              
    
│ │ │          
    
│ │ │            
    The minimum number of minors computed before checking the codimension can also be controlled by an option MinMinorsFunction.  This is should be a function of a single variable, the number of minors computed.  Finally, via the option CodimCheckFunction, you can pass the regularInCodimension a function which controls how frequently the codimension of the partial Jacobian ideal is computed.  By default this is the floor of 1.3^k. Finally, passing the option Modulus => p will do the computation after changing the coefficient ring to ZZ/p.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -72,19 +72,19 @@
│ │ │ │  
│ │ │ │  o5 : Ideal of T
│ │ │ │  i6 : S = T/I;
│ │ │ │  i7 : dim S
│ │ │ │  
│ │ │ │  o7 = 3
│ │ │ │  i8 : time regularInCodimension(1, S)
│ │ │ │ - -- used 0.654392s (cpu); 0.499543s (thread); 0s (gc)
│ │ │ │ + -- used 0.734378s (cpu); 0.588417s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  i9 : time regularInCodimension(2, S)
│ │ │ │ - -- used 7.00197s (cpu); 5.27802s (thread); 0s (gc)
│ │ │ │ + -- used 7.07038s (cpu); 5.35834s (thread); 0s (gc)
│ │ │ │  There are numerous examples where regularInCodimension is several orders of
│ │ │ │  magnitude faster that calls of dim singularLocus.
│ │ │ │  The following is a (pruned) affine chart on an Abelian surface obtained as a
│ │ │ │  product of two elliptic curves. It is nonsingular, as our function verifies. If
│ │ │ │  one does not prune it, then the dim singularLocus call takes an enormous amount
│ │ │ │  of time, otherwise the running times of dim singularLocus and our function are
│ │ │ │  frequently about the same.
│ │ │ │ @@ -92,27 +92,27 @@
│ │ │ │  (g^3+h^3+1,f*g^3+f*h^3+f,c*g^3+c*h^3+c,f^2*g^3+f^2*h^3+f^2,c*f*g^3+c*f*h^3+c*f,c^2*g^3+c^2*h^3+c^2,f^3*g^3+f^3*h^3+f^3,c*f^2*g^3+c*f^2*h^3+c*f^2,c^2*f*g^3+c^2*f*h^3+c^2*f,c^3-
│ │ │ │  f^2-c,c^3*h-f^2*h-c*h,c^3*g-f^2*g-c*g,c^3*h^2-f^2*h^2-c*h^2,c^3*g*h-f^2*g*h-c*g*h,c^3*g^2-f^2*g^2-c*g^2,c^3*h^3-f^2*h^3-c*h^3,c^3*g*h^2-f^2*g*h^2-c*g*h^2,c^3*g^2*h-f^2*g^2*h-
│ │ │ │  c*g^2*h,c^3*g^3+f^2*h^3+c*h^3+f^2+c);
│ │ │ │  i11 : dim(R)
│ │ │ │  
│ │ │ │  o11 = 2
│ │ │ │  i12 : time (dim singularLocus (R))
│ │ │ │ - -- used 0.0199988s (cpu); 0.0198096s (thread); 0s (gc)
│ │ │ │ + -- used 0.0239444s (cpu); 0.0211436s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o12 = -1
│ │ │ │  i13 : time regularInCodimension(2, R)
│ │ │ │ - -- used 0.182885s (cpu); 0.135147s (thread); 0s (gc)
│ │ │ │ + -- used 0.198059s (cpu); 0.150608s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 = true
│ │ │ │  i14 : time regularInCodimension(2, R)
│ │ │ │ - -- used 0.91949s (cpu); 0.572332s (thread); 0s (gc)
│ │ │ │ + -- used 1.05492s (cpu); 0.672474s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 = true
│ │ │ │  i15 : time regularInCodimension(2, R)
│ │ │ │ - -- used 1.24103s (cpu); 0.866303s (thread); 0s (gc)
│ │ │ │ + -- used 1.46998s (cpu); 1.00868s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = true
│ │ │ │  The function works by choosing interesting looking submatrices, computing their
│ │ │ │  determinants, and periodically (based on a logarithmic growth setting),
│ │ │ │  computing the dimension of a subideal of the Jacobian. The option Verbose can
│ │ │ │  be used to see this in action.
│ │ │ │  i16 : time regularInCodimension(2, S, Verbose=>true)
│ │ │ │ @@ -461,15 +461,15 @@
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │ -internalChooseMinor: Ch -- used 6.56865s (cpu); 4.94928s (thread); 0s (gc)
│ │ │ │ +internalChooseMinor: Ch -- used 7.64161s (cpu); 5.64553s (thread); 0s (gc)
│ │ │ │  oosing GRevLexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │ @@ -515,15 +515,15 @@
│ │ │ │  a function to the option MaxMinors. This function should have two inputs; the
│ │ │ │  first is minimum number of minors needed to determine whether the ring is
│ │ │ │  regular in codimension n, and the second is the total number of minors
│ │ │ │  available in the Jacobian. The function regularInCodimension does not recompute
│ │ │ │  determinants, so MaxMinors or is only an upper bound on the number of minors
│ │ │ │  computed.
│ │ │ │  i17 : time regularInCodimension(2, S, Verbose=>true, MaxMinors=>30)
│ │ │ │ - -- used 1.27203s (cpu); 0.95852s (thread); 0s (gc)
│ │ │ │ + -- used 1.61781s (cpu); 1.2258s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =3, there are 17325 possible 4 by 4
│ │ │ │  minors, we will compute up to 30 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │ @@ -590,51 +590,51 @@
│ │ │ │  because there are a small number of entries with nonzero constant terms, which
│ │ │ │  are selected repeatedly). However, in our first example, the LexSmallestTerm is
│ │ │ │  much faster, and Random does not perform well at all.
│ │ │ │  i18 : StrategyCurrent#Random = 0;
│ │ │ │  i19 : StrategyCurrent#LexSmallest = 100;
│ │ │ │  i20 : StrategyCurrent#LexSmallestTerm = 0;
│ │ │ │  i21 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 0.30306s (cpu); 0.217176s (thread); 0s (gc)
│ │ │ │ + -- used 0.377725s (cpu); 0.244506s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = true
│ │ │ │  i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 0.113658s (cpu); 0.0792881s (thread); 0s (gc)
│ │ │ │ + -- used 0.146987s (cpu); 0.083492s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o22 = true
│ │ │ │  i23 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 0.366044s (cpu); 0.272723s (thread); 0s (gc)
│ │ │ │ + -- used 0.452641s (cpu); 0.318563s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = true
│ │ │ │  i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 1.73381s (cpu); 1.2512s (thread); 0s (gc)
│ │ │ │ + -- used 1.94543s (cpu); 1.40704s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o24 = true
│ │ │ │  i25 : StrategyCurrent#LexSmallest = 0;
│ │ │ │  i26 : StrategyCurrent#LexSmallestTerm = 100;
│ │ │ │  i27 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 2.32962s (cpu); 1.64458s (thread); 0s (gc)
│ │ │ │ + -- used 2.58283s (cpu); 1.8019s (thread); 0s (gc)
│ │ │ │  i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 2.38741s (cpu); 1.61396s (thread); 0s (gc)
│ │ │ │ + -- used 2.72717s (cpu); 1.8169s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o28 = true
│ │ │ │  i29 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 0.462956s (cpu); 0.370163s (thread); 0s (gc)
│ │ │ │ + -- used 0.477227s (cpu); 0.356693s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o29 = true
│ │ │ │  i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 0.770202s (cpu); 0.604272s (thread); 0s (gc)
│ │ │ │ + -- used 0.988081s (cpu); 0.792663s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o30 = true
│ │ │ │  i31 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ │ - -- used 1.07872s (cpu); 0.871235s (thread); 0s (gc)
│ │ │ │ + -- used 1.22464s (cpu); 0.967794s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o31 = true
│ │ │ │  i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ │ - -- used 1.85936s (cpu); 1.48073s (thread); 0s (gc)
│ │ │ │ + -- used 1.98063s (cpu); 1.49924s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o32 = true
│ │ │ │  The minimum number of minors computed before checking the codimension can also
│ │ │ │  be controlled by an option MinMinorsFunction. This is should be a function of a
│ │ │ │  single variable, the number of minors computed. Finally, via the option
│ │ │ │  CodimCheckFunction, you can pass the regularInCodimension a function which
│ │ │ │  controls how frequently the codimension of the partial Jacobian ideal is
│ │ ├── ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/___Fitting_spideals_spof_spfinite_spmodules.out
│ │ │ @@ -81,23 +81,23 @@
│ │ │  
│ │ │  i14 : K3=nextDegree(gens ker Q2,2,S);
│ │ │  
│ │ │                8      8
│ │ │  o14 : Matrix R  <-- R
│ │ │  
│ │ │  i15 : time I=co1Fitting(K3)
│ │ │ - -- used 0.00282615s (cpu); 0.00282256s (thread); 0s (gc)
│ │ │ + -- used 0.00288562s (cpu); 0.00288218s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = ideal (a a   + a  - a  , a a   - a , a a   + a  - a  , a a   - a )
│ │ │                9 11    5    12   3 11    6   9 10    4    11   3 10    5
│ │ │  
│ │ │  o15 : Ideal of R
│ │ │  
│ │ │  i16 : time J=fittingIdeal(2-1,coker K3);
│ │ │ - -- used 0.00631438s (cpu); 0.00631331s (thread); 0s (gc)
│ │ │ + -- used 0.00652464s (cpu); 0.0065296s (thread); 0s (gc)
│ │ │  
│ │ │  o16 : Ideal of R
│ │ │  
│ │ │  i17 : I==J
│ │ │  
│ │ │  o17 = true
│ │ ├── ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/___Fitting_spideals_spof_spfinite_spmodules.html
│ │ │ @@ -202,26 +202,26 @@
│ │ │                8      8
│ │ │  o14 : Matrix R  <-- R
    
│ │ │              
    
│ │ │                
    i15 : time I=co1Fitting(K3)
│ │ │ - -- used 0.00282615s (cpu); 0.00282256s (thread); 0s (gc)
│ │ │ + -- used 0.00288562s (cpu); 0.00288218s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = ideal (a a   + a  - a  , a a   - a , a a   + a  - a  , a a   - a )
│ │ │                9 11    5    12   3 11    6   9 10    4    11   3 10    5
│ │ │  
│ │ │  o15 : Ideal of R
    
│ │ │              
    
│ │ │                
    i16 : time J=fittingIdeal(2-1,coker K3);
│ │ │ - -- used 0.00631438s (cpu); 0.00631331s (thread); 0s (gc)
│ │ │ + -- used 0.00652464s (cpu); 0.0065296s (thread); 0s (gc)
│ │ │  
│ │ │  o16 : Ideal of R
    
│ │ │              
    
│ │ │                
    i17 : I==J
│ │ │ ├── html2text {}
│ │ │ │ @@ -95,22 +95,22 @@
│ │ │ │                2      6
│ │ │ │  o13 : Matrix R  <-- R
│ │ │ │  i14 : K3=nextDegree(gens ker Q2,2,S);
│ │ │ │  
│ │ │ │                8      8
│ │ │ │  o14 : Matrix R  <-- R
│ │ │ │  i15 : time I=co1Fitting(K3)
│ │ │ │ - -- used 0.00282615s (cpu); 0.00282256s (thread); 0s (gc)
│ │ │ │ + -- used 0.00288562s (cpu); 0.00288218s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = ideal (a a   + a  - a  , a a   - a , a a   + a  - a  , a a   - a )
│ │ │ │                9 11    5    12   3 11    6   9 10    4    11   3 10    5
│ │ │ │  
│ │ │ │  o15 : Ideal of R
│ │ │ │  i16 : time J=fittingIdeal(2-1,coker K3);
│ │ │ │ - -- used 0.00631438s (cpu); 0.00631331s (thread); 0s (gc)
│ │ │ │ + -- used 0.00652464s (cpu); 0.0065296s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 : Ideal of R
│ │ │ │  i17 : I==J
│ │ │ │  
│ │ │ │  o17 = true
│ │ │ │  Note that our method is a bit faster for this small example, and for rank 2
│ │ │ │  quotients of S^3=\mathbb{Z}[x,y]^3 the time difference is massive.
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Object.out
│ │ │ @@ -4,19 +4,19 @@
│ │ │  
│ │ │  o1 = 5
│ │ │  
│ │ │  o1 : ForeignObject of type int32
│ │ │  
│ │ │  i2 : peek x
│ │ │  
│ │ │ -o2 = int32{Address => 0x7f7f285265f0}
│ │ │ +o2 = int32{Address => 0x7f44129bc560}
│ │ │  
│ │ │  i3 : address x
│ │ │  
│ │ │ -o3 = 0x7f7f285265f0
│ │ │ +o3 = 0x7f44129bc560
│ │ │  
│ │ │  o3 : Pointer
│ │ │  
│ │ │  i4 : class x
│ │ │  
│ │ │  o4 = int32
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Pointer__Array__Type.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │  
│ │ │  o2 = {the, quick, brown, fox, jumps, over, the, lazy, dog}
│ │ │  
│ │ │  o2 : ForeignObject of type char**
│ │ │  
│ │ │  i3 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │  
│ │ │ -o3 = {0x7f7f2854a030, 0x7f7f2854a020, 0x7f7f2854a010}
│ │ │ +o3 = {0x7f4412a030a0, 0x7f4412a03090, 0x7f4412a03080}
│ │ │  
│ │ │  o3 : ForeignObject of type void**
│ │ │  
│ │ │  i4 : x = charstarstar {"foo", "bar", "baz"}
│ │ │  
│ │ │  o4 = {foo, bar, baz}
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Pointer__Array__Type_sp__Visible__List.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  
│ │ │  o1 = {foo, bar}
│ │ │  
│ │ │  o1 : ForeignObject of type char**
│ │ │  
│ │ │  i2 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │  
│ │ │ -o2 = {0x7f7f28567e20, 0x7f7f28567e10, 0x7f7f28567e00}
│ │ │ +o2 = {0x7f4412a4bee0, 0x7f4412a4bed0, 0x7f4412a4bec0}
│ │ │  
│ │ │  o2 : ForeignObject of type void**
│ │ │  
│ │ │  i3 : int2star = foreignPointerArrayType(2 * int)
│ │ │  
│ │ │  o3 = int32[2]*
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Pointer__Type_sp__Pointer.out
│ │ │ @@ -1,15 +1,15 @@
│ │ │  -- -*- M2-comint -*- hash: 1730835169888399450
│ │ │  
│ │ │  i1 : ptr = address int 0
│ │ │  
│ │ │ -o1 = 0x7f7f1f69e700
│ │ │ +o1 = 0x7f44151686f0
│ │ │  
│ │ │  o1 : Pointer
│ │ │  
│ │ │  i2 : voidstar ptr
│ │ │  
│ │ │ -o2 = 0x7f7f1f69e700
│ │ │ +o2 = 0x7f44151686f0
│ │ │  
│ │ │  o2 : ForeignObject of type void*
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Type_sp__Pointer.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  
│ │ │  o1 = 5
│ │ │  
│ │ │  o1 : ForeignObject of type int32
│ │ │  
│ │ │  i2 : ptr = address x
│ │ │  
│ │ │ -o2 = 0x7f7f28526ca0
│ │ │ +o2 = 0x7f44129bc730
│ │ │  
│ │ │  o2 : Pointer
│ │ │  
│ │ │  i3 : int ptr
│ │ │  
│ │ │  o3 = 5
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Type_sp_st_spvoidstar.out
│ │ │ @@ -1,12 +1,12 @@
│ │ │  -- -*- M2-comint -*- hash: 1731230829183683930
│ │ │  
│ │ │  i1 : ptr = voidstar address int 5
│ │ │  
│ │ │ -o1 = 0x7f7f2854aa70
│ │ │ +o1 = 0x7f4412a03b60
│ │ │  
│ │ │  o1 : ForeignObject of type void*
│ │ │  
│ │ │  i2 : int * ptr
│ │ │  
│ │ │  o2 = 5
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Union__Type_sp__Thing.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  
│ │ │  o1 = myunion
│ │ │  
│ │ │  o1 : ForeignUnionType
│ │ │  
│ │ │  i2 : myunion 27
│ │ │  
│ │ │ -o2 = HashTable{"bar" => 6.92598e-310}
│ │ │ +o2 = HashTable{"bar" => 6.91346e-310}
│ │ │                 "foo" => 27
│ │ │  
│ │ │  o2 : ForeignObject of type myunion
│ │ │  
│ │ │  i3 : myunion pi
│ │ │  
│ │ │  o3 = HashTable{"bar" => 3.14159   }
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Pointer.out
│ │ │ @@ -4,28 +4,28 @@
│ │ │  
│ │ │  o1 = 20
│ │ │  
│ │ │  o1 : ForeignObject of type int32
│ │ │  
│ │ │  i2 : peek x
│ │ │  
│ │ │ -o2 = int32{Address => 0x7f7f285264f0}
│ │ │ +o2 = int32{Address => 0x7f44129bc5a0}
│ │ │  
│ │ │  i3 : ptr = address x
│ │ │  
│ │ │ -o3 = 0x7f7f285264f0
│ │ │ +o3 = 0x7f44129bc5a0
│ │ │  
│ │ │  o3 : Pointer
│ │ │  
│ │ │  i4 : ptr + 5
│ │ │  
│ │ │ -o4 = 0x7f7f285264f5
│ │ │ +o4 = 0x7f44129bc5a5
│ │ │  
│ │ │  o4 : Pointer
│ │ │  
│ │ │  i5 : ptr - 3
│ │ │  
│ │ │ -o5 = 0x7f7f285264ed
│ │ │ +o5 = 0x7f44129bc59d
│ │ │  
│ │ │  o5 : Pointer
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Shared__Library.out
│ │ │ @@ -4,10 +4,10 @@
│ │ │  
│ │ │  o1 = mpfr
│ │ │  
│ │ │  o1 : SharedLibrary
│ │ │  
│ │ │  i2 : peek mpfr
│ │ │  
│ │ │ -o2 = SharedLibrary{0x7f7f2f6e4550, mpfr}
│ │ │ +o2 = SharedLibrary{0x7f4426873550, mpfr}
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/__st_spvoidstar_sp_eq_sp__Thing.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  
│ │ │  o1 = 5
│ │ │  
│ │ │  o1 : ForeignObject of type int32
│ │ │  
│ │ │  i2 : ptr = address x
│ │ │  
│ │ │ -o2 = 0x7f7f285268c0
│ │ │ +o2 = 0x7f44129bcb20
│ │ │  
│ │ │  o2 : Pointer
│ │ │  
│ │ │  i3 : *ptr = int 6
│ │ │  
│ │ │  o3 = 6
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_address.out
│ │ │ @@ -1,15 +1,15 @@
│ │ │  -- -*- M2-comint -*- hash: 1730181884377373595
│ │ │  
│ │ │  i1 : address int
│ │ │  
│ │ │ -o1 = 0x563c19575b40
│ │ │ +o1 = 0x55a43bc37b40
│ │ │  
│ │ │  o1 : Pointer
│ │ │  
│ │ │  i2 : address int 5
│ │ │  
│ │ │ -o2 = 0x7f7f28526a20
│ │ │ +o2 = 0x7f44129bc990
│ │ │  
│ │ │  o2 : Pointer
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_foreign__Function.out
│ │ │ @@ -78,14 +78,14 @@
│ │ │  
│ │ │  o16 = free
│ │ │  
│ │ │  o16 : ForeignFunction
│ │ │  
│ │ │  i17 : x = malloc 8
│ │ │  
│ │ │ -o17 = 0x7f443c06a4f0
│ │ │ +o17 = 0x7f9fd406a4f0
│ │ │  
│ │ │  o17 : ForeignObject of type void*
│ │ │  
│ │ │  i18 : registerFinalizer(x, free)
│ │ │  
│ │ │  i19 :
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_get__Memory.out
│ │ │ @@ -1,21 +1,21 @@
│ │ │  -- -*- M2-comint -*- hash: 10647988412767280310
│ │ │  
│ │ │  i1 : ptr = getMemory 8
│ │ │  
│ │ │ -o1 = 0x7f7f2b093240
│ │ │ +o1 = 0x7f44213c1490
│ │ │  
│ │ │  o1 : ForeignObject of type void*
│ │ │  
│ │ │  i2 : ptr = getMemory(8, Atomic => true)
│ │ │  
│ │ │ -o2 = 0x7f7f285260a0
│ │ │ +o2 = 0x7f44129bc230
│ │ │  
│ │ │  o2 : ForeignObject of type void*
│ │ │  
│ │ │  i3 : ptr = getMemory int
│ │ │  
│ │ │ -o3 = 0x7f7f2854afc0
│ │ │ +o3 = 0x7f44129bc120
│ │ │  
│ │ │  o3 : ForeignObject of type void*
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_register__Finalizer_lp__Foreign__Object_cm__Function_rp.out
│ │ │ @@ -17,18 +17,18 @@
│ │ │  o3 = finalizer
│ │ │  
│ │ │  o3 : FunctionClosure
│ │ │  
│ │ │  i4 : for i to 9 do (x := malloc 8; registerFinalizer(x, finalizer))
│ │ │  
│ │ │  i5 : collectGarbage()
│ │ │ -freeing memory at 0x7f7f1407f910
│ │ │ -freeing memory at 0x7f7f1407f930
│ │ │ -freeing memory at 0x7f7f1407f950
│ │ │ -freeing memory at 0x7f7f1407f990
│ │ │ -freeing memory at 0x7f7f1407f250
│ │ │ -freeing memory at 0x7f7f1407f230
│ │ │ -freeing memory at 0x7f7f1407f9b0
│ │ │ -freeing memory at 0x7f7f1407f970
│ │ │ -freeing memory at 0x7f7f1407f8f0
│ │ │ +freeing memory at 0x7f43fc07f250
│ │ │ +freeing memory at 0x7f43fc07f930
│ │ │ +freeing memory at 0x7f43fc07f8f0
│ │ │ +freeing memory at 0x7f43fc07f230
│ │ │ +freeing memory at 0x7f43fc07f990
│ │ │ +freeing memory at 0x7f43fc07f970
│ │ │ +freeing memory at 0x7f43fc07f9b0
│ │ │ +freeing memory at 0x7f43fc07f910
│ │ │ +freeing memory at 0x7f43fc07f950
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_value_lp__Foreign__Object_rp.out
│ │ │ @@ -20,21 +20,21 @@
│ │ │  
│ │ │  o4 = 5
│ │ │  
│ │ │  o4 : RR (of precision 53)
│ │ │  
│ │ │  i5 : x = voidstar address int 5
│ │ │  
│ │ │ -o5 = 0x7f7f2854ad20
│ │ │ +o5 = 0x7f4412a03bb0
│ │ │  
│ │ │  o5 : ForeignObject of type void*
│ │ │  
│ │ │  i6 : value x
│ │ │  
│ │ │ -o6 = 0x7f7f2854ad20
│ │ │ +o6 = 0x7f4412a03bb0
│ │ │  
│ │ │  o6 : Pointer
│ │ │  
│ │ │  i7 : x = charstar "Hello, world!"
│ │ │  
│ │ │  o7 = Hello, world!
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Object.html
│ │ │ @@ -64,27 +64,27 @@
│ │ │  o1 : ForeignObject of type int32
    
│ │ │              
    
│ │ │                
    i2 : peek x
│ │ │  
│ │ │ -o2 = int32{Address => 0x7f7f285265f0}
    
│ │ │ +o2 = int32{Address => 0x7f44129bc560}
│ │ │              
    
│ │ │          
    
│ │ │            
    To get this, use address.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i3 : address x
│ │ │  
│ │ │ -o3 = 0x7f7f285265f0
│ │ │ +o3 = 0x7f44129bc560
│ │ │  
│ │ │  o3 : Pointer
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Use class to determine the type of the object.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,19 +10,19 @@
│ │ │ │  i1 : x = int 5
│ │ │ │  
│ │ │ │  o1 = 5
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type int32
│ │ │ │  i2 : peek x
│ │ │ │  
│ │ │ │ -o2 = int32{Address => 0x7f7f285265f0}
│ │ │ │ +o2 = int32{Address => 0x7f44129bc560}
│ │ │ │  To get this, use _a_d_d_r_e_s_s.
│ │ │ │  i3 : address x
│ │ │ │  
│ │ │ │ -o3 = 0x7f7f285265f0
│ │ │ │ +o3 = 0x7f44129bc560
│ │ │ │  
│ │ │ │  o3 : Pointer
│ │ │ │  Use _c_l_a_s_s to determine the type of the object.
│ │ │ │  i4 : class x
│ │ │ │  
│ │ │ │  o4 = int32
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Pointer__Array__Type.html
│ │ │ @@ -74,15 +74,15 @@
│ │ │  o2 : ForeignObject of type char**
    
│ │ │              
    
│ │ │                
    i3 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │  
│ │ │ -o3 = {0x7f7f2854a030, 0x7f7f2854a020, 0x7f7f2854a010}
│ │ │ +o3 = {0x7f4412a030a0, 0x7f4412a03090, 0x7f4412a03080}
│ │ │  
│ │ │  o3 : ForeignObject of type void**
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Foreign pointer arrays may be subscripted using _.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │           "lazy", "dog"}
│ │ │ │  
│ │ │ │  o2 = {the, quick, brown, fox, jumps, over, the, lazy, dog}
│ │ │ │  
│ │ │ │  o2 : ForeignObject of type char**
│ │ │ │  i3 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │ │  
│ │ │ │ -o3 = {0x7f7f2854a030, 0x7f7f2854a020, 0x7f7f2854a010}
│ │ │ │ +o3 = {0x7f4412a030a0, 0x7f4412a03090, 0x7f4412a03080}
│ │ │ │  
│ │ │ │  o3 : ForeignObject of type void**
│ │ │ │  Foreign pointer arrays may be subscripted using __.
│ │ │ │  i4 : x = charstarstar {"foo", "bar", "baz"}
│ │ │ │  
│ │ │ │  o4 = {foo, bar, baz}
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Pointer__Array__Type_sp__Visible__List.html
│ │ │ @@ -82,15 +82,15 @@
│ │ │  o1 : ForeignObject of type char**
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │  
│ │ │ -o2 = {0x7f7f28567e20, 0x7f7f28567e10, 0x7f7f28567e00}
│ │ │ +o2 = {0x7f4412a4bee0, 0x7f4412a4bed0, 0x7f4412a4bec0}
│ │ │  
│ │ │  o2 : ForeignObject of type void**
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : int2star = foreignPointerArrayType(2 * int)
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │  i1 : charstarstar {"foo", "bar"}
│ │ │ │  
│ │ │ │  o1 = {foo, bar}
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type char**
│ │ │ │  i2 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │ │  
│ │ │ │ -o2 = {0x7f7f28567e20, 0x7f7f28567e10, 0x7f7f28567e00}
│ │ │ │ +o2 = {0x7f4412a4bee0, 0x7f4412a4bed0, 0x7f4412a4bec0}
│ │ │ │  
│ │ │ │  o2 : ForeignObject of type void**
│ │ │ │  i3 : int2star = foreignPointerArrayType(2 * int)
│ │ │ │  
│ │ │ │  o3 = int32[2]*
│ │ │ │  
│ │ │ │  o3 : ForeignPointerArrayType
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Pointer__Type_sp__Pointer.html
│ │ │ @@ -73,24 +73,24 @@
│ │ │            
    To cast a Macaulay2 pointer to a foreign object with a pointer type, give the type followed by the pointer.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : ptr = address int 0
│ │ │  
│ │ │ -o1 = 0x7f7f1f69e700
│ │ │ +o1 = 0x7f44151686f0
│ │ │  
│ │ │  o1 : Pointer
    
│ │ │              
    
│ │ │                
    i2 : voidstar ptr
│ │ │  
│ │ │ -o2 = 0x7f7f1f69e700
│ │ │ +o2 = 0x7f44151686f0
│ │ │  
│ │ │  o2 : ForeignObject of type void*
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,20 +15,20 @@
│ │ │ │      * Outputs:
│ │ │ │            o a _f_o_r_e_i_g_n_ _o_b_j_e_c_t,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  To cast a Macaulay2 pointer to a foreign object with a pointer type, give the
│ │ │ │  type followed by the pointer.
│ │ │ │  i1 : ptr = address int 0
│ │ │ │  
│ │ │ │ -o1 = 0x7f7f1f69e700
│ │ │ │ +o1 = 0x7f44151686f0
│ │ │ │  
│ │ │ │  o1 : Pointer
│ │ │ │  i2 : voidstar ptr
│ │ │ │  
│ │ │ │ -o2 = 0x7f7f1f69e700
│ │ │ │ +o2 = 0x7f44151686f0
│ │ │ │  
│ │ │ │  o2 : ForeignObject of type void*
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _F_o_r_e_i_g_n_P_o_i_n_t_e_r_T_y_p_e_ _P_o_i_n_t_e_r -- cast a Macaulay2 pointer to a foreign
│ │ │ │        pointer
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Type_sp__Pointer.html
│ │ │ @@ -82,15 +82,15 @@
│ │ │  o1 : ForeignObject of type int32
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : ptr = address x
│ │ │  
│ │ │ -o2 = 0x7f7f28526ca0
│ │ │ +o2 = 0x7f44129bc730
│ │ │  
│ │ │  o2 : Pointer
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : int ptr
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,15 +18,15 @@
│ │ │ │  i1 : x = int 5
│ │ │ │  
│ │ │ │  o1 = 5
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type int32
│ │ │ │  i2 : ptr = address x
│ │ │ │  
│ │ │ │ -o2 = 0x7f7f28526ca0
│ │ │ │ +o2 = 0x7f44129bc730
│ │ │ │  
│ │ │ │  o2 : Pointer
│ │ │ │  i3 : int ptr
│ │ │ │  
│ │ │ │  o3 = 5
│ │ │ │  
│ │ │ │  o3 : ForeignObject of type int32
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Type_sp_st_spvoidstar.html
│ │ │ @@ -73,15 +73,15 @@
│ │ │            
    This is syntactic sugar for T value ptr (see ForeignType Pointer) for dereferencing pointers.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : ptr = voidstar address int 5
│ │ │  
│ │ │ -o1 = 0x7f7f2854aa70
│ │ │ +o1 = 0x7f4412a03b60
│ │ │  
│ │ │  o1 : ForeignObject of type void*
    
│ │ │              
    
│ │ │                
    i2 : int * ptr
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,15 +14,15 @@
│ │ │ │      * Outputs:
│ │ │ │            o a _f_o_r_e_i_g_n_ _o_b_j_e_c_t, of type T;
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This is syntactic sugar for T value ptr (see _F_o_r_e_i_g_n_T_y_p_e_ _P_o_i_n_t_e_r) for
│ │ │ │  dereferencing pointers.
│ │ │ │  i1 : ptr = voidstar address int 5
│ │ │ │  
│ │ │ │ -o1 = 0x7f7f2854aa70
│ │ │ │ +o1 = 0x7f4412a03b60
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type void*
│ │ │ │  i2 : int * ptr
│ │ │ │  
│ │ │ │  o2 = 5
│ │ │ │  
│ │ │ │  o2 : ForeignObject of type int32
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Union__Type_sp__Thing.html
│ │ │ @@ -82,15 +82,15 @@
│ │ │  o1 : ForeignUnionType
    
│ │ │              
    
│ │ │                
    i2 : myunion 27
│ │ │  
│ │ │ -o2 = HashTable{"bar" => 6.92598e-310}
│ │ │ +o2 = HashTable{"bar" => 6.91346e-310}
│ │ │                 "foo" => 27
│ │ │  
│ │ │  o2 : ForeignObject of type myunion
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │  i1 : myunion = foreignUnionType("myunion", {"foo" => int, "bar" => double})
│ │ │ │  
│ │ │ │  o1 = myunion
│ │ │ │  
│ │ │ │  o1 : ForeignUnionType
│ │ │ │  i2 : myunion 27
│ │ │ │  
│ │ │ │ -o2 = HashTable{"bar" => 6.92598e-310}
│ │ │ │ +o2 = HashTable{"bar" => 6.91346e-310}
│ │ │ │                 "foo" => 27
│ │ │ │  
│ │ │ │  o2 : ForeignObject of type myunion
│ │ │ │  i3 : myunion pi
│ │ │ │  
│ │ │ │  o3 = HashTable{"bar" => 3.14159   }
│ │ │ │                 "foo" => 1413754136
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Pointer.html
│ │ │ @@ -64,50 +64,50 @@
│ │ │  o1 : ForeignObject of type int32
│ │ │              
    
│ │ │                
    i2 : peek x
│ │ │  
│ │ │ -o2 = int32{Address => 0x7f7f285264f0}
    
│ │ │ +o2 = int32{Address => 0x7f44129bc5a0}
│ │ │              
    
│ │ │          
    
│ │ │            
    These pointers can be accessed using address.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i3 : ptr = address x
│ │ │  
│ │ │ -o3 = 0x7f7f285264f0
│ │ │ +o3 = 0x7f44129bc5a0
│ │ │  
│ │ │  o3 : Pointer
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Simple arithmetic can be performed on pointers.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : ptr + 5
│ │ │  
│ │ │ -o4 = 0x7f7f285264f5
│ │ │ +o4 = 0x7f44129bc5a5
│ │ │  
│ │ │  o4 : Pointer
    
│ │ │              
    
│ │ │                
    i5 : ptr - 3
│ │ │  
│ │ │ -o5 = 0x7f7f285264ed
│ │ │ +o5 = 0x7f44129bc59d
│ │ │  
│ │ │  o5 : Pointer
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,30 +10,30 @@
│ │ │ │  i1 : x = int 20
│ │ │ │  
│ │ │ │  o1 = 20
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type int32
│ │ │ │  i2 : peek x
│ │ │ │  
│ │ │ │ -o2 = int32{Address => 0x7f7f285264f0}
│ │ │ │ +o2 = int32{Address => 0x7f44129bc5a0}
│ │ │ │  These pointers can be accessed using _a_d_d_r_e_s_s.
│ │ │ │  i3 : ptr = address x
│ │ │ │  
│ │ │ │ -o3 = 0x7f7f285264f0
│ │ │ │ +o3 = 0x7f44129bc5a0
│ │ │ │  
│ │ │ │  o3 : Pointer
│ │ │ │  Simple arithmetic can be performed on pointers.
│ │ │ │  i4 : ptr + 5
│ │ │ │  
│ │ │ │ -o4 = 0x7f7f285264f5
│ │ │ │ +o4 = 0x7f44129bc5a5
│ │ │ │  
│ │ │ │  o4 : Pointer
│ │ │ │  i5 : ptr - 3
│ │ │ │  
│ │ │ │ -o5 = 0x7f7f285264ed
│ │ │ │ +o5 = 0x7f44129bc59d
│ │ │ │  
│ │ │ │  o5 : Pointer
│ │ │ │  ******** MMeennuu ********
│ │ │ │      * _n_u_l_l_P_o_i_n_t_e_r -- the null pointer
│ │ │ │      * _a_d_d_r_e_s_s -- pointer to type or object
│ │ │ │      * _F_o_r_e_i_g_n_T_y_p_e_ _P_o_i_n_t_e_r -- dereference a pointer
│ │ │ │  ********** FFuunnccttiioonnss aanndd mmeetthhooddss rreettuurrnniinngg aa ppooiinntteerr:: **********
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Shared__Library.html
│ │ │ @@ -64,15 +64,15 @@
│ │ │  o1 : SharedLibrary
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : peek mpfr
│ │ │  
│ │ │ -o2 = SharedLibrary{0x7f7f2f6e4550, mpfr}
    
│ │ │ +o2 = SharedLibrary{0x7f4426873550, mpfr}
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    Menu
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : address int
│ │ │  
│ │ │ -o1 = 0x563c19575b40
│ │ │ +o1 = 0x55a43bc37b40
│ │ │  
│ │ │  o1 : Pointer
    
│ │ │              
    
│ │ │          
    
│ │ │            
    If x is a foreign object, then this returns the address to the object.  It behaves like the & "address-of" operator in C.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i2 : address int 5
│ │ │  
│ │ │ -o2 = 0x7f7f28526a20
│ │ │ +o2 = 0x7f44129bc990
│ │ │  
│ │ │  o2 : Pointer
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,22 +11,22 @@
│ │ │ │      * Outputs:
│ │ │ │            o a _p_o_i_n_t_e_r,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  If x is a foreign type, then this returns the address to the ffi_type struct
│ │ │ │  used by libffi to identify the type.
│ │ │ │  i1 : address int
│ │ │ │  
│ │ │ │ -o1 = 0x563c19575b40
│ │ │ │ +o1 = 0x55a43bc37b40
│ │ │ │  
│ │ │ │  o1 : Pointer
│ │ │ │  If x is a foreign object, then this returns the address to the object. It
│ │ │ │  behaves like the & "address-of" operator in C.
│ │ │ │  i2 : address int 5
│ │ │ │  
│ │ │ │ -o2 = 0x7f7f28526a20
│ │ │ │ +o2 = 0x7f44129bc990
│ │ │ │  
│ │ │ │  o2 : Pointer
│ │ │ │  ********** WWaayyss ttoo uussee aaddddrreessss:: **********
│ │ │ │      * address(ForeignObject)
│ │ │ │      * address(ForeignType)
│ │ │ │      * address(Nothing) (missing documentation)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/_foreign__Function.html
│ │ │ @@ -232,15 +232,15 @@
│ │ │  o16 : ForeignFunction
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i17 : x = malloc 8
│ │ │  
│ │ │ -o17 = 0x7f443c06a4f0
│ │ │ +o17 = 0x7f9fd406a4f0
│ │ │  
│ │ │  o17 : ForeignObject of type void*
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i18 : registerFinalizer(x, free)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -95,15 +95,15 @@
│ │ │ │  i16 : free = foreignFunction("free", void, voidstar)
│ │ │ │  
│ │ │ │  o16 = free
│ │ │ │  
│ │ │ │  o16 : ForeignFunction
│ │ │ │  i17 : x = malloc 8
│ │ │ │  
│ │ │ │ -o17 = 0x7f443c06a4f0
│ │ │ │ +o17 = 0x7f9fd406a4f0
│ │ │ │  
│ │ │ │  o17 : ForeignObject of type void*
│ │ │ │  i18 : registerFinalizer(x, free)
│ │ │ │  ********** WWaayyss ttoo uussee ffoorreeiiggnnFFuunnccttiioonn:: **********
│ │ │ │      * foreignFunction(Pointer,String,ForeignType,VisibleList)
│ │ │ │      * foreignFunction(SharedLibrary,String,ForeignType,ForeignType)
│ │ │ │      * foreignFunction(SharedLibrary,String,ForeignType,VisibleList)
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/_get__Memory.html
│ │ │ @@ -77,43 +77,43 @@
│ │ │            
    Allocate n bytes of memory using the GC garbage collector.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : ptr = getMemory 8
│ │ │  
│ │ │ -o1 = 0x7f7f2b093240
│ │ │ +o1 = 0x7f44213c1490
│ │ │  
│ │ │  o1 : ForeignObject of type void*
    
│ │ │              
    
│ │ │          
    
│ │ │            
    If the memory will not contain any pointers, then set the Atomic option to true.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i2 : ptr = getMemory(8, Atomic => true)
│ │ │  
│ │ │ -o2 = 0x7f7f285260a0
│ │ │ +o2 = 0x7f44129bc230
│ │ │  
│ │ │  o2 : ForeignObject of type void*
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Alternatively, a foreign object type T may be specified.  In this case, the number of bytes and whether the Atomic option should be set will be determined automatically.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i3 : ptr = getMemory int
│ │ │  
│ │ │ -o3 = 0x7f7f2854afc0
│ │ │ +o3 = 0x7f44129bc120
│ │ │  
│ │ │  o3 : ForeignObject of type void*
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,30 +14,30 @@
│ │ │ │            o Atomic => ..., default value false
│ │ │ │      * Outputs:
│ │ │ │            o an instance of the type _v_o_i_d_s_t_a_r,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Allocate n bytes of memory using the _G_C_ _g_a_r_b_a_g_e_ _c_o_l_l_e_c_t_o_r.
│ │ │ │  i1 : ptr = getMemory 8
│ │ │ │  
│ │ │ │ -o1 = 0x7f7f2b093240
│ │ │ │ +o1 = 0x7f44213c1490
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type void*
│ │ │ │  If the memory will not contain any pointers, then set the Atomic option to
│ │ │ │  _t_r_u_e.
│ │ │ │  i2 : ptr = getMemory(8, Atomic => true)
│ │ │ │  
│ │ │ │ -o2 = 0x7f7f285260a0
│ │ │ │ +o2 = 0x7f44129bc230
│ │ │ │  
│ │ │ │  o2 : ForeignObject of type void*
│ │ │ │  Alternatively, a foreign object type T may be specified. In this case, the
│ │ │ │  number of bytes and whether the Atomic option should be set will be determined
│ │ │ │  automatically.
│ │ │ │  i3 : ptr = getMemory int
│ │ │ │  
│ │ │ │ -o3 = 0x7f7f2854afc0
│ │ │ │ +o3 = 0x7f44129bc120
│ │ │ │  
│ │ │ │  o3 : ForeignObject of type void*
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_g_i_s_t_e_r_F_i_n_a_l_i_z_e_r_(_F_o_r_e_i_g_n_O_b_j_e_c_t_,_F_u_n_c_t_i_o_n_) -- register a finalizer for a
│ │ │ │        foreign object
│ │ │ │  ********** WWaayyss ttoo uussee ggeettMMeemmoorryy:: **********
│ │ │ │      * getMemory(ForeignType)
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/_register__Finalizer_lp__Foreign__Object_cm__Function_rp.html
│ │ │ @@ -100,23 +100,23 @@
│ │ │              
│ │ │                
    i4 : for i to 9 do (x := malloc 8; registerFinalizer(x, finalizer))
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : collectGarbage()
│ │ │ -freeing memory at 0x7f7f1407f910
│ │ │ -freeing memory at 0x7f7f1407f930
│ │ │ -freeing memory at 0x7f7f1407f950
│ │ │ -freeing memory at 0x7f7f1407f990
│ │ │ -freeing memory at 0x7f7f1407f250
│ │ │ -freeing memory at 0x7f7f1407f230
│ │ │ -freeing memory at 0x7f7f1407f9b0
│ │ │ -freeing memory at 0x7f7f1407f970
│ │ │ -freeing memory at 0x7f7f1407f8f0
    
│ │ │ +freeing memory at 0x7f43fc07f250
│ │ │ +freeing memory at 0x7f43fc07f930
│ │ │ +freeing memory at 0x7f43fc07f8f0
│ │ │ +freeing memory at 0x7f43fc07f230
│ │ │ +freeing memory at 0x7f43fc07f990
│ │ │ +freeing memory at 0x7f43fc07f970
│ │ │ +freeing memory at 0x7f43fc07f9b0
│ │ │ +freeing memory at 0x7f43fc07f910
│ │ │ +freeing memory at 0x7f43fc07f950
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : x = voidstar address int 5
│ │ │  
│ │ │ -o5 = 0x7f7f2854ad20
│ │ │ +o5 = 0x7f4412a03bb0
│ │ │  
│ │ │  o5 : ForeignObject of type void*
    
│ │ │              
    
│ │ │                
    i6 : value x
│ │ │  
│ │ │ -o6 = 0x7f7f2854ad20
│ │ │ +o6 = 0x7f4412a03bb0
│ │ │  
│ │ │  o6 : Pointer
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Foreign string objects are converted to strings.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,20 +34,20 @@
│ │ │ │  
│ │ │ │  o4 = 5
│ │ │ │  
│ │ │ │  o4 : RR (of precision 53)
│ │ │ │  Foreign pointer objects are converted to _P_o_i_n_t_e_r objects.
│ │ │ │  i5 : x = voidstar address int 5
│ │ │ │  
│ │ │ │ -o5 = 0x7f7f2854ad20
│ │ │ │ +o5 = 0x7f4412a03bb0
│ │ │ │  
│ │ │ │  o5 : ForeignObject of type void*
│ │ │ │  i6 : value x
│ │ │ │  
│ │ │ │ -o6 = 0x7f7f2854ad20
│ │ │ │ +o6 = 0x7f4412a03bb0
│ │ │ │  
│ │ │ │  o6 : Pointer
│ │ │ │  Foreign string objects are converted to strings.
│ │ │ │  i7 : x = charstar "Hello, world!"
│ │ │ │  
│ │ │ │  o7 = Hello, world!
│ │ ├── ./usr/share/doc/Macaulay2/FourTiTwo/example-output/_put__Matrix.out
│ │ │ @@ -6,27 +6,27 @@
│ │ │       | 1 2 3 4 |
│ │ │  
│ │ │                2       4
│ │ │  o1 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i2 : s = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-15935-0/0
│ │ │ +o2 = /tmp/M2-20955-0/0
│ │ │  
│ │ │  i3 : F = openOut(s)
│ │ │  
│ │ │ -o3 = /tmp/M2-15935-0/0
│ │ │ +o3 = /tmp/M2-20955-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : putMatrix(F,A)
│ │ │  
│ │ │  i5 : close(F)
│ │ │  
│ │ │ -o5 = /tmp/M2-15935-0/0
│ │ │ +o5 = /tmp/M2-20955-0/0
│ │ │  
│ │ │  o5 : File
│ │ │  
│ │ │  i6 : getMatrix(s)
│ │ │  
│ │ │  o6 = | 1 1 1 1 |
│ │ │       | 1 2 3 4 |
│ │ ├── ./usr/share/doc/Macaulay2/FourTiTwo/html/_put__Matrix.html
│ │ │ @@ -79,36 +79,36 @@
│ │ │  o1 : Matrix ZZ  <-- ZZ
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : s = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-15935-0/0
    
│ │ │ +o2 = /tmp/M2-20955-0/0
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : F = openOut(s)
│ │ │  
│ │ │ -o3 = /tmp/M2-15935-0/0
│ │ │ +o3 = /tmp/M2-20955-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : putMatrix(F,A)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : close(F)
│ │ │  
│ │ │ -o5 = /tmp/M2-15935-0/0
│ │ │ +o5 = /tmp/M2-20955-0/0
│ │ │  
│ │ │  o5 : File
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : getMatrix(s)
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,24 +16,24 @@
│ │ │ │  o1 = | 1 1 1 1 |
│ │ │ │       | 1 2 3 4 |
│ │ │ │  
│ │ │ │                2       4
│ │ │ │  o1 : Matrix ZZ  <-- ZZ
│ │ │ │  i2 : s = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-15935-0/0
│ │ │ │ +o2 = /tmp/M2-20955-0/0
│ │ │ │  i3 : F = openOut(s)
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-15935-0/0
│ │ │ │ +o3 = /tmp/M2-20955-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : putMatrix(F,A)
│ │ │ │  i5 : close(F)
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-15935-0/0
│ │ │ │ +o5 = /tmp/M2-20955-0/0
│ │ │ │  
│ │ │ │  o5 : File
│ │ │ │  i6 : getMatrix(s)
│ │ │ │  
│ │ │ │  o6 = | 1 1 1 1 |
│ │ │ │       | 1 2 3 4 |
│ │ ├── ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_fpt.out
│ │ │ @@ -155,31 +155,31 @@
│ │ │  i26 : numeric fpt(f, DepthOfSearch => 3, FinalAttempt => true) -- FinalAttempt improves the estimate slightly
│ │ │  
│ │ │  o26 = {.142067, .144}
│ │ │  
│ │ │  o26 : List
│ │ │  
│ │ │  i27 : time numeric fpt(f, DepthOfSearch => 3, FinalAttempt => true)
│ │ │ - -- used 2.21552s (cpu); 1.26312s (thread); 0s (gc)
│ │ │ + -- used 2.41009s (cpu); 1.42064s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = {.142067, .144}
│ │ │  
│ │ │  o27 : List
│ │ │  
│ │ │  i28 : time fpt(f, DepthOfSearch => 3, Attempts => 7)
│ │ │ - -- used 1.34032s (cpu); 0.819238s (thread); 0s (gc)
│ │ │ + -- used 1.38069s (cpu); 0.852285s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o28 = -
│ │ │        7
│ │ │  
│ │ │  o28 : QQ
│ │ │  
│ │ │  i29 : time fpt(f, DepthOfSearch => 4)
│ │ │ - -- used 1.09779s (cpu); 0.697093s (thread); 0s (gc)
│ │ │ + -- used 1.12473s (cpu); 0.686062s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o29 = -
│ │ │        7
│ │ │  
│ │ │  o29 : QQ
│ │ ├── ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_frobenius__Nu.out
│ │ │ @@ -43,34 +43,34 @@
│ │ │  o12 = 220
│ │ │  
│ │ │  i13 : R = ZZ/17[x,y,z];
│ │ │  
│ │ │  i14 : f = x^3 + y^4 + z^5; -- a diagonal polynomial
│ │ │  
│ │ │  i15 : time frobeniusNu(3, f)
│ │ │ - -- used 0.00399957s (cpu); 0.00414499s (thread); 0s (gc)
│ │ │ + -- used 0.00404505s (cpu); 0.00549768s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3756
│ │ │  
│ │ │  i16 : time frobeniusNu(3, f, UseSpecialAlgorithms => false)
│ │ │ - -- used 0.512037s (cpu); 0.349891s (thread); 0s (gc)
│ │ │ + -- used 0.505327s (cpu); 0.30174s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 3756
│ │ │  
│ │ │  i17 : R = ZZ/5[x,y,z];
│ │ │  
│ │ │  i18 : f = x^3 + y^3 + z^3 + x*y*z;
│ │ │  
│ │ │  i19 : time frobeniusNu(4, f) -- ContainmentTest is set to FrobeniusRoot, by default
│ │ │ - -- used 0.302679s (cpu); 0.200173s (thread); 0s (gc)
│ │ │ + -- used 0.359796s (cpu); 0.22315s (thread); 0s (gc)
│ │ │  
│ │ │  o19 = 499
│ │ │  
│ │ │  i20 : time frobeniusNu(4, f, ContainmentTest => StandardPower)
│ │ │ - -- used 1.56248s (cpu); 1.20583s (thread); 0s (gc)
│ │ │ + -- used 1.4176s (cpu); 1.20019s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = 499
│ │ │  
│ │ │  i21 : R = ZZ/3[x,y];
│ │ │  
│ │ │  i22 : M = ideal(x, y);
│ │ │  
│ │ │ @@ -85,34 +85,34 @@
│ │ │  o24 = 8
│ │ │  
│ │ │  i25 : R = ZZ/5[x,y,z];
│ │ │  
│ │ │  i26 : f = x^2*y^4 + y^2*z^7 + z^2*x^8;
│ │ │  
│ │ │  i27 : time frobeniusNu(5, f) -- uses binary search (default)
│ │ │ - -- used 1.02964s (cpu); 0.633188s (thread); 0s (gc)
│ │ │ + -- used 1.27629s (cpu); 0.733774s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = 1124
│ │ │  
│ │ │  i28 : time frobeniusNu(5, f, Search => Linear)
│ │ │ - -- used 1.53299s (cpu); 0.899133s (thread); 0s (gc)
│ │ │ + -- used 1.84728s (cpu); 1.07549s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 1124
│ │ │  
│ │ │  i29 : M = ideal(x, y, z);
│ │ │  
│ │ │  o29 : Ideal of R
│ │ │  
│ │ │  i30 : time frobeniusNu(2, M, M^2) -- uses binary search (default)
│ │ │ - -- used 2.15426s (cpu); 1.79821s (thread); 0s (gc)
│ │ │ + -- used 1.74161s (cpu); 1.43506s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 97
│ │ │  
│ │ │  i31 : time frobeniusNu(2, M, M^2, Search => Linear) -- but linear search gets luckier
│ │ │ - -- used 0.692075s (cpu); 0.57169s (thread); 0s (gc)
│ │ │ + -- used 0.553469s (cpu); 0.492482s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 97
│ │ │  
│ │ │  i32 : R = ZZ/7[x,y];
│ │ │  
│ │ │  i33 : f = (x - 1)^3 - (y - 2)^2;
│ │ ├── ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_fpt.html
│ │ │ @@ -363,37 +363,37 @@
│ │ │          
    
│ │ │            
    The computations performed when FinalAttempt is set to true are often slow, and often fail to improve the estimate, and for this reason, this option should be used sparingly. It is often more effective to increase the values of Attempts or DepthOfSearch, instead.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -228,29 +228,29 @@
│ │ │ │  
│ │ │ │  o26 : List
│ │ │ │  The computations performed when FinalAttempt is set to true are often slow, and
│ │ │ │  often fail to improve the estimate, and for this reason, this option should be
│ │ │ │  used sparingly. It is often more effective to increase the values of Attempts
│ │ │ │  or DepthOfSearch, instead.
│ │ │ │  i27 : time numeric fpt(f, DepthOfSearch => 3, FinalAttempt => true)
│ │ │ │ - -- used 2.21552s (cpu); 1.26312s (thread); 0s (gc)
│ │ │ │ + -- used 2.41009s (cpu); 1.42064s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = {.142067, .144}
│ │ │ │  
│ │ │ │  o27 : List
│ │ │ │  i28 : time fpt(f, DepthOfSearch => 3, Attempts => 7)
│ │ │ │ - -- used 1.34032s (cpu); 0.819238s (thread); 0s (gc)
│ │ │ │ + -- used 1.38069s (cpu); 0.852285s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        1
│ │ │ │  o28 = -
│ │ │ │        7
│ │ │ │  
│ │ │ │  o28 : QQ
│ │ │ │  i29 : time fpt(f, DepthOfSearch => 4)
│ │ │ │ - -- used 1.09779s (cpu); 0.697093s (thread); 0s (gc)
│ │ │ │ + -- used 1.12473s (cpu); 0.686062s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        1
│ │ │ │  o29 = -
│ │ │ │        7
│ │ │ │  
│ │ │ │  o29 : QQ
│ │ │ │  As seen in several examples above, when the exact answer is not found, a list
│ │ ├── ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_frobenius__Nu.html
│ │ │ @@ -192,23 +192,23 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i27 : time numeric fpt(f, DepthOfSearch => 3, FinalAttempt => true)
│ │ │ - -- used 2.21552s (cpu); 1.26312s (thread); 0s (gc)
│ │ │ + -- used 2.41009s (cpu); 1.42064s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = {.142067, .144}
│ │ │  
│ │ │  o27 : List
    
│ │ │              
    
│ │ │                
    i28 : time fpt(f, DepthOfSearch => 3, Attempts => 7)
│ │ │ - -- used 1.34032s (cpu); 0.819238s (thread); 0s (gc)
│ │ │ + -- used 1.38069s (cpu); 0.852285s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o28 = -
│ │ │        7
│ │ │  
│ │ │  o28 : QQ
    
│ │ │              
    
│ │ │                
    i29 : time fpt(f, DepthOfSearch => 4)
│ │ │ - -- used 1.09779s (cpu); 0.697093s (thread); 0s (gc)
│ │ │ + -- used 1.12473s (cpu); 0.686062s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o29 = -
│ │ │        7
│ │ │  
│ │ │  o29 : QQ
    
│ │ │                  		
│ │ │                
    i14 : f = x^3 + y^4 + z^5; -- a diagonal polynomial
    
│ │ │              
    
│ │ │                
    i15 : time frobeniusNu(3, f)
│ │ │ - -- used 0.00399957s (cpu); 0.00414499s (thread); 0s (gc)
│ │ │ + -- used 0.00404505s (cpu); 0.00549768s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3756
    
│ │ │              
    
│ │ │                
    i16 : time frobeniusNu(3, f, UseSpecialAlgorithms => false)
│ │ │ - -- used 0.512037s (cpu); 0.349891s (thread); 0s (gc)
│ │ │ + -- used 0.505327s (cpu); 0.30174s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 3756
    
│ │ │              
    
│ │ │          
    
│ │ │            
    The valid values for the option ContainmentTest are FrobeniusPower, FrobeniusRoot, and StandardPower. The default value of this option depends on what is passed to frobeniusNu. Indeed, by default, ContainmentTest is set to FrobeniusRoot if frobeniusNu is passed a ring element $f$, and is set to StandardPower if frobeniusNu is passed an ideal $I$. We describe the consequences of setting ContainmentTest to each of these values below.
    
│ │ │ @@ -225,23 +225,23 @@
│ │ │              
│ │ │                
    i18 : f = x^3 + y^3 + z^3 + x*y*z;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i19 : time frobeniusNu(4, f) -- ContainmentTest is set to FrobeniusRoot, by default
│ │ │ - -- used 0.302679s (cpu); 0.200173s (thread); 0s (gc)
│ │ │ + -- used 0.359796s (cpu); 0.22315s (thread); 0s (gc)
│ │ │  
│ │ │  o19 = 499
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i20 : time frobeniusNu(4, f, ContainmentTest => StandardPower)
│ │ │ - -- used 1.56248s (cpu); 1.20583s (thread); 0s (gc)
│ │ │ + -- used 1.4176s (cpu); 1.20019s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = 499
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    Finally, when ContainmentTest is set to FrobeniusPower, then instead of producing the invariant $\nu_I^J(p^e)$ as defined above, frobeniusNu instead outputs the maximal integer $n$ such that the $n$^{th} (generalized) Frobenius power of $I$ is not contained in the $p^e$-th Frobenius power of $J$. Here, the $n$^{th} Frobenius power of $I$, when $n$ is a nonnegative integer, is as defined in the paper Frobenius Powers by Hernández, Teixeira, and Witt, which can be computed with the function frobeniusPower, from the TestIdeals package. In particular, frobeniusNu(e,I,J) and frobeniusNu(e,I,J,ContainmentTest=>FrobeniusPower) need not agree. However, they will agree when $I$ is a principal ideal.
    
│ │ │ @@ -287,46 +287,46 @@
│ │ │              
│ │ │                
    i26 : f = x^2*y^4 + y^2*z^7 + z^2*x^8;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i27 : time frobeniusNu(5, f) -- uses binary search (default)
│ │ │ - -- used 1.02964s (cpu); 0.633188s (thread); 0s (gc)
│ │ │ + -- used 1.27629s (cpu); 0.733774s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = 1124
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i28 : time frobeniusNu(5, f, Search => Linear)
│ │ │ - -- used 1.53299s (cpu); 0.899133s (thread); 0s (gc)
│ │ │ + -- used 1.84728s (cpu); 1.07549s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 1124
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i29 : M = ideal(x, y, z);
│ │ │  
│ │ │  o29 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i30 : time frobeniusNu(2, M, M^2) -- uses binary search (default)
│ │ │ - -- used 2.15426s (cpu); 1.79821s (thread); 0s (gc)
│ │ │ + -- used 1.74161s (cpu); 1.43506s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 97
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i31 : time frobeniusNu(2, M, M^2, Search => Linear) -- but linear search gets luckier
│ │ │ - -- used 0.692075s (cpu); 0.57169s (thread); 0s (gc)
│ │ │ + -- used 0.553469s (cpu); 0.492482s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 97
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    The option AtOrigin (default value true) can be turned off to tell frobeniusNu to effectively do the computation over all possible maximal ideals $J$ and take the minimum.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -106,19 +106,19 @@
│ │ │ │  algorithms, namely diagonal polynomials, binomials, forms in two variables, and
│ │ │ │  polynomials whose factors are in simple normal crossing. This feature can be
│ │ │ │  disabled by setting the option UseSpecialAlgorithms (default value true) to
│ │ │ │  false.
│ │ │ │  i13 : R = ZZ/17[x,y,z];
│ │ │ │  i14 : f = x^3 + y^4 + z^5; -- a diagonal polynomial
│ │ │ │  i15 : time frobeniusNu(3, f)
│ │ │ │ - -- used 0.00399957s (cpu); 0.00414499s (thread); 0s (gc)
│ │ │ │ + -- used 0.00404505s (cpu); 0.00549768s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 3756
│ │ │ │  i16 : time frobeniusNu(3, f, UseSpecialAlgorithms => false)
│ │ │ │ - -- used 0.512037s (cpu); 0.349891s (thread); 0s (gc)
│ │ │ │ + -- used 0.505327s (cpu); 0.30174s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = 3756
│ │ │ │  The valid values for the option ContainmentTest are FrobeniusPower,
│ │ │ │  FrobeniusRoot, and StandardPower. The default value of this option depends on
│ │ │ │  what is passed to frobeniusNu. Indeed, by default, ContainmentTest is set to
│ │ │ │  FrobeniusRoot if frobeniusNu is passed a ring element $f$, and is set to
│ │ │ │  StandardPower if frobeniusNu is passed an ideal $I$. We describe the
│ │ │ │ @@ -133,19 +133,19 @@
│ │ │ │  is contained in $J$. The output is unaffected, but this option often speeds up
│ │ │ │  computations, specially when a polynomial or principal ideal is passed as the
│ │ │ │  second argument.
│ │ │ │  i17 : R = ZZ/5[x,y,z];
│ │ │ │  i18 : f = x^3 + y^3 + z^3 + x*y*z;
│ │ │ │  i19 : time frobeniusNu(4, f) -- ContainmentTest is set to FrobeniusRoot, by
│ │ │ │  default
│ │ │ │ - -- used 0.302679s (cpu); 0.200173s (thread); 0s (gc)
│ │ │ │ + -- used 0.359796s (cpu); 0.22315s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o19 = 499
│ │ │ │  i20 : time frobeniusNu(4, f, ContainmentTest => StandardPower)
│ │ │ │ - -- used 1.56248s (cpu); 1.20583s (thread); 0s (gc)
│ │ │ │ + -- used 1.4176s (cpu); 1.20019s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o20 = 499
│ │ │ │  Finally, when ContainmentTest is set to FrobeniusPower, then instead of
│ │ │ │  producing the invariant $\nu_I^J(p^e)$ as defined above, frobeniusNu instead
│ │ │ │  outputs the maximal integer $n$ such that the $n$^{th} (generalized) Frobenius
│ │ │ │  power of $I$ is not contained in the $p^e$-th Frobenius power of $J$. Here, the
│ │ │ │  $n$^{th} Frobenius power of $I$, when $n$ is a nonnegative integer, is as
│ │ │ │ @@ -167,31 +167,31 @@
│ │ │ │  The function frobeniusNu works by searching through the list of potential
│ │ │ │  integers $n$ and checking containments of $I^n$ in the specified Frobenius
│ │ │ │  power of $J$. The way this search is approached is specified by the option
│ │ │ │  Search, which can be set to Binary (the default value) or Linear.
│ │ │ │  i25 : R = ZZ/5[x,y,z];
│ │ │ │  i26 : f = x^2*y^4 + y^2*z^7 + z^2*x^8;
│ │ │ │  i27 : time frobeniusNu(5, f) -- uses binary search (default)
│ │ │ │ - -- used 1.02964s (cpu); 0.633188s (thread); 0s (gc)
│ │ │ │ + -- used 1.27629s (cpu); 0.733774s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = 1124
│ │ │ │  i28 : time frobeniusNu(5, f, Search => Linear)
│ │ │ │ - -- used 1.53299s (cpu); 0.899133s (thread); 0s (gc)
│ │ │ │ + -- used 1.84728s (cpu); 1.07549s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o28 = 1124
│ │ │ │  i29 : M = ideal(x, y, z);
│ │ │ │  
│ │ │ │  o29 : Ideal of R
│ │ │ │  i30 : time frobeniusNu(2, M, M^2) -- uses binary search (default)
│ │ │ │ - -- used 2.15426s (cpu); 1.79821s (thread); 0s (gc)
│ │ │ │ + -- used 1.74161s (cpu); 1.43506s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o30 = 97
│ │ │ │  i31 : time frobeniusNu(2, M, M^2, Search => Linear) -- but linear search gets
│ │ │ │  luckier
│ │ │ │ - -- used 0.692075s (cpu); 0.57169s (thread); 0s (gc)
│ │ │ │ + -- used 0.553469s (cpu); 0.492482s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o31 = 97
│ │ │ │  The option AtOrigin (default value true) can be turned off to tell frobeniusNu
│ │ │ │  to effectively do the computation over all possible maximal ideals $J$ and take
│ │ │ │  the minimum.
│ │ │ │  i32 : R = ZZ/7[x,y];
│ │ │ │  i33 : f = (x - 1)^3 - (y - 2)^2;
│ │ ├── ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_orbit__Closure.out
│ │ │ @@ -208,21 +208,21 @@
│ │ │        | 3/7 5/4 3/7  10   |
│ │ │        | 6/7 2/9 5    3/2  |
│ │ │  
│ │ │                 3       4
│ │ │  o26 : Matrix QQ  <-- QQ
│ │ │  
│ │ │  i27 : time C = orbitClosure(X,Mat)
│ │ │ - -- used 0.595776s (cpu); 0.359545s (thread); 0s (gc)
│ │ │ + -- used 2.01368s (cpu); 0.536994s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o27 : KClass
│ │ │  
│ │ │  i28 : time C = orbitClosure(X,Mat, RREFMethod => true)
│ │ │ - -- used 1.80107s (cpu); 1.03657s (thread); 0s (gc)
│ │ │ + -- used 3.22205s (cpu); 1.04064s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o28 : KClass
│ │ │  
│ │ │  i29 :
│ │ ├── ./usr/share/doc/Macaulay2/GKMVarieties/html/_orbit__Closure.html
│ │ │ @@ -386,25 +386,25 @@
│ │ │                 3       4
│ │ │  o26 : Matrix QQ  <-- QQ
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i27 : time C = orbitClosure(X,Mat)
│ │ │ - -- used 0.595776s (cpu); 0.359545s (thread); 0s (gc)
│ │ │ + -- used 2.01368s (cpu); 0.536994s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o27 : KClass
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i28 : time C = orbitClosure(X,Mat, RREFMethod => true)
│ │ │ - -- used 1.80107s (cpu); 1.03657s (thread); 0s (gc)
│ │ │ + -- used 3.22205s (cpu); 1.04064s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o28 : KClass
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -241,21 +241,21 @@
│ │ │ │  o26 = | 7   6   3/10 10/9 |
│ │ │ │        | 3/7 5/4 3/7  10   |
│ │ │ │        | 6/7 2/9 5    3/2  |
│ │ │ │  
│ │ │ │                 3       4
│ │ │ │  o26 : Matrix QQ  <-- QQ
│ │ │ │  i27 : time C = orbitClosure(X,Mat)
│ │ │ │ - -- used 0.595776s (cpu); 0.359545s (thread); 0s (gc)
│ │ │ │ + -- used 2.01368s (cpu); 0.536994s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = an "equivariant K-class" on a GKM variety
│ │ │ │  
│ │ │ │  o27 : KClass
│ │ │ │  i28 : time C = orbitClosure(X,Mat, RREFMethod => true)
│ │ │ │ - -- used 1.80107s (cpu); 1.03657s (thread); 0s (gc)
│ │ │ │ + -- used 3.22205s (cpu); 1.04064s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o28 = an "equivariant K-class" on a GKM variety
│ │ │ │  
│ │ │ │  o28 : KClass
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _g_e_n_e_r_a_l_i_z_e_d_F_l_a_g_V_a_r_i_e_t_y -- makes a generalized flag variety as a GKM
│ │ │ │        variety
│ │ ├── ./usr/share/doc/Macaulay2/Graphs/example-output/_new__Digraph.out
│ │ │ @@ -32,12 +32,12 @@
│ │ │                                           5 => {6}
│ │ │                                           6 => {}
│ │ │  
│ │ │  o2 : SortedDigraph
│ │ │  
│ │ │  i3 : keys H
│ │ │  
│ │ │ -o3 = {map, newDigraph, digraph}
│ │ │ +o3 = {map, digraph, newDigraph}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Graphs/html/_new__Digraph.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │  o2 : SortedDigraph
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : keys H
│ │ │  
│ │ │ -o3 = {map, newDigraph, digraph}
│ │ │ +o3 = {map, digraph, newDigraph}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -36,15 +36,15 @@
│ │ │ │                                           4 => {}
│ │ │ │                                           5 => {6}
│ │ │ │                                           6 => {}
│ │ │ │  
│ │ │ │  o2 : SortedDigraph
│ │ │ │  i3 : keys H
│ │ │ │  
│ │ │ │ -o3 = {map, newDigraph, digraph}
│ │ │ │ +o3 = {map, digraph, newDigraph}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_o_p_S_o_r_t -- topologically sort the vertices of a digraph
│ │ │ │      * _S_o_r_t_e_d_D_i_g_r_a_p_h -- hashtable used in topSort
│ │ │ │      * _t_o_p_o_l_o_g_i_c_a_l_S_o_r_t -- outputs a list of vertices in a topologically sorted
│ │ │ │        order of a DAG.
│ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_nonminimal__Maps.out
│ │ │ @@ -103,46 +103,46 @@
│ │ │  
│ │ │  i13 : #compsJ
│ │ │  
│ │ │  o13 = 2
│ │ │  
│ │ │  i14 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │  
│ │ │ -o14 = | 42 9 39 9 34 7 -12 -17 -29 -35 50 2 13 19 -44 50 2 -29 15 2 -27 21
│ │ │ +o14 = | -6 48 44 -23 -2 -11 -35 -26 27 -43 48 27 15 -22 25 -16 34 -29 46 -20
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -36 -29 -39 -10 24 -16 19 -29 39 -38 -22 -8 -30 -24 |
│ │ │ +      40 21 -30 -38 -19 -8 -36 39 19 -29 -16 -29 -10 19 24 -24 |
│ │ │  
│ │ │                 1       36
│ │ │  o14 : Matrix kk  <-- kk
│ │ │  
│ │ │  i15 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │  
│ │ │ -o15 = | 30 10 43 20 -39 23 -30 40 -34 22 46 -25 21 -18 -35 -1 21 -39 -45 16
│ │ │ +o15 = | -48 -46 16 17 -1 -43 15 -1 12 -18 -6 -28 14 -28 -9 32 -22 -39 6 -47
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -35 -5 19 -47 -20 -13 34 33 -28 -43 22 2 0 -15 -47 38 |
│ │ │ +      28 -37 -47 38 -16 -15 34 27 -13 -43 22 16 0 -18 19 2 |
│ │ │  
│ │ │                 1       36
│ │ │  o15 : Matrix kk  <-- kk
│ │ │  
│ │ │  i16 : F1 = sub(F, (vars S)|pt1)
│ │ │  
│ │ │ -              2              2                            2               
│ │ │ -o16 = ideal (a  - 44b*c - 35c  + 2a*d + 7b*d + 39c*d + 42d , a*b - 39b*c +
│ │ │ +              2              2                             2               
│ │ │ +o16 = ideal (a  + 25b*c - 43c  + 27a*d - 11b*d + 44c*d - 6d , a*b - 19b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2                   2                 
│ │ │ -      15c  - 27a*d + 13b*d - 29c*d + 9d , a*c - 38b*c - 10c  - 16a*d + 2b*d +
│ │ │ +         2                              2                  2                
│ │ │ +      46c  + 40a*d + 15b*d + 27c*d + 48d , a*c - 29b*c - 8c  + 39a*d - 20b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2   2              2                              2     2  
│ │ │ -      19c*d + 34d , b  - 30b*c + 19c  - 22a*d + 21b*d + 50c*d - 12d , b*c  -
│ │ │ +                  2   2              2                              2     2  
│ │ │ +      - 22c*d - 2d , b  + 24b*c + 19c  - 10a*d + 21b*d - 16c*d - 35d , b*c  -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2         2       2        2     3   3                2   
│ │ │ -      29b*c*d - 36c d + 24a*d  + 2b*d  + 50c*d  + 9d , c  - 24b*c*d + 39c d -
│ │ │ +                   2         2        2        2      3   3                2 
│ │ │ +      29b*c*d - 30c d - 36a*d  + 34b*d  + 48c*d  - 23d , c  - 24b*c*d - 16c d
│ │ │        -----------------------------------------------------------------------
│ │ │ -          2        2        2      3
│ │ │ -      8a*d  - 29b*d  - 29c*d  - 17d )
│ │ │ +             2        2        2      3
│ │ │ +      + 19a*d  - 38b*d  - 29c*d  - 26d )
│ │ │  
│ │ │  o16 : Ideal of S
│ │ │  
│ │ │  i17 : betti res F1
│ │ │  
│ │ │               0 1 2 3
│ │ │  o17 = total: 1 6 8 3
│ │ │ @@ -150,28 +150,28 @@
│ │ │            1: . 4 4 1
│ │ │            2: . 2 4 2
│ │ │  
│ │ │  o17 : BettiTally
│ │ │  
│ │ │  i18 : F2 = sub(F, (vars S)|pt2)
│ │ │  
│ │ │ -              2              2                              2               
│ │ │ -o18 = ideal (a  - 35b*c + 22c  - 25a*d + 23b*d + 43c*d + 30d , a*b - 20b*c -
│ │ │ +              2             2                              2               
│ │ │ +o18 = ideal (a  - 9b*c - 18c  - 28a*d - 43b*d + 16c*d - 48d , a*b - 16b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                              2                  2                
│ │ │ -      45c  - 35a*d + 21b*d - 34c*d + 10d , a*c + 2b*c - 13c  + 33a*d + 16b*d
│ │ │ +        2                              2                   2                
│ │ │ +      6c  + 28a*d + 14b*d + 12c*d - 46d , a*c + 16b*c - 15c  + 27a*d - 47b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2   2              2                   2     2            
│ │ │ -      - 18c*d - 39d , b  - 47b*c - 28c  - 5b*d - c*d - 30d , b*c  - 43b*c*d +
│ │ │ +                 2   2              2                      2     2          
│ │ │ +      - 28c*d - d , b  + 19b*c - 13c  - 37b*d + 32c*d + 15d , b*c  - 43b*c*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2         2        2        2      3   3                2         2
│ │ │ -      19c d + 34a*d  + 21b*d  + 46c*d  + 20d , c  + 38b*c*d + 22c d - 15a*d 
│ │ │ +           2         2        2       2      3   3               2         2
│ │ │ +      - 47c d + 34a*d  - 22b*d  - 6c*d  + 17d , c  + 2b*c*d + 22c d - 18a*d 
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2      3
│ │ │ -      - 47b*d  - 39c*d  + 40d )
│ │ │ +             2        2    3
│ │ │ +      + 38b*d  - 39c*d  - d )
│ │ │  
│ │ │  o18 : Ideal of S
│ │ │  
│ │ │  i19 : betti res F2
│ │ │  
│ │ │               0 1 2 3
│ │ │  o19 = total: 1 6 8 3
│ │ │ @@ -179,27 +179,30 @@
│ │ │            1: . 4 4 1
│ │ │            2: . 2 4 2
│ │ │  
│ │ │  o19 : BettiTally
│ │ │  
│ │ │  i20 : netList decompose F1
│ │ │  
│ │ │ -      +------------------------------------------------------+
│ │ │ -o20 = |ideal (c - 13d, b + 32d, a + 36d)                     |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |ideal (c - 16d, b + d, a + 16d)                       |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |                                    2            2    |
│ │ │ -      |ideal (b - 6c + 33d, a - 36c + 2d, c  + 43c*d - d )   |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |                                     2              2 |
│ │ │ -      |ideal (b + 29c + 7d, a - 19c + 24d, c  - 20c*d - 30d )|
│ │ │ -      +------------------------------------------------------+
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +o20 = |ideal (c + 39d, b + 27d, a - 18d)                                                                                                                        |
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                            2              2                     2   3                2        2        2      3     2                2         2      3 |
│ │ │ +      |ideal (a - 29b - 8c - 13d, b  + 24b*c + 19c  + 34b*d + 5c*d + 37d , c  - 24b*c*d - 16c d + 8b*d  + 22c*d  + 19d , b*c  - 29b*c*d - 30c d - 38c*d  + 14d )|
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  
│ │ │  i21 : netList decompose F2
│ │ │  
│ │ │ -      +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                       2                              2   2              2                   2                   2                              2   2              2                              2   3                2         2        2        2      3     2                2         2        2        2      3 |
│ │ │ -o21 = |ideal (a*c + 2b*c - 13c  + 33a*d + 16b*d - 18c*d - 39d , b  - 47b*c - 28c  - 5b*d - c*d - 30d , a*b - 20b*c - 45c  - 35a*d + 21b*d - 34c*d + 10d , a  - 35b*c + 22c  - 25a*d + 23b*d + 43c*d + 30d , c  + 38b*c*d + 22c d - 15a*d  - 47b*d  - 39c*d  + 40d , b*c  - 43b*c*d + 19c d + 34a*d  + 21b*d  + 46c*d  + 20d )|
│ │ │ -      +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      +-------------------------------------------------------+
│ │ │ +o21 = |ideal (c - 32d, b - 5d, a - 29d)                       |
│ │ │ +      +-------------------------------------------------------+
│ │ │ +      |ideal (c + 43d, b - 47d, a - 27d)                      |
│ │ │ +      +-------------------------------------------------------+
│ │ │ +      |ideal (c + 24d, b - 49d, a)                            |
│ │ │ +      +-------------------------------------------------------+
│ │ │ +      |ideal (c + 14d, b + 31d, a - 16d)                      |
│ │ │ +      +-------------------------------------------------------+
│ │ │ +      |                                      2              2 |
│ │ │ +      |ideal (b + 11c + 22d, a + 11c + 42d, c  - 43c*d + 31d )|
│ │ │ +      +-------------------------------------------------------+
│ │ │  
│ │ │  i22 :
│ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_random__Point__On__Rational__Variety_lp__Ideal_rp.out
│ │ │ @@ -200,67 +200,72 @@
│ │ │  
│ │ │  o12 = {11, 8}
│ │ │  
│ │ │  o12 : List
│ │ │  
│ │ │  i13 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │  
│ │ │ -o13 = | 50 15 46 -33 2 -43 -46 8 33 19 -2 -18 -8 -22 43 -29 19 3 -16 -29 -38
│ │ │ +o13 = | 13 48 43 23 41 36 -4 -12 -30 -16 -33 -36 19 19 30 -10 -38 32 -29 -8
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -24 -10 -29 |
│ │ │ +      -29 -22 -29 -24 |
│ │ │  
│ │ │                 1       24
│ │ │  o13 : Matrix kk  <-- kk
│ │ │  
│ │ │  i14 : F1 = sub(F, (vars S)|pt1)
│ │ │  
│ │ │ -              2              2                             2               
│ │ │ -o14 = ideal (a  + 33b*c - 33c  + 19a*d + 2b*d + 15c*d + 50d , a*b + 43b*c -
│ │ │ +              2              2                              2               
│ │ │ +o14 = ideal (a  - 30b*c + 23c  - 16a*d + 41b*d + 48c*d + 13d , a*b + 30b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -        2                              2                   2                 
│ │ │ -      2c  - 29a*d - 18b*d - 43c*d + 46d , a*c - 38b*c + 19c  - 24a*d + 3b*d -
│ │ │ +         2                              2                   2                
│ │ │ +      33c  - 10a*d - 36b*d + 36c*d + 43d , a*c - 29b*c - 38c  - 22a*d + 32b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                2   2              2                             2
│ │ │ -      8c*d - 46d , b  - 10b*c - 16c  - 29a*d - 29b*d - 22c*d + 8d )
│ │ │ +                  2   2              2                             2
│ │ │ +      + 19c*d - 4d , b  - 29b*c - 29c  - 24a*d - 8b*d + 19c*d - 12d )
│ │ │  
│ │ │  o14 : Ideal of S
│ │ │  
│ │ │  i15 : decompose F1
│ │ │  
│ │ │ -                                    2              2                      2
│ │ │ -o15 = {ideal (a - 38b + 19c + 44d, b  - 10b*c - 16c  - 20b*d + 24c*d - 29d ),
│ │ │ +                                   2              2                     2
│ │ │ +o15 = {ideal (a - 29b - 38c - 9d, b  - 29b*c - 29c  + 3b*d + 16c*d - 26d ),
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ideal (c - 24d, b - 38d, a + 15d)}
│ │ │ +      ideal (c - 22d, b - 21d, a + 8d)}
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │  
│ │ │ -o16 = | -14 40 -5 26 -48 -26 -35 41 -8 -15 -38 31 -13 29 21 16 39 21 -18 19
│ │ │ +o16 = | 46 -2 16 -20 -1 -30 -43 -41 17 -4 -16 -29 -39 40 49 -39 -18 -13 -47
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -47 -39 34 0 |
│ │ │ +      34 19 21 39 0 |
│ │ │  
│ │ │                 1       24
│ │ │  o16 : Matrix kk  <-- kk
│ │ │  
│ │ │  i17 : F2 = sub(F, (vars S)|pt2)
│ │ │  
│ │ │ -              2             2                              2               
│ │ │ -o17 = ideal (a  - 8b*c + 26c  - 15a*d - 48b*d + 40c*d - 14d , a*b + 21b*c -
│ │ │ +              2              2                          2                   2
│ │ │ +o17 = ideal (a  + 17b*c - 20c  - 4a*d - b*d - 2c*d + 46d , a*b + 49b*c - 16c 
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2                   2                
│ │ │ -      38c  + 16a*d + 31b*d - 26c*d - 5d , a*c - 47b*c + 39c  - 39a*d + 21b*d
│ │ │ +                                   2                   2                  
│ │ │ +      - 39a*d - 29b*d - 30c*d + 16d , a*c + 19b*c - 18c  + 21a*d - 13b*d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2   2              2                      2
│ │ │ -      - 13c*d - 35d , b  + 34b*c - 18c  + 19b*d + 29c*d + 41d )
│ │ │ +                 2   2              2                      2
│ │ │ +      39c*d - 43d , b  + 39b*c - 47c  + 34b*d + 40c*d - 41d )
│ │ │  
│ │ │  o17 : Ideal of S
│ │ │  
│ │ │  i18 : decompose F2
│ │ │  
│ │ │ -o18 = {ideal (b + 19c - 18d, a + 23c + 43d), ideal (b + 15c + 37d, a + 37c +
│ │ │ +                               2                              2   2          
│ │ │ +o18 = {ideal (a*c + 19b*c - 18c  + 21a*d - 13b*d - 39c*d - 43d , b  + 39b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -      26d)}
│ │ │ +         2                      2                   2                        
│ │ │ +      47c  + 34b*d + 40c*d - 41d , a*b + 49b*c - 16c  - 39a*d - 29b*d - 30c*d
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +           2   2              2                          2
│ │ │ +      + 16d , a  + 17b*c - 20c  - 4a*d - b*d - 2c*d + 46d )}
│ │ │  
│ │ │  o18 : List
│ │ │  
│ │ │  i19 :
│ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_random__Points__On__Rational__Variety_lp__Ideal_cm__Z__Z_rp.out
│ │ │ @@ -68,54 +68,54 @@
│ │ │  
│ │ │  o12 = {11, 8}
│ │ │  
│ │ │  o12 : List
│ │ │  
│ │ │  i13 : netList randomPointsOnRationalVariety(compsJ_0, 10)
│ │ │  
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -o13 = || 29 -40 15 -49 3 -13 -6 -39 2 39 47 15 19 -47 -46 -39 -16 32 -43 34 -13 -18 21 -38 | |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || 37 -7 -24 8 -26 38 9 -31 24 -47 -34 12 16 22 -22 45 -28 16 -47 2 -48 -34 38 -15 |   |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || 6 1 -31 -7 44 8 -50 24 -48 -16 23 23 -23 39 -5 43 19 -15 48 15 -11 -17 7 47 |       |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || -41 -49 6 -16 -12 31 23 6 -7 11 3 -42 40 11 -28 46 35 -28 -3 33 1 -28 -38 36 |      |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || -11 -27 -4 40 -34 6 44 -2 19 -23 -29 21 29 -47 -37 15 -47 -24 -10 2 -13 -37 -7 22 | |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || -50 42 20 -30 -46 -48 -5 40 -47 39 13 47 32 -9 41 -32 -18 25 -30 -22 24 -20 27 30 | |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || 50 22 -30 3 -43 -29 -33 -18 6 39 -29 24 -49 -33 -15 -19 -15 -37 44 33 -20 17 0 -48 ||
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || -9 31 -37 -42 -7 -8 -11 -21 12 9 13 -9 13 -26 11 22 36 34 -8 4 -11 -49 -39 -39 |    |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || 47 14 -11 -16 -20 -40 42 5 -2 36 8 -45 -30 41 -26 16 -8 -34 35 -22 -6 -28 -3 43 |   |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || 23 -8 -3 -17 38 0 11 -33 -7 6 -31 -4 -31 25 6 -2 -35 -11 -13 3 -49 -41 40 -9 |      |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -
│ │ │ -i14 : netList randomPointsOnRationalVariety(compsJ_1, 10)
│ │ │ -
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -o14 = || 38 -31 49 39 4 46 -29 -5 -39 -40 14 -11 -31 46 43 -26 4 30 -35 27 -40 37 -47 0 |   |
│ │ │ +o13 = || 13 15 3 36 2 48 44 -35 -34 39 5 -32 34 19 -42 -47 -16 -34 -39 -13 -18 -43 21 -38 | |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || -1 -5 -10 -10 -11 42 6 46 -4 47 42 -40 47 -27 -20 49 -39 -31 -37 -29 -48 30 -48 0 ||
│ │ │ +      || -43 48 14 29 -47 -10 47 22 8 -47 15 -26 2 16 -49 22 -28 -18 45 -48 -34 -47 38 -15 ||
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || 29 18 20 1 18 26 -31 -45 -21 10 22 -30 10 32 -31 -21 -49 28 -22 46 1 40 -18 0 |    |
│ │ │ +      || -3 45 42 47 -50 16 -30 28 43 -16 24 19 15 -23 37 39 19 -8 43 -11 -17 48 7 47 |     |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || -17 3 17 -9 -36 -45 49 30 -45 24 -28 41 8 -4 -26 -28 7 30 -41 -17 -13 3 13 0 |     |
│ │ │ +      || -49 7 32 -6 -30 -41 -10 2 44 11 -25 4 33 40 -19 11 35 -17 46 1 -28 -3 -38 36 |     |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || 37 33 -47 -20 -49 45 29 19 41 13 -38 44 23 40 -48 45 8 -29 42 -46 49 -18 30 0 |    |
│ │ │ +      || 35 -48 -2 45 -35 29 34 12 -32 -23 50 2 2 29 -3 -47 -47 -34 15 -13 -37 -10 -7 22 |  |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || -9 -3 -26 13 35 49 -8 49 -40 13 -20 9 27 5 -8 -15 -28 15 -18 -16 -46 12 18 0 |     |
│ │ │ +      || 47 8 -14 6 -1 -13 -7 16 -20 39 -34 -22 -22 32 17 -9 -18 -6 -32 24 -20 -30 27 30 |  |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || 28 32 0 0 -17 -44 25 42 7 -35 29 -17 19 8 -9 -26 -21 23 20 -23 44 -39 -37 0 |      |
│ │ │ +      || -2 -36 -39 41 -6 34 -10 42 5 39 20 33 33 -49 -15 -33 -15 41 -19 -20 17 44 0 -48 |  |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || -30 -29 27 14 17 39 33 15 -35 50 -50 45 -33 13 24 -44 0 -47 -9 47 -28 6 -28 0 |    |
│ │ │ +      || -30 37 -9 16 -36 19 -13 -14 -19 9 -33 5 4 13 44 -26 36 -12 22 -11 -49 -8 -39 -39 | |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || 7 -12 42 -29 30 1 3 -28 -7 36 -26 -40 42 38 -20 -23 28 -29 -28 5 -37 -33 26 0 |    |
│ │ │ +      || 27 41 32 -44 40 -20 41 33 28 36 44 31 -22 -30 9 41 -8 30 16 -6 -28 35 -3 43 |      |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || 28 -10 13 -39 -20 11 13 -13 -37 8 -36 -29 -29 17 24 -50 44 30 -13 22 5 -20 4 0 |   |
│ │ │ +      || 37 -2 17 -42 -42 -12 18 -31 33 6 19 -31 3 -31 -11 25 -35 28 -2 -49 -41 -13 40 -9 | |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │  
│ │ │ +i14 : netList randomPointsOnRationalVariety(compsJ_1, 10)
│ │ │ +
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +o14 = || -41 -1 -48 25 40 4 35 16 26 -41 -28 -16 27 -14 -39 4 4 30 -40 37 -31 -35 -47 0 |     |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || -1 19 -3 12 50 3 4 25 48 50 34 -6 -29 6 -5 36 -39 -31 -48 30 47 -37 -48 0 |          |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || -27 -3 -40 22 27 3 -28 -41 -12 -34 -10 40 46 29 30 24 -49 28 1 40 10 -22 -18 0 |     |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || -26 -6 24 28 -27 26 34 47 13 50 3 -42 -17 5 4 -35 7 30 -13 3 8 -41 13 0 |            |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || 49 -7 48 1 48 25 25 -10 49 36 -16 35 -46 -5 25 -33 8 -29 49 -18 23 42 30 0 |         |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || -35 28 -6 22 50 -49 2 -5 -11 -39 30 27 -16 34 -9 -34 -28 15 -46 12 27 -18 18 0 |     |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || -49 -44 -16 -10 48 18 22 33 -35 -48 -28 -8 -23 -48 -25 -3 -21 23 44 -39 19 20 -37 0 ||
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || -33 -14 -18 10 2 -43 -26 45 10 19 -15 25 47 9 -15 -22 0 -47 -28 6 -33 -9 -28 0 |     |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || 20 -27 -17 2 -47 -23 13 40 -19 -13 39 -23 5 -3 47 -6 28 -29 -37 -33 42 -28 26 0 |    |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || 19 10 -10 47 41 20 -43 -34 -43 2 44 29 22 35 -42 16 44 30 5 -20 -29 -13 4 0 |        |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +
│ │ │  i15 :
│ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/html/_nonminimal__Maps.html
│ │ │ @@ -232,52 +232,52 @@
│ │ │  o13 = 2
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │  
│ │ │ -o14 = | 42 9 39 9 34 7 -12 -17 -29 -35 50 2 13 19 -44 50 2 -29 15 2 -27 21
│ │ │ +o14 = | -6 48 44 -23 -2 -11 -35 -26 27 -43 48 27 15 -22 25 -16 34 -29 46 -20
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -36 -29 -39 -10 24 -16 19 -29 39 -38 -22 -8 -30 -24 |
│ │ │ +      40 21 -30 -38 -19 -8 -36 39 19 -29 -16 -29 -10 19 24 -24 |
│ │ │  
│ │ │                 1       36
│ │ │  o14 : Matrix kk  <-- kk
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │  
│ │ │ -o15 = | 30 10 43 20 -39 23 -30 40 -34 22 46 -25 21 -18 -35 -1 21 -39 -45 16
│ │ │ +o15 = | -48 -46 16 17 -1 -43 15 -1 12 -18 -6 -28 14 -28 -9 32 -22 -39 6 -47
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -35 -5 19 -47 -20 -13 34 33 -28 -43 22 2 0 -15 -47 38 |
│ │ │ +      28 -37 -47 38 -16 -15 34 27 -13 -43 22 16 0 -18 19 2 |
│ │ │  
│ │ │                 1       36
│ │ │  o15 : Matrix kk  <-- kk
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i16 : F1 = sub(F, (vars S)|pt1)
│ │ │  
│ │ │ -              2              2                            2               
│ │ │ -o16 = ideal (a  - 44b*c - 35c  + 2a*d + 7b*d + 39c*d + 42d , a*b - 39b*c +
│ │ │ +              2              2                             2               
│ │ │ +o16 = ideal (a  + 25b*c - 43c  + 27a*d - 11b*d + 44c*d - 6d , a*b - 19b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2                   2                 
│ │ │ -      15c  - 27a*d + 13b*d - 29c*d + 9d , a*c - 38b*c - 10c  - 16a*d + 2b*d +
│ │ │ +         2                              2                  2                
│ │ │ +      46c  + 40a*d + 15b*d + 27c*d + 48d , a*c - 29b*c - 8c  + 39a*d - 20b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2   2              2                              2     2  
│ │ │ -      19c*d + 34d , b  - 30b*c + 19c  - 22a*d + 21b*d + 50c*d - 12d , b*c  -
│ │ │ +                  2   2              2                              2     2  
│ │ │ +      - 22c*d - 2d , b  + 24b*c + 19c  - 10a*d + 21b*d - 16c*d - 35d , b*c  -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2         2       2        2     3   3                2   
│ │ │ -      29b*c*d - 36c d + 24a*d  + 2b*d  + 50c*d  + 9d , c  - 24b*c*d + 39c d -
│ │ │ +                   2         2        2        2      3   3                2 
│ │ │ +      29b*c*d - 30c d - 36a*d  + 34b*d  + 48c*d  - 23d , c  - 24b*c*d - 16c d
│ │ │        -----------------------------------------------------------------------
│ │ │ -          2        2        2      3
│ │ │ -      8a*d  - 29b*d  - 29c*d  - 17d )
│ │ │ +             2        2        2      3
│ │ │ +      + 19a*d  - 38b*d  - 29c*d  - 26d )
│ │ │  
│ │ │  o16 : Ideal of S
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i17 : betti res F1
│ │ │ @@ -291,28 +291,28 @@
│ │ │  o17 : BettiTally
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i18 : F2 = sub(F, (vars S)|pt2)
│ │ │  
│ │ │ -              2              2                              2               
│ │ │ -o18 = ideal (a  - 35b*c + 22c  - 25a*d + 23b*d + 43c*d + 30d , a*b - 20b*c -
│ │ │ +              2             2                              2               
│ │ │ +o18 = ideal (a  - 9b*c - 18c  - 28a*d - 43b*d + 16c*d - 48d , a*b - 16b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                              2                  2                
│ │ │ -      45c  - 35a*d + 21b*d - 34c*d + 10d , a*c + 2b*c - 13c  + 33a*d + 16b*d
│ │ │ +        2                              2                   2                
│ │ │ +      6c  + 28a*d + 14b*d + 12c*d - 46d , a*c + 16b*c - 15c  + 27a*d - 47b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2   2              2                   2     2            
│ │ │ -      - 18c*d - 39d , b  - 47b*c - 28c  - 5b*d - c*d - 30d , b*c  - 43b*c*d +
│ │ │ +                 2   2              2                      2     2          
│ │ │ +      - 28c*d - d , b  + 19b*c - 13c  - 37b*d + 32c*d + 15d , b*c  - 43b*c*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2         2        2        2      3   3                2         2
│ │ │ -      19c d + 34a*d  + 21b*d  + 46c*d  + 20d , c  + 38b*c*d + 22c d - 15a*d 
│ │ │ +           2         2        2       2      3   3               2         2
│ │ │ +      - 47c d + 34a*d  - 22b*d  - 6c*d  + 17d , c  + 2b*c*d + 22c d - 18a*d 
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2      3
│ │ │ -      - 47b*d  - 39c*d  + 40d )
│ │ │ +             2        2    3
│ │ │ +      + 38b*d  - 39c*d  - d )
│ │ │  
│ │ │  o18 : Ideal of S
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i19 : betti res F2
│ │ │ @@ -331,35 +331,38 @@
│ │ │            
    What are the ideals F1 and F2?
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i20 : netList decompose F1
│ │ │  
│ │ │ -      +------------------------------------------------------+
│ │ │ -o20 = |ideal (c - 13d, b + 32d, a + 36d)                     |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |ideal (c - 16d, b + d, a + 16d)                       |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |                                    2            2    |
│ │ │ -      |ideal (b - 6c + 33d, a - 36c + 2d, c  + 43c*d - d )   |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |                                     2              2 |
│ │ │ -      |ideal (b + 29c + 7d, a - 19c + 24d, c  - 20c*d - 30d )|
│ │ │ -      +------------------------------------------------------+
    
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +o20 = |ideal (c + 39d, b + 27d, a - 18d)                                                                                                                        |
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                            2              2                     2   3                2        2        2      3     2                2         2      3 |
│ │ │ +      |ideal (a - 29b - 8c - 13d, b  + 24b*c + 19c  + 34b*d + 5c*d + 37d , c  - 24b*c*d - 16c d + 8b*d  + 22c*d  + 19d , b*c  - 29b*c*d - 30c d - 38c*d  + 14d )|
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │              
    
│ │ │                
    i21 : netList decompose F2
│ │ │  
│ │ │ -      +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                       2                              2   2              2                   2                   2                              2   2              2                              2   3                2         2        2        2      3     2                2         2        2        2      3 |
│ │ │ -o21 = |ideal (a*c + 2b*c - 13c  + 33a*d + 16b*d - 18c*d - 39d , b  - 47b*c - 28c  - 5b*d - c*d - 30d , a*b - 20b*c - 45c  - 35a*d + 21b*d - 34c*d + 10d , a  - 35b*c + 22c  - 25a*d + 23b*d + 43c*d + 30d , c  + 38b*c*d + 22c d - 15a*d  - 47b*d  - 39c*d  + 40d , b*c  - 43b*c*d + 19c d + 34a*d  + 21b*d  + 46c*d  + 20d )|
│ │ │ -      +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
    
│ │ │ +      +-------------------------------------------------------+
│ │ │ +o21 = |ideal (c - 32d, b - 5d, a - 29d)                       |
│ │ │ +      +-------------------------------------------------------+
│ │ │ +      |ideal (c + 43d, b - 47d, a - 27d)                      |
│ │ │ +      +-------------------------------------------------------+
│ │ │ +      |ideal (c + 24d, b - 49d, a)                            |
│ │ │ +      +-------------------------------------------------------+
│ │ │ +      |ideal (c + 14d, b + 31d, a - 16d)                      |
│ │ │ +      +-------------------------------------------------------+
│ │ │ +      |                                      2              2 |
│ │ │ +      |ideal (b + 11c + 22d, a + 11c + 42d, c  - 43c*d + 31d )|
│ │ │ +      +-------------------------------------------------------+
│ │ │              
    
│ │ │          
    
│ │ │            
    We can determine what these represent.  One should be a set of 6 points, where 5 lie on a plane.  The other should be 6 points with 3 points on one line, and the other 3 points on a skew line.
    
│ │ │          
    
│ │ │        
│ │ │ ├── html2text {}
│ │ │ │ @@ -170,117 +170,115 @@
│ │ │ │  32   13   21   33   19   31
│ │ │ │  i12 : compsJ = decompose J;
│ │ │ │  i13 : #compsJ
│ │ │ │  
│ │ │ │  o13 = 2
│ │ │ │  i14 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │ │  
│ │ │ │ -o14 = | 42 9 39 9 34 7 -12 -17 -29 -35 50 2 13 19 -44 50 2 -29 15 2 -27 21
│ │ │ │ +o14 = | -6 48 44 -23 -2 -11 -35 -26 27 -43 48 27 15 -22 25 -16 34 -29 46 -20
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -36 -29 -39 -10 24 -16 19 -29 39 -38 -22 -8 -30 -24 |
│ │ │ │ +      40 21 -30 -38 -19 -8 -36 39 19 -29 -16 -29 -10 19 24 -24 |
│ │ │ │  
│ │ │ │                 1       36
│ │ │ │  o14 : Matrix kk  <-- kk
│ │ │ │  i15 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │ │  
│ │ │ │ -o15 = | 30 10 43 20 -39 23 -30 40 -34 22 46 -25 21 -18 -35 -1 21 -39 -45 16
│ │ │ │ +o15 = | -48 -46 16 17 -1 -43 15 -1 12 -18 -6 -28 14 -28 -9 32 -22 -39 6 -47
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -35 -5 19 -47 -20 -13 34 33 -28 -43 22 2 0 -15 -47 38 |
│ │ │ │ +      28 -37 -47 38 -16 -15 34 27 -13 -43 22 16 0 -18 19 2 |
│ │ │ │  
│ │ │ │                 1       36
│ │ │ │  o15 : Matrix kk  <-- kk
│ │ │ │  i16 : F1 = sub(F, (vars S)|pt1)
│ │ │ │  
│ │ │ │ -              2              2                            2
│ │ │ │ -o16 = ideal (a  - 44b*c - 35c  + 2a*d + 7b*d + 39c*d + 42d , a*b - 39b*c +
│ │ │ │ +              2              2                             2
│ │ │ │ +o16 = ideal (a  + 25b*c - 43c  + 27a*d - 11b*d + 44c*d - 6d , a*b - 19b*c +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -         2                             2                   2
│ │ │ │ -      15c  - 27a*d + 13b*d - 29c*d + 9d , a*c - 38b*c - 10c  - 16a*d + 2b*d +
│ │ │ │ +         2                              2                  2
│ │ │ │ +      46c  + 40a*d + 15b*d + 27c*d + 48d , a*c - 29b*c - 8c  + 39a*d - 20b*d
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                 2   2              2                              2     2
│ │ │ │ -      19c*d + 34d , b  - 30b*c + 19c  - 22a*d + 21b*d + 50c*d - 12d , b*c  -
│ │ │ │ +                  2   2              2                              2     2
│ │ │ │ +      - 22c*d - 2d , b  + 24b*c + 19c  - 10a*d + 21b*d - 16c*d - 35d , b*c  -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                   2         2       2        2     3   3                2
│ │ │ │ -      29b*c*d - 36c d + 24a*d  + 2b*d  + 50c*d  + 9d , c  - 24b*c*d + 39c d -
│ │ │ │ +                   2         2        2        2      3   3                2
│ │ │ │ +      29b*c*d - 30c d - 36a*d  + 34b*d  + 48c*d  - 23d , c  - 24b*c*d - 16c d
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -          2        2        2      3
│ │ │ │ -      8a*d  - 29b*d  - 29c*d  - 17d )
│ │ │ │ +             2        2        2      3
│ │ │ │ +      + 19a*d  - 38b*d  - 29c*d  - 26d )
│ │ │ │  
│ │ │ │  o16 : Ideal of S
│ │ │ │  i17 : betti res F1
│ │ │ │  
│ │ │ │               0 1 2 3
│ │ │ │  o17 = total: 1 6 8 3
│ │ │ │            0: 1 . . .
│ │ │ │            1: . 4 4 1
│ │ │ │            2: . 2 4 2
│ │ │ │  
│ │ │ │  o17 : BettiTally
│ │ │ │  i18 : F2 = sub(F, (vars S)|pt2)
│ │ │ │  
│ │ │ │ -              2              2                              2
│ │ │ │ -o18 = ideal (a  - 35b*c + 22c  - 25a*d + 23b*d + 43c*d + 30d , a*b - 20b*c -
│ │ │ │ +              2             2                              2
│ │ │ │ +o18 = ideal (a  - 9b*c - 18c  - 28a*d - 43b*d + 16c*d - 48d , a*b - 16b*c +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -         2                              2                  2
│ │ │ │ -      45c  - 35a*d + 21b*d - 34c*d + 10d , a*c + 2b*c - 13c  + 33a*d + 16b*d
│ │ │ │ +        2                              2                   2
│ │ │ │ +      6c  + 28a*d + 14b*d + 12c*d - 46d , a*c + 16b*c - 15c  + 27a*d - 47b*d
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                   2   2              2                   2     2
│ │ │ │ -      - 18c*d - 39d , b  - 47b*c - 28c  - 5b*d - c*d - 30d , b*c  - 43b*c*d +
│ │ │ │ +                 2   2              2                      2     2
│ │ │ │ +      - 28c*d - d , b  + 19b*c - 13c  - 37b*d + 32c*d + 15d , b*c  - 43b*c*d
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -         2         2        2        2      3   3                2         2
│ │ │ │ -      19c d + 34a*d  + 21b*d  + 46c*d  + 20d , c  + 38b*c*d + 22c d - 15a*d
│ │ │ │ +           2         2        2       2      3   3               2         2
│ │ │ │ +      - 47c d + 34a*d  - 22b*d  - 6c*d  + 17d , c  + 2b*c*d + 22c d - 18a*d
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -             2        2      3
│ │ │ │ -      - 47b*d  - 39c*d  + 40d )
│ │ │ │ +             2        2    3
│ │ │ │ +      + 38b*d  - 39c*d  - d )
│ │ │ │  
│ │ │ │  o18 : Ideal of S
│ │ │ │  i19 : betti res F2
│ │ │ │  
│ │ │ │               0 1 2 3
│ │ │ │  o19 = total: 1 6 8 3
│ │ │ │            0: 1 . . .
│ │ │ │            1: . 4 4 1
│ │ │ │            2: . 2 4 2
│ │ │ │  
│ │ │ │  o19 : BettiTally
│ │ │ │  What are the ideals F1 and F2?
│ │ │ │  i20 : netList decompose F1
│ │ │ │  
│ │ │ │ -      +------------------------------------------------------+
│ │ │ │ -o20 = |ideal (c - 13d, b + 32d, a + 36d)                     |
│ │ │ │ -      +------------------------------------------------------+
│ │ │ │ -      |ideal (c - 16d, b + d, a + 16d)                       |
│ │ │ │ -      +------------------------------------------------------+
│ │ │ │ -      |                                    2            2    |
│ │ │ │ -      |ideal (b - 6c + 33d, a - 36c + 2d, c  + 43c*d - d )   |
│ │ │ │ -      +------------------------------------------------------+
│ │ │ │ -      |                                     2              2 |
│ │ │ │ -      |ideal (b + 29c + 7d, a - 19c + 24d, c  - 20c*d - 30d )|
│ │ │ │ -      +------------------------------------------------------+
│ │ │ │ -i21 : netList decompose F2
│ │ │ │ -
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │ +--+
│ │ │ │ +o20 = |ideal (c + 39d, b + 27d, a - 18d)
│ │ │ │ +|
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------------------------
│ │ │ │ --+
│ │ │ │ -      |                       2                              2   2
│ │ │ │ -2                   2                   2                              2   2
│ │ │ │ -2                              2   3                2         2        2
│ │ │ │ -2      3     2                2         2        2        2      3 |
│ │ │ │ -o21 = |ideal (a*c + 2b*c - 13c  + 33a*d + 16b*d - 18c*d - 39d , b  - 47b*c -
│ │ │ │ -28c  - 5b*d - c*d - 30d , a*b - 20b*c - 45c  - 35a*d + 21b*d - 34c*d + 10d , a
│ │ │ │ -- 35b*c + 22c  - 25a*d + 23b*d + 43c*d + 30d , c  + 38b*c*d + 22c d - 15a*d  -
│ │ │ │ -47b*d  - 39c*d  + 40d , b*c  - 43b*c*d + 19c d + 34a*d  + 21b*d  + 46c*d  + 20d
│ │ │ │ +--+
│ │ │ │ +      |                            2              2                     2   3
│ │ │ │ +2        2        2      3     2                2         2      3 |
│ │ │ │ +      |ideal (a - 29b - 8c - 13d, b  + 24b*c + 19c  + 34b*d + 5c*d + 37d , c  -
│ │ │ │ +24b*c*d - 16c d + 8b*d  + 22c*d  + 19d , b*c  - 29b*c*d - 30c d - 38c*d  + 14d
│ │ │ │  )|
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------------------------
│ │ │ │ --+
│ │ │ │ +--+
│ │ │ │ +i21 : netList decompose F2
│ │ │ │ +
│ │ │ │ +      +-------------------------------------------------------+
│ │ │ │ +o21 = |ideal (c - 32d, b - 5d, a - 29d)                       |
│ │ │ │ +      +-------------------------------------------------------+
│ │ │ │ +      |ideal (c + 43d, b - 47d, a - 27d)                      |
│ │ │ │ +      +-------------------------------------------------------+
│ │ │ │ +      |ideal (c + 24d, b - 49d, a)                            |
│ │ │ │ +      +-------------------------------------------------------+
│ │ │ │ +      |ideal (c + 14d, b + 31d, a - 16d)                      |
│ │ │ │ +      +-------------------------------------------------------+
│ │ │ │ +      |                                      2              2 |
│ │ │ │ +      |ideal (b + 11c + 22d, a + 11c + 42d, c  - 43c*d + 31d )|
│ │ │ │ +      +-------------------------------------------------------+
│ │ │ │  We can determine what these represent. One should be a set of 6 points, where 5
│ │ │ │  lie on a plane. The other should be 6 points with 3 points on one line, and the
│ │ │ │  other 3 points on a skew line.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_a_n_d_o_m_P_o_i_n_t_O_n_R_a_t_i_o_n_a_l_V_a_r_i_e_t_y -- find a random point on a variety that can
│ │ │ │        be detected to be rational
│ │ │ │  ********** WWaayyss ttoo uussee nnoonnmmiinniimmaallMMaappss:: **********
│ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/html/_random__Point__On__Rational__Variety_lp__Ideal_rp.html
│ │ │ @@ -318,90 +318,95 @@
│ │ │            
    There are 2 components.  We attempt to find a point on the first component
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i13 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │  
│ │ │ -o13 = | 50 15 46 -33 2 -43 -46 8 33 19 -2 -18 -8 -22 43 -29 19 3 -16 -29 -38
│ │ │ +o13 = | 13 48 43 23 41 36 -4 -12 -30 -16 -33 -36 19 19 30 -10 -38 32 -29 -8
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -24 -10 -29 |
│ │ │ +      -29 -22 -29 -24 |
│ │ │  
│ │ │                 1       24
│ │ │  o13 : Matrix kk  <-- kk
    
│ │ │              
    
│ │ │                
    i14 : F1 = sub(F, (vars S)|pt1)
│ │ │  
│ │ │ -              2              2                             2               
│ │ │ -o14 = ideal (a  + 33b*c - 33c  + 19a*d + 2b*d + 15c*d + 50d , a*b + 43b*c -
│ │ │ +              2              2                              2               
│ │ │ +o14 = ideal (a  - 30b*c + 23c  - 16a*d + 41b*d + 48c*d + 13d , a*b + 30b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -        2                              2                   2                 
│ │ │ -      2c  - 29a*d - 18b*d - 43c*d + 46d , a*c - 38b*c + 19c  - 24a*d + 3b*d -
│ │ │ +         2                              2                   2                
│ │ │ +      33c  - 10a*d - 36b*d + 36c*d + 43d , a*c - 29b*c - 38c  - 22a*d + 32b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                2   2              2                             2
│ │ │ -      8c*d - 46d , b  - 10b*c - 16c  - 29a*d - 29b*d - 22c*d + 8d )
│ │ │ +                  2   2              2                             2
│ │ │ +      + 19c*d - 4d , b  - 29b*c - 29c  - 24a*d - 8b*d + 19c*d - 12d )
│ │ │  
│ │ │  o14 : Ideal of S
    
│ │ │              
    
│ │ │                
    i15 : decompose F1
│ │ │  
│ │ │ -                                    2              2                      2
│ │ │ -o15 = {ideal (a - 38b + 19c + 44d, b  - 10b*c - 16c  - 20b*d + 24c*d - 29d ),
│ │ │ +                                   2              2                     2
│ │ │ +o15 = {ideal (a - 29b - 38c - 9d, b  - 29b*c - 29c  + 3b*d + 16c*d - 26d ),
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ideal (c - 24d, b - 38d, a + 15d)}
│ │ │ +      ideal (c - 22d, b - 21d, a + 8d)}
│ │ │  
│ │ │  o15 : List
    
│ │ │              
    
│ │ │          
    
│ │ │            
    We attempt to find a point on the second component in parameter space, and its corresponding ideal.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i16 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │  
│ │ │ -o16 = | -14 40 -5 26 -48 -26 -35 41 -8 -15 -38 31 -13 29 21 16 39 21 -18 19
│ │ │ +o16 = | 46 -2 16 -20 -1 -30 -43 -41 17 -4 -16 -29 -39 40 49 -39 -18 -13 -47
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -47 -39 34 0 |
│ │ │ +      34 19 21 39 0 |
│ │ │  
│ │ │                 1       24
│ │ │  o16 : Matrix kk  <-- kk
    
│ │ │              
    
│ │ │                
    i17 : F2 = sub(F, (vars S)|pt2)
│ │ │  
│ │ │ -              2             2                              2               
│ │ │ -o17 = ideal (a  - 8b*c + 26c  - 15a*d - 48b*d + 40c*d - 14d , a*b + 21b*c -
│ │ │ +              2              2                          2                   2
│ │ │ +o17 = ideal (a  + 17b*c - 20c  - 4a*d - b*d - 2c*d + 46d , a*b + 49b*c - 16c 
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2                   2                
│ │ │ -      38c  + 16a*d + 31b*d - 26c*d - 5d , a*c - 47b*c + 39c  - 39a*d + 21b*d
│ │ │ +                                   2                   2                  
│ │ │ +      - 39a*d - 29b*d - 30c*d + 16d , a*c + 19b*c - 18c  + 21a*d - 13b*d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2   2              2                      2
│ │ │ -      - 13c*d - 35d , b  + 34b*c - 18c  + 19b*d + 29c*d + 41d )
│ │ │ +                 2   2              2                      2
│ │ │ +      39c*d - 43d , b  + 39b*c - 47c  + 34b*d + 40c*d - 41d )
│ │ │  
│ │ │  o17 : Ideal of S
    
│ │ │              
    
│ │ │                
    i18 : decompose F2
│ │ │  
│ │ │ -o18 = {ideal (b + 19c - 18d, a + 23c + 43d), ideal (b + 15c + 37d, a + 37c +
│ │ │ +                               2                              2   2          
│ │ │ +o18 = {ideal (a*c + 19b*c - 18c  + 21a*d - 13b*d - 39c*d - 43d , b  + 39b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -      26d)}
│ │ │ +         2                      2                   2                        
│ │ │ +      47c  + 34b*d + 40c*d - 41d , a*b + 49b*c - 16c  - 39a*d - 29b*d - 30c*d
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +           2   2              2                          2
│ │ │ +      + 16d , a  + 17b*c - 20c  - 4a*d - b*d - 2c*d + 46d )}
│ │ │  
│ │ │  o18 : List
    
│ │ │              
    
│ │ │          
    
│ │ │            
    It turns out that this is the ideal of 2 skew lines, just not defined over this field.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -212,67 +212,72 @@
│ │ │ │  
│ │ │ │  o12 = {11, 8}
│ │ │ │  
│ │ │ │  o12 : List
│ │ │ │  There are 2 components. We attempt to find a point on the first component
│ │ │ │  i13 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │ │  
│ │ │ │ -o13 = | 50 15 46 -33 2 -43 -46 8 33 19 -2 -18 -8 -22 43 -29 19 3 -16 -29 -38
│ │ │ │ +o13 = | 13 48 43 23 41 36 -4 -12 -30 -16 -33 -36 19 19 30 -10 -38 32 -29 -8
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -24 -10 -29 |
│ │ │ │ +      -29 -22 -29 -24 |
│ │ │ │  
│ │ │ │                 1       24
│ │ │ │  o13 : Matrix kk  <-- kk
│ │ │ │  i14 : F1 = sub(F, (vars S)|pt1)
│ │ │ │  
│ │ │ │ -              2              2                             2
│ │ │ │ -o14 = ideal (a  + 33b*c - 33c  + 19a*d + 2b*d + 15c*d + 50d , a*b + 43b*c -
│ │ │ │ +              2              2                              2
│ │ │ │ +o14 = ideal (a  - 30b*c + 23c  - 16a*d + 41b*d + 48c*d + 13d , a*b + 30b*c -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -        2                              2                   2
│ │ │ │ -      2c  - 29a*d - 18b*d - 43c*d + 46d , a*c - 38b*c + 19c  - 24a*d + 3b*d -
│ │ │ │ +         2                              2                   2
│ │ │ │ +      33c  - 10a*d - 36b*d + 36c*d + 43d , a*c - 29b*c - 38c  - 22a*d + 32b*d
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                2   2              2                             2
│ │ │ │ -      8c*d - 46d , b  - 10b*c - 16c  - 29a*d - 29b*d - 22c*d + 8d )
│ │ │ │ +                  2   2              2                             2
│ │ │ │ +      + 19c*d - 4d , b  - 29b*c - 29c  - 24a*d - 8b*d + 19c*d - 12d )
│ │ │ │  
│ │ │ │  o14 : Ideal of S
│ │ │ │  i15 : decompose F1
│ │ │ │  
│ │ │ │ -                                    2              2                      2
│ │ │ │ -o15 = {ideal (a - 38b + 19c + 44d, b  - 10b*c - 16c  - 20b*d + 24c*d - 29d ),
│ │ │ │ +                                   2              2                     2
│ │ │ │ +o15 = {ideal (a - 29b - 38c - 9d, b  - 29b*c - 29c  + 3b*d + 16c*d - 26d ),
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      ideal (c - 24d, b - 38d, a + 15d)}
│ │ │ │ +      ideal (c - 22d, b - 21d, a + 8d)}
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  We attempt to find a point on the second component in parameter space, and its
│ │ │ │  corresponding ideal.
│ │ │ │  i16 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │ │  
│ │ │ │ -o16 = | -14 40 -5 26 -48 -26 -35 41 -8 -15 -38 31 -13 29 21 16 39 21 -18 19
│ │ │ │ +o16 = | 46 -2 16 -20 -1 -30 -43 -41 17 -4 -16 -29 -39 40 49 -39 -18 -13 -47
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -47 -39 34 0 |
│ │ │ │ +      34 19 21 39 0 |
│ │ │ │  
│ │ │ │                 1       24
│ │ │ │  o16 : Matrix kk  <-- kk
│ │ │ │  i17 : F2 = sub(F, (vars S)|pt2)
│ │ │ │  
│ │ │ │ -              2             2                              2
│ │ │ │ -o17 = ideal (a  - 8b*c + 26c  - 15a*d - 48b*d + 40c*d - 14d , a*b + 21b*c -
│ │ │ │ +              2              2                          2                   2
│ │ │ │ +o17 = ideal (a  + 17b*c - 20c  - 4a*d - b*d - 2c*d + 46d , a*b + 49b*c - 16c
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -         2                             2                   2
│ │ │ │ -      38c  + 16a*d + 31b*d - 26c*d - 5d , a*c - 47b*c + 39c  - 39a*d + 21b*d
│ │ │ │ +                                   2                   2
│ │ │ │ +      - 39a*d - 29b*d - 30c*d + 16d , a*c + 19b*c - 18c  + 21a*d - 13b*d -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                   2   2              2                      2
│ │ │ │ -      - 13c*d - 35d , b  + 34b*c - 18c  + 19b*d + 29c*d + 41d )
│ │ │ │ +                 2   2              2                      2
│ │ │ │ +      39c*d - 43d , b  + 39b*c - 47c  + 34b*d + 40c*d - 41d )
│ │ │ │  
│ │ │ │  o17 : Ideal of S
│ │ │ │  i18 : decompose F2
│ │ │ │  
│ │ │ │ -o18 = {ideal (b + 19c - 18d, a + 23c + 43d), ideal (b + 15c + 37d, a + 37c +
│ │ │ │ +                               2                              2   2
│ │ │ │ +o18 = {ideal (a*c + 19b*c - 18c  + 21a*d - 13b*d - 39c*d - 43d , b  + 39b*c -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      26d)}
│ │ │ │ +         2                      2                   2
│ │ │ │ +      47c  + 34b*d + 40c*d - 41d , a*b + 49b*c - 16c  - 39a*d - 29b*d - 30c*d
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +           2   2              2                          2
│ │ │ │ +      + 16d , a  + 17b*c - 20c  - 4a*d - b*d - 2c*d + 46d )}
│ │ │ │  
│ │ │ │  o18 : List
│ │ │ │  It turns out that this is the ideal of 2 skew lines, just not defined over this
│ │ │ │  field.
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  This routine expects the input to represent an irreducible variety
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/html/_random__Points__On__Rational__Variety_lp__Ideal_cm__Z__Z_rp.html
│ │ │ @@ -186,64 +186,64 @@
│ │ │            
    There are 2 components.  We attempt to find points on each of these two components. We are successful.  This indicates that the corresponding varieties are both rational. Also, if we can find one point, we can find as many as we want.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │ -          
│ │ │ -          
│ │ │ -            
│ │ │            
│ │ │ +          
│ │ │ +            
│ │ │ +          
│ │ │          
    
│ │ │                
    i13 : netList randomPointsOnRationalVariety(compsJ_0, 10)
│ │ │  
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -o13 = || 29 -40 15 -49 3 -13 -6 -39 2 39 47 15 19 -47 -46 -39 -16 32 -43 34 -13 -18 21 -38 | |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || 37 -7 -24 8 -26 38 9 -31 24 -47 -34 12 16 22 -22 45 -28 16 -47 2 -48 -34 38 -15 |   |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || 6 1 -31 -7 44 8 -50 24 -48 -16 23 23 -23 39 -5 43 19 -15 48 15 -11 -17 7 47 |       |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || -41 -49 6 -16 -12 31 23 6 -7 11 3 -42 40 11 -28 46 35 -28 -3 33 1 -28 -38 36 |      |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || -11 -27 -4 40 -34 6 44 -2 19 -23 -29 21 29 -47 -37 15 -47 -24 -10 2 -13 -37 -7 22 | |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || -50 42 20 -30 -46 -48 -5 40 -47 39 13 47 32 -9 41 -32 -18 25 -30 -22 24 -20 27 30 | |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || 50 22 -30 3 -43 -29 -33 -18 6 39 -29 24 -49 -33 -15 -19 -15 -37 44 33 -20 17 0 -48 ||
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || -9 31 -37 -42 -7 -8 -11 -21 12 9 13 -9 13 -26 11 22 36 34 -8 4 -11 -49 -39 -39 |    |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || 47 14 -11 -16 -20 -40 42 5 -2 36 8 -45 -30 41 -26 16 -8 -34 35 -22 -6 -28 -3 43 |   |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || 23 -8 -3 -17 38 0 11 -33 -7 6 -31 -4 -31 25 6 -2 -35 -11 -13 3 -49 -41 40 -9 |      |
│ │ │ -      +--------------------------------------------------------------------------------------+
    
│ │ │ -            
    
│ │ │ -              
    i14 : netList randomPointsOnRationalVariety(compsJ_1, 10)
│ │ │ -
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -o14 = || 38 -31 49 39 4 46 -29 -5 -39 -40 14 -11 -31 46 43 -26 4 30 -35 27 -40 37 -47 0 |   |
│ │ │ +o13 = || 13 15 3 36 2 48 44 -35 -34 39 5 -32 34 19 -42 -47 -16 -34 -39 -13 -18 -43 21 -38 | |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || -1 -5 -10 -10 -11 42 6 46 -4 47 42 -40 47 -27 -20 49 -39 -31 -37 -29 -48 30 -48 0 ||
│ │ │ +      || -43 48 14 29 -47 -10 47 22 8 -47 15 -26 2 16 -49 22 -28 -18 45 -48 -34 -47 38 -15 ||
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || 29 18 20 1 18 26 -31 -45 -21 10 22 -30 10 32 -31 -21 -49 28 -22 46 1 40 -18 0 |    |
│ │ │ +      || -3 45 42 47 -50 16 -30 28 43 -16 24 19 15 -23 37 39 19 -8 43 -11 -17 48 7 47 |     |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || -17 3 17 -9 -36 -45 49 30 -45 24 -28 41 8 -4 -26 -28 7 30 -41 -17 -13 3 13 0 |     |
│ │ │ +      || -49 7 32 -6 -30 -41 -10 2 44 11 -25 4 33 40 -19 11 35 -17 46 1 -28 -3 -38 36 |     |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || 37 33 -47 -20 -49 45 29 19 41 13 -38 44 23 40 -48 45 8 -29 42 -46 49 -18 30 0 |    |
│ │ │ +      || 35 -48 -2 45 -35 29 34 12 -32 -23 50 2 2 29 -3 -47 -47 -34 15 -13 -37 -10 -7 22 |  |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || -9 -3 -26 13 35 49 -8 49 -40 13 -20 9 27 5 -8 -15 -28 15 -18 -16 -46 12 18 0 |     |
│ │ │ +      || 47 8 -14 6 -1 -13 -7 16 -20 39 -34 -22 -22 32 17 -9 -18 -6 -32 24 -20 -30 27 30 |  |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || 28 32 0 0 -17 -44 25 42 7 -35 29 -17 19 8 -9 -26 -21 23 20 -23 44 -39 -37 0 |      |
│ │ │ +      || -2 -36 -39 41 -6 34 -10 42 5 39 20 33 33 -49 -15 -33 -15 41 -19 -20 17 44 0 -48 |  |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || -30 -29 27 14 17 39 33 15 -35 50 -50 45 -33 13 24 -44 0 -47 -9 47 -28 6 -28 0 |    |
│ │ │ +      || -30 37 -9 16 -36 19 -13 -14 -19 9 -33 5 4 13 44 -26 36 -12 22 -11 -49 -8 -39 -39 | |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || 7 -12 42 -29 30 1 3 -28 -7 36 -26 -40 42 38 -20 -23 28 -29 -28 5 -37 -33 26 0 |    |
│ │ │ +      || 27 41 32 -44 40 -20 41 33 28 36 44 31 -22 -30 9 41 -8 30 16 -6 -28 35 -3 43 |      |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || 28 -10 13 -39 -20 11 13 -13 -37 8 -36 -29 -29 17 24 -50 44 30 -13 22 5 -20 4 0 |   |
│ │ │ +      || 37 -2 17 -42 -42 -12 18 -31 33 6 19 -31 3 -31 -11 25 -35 28 -2 -49 -41 -13 40 -9 | |
│ │ │        +-------------------------------------------------------------------------------------+
    
│ │ │              
    
│ │ │ +              
    i14 : netList randomPointsOnRationalVariety(compsJ_1, 10)
│ │ │ +
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +o14 = || -41 -1 -48 25 40 4 35 16 26 -41 -28 -16 27 -14 -39 4 4 30 -40 37 -31 -35 -47 0 |     |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || -1 19 -3 12 50 3 4 25 48 50 34 -6 -29 6 -5 36 -39 -31 -48 30 47 -37 -48 0 |          |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || -27 -3 -40 22 27 3 -28 -41 -12 -34 -10 40 46 29 30 24 -49 28 1 40 10 -22 -18 0 |     |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || -26 -6 24 28 -27 26 34 47 13 50 3 -42 -17 5 4 -35 7 30 -13 3 8 -41 13 0 |            |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || 49 -7 48 1 48 25 25 -10 49 36 -16 35 -46 -5 25 -33 8 -29 49 -18 23 42 30 0 |         |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || -35 28 -6 22 50 -49 2 -5 -11 -39 30 27 -16 34 -9 -34 -28 15 -46 12 27 -18 18 0 |     |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || -49 -44 -16 -10 48 18 22 33 -35 -48 -28 -8 -23 -48 -25 -3 -21 23 44 -39 19 20 -37 0 ||
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || -33 -14 -18 10 2 -43 -26 45 10 19 -15 25 47 9 -15 -22 0 -47 -28 6 -33 -9 -28 0 |     |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || 20 -27 -17 2 -47 -23 13 40 -19 -13 39 -23 5 -3 47 -6 28 -29 -37 -33 42 -28 26 0 |    |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || 19 10 -10 47 41 20 -43 -34 -43 2 44 29 22 35 -42 16 44 30 5 -20 -29 -13 4 0 |        |
│ │ │ +      +---------------------------------------------------------------------------------------+
    
│ │ │ +            
    
│ │ │        
│ │ │        
    
│ │ │          
    Caveat
    
│ │ │          
    
│ │ │            
    This routine expects the input to represent an irreducible variety
    
│ │ │          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -84,99 +84,99 @@
│ │ │ │  There are 2 components. We attempt to find points on each of these two
│ │ │ │  components. We are successful. This indicates that the corresponding varieties
│ │ │ │  are both rational. Also, if we can find one point, we can find as many as we
│ │ │ │  want.
│ │ │ │  i13 : netList randomPointsOnRationalVariety(compsJ_0, 10)
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ ---------------+
│ │ │ │ -o13 = || 29 -40 15 -49 3 -13 -6 -39 2 39 47 15 19 -47 -46 -39 -16 32 -43 34 -13
│ │ │ │ --18 21 -38 | |
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ ---------------+
│ │ │ │ -      || 37 -7 -24 8 -26 38 9 -31 24 -47 -34 12 16 22 -22 45 -28 16 -47 2 -48 -
│ │ │ │ -34 38 -15 |   |
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ ---------------+
│ │ │ │ -      || 6 1 -31 -7 44 8 -50 24 -48 -16 23 23 -23 39 -5 43 19 -15 48 15 -11 -17
│ │ │ │ -7 47 |       |
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ ---------------+
│ │ │ │ -      || -41 -49 6 -16 -12 31 23 6 -7 11 3 -42 40 11 -28 46 35 -28 -3 33 1 -28
│ │ │ │ --38 36 |      |
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ ---------------+
│ │ │ │ -      || -11 -27 -4 40 -34 6 44 -2 19 -23 -29 21 29 -47 -37 15 -47 -24 -10 2 -
│ │ │ │ -13 -37 -7 22 | |
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ ---------------+
│ │ │ │ -      || -50 42 20 -30 -46 -48 -5 40 -47 39 13 47 32 -9 41 -32 -18 25 -30 -22
│ │ │ │ -24 -20 27 30 | |
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ ---------------+
│ │ │ │ -      || 50 22 -30 3 -43 -29 -33 -18 6 39 -29 24 -49 -33 -15 -19 -15 -37 44 33
│ │ │ │ --20 17 0 -48 ||
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ ---------------+
│ │ │ │ -      || -9 31 -37 -42 -7 -8 -11 -21 12 9 13 -9 13 -26 11 22 36 34 -8 4 -11 -49
│ │ │ │ --39 -39 |    |
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ ---------------+
│ │ │ │ -      || 47 14 -11 -16 -20 -40 42 5 -2 36 8 -45 -30 41 -26 16 -8 -34 35 -22 -
│ │ │ │ -6 -28 -3 43 |   |
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ ---------------+
│ │ │ │ -      || 23 -8 -3 -17 38 0 11 -33 -7 6 -31 -4 -31 25 6 -2 -35 -11 -13 3 -49 -41
│ │ │ │ -40 -9 |      |
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ ---------------+
│ │ │ │ -i14 : netList randomPointsOnRationalVariety(compsJ_1, 10)
│ │ │ │ -
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │  -------------+
│ │ │ │ -o14 = || 38 -31 49 39 4 46 -29 -5 -39 -40 14 -11 -31 46 43 -26 4 30 -35 27 -40
│ │ │ │ -37 -47 0 |   |
│ │ │ │ +o13 = || 13 15 3 36 2 48 44 -35 -34 39 5 -32 34 19 -42 -47 -16 -34 -39 -13 -18
│ │ │ │ +-43 21 -38 | |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------+
│ │ │ │ -      || -1 -5 -10 -10 -11 42 6 46 -4 47 42 -40 47 -27 -20 49 -39 -31 -37 -29 -
│ │ │ │ -48 30 -48 0 ||
│ │ │ │ +      || -43 48 14 29 -47 -10 47 22 8 -47 15 -26 2 16 -49 22 -28 -18 45 -48 -34
│ │ │ │ +-47 38 -15 ||
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------+
│ │ │ │ -      || 29 18 20 1 18 26 -31 -45 -21 10 22 -30 10 32 -31 -21 -49 28 -22 46 1
│ │ │ │ -40 -18 0 |    |
│ │ │ │ +      || -3 45 42 47 -50 16 -30 28 43 -16 24 19 15 -23 37 39 19 -8 43 -11 -17
│ │ │ │ +48 7 47 |     |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------+
│ │ │ │ -      || -17 3 17 -9 -36 -45 49 30 -45 24 -28 41 8 -4 -26 -28 7 30 -41 -17 -13
│ │ │ │ -3 13 0 |     |
│ │ │ │ +      || -49 7 32 -6 -30 -41 -10 2 44 11 -25 4 33 40 -19 11 35 -17 46 1 -28 -
│ │ │ │ +3 -38 36 |     |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------+
│ │ │ │ -      || 37 33 -47 -20 -49 45 29 19 41 13 -38 44 23 40 -48 45 8 -29 42 -46 49 -
│ │ │ │ -18 30 0 |    |
│ │ │ │ +      || 35 -48 -2 45 -35 29 34 12 -32 -23 50 2 2 29 -3 -47 -47 -34 15 -13 -37
│ │ │ │ +-10 -7 22 |  |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------+
│ │ │ │ -      || -9 -3 -26 13 35 49 -8 49 -40 13 -20 9 27 5 -8 -15 -28 15 -18 -16 -46
│ │ │ │ -12 18 0 |     |
│ │ │ │ +      || 47 8 -14 6 -1 -13 -7 16 -20 39 -34 -22 -22 32 17 -9 -18 -6 -32 24 -20
│ │ │ │ +-30 27 30 |  |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------+
│ │ │ │ -      || 28 32 0 0 -17 -44 25 42 7 -35 29 -17 19 8 -9 -26 -21 23 20 -23 44 -39
│ │ │ │ --37 0 |      |
│ │ │ │ +      || -2 -36 -39 41 -6 34 -10 42 5 39 20 33 33 -49 -15 -33 -15 41 -19 -20 17
│ │ │ │ +44 0 -48 |  |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------+
│ │ │ │ -      || -30 -29 27 14 17 39 33 15 -35 50 -50 45 -33 13 24 -44 0 -47 -9 47 -28
│ │ │ │ -6 -28 0 |    |
│ │ │ │ +      || -30 37 -9 16 -36 19 -13 -14 -19 9 -33 5 4 13 44 -26 36 -12 22 -11 -49
│ │ │ │ +-8 -39 -39 | |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------+
│ │ │ │ -      || 7 -12 42 -29 30 1 3 -28 -7 36 -26 -40 42 38 -20 -23 28 -29 -28 5 -37 -
│ │ │ │ -33 26 0 |    |
│ │ │ │ +      || 27 41 32 -44 40 -20 41 33 28 36 44 31 -22 -30 9 41 -8 30 16 -6 -28 35
│ │ │ │ +-3 43 |      |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------+
│ │ │ │ -      || 28 -10 13 -39 -20 11 13 -13 -37 8 -36 -29 -29 17 24 -50 44 30 -13 22 5
│ │ │ │ --20 4 0 |   |
│ │ │ │ +      || 37 -2 17 -42 -42 -12 18 -31 33 6 19 -31 3 -31 -11 25 -35 28 -2 -49 -41
│ │ │ │ +-13 40 -9 | |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------+
│ │ │ │ +i14 : netList randomPointsOnRationalVariety(compsJ_1, 10)
│ │ │ │ +
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +---------------+
│ │ │ │ +o14 = || -41 -1 -48 25 40 4 35 16 26 -41 -28 -16 27 -14 -39 4 4 30 -40 37 -31 -
│ │ │ │ +35 -47 0 |     |
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +---------------+
│ │ │ │ +      || -1 19 -3 12 50 3 4 25 48 50 34 -6 -29 6 -5 36 -39 -31 -48 30 47 -37 -
│ │ │ │ +48 0 |          |
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +---------------+
│ │ │ │ +      || -27 -3 -40 22 27 3 -28 -41 -12 -34 -10 40 46 29 30 24 -49 28 1 40 10 -
│ │ │ │ +22 -18 0 |     |
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +---------------+
│ │ │ │ +      || -26 -6 24 28 -27 26 34 47 13 50 3 -42 -17 5 4 -35 7 30 -13 3 8 -41 13
│ │ │ │ +0 |            |
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +---------------+
│ │ │ │ +      || 49 -7 48 1 48 25 25 -10 49 36 -16 35 -46 -5 25 -33 8 -29 49 -18 23 42
│ │ │ │ +30 0 |         |
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +---------------+
│ │ │ │ +      || -35 28 -6 22 50 -49 2 -5 -11 -39 30 27 -16 34 -9 -34 -28 15 -46 12 27
│ │ │ │ +-18 18 0 |     |
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +---------------+
│ │ │ │ +      || -49 -44 -16 -10 48 18 22 33 -35 -48 -28 -8 -23 -48 -25 -3 -21 23 44 -
│ │ │ │ +39 19 20 -37 0 ||
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +---------------+
│ │ │ │ +      || -33 -14 -18 10 2 -43 -26 45 10 19 -15 25 47 9 -15 -22 0 -47 -28 6 -33
│ │ │ │ +-9 -28 0 |     |
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +---------------+
│ │ │ │ +      || 20 -27 -17 2 -47 -23 13 40 -19 -13 39 -23 5 -3 47 -6 28 -29 -37 -33 42
│ │ │ │ +-28 26 0 |    |
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +---------------+
│ │ │ │ +      || 19 10 -10 47 41 20 -43 -34 -43 2 44 29 22 35 -42 16 44 30 5 -20 -29 -
│ │ │ │ +13 4 0 |        |
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +---------------+
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  This routine expects the input to represent an irreducible variety
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_a_n_d_o_m_P_o_i_n_t_O_n_R_a_t_i_o_n_a_l_V_a_r_i_e_t_y_(_I_d_e_a_l_) -- find a random point on a variety
│ │ │ │        that can be detected to be rational
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _r_a_n_d_o_m_P_o_i_n_t_s_O_n_R_a_t_i_o_n_a_l_V_a_r_i_e_t_y_(_I_d_e_a_l_,_Z_Z_) -- find random points on a
│ │ ├── ./usr/share/doc/Macaulay2/GroebnerWalk/example-output/___Groebner__Walk.out
│ │ │ @@ -11,21 +11,21 @@
│ │ │  i3 : R2 = QQ[x,y,z,u,v, MonomialOrder=>Weights=>{0,0,0,1,1}];
│ │ │  
│ │ │  i4 : I2 = sub(I1, R2);
│ │ │  
│ │ │  o4 : Ideal of R2
│ │ │  
│ │ │  i5 : elapsedTime gb I2
│ │ │ - -- 3.00191s elapsed
│ │ │ + -- 1.98644s elapsed
│ │ │  
│ │ │  o5 = GroebnerBasis[status: done; S-pairs encountered up to degree 16]
│ │ │  
│ │ │  o5 : GroebnerBasis
│ │ │  
│ │ │  i6 : elapsedTime groebnerWalk(gb I1, R2)
│ │ │ - -- 2.08916s elapsed
│ │ │ + -- 1.63602s elapsed
│ │ │  
│ │ │  o6 = GroebnerBasis[status: done; S-pairs encountered up to degree 0]
│ │ │  
│ │ │  o6 : GroebnerBasis
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/GroebnerWalk/html/index.html
│ │ │ @@ -92,30 +92,30 @@
│ │ │  
│ │ │  o4 : Ideal of R2    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime gb I2
│ │ │ - -- 3.00191s elapsed
│ │ │ + -- 1.98644s elapsed
│ │ │  
│ │ │  o5 = GroebnerBasis[status: done; S-pairs encountered up to degree 16]
│ │ │  
│ │ │  o5 : GroebnerBasis
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    but it is faster to compute directly in the first order and then use the Groebner walk.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i6 : elapsedTime groebnerWalk(gb I1, R2)
│ │ │ - -- 2.08916s elapsed
│ │ │ + -- 1.63602s elapsed
│ │ │  
│ │ │  o6 = GroebnerBasis[status: done; S-pairs encountered up to degree 0]
│ │ │  
│ │ │  o6 : GroebnerBasis
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -38,23 +38,23 @@
│ │ │ │  using a different weight vector and then graded reverse lexicographic we could
│ │ │ │  substitute and compute directly,
│ │ │ │  i3 : R2 = QQ[x,y,z,u,v, MonomialOrder=>Weights=>{0,0,0,1,1}];
│ │ │ │  i4 : I2 = sub(I1, R2);
│ │ │ │  
│ │ │ │  o4 : Ideal of R2
│ │ │ │  i5 : elapsedTime gb I2
│ │ │ │ - -- 3.00191s elapsed
│ │ │ │ + -- 1.98644s elapsed
│ │ │ │  
│ │ │ │  o5 = GroebnerBasis[status: done; S-pairs encountered up to degree 16]
│ │ │ │  
│ │ │ │  o5 : GroebnerBasis
│ │ │ │  but it is faster to compute directly in the first order and then use the
│ │ │ │  Groebner walk.
│ │ │ │  i6 : elapsedTime groebnerWalk(gb I1, R2)
│ │ │ │ - -- 2.08916s elapsed
│ │ │ │ + -- 1.63602s elapsed
│ │ │ │  
│ │ │ │  o6 = GroebnerBasis[status: done; S-pairs encountered up to degree 0]
│ │ │ │  
│ │ │ │  o6 : GroebnerBasis
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  The target ring must be the same ring as the ring of the starting ideal, except
│ │ │ │  with different monomial order. The ring must be a polynomial ring over a field.
│ │ ├── ./usr/share/doc/Macaulay2/Hadamard/example-output/_hadamard__Power_lp__List_cm__Z__Z_rp.out
│ │ │ @@ -6,20 +6,22 @@
│ │ │  o1 = {Point{1, 1, -}, Point{1, 0, 1}, Point{1, 2, 4}}
│ │ │                    2
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : hadamardPower(L,3)
│ │ │  
│ │ │ -                  1                                                    
│ │ │ -o2 = {Point{1, 0, -}, Point{1, 8, 64}, Point{1, 4, 8}, Point{1, 0, 16},
│ │ │ -                  4                                                    
│ │ │ +                  1                                                  
│ │ │ +o2 = {Point{1, 0, -}, Point{1, 0, 2}, Point{1, 2, 1}, Point{1, 0, 4},
│ │ │ +                  2                                                  
│ │ │       ------------------------------------------------------------------------
│ │ │ -                                 1                               1
│ │ │ -     Point{1, 0, 1}, Point{1, 1, -}, Point{1, 0, 2}, Point{1, 0, -}, Point{1,
│ │ │ -                                 8                               2
│ │ │ +                 1                                                    
│ │ │ +     Point{1, 0, -}, Point{1, 8, 64}, Point{1, 4, 8}, Point{1, 0, 16},
│ │ │ +                 4                                                    
│ │ │       ------------------------------------------------------------------------
│ │ │ -     2, 1}, Point{1, 0, 4}}
│ │ │ +                                 1
│ │ │ +     Point{1, 0, 1}, Point{1, 1, -}}
│ │ │ +                                 8
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Hadamard/example-output/_hadamard__Product_lp__List_cm__List_rp.out
│ │ │ @@ -2,12 +2,12 @@
│ │ │  
│ │ │  i1 : L = {point{0,1}, point{1,2}};
│ │ │  
│ │ │  i2 : M = {point{1,0}, point{2,2}};
│ │ │  
│ │ │  i3 : hadamardProduct(L,M)
│ │ │  
│ │ │ -o3 = {Point{1, 0}, Point{0, 2}, Point{2, 4}}
│ │ │ +o3 = {Point{2, 4}, Point{1, 0}, Point{0, 2}}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Hadamard/html/_hadamard__Power_lp__List_cm__Z__Z_rp.html
│ │ │ @@ -84,23 +84,25 @@
│ │ │  o1 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : hadamardPower(L,3)
│ │ │  
│ │ │ -                  1                                                    
│ │ │ -o2 = {Point{1, 0, -}, Point{1, 8, 64}, Point{1, 4, 8}, Point{1, 0, 16},
│ │ │ -                  4                                                    
│ │ │ +                  1                                                  
│ │ │ +o2 = {Point{1, 0, -}, Point{1, 0, 2}, Point{1, 2, 1}, Point{1, 0, 4},
│ │ │ +                  2                                                  
│ │ │       ------------------------------------------------------------------------
│ │ │ -                                 1                               1
│ │ │ -     Point{1, 0, 1}, Point{1, 1, -}, Point{1, 0, 2}, Point{1, 0, -}, Point{1,
│ │ │ -                                 8                               2
│ │ │ +                 1                                                    
│ │ │ +     Point{1, 0, -}, Point{1, 8, 64}, Point{1, 4, 8}, Point{1, 0, 16},
│ │ │ +                 4                                                    
│ │ │       ------------------------------------------------------------------------
│ │ │ -     2, 1}, Point{1, 0, 4}}
│ │ │ +                                 1
│ │ │ +     Point{1, 0, 1}, Point{1, 1, -}}
│ │ │ +                                 8
│ │ │  
│ │ │  o2 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,22 +22,24 @@
│ │ │ │  o1 = {Point{1, 1, -}, Point{1, 0, 1}, Point{1, 2, 4}}
│ │ │ │                    2
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : hadamardPower(L,3)
│ │ │ │  
│ │ │ │                    1
│ │ │ │ -o2 = {Point{1, 0, -}, Point{1, 8, 64}, Point{1, 4, 8}, Point{1, 0, 16},
│ │ │ │ -                  4
│ │ │ │ +o2 = {Point{1, 0, -}, Point{1, 0, 2}, Point{1, 2, 1}, Point{1, 0, 4},
│ │ │ │ +                  2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -                                 1                               1
│ │ │ │ -     Point{1, 0, 1}, Point{1, 1, -}, Point{1, 0, 2}, Point{1, 0, -}, Point{1,
│ │ │ │ -                                 8                               2
│ │ │ │ +                 1
│ │ │ │ +     Point{1, 0, -}, Point{1, 8, 64}, Point{1, 4, 8}, Point{1, 0, 16},
│ │ │ │ +                 4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     2, 1}, Point{1, 0, 4}}
│ │ │ │ +                                 1
│ │ │ │ +     Point{1, 0, 1}, Point{1, 1, -}}
│ │ │ │ +                                 8
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _h_a_d_a_m_a_r_d_P_o_w_e_r_(_L_i_s_t_,_Z_Z_) -- computes the $r$-th Hadmard powers of a set
│ │ │ │        points
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Hadamard/html/_hadamard__Product_lp__List_cm__List_rp.html
│ │ │ @@ -83,15 +83,15 @@
│ │ │                
    i2 : M = {point{1,0}, point{2,2}};
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : hadamardProduct(L,M)
│ │ │  
│ │ │ -o3 = {Point{1, 0}, Point{0, 2}, Point{2, 4}}
│ │ │ +o3 = {Point{2, 4}, Point{1, 0}, Point{0, 2}}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,15 +17,15 @@
│ │ │ │  Given two sets of points $L$ and $M$ returns the list of (well-defined)
│ │ │ │  entrywise multiplication of pairs of points in the cartesian product $L\times
│ │ │ │  M$.
│ │ │ │  i1 : L = {point{0,1}, point{1,2}};
│ │ │ │  i2 : M = {point{1,0}, point{2,2}};
│ │ │ │  i3 : hadamardProduct(L,M)
│ │ │ │  
│ │ │ │ -o3 = {Point{1, 0}, Point{0, 2}, Point{2, 4}}
│ │ │ │ +o3 = {Point{2, 4}, Point{1, 0}, Point{0, 2}}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _h_a_d_a_m_a_r_d_P_r_o_d_u_c_t_(_L_i_s_t_,_L_i_s_t_) -- Hadamard product of two sets of points
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/Hadamard.m2:345:0.
│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_css__Lead__Term.out
│ │ │ @@ -44,19 +44,19 @@
│ │ │  o5 = {9, 1, 99999, 9999999, 3, 999}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : netList cssLeadTerm(Hbeta, w)
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │ - -- .000004479s elapsed
│ │ │ - -- .000004238s elapsed
│ │ │ - -- .000003977s elapsed
│ │ │ - -- .000001883s elapsed
│ │ │ - -- .000001392s elapsed
│ │ │ + -- .000009152s elapsed
│ │ │ + -- .000007958s elapsed
│ │ │ + -- .00000841s elapsed
│ │ │ + -- .000008402s elapsed
│ │ │ + -- .000009382s elapsed
│ │ │  
│ │ │       +----------------------------------------------------+
│ │ │       |   1 5   5 5                                        |
│ │ │       | - - - - - -                                        |
│ │ │       |   2 2   2 2                                        |
│ │ │  o6 = |x   x x   x                                         |
│ │ │       | 1   2 4   5                                        |
│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_solve__Frobenius__Ideal.out
│ │ │ @@ -3,15 +3,15 @@
│ │ │  i1 : R = QQ[t_1..t_5];
│ │ │  
│ │ │  i2 : I = ideal(t_1+t_2+t_3+t_4+t_5, t_1+t_2-t_4, t_2+t_3-t_4, t_1*t_3, t_2*t_4);
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : solveFrobeniusIdeal I
│ │ │ - -- .000004088s elapsed
│ │ │ + -- .000006091s elapsed
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │  
│ │ │                                                                               
│ │ │  o3 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │                  0        1        2       3        0       1       2       4 
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -24,15 +24,15 @@
│ │ │         2    4    0   4    4    1   2    4    2   4    4    3       4
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : W = makeWeylAlgebra(QQ[x_1..x_5]);
│ │ │  
│ │ │  i5 : solveFrobeniusIdeal(I, W)
│ │ │ - -- .00000535s elapsed
│ │ │ + -- .000006339s elapsed
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │  
│ │ │                                                                               
│ │ │  o5 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │                  0        1        2       3        0       1       2       4 
│ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/html/_css__Lead__Term.html
│ │ │ @@ -134,19 +134,19 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : netList cssLeadTerm(Hbeta, w)
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │ - -- .000004479s elapsed
│ │ │ - -- .000004238s elapsed
│ │ │ - -- .000003977s elapsed
│ │ │ - -- .000001883s elapsed
│ │ │ - -- .000001392s elapsed
│ │ │ + -- .000009152s elapsed
│ │ │ + -- .000007958s elapsed
│ │ │ + -- .00000841s elapsed
│ │ │ + -- .000008402s elapsed
│ │ │ + -- .000009382s elapsed
│ │ │  
│ │ │       +----------------------------------------------------+
│ │ │       |   1 5   5 5                                        |
│ │ │       | - - - - - -                                        |
│ │ │       |   2 2   2 2                                        |
│ │ │  o6 = |x   x x   x                                         |
│ │ │       | 1   2 4   5                                        |
│ │ │ ├── html2text {}
│ │ │ │ @@ -57,19 +57,19 @@
│ │ │ │  o5 = {9, 1, 99999, 9999999, 3, 999}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : netList cssLeadTerm(Hbeta, w)
│ │ │ │  Warning:  F4 Algorithm not available over current coefficient ring or
│ │ │ │  inhomogeneous ideal.
│ │ │ │  Converting to Naive algorithm.
│ │ │ │ - -- .000004479s elapsed
│ │ │ │ - -- .000004238s elapsed
│ │ │ │ - -- .000003977s elapsed
│ │ │ │ - -- .000001883s elapsed
│ │ │ │ - -- .000001392s elapsed
│ │ │ │ + -- .000009152s elapsed
│ │ │ │ + -- .000007958s elapsed
│ │ │ │ + -- .00000841s elapsed
│ │ │ │ + -- .000008402s elapsed
│ │ │ │ + -- .000009382s elapsed
│ │ │ │  
│ │ │ │       +----------------------------------------------------+
│ │ │ │       |   1 5   5 5                                        |
│ │ │ │       | - - - - - -                                        |
│ │ │ │       |   2 2   2 2                                        |
│ │ │ │  o6 = |x   x x   x                                         |
│ │ │ │       | 1   2 4   5                                        |
│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/html/_solve__Frobenius__Ideal.html
│ │ │ @@ -84,15 +84,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : solveFrobeniusIdeal I
│ │ │ - -- .000004088s elapsed
│ │ │ + -- .000006091s elapsed
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │  
│ │ │                                                                               
│ │ │  o3 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │                  0        1        2       3        0       1       2       4 
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -113,15 +113,15 @@
│ │ │              
│ │ │                
    i4 : W = makeWeylAlgebra(QQ[x_1..x_5]);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : solveFrobeniusIdeal(I, W)
│ │ │ - -- .00000535s elapsed
│ │ │ + -- .000006339s elapsed
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │  
│ │ │                                                                               
│ │ │  o5 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │                  0        1        2       3        0       1       2       4 
│ │ │       ------------------------------------------------------------------------
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,15 +17,15 @@
│ │ │ │  Here is [_S_S_T, Example 2.3.16]:
│ │ │ │  i1 : R = QQ[t_1..t_5];
│ │ │ │  i2 : I = ideal(t_1+t_2+t_3+t_4+t_5, t_1+t_2-t_4, t_2+t_3-t_4, t_1*t_3,
│ │ │ │  t_2*t_4);
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : solveFrobeniusIdeal I
│ │ │ │ - -- .000004088s elapsed
│ │ │ │ + -- .000006091s elapsed
│ │ │ │  Warning:  F4 Algorithm not available over current coefficient ring or
│ │ │ │  inhomogeneous ideal.
│ │ │ │  Converting to Naive algorithm.
│ │ │ │  
│ │ │ │  
│ │ │ │  o3 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │ │                  0        1        2       3        0       1       2       4
│ │ │ │ @@ -37,15 +37,15 @@
│ │ │ │         1             1             1             3                 2
│ │ │ │       - -logX logX  - -logX logX  - -logX logX  - -logX logX  + logX }
│ │ │ │         2    4    0   4    4    1   2    4    2   4    4    3       4
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : W = makeWeylAlgebra(QQ[x_1..x_5]);
│ │ │ │  i5 : solveFrobeniusIdeal(I, W)
│ │ │ │ - -- .00000535s elapsed
│ │ │ │ + -- .000006339s elapsed
│ │ │ │  Warning:  F4 Algorithm not available over current coefficient ring or
│ │ │ │  inhomogeneous ideal.
│ │ │ │  Converting to Naive algorithm.
│ │ │ │  
│ │ │ │  
│ │ │ │  o5 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │ │                  0        1        2       3        0       1       2       4
│ │ ├── ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/example-output/_bracket.out
│ │ │ @@ -85,19 +85,16 @@
│ │ │  
│ │ │  o13 = 600
│ │ │  
│ │ │  i14 : H' = select(keys H, k->H#k != 0);
│ │ │  
│ │ │  i15 : H'
│ │ │  
│ │ │ -o15 = {({T , T }, T T  + T T  - z*T   + y*T  ), ({T , T }, - T T  + y*T  ),
│ │ │ -          3   7    4 6    3 7      11      13      2   9      2 9      16  
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      ({T , T }, - T T  - T T  - z*T   + x*T  ), ({T , T }, - T T  + z*T  ),
│ │ │ -         1   9      5 6    1 9      14      17      4   7      4 7      13  
│ │ │ +o15 = {({T , T }, - T T  - T T  - z*T   + x*T  ), ({T , T }, - T T  + z*T  ),
│ │ │ +          1   9      5 6    1 9      14      17      4   7      4 7      13  
│ │ │        -----------------------------------------------------------------------
│ │ │        ({T , T  }, T T  - T T   + x*T  ), ({T , T }, - T T  + z*T  ), ({T ,
│ │ │           1   10    4 6    1 10      20      5   9      5 9      16      3 
│ │ │        -----------------------------------------------------------------------
│ │ │        T }, T T  - z*T   + x*T  ), ({T , T }, - T T  - T T  - z*T   + x*T  ),
│ │ │         7    3 7      11      12      5   6      5 6    1 9      14      17  
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -151,22 +148,25 @@
│ │ │        -----------------------------------------------------------------------
│ │ │        z*T  ), ({T , T  }, T T  - T T   - z*T   + z*T  ), ({T , T }, T T  +
│ │ │           17      5   10    4 9    5 10      17      19      3   8    2 6  
│ │ │        -----------------------------------------------------------------------
│ │ │        T T  + T T  + y*T   - z*T  ), ({T , T }, T T  + y*T   - z*T  ), ({T ,
│ │ │         3 8    4 9      14      17      3   6    3 6      11      12      5 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T }, - T T  - T T   + z*T   + z*T  )}
│ │ │ -       7      5 7    4 10      12      20
│ │ │ +      T }, - T T  - T T   + z*T   + z*T  ), ({T , T }, T T  + T T  - z*T   +
│ │ │ +       7      5 7    4 10      12      20      3   7    4 6    3 7      11  
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      y*T  ), ({T , T }, - T T  + y*T  )}
│ │ │ +         13      2   9      2 9      16
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : H#(H'_0)
│ │ │  
│ │ │ -o16 = 1
│ │ │ +o16 = -1
│ │ │  
│ │ │  o16 : S[T ..T  ]
│ │ │           1   99
│ │ │  
│ │ │  i17 : bracketMatrix(A,1,2)
│ │ │  
│ │ │  o17 = | 0    -T_8 -T_6 -T_7 -T_10 |
│ │ ├── ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/html/_bracket.html
│ │ │ @@ -215,19 +215,16 @@
│ │ │                
    i14 : H' = select(keys H, k->H#k != 0);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : H'
│ │ │  
│ │ │ -o15 = {({T , T }, T T  + T T  - z*T   + y*T  ), ({T , T }, - T T  + y*T  ),
│ │ │ -          3   7    4 6    3 7      11      13      2   9      2 9      16  
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      ({T , T }, - T T  - T T  - z*T   + x*T  ), ({T , T }, - T T  + z*T  ),
│ │ │ -         1   9      5 6    1 9      14      17      4   7      4 7      13  
│ │ │ +o15 = {({T , T }, - T T  - T T  - z*T   + x*T  ), ({T , T }, - T T  + z*T  ),
│ │ │ +          1   9      5 6    1 9      14      17      4   7      4 7      13  
│ │ │        -----------------------------------------------------------------------
│ │ │        ({T , T  }, T T  - T T   + x*T  ), ({T , T }, - T T  + z*T  ), ({T ,
│ │ │           1   10    4 6    1 10      20      5   9      5 9      16      3 
│ │ │        -----------------------------------------------------------------------
│ │ │        T }, T T  - z*T   + x*T  ), ({T , T }, - T T  - T T  - z*T   + x*T  ),
│ │ │         7    3 7      11      12      5   6      5 6    1 9      14      17  
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -281,25 +278,28 @@
│ │ │        -----------------------------------------------------------------------
│ │ │        z*T  ), ({T , T  }, T T  - T T   - z*T   + z*T  ), ({T , T }, T T  +
│ │ │           17      5   10    4 9    5 10      17      19      3   8    2 6  
│ │ │        -----------------------------------------------------------------------
│ │ │        T T  + T T  + y*T   - z*T  ), ({T , T }, T T  + y*T   - z*T  ), ({T ,
│ │ │         3 8    4 9      14      17      3   6    3 6      11      12      5 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T }, - T T  - T T   + z*T   + z*T  )}
│ │ │ -       7      5 7    4 10      12      20
│ │ │ +      T }, - T T  - T T   + z*T   + z*T  ), ({T , T }, T T  + T T  - z*T   +
│ │ │ +       7      5 7    4 10      12      20      3   7    4 6    3 7      11  
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      y*T  ), ({T , T }, - T T  + y*T  )}
│ │ │ +         13      2   9      2 9      16
│ │ │  
│ │ │  o15 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i16 : H#(H'_0)
│ │ │  
│ │ │ -o16 = 1
│ │ │ +o16 = -1
│ │ │  
│ │ │  o16 : S[T ..T  ]
│ │ │           1   99
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -118,19 +118,16 @@
│ │ │ │  37   38   39   40   41   42   43   44
│ │ │ │  i13 : #keys H
│ │ │ │  
│ │ │ │  o13 = 600
│ │ │ │  i14 : H' = select(keys H, k->H#k != 0);
│ │ │ │  i15 : H'
│ │ │ │  
│ │ │ │ -o15 = {({T , T }, T T  + T T  - z*T   + y*T  ), ({T , T }, - T T  + y*T  ),
│ │ │ │ -          3   7    4 6    3 7      11      13      2   9      2 9      16
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      ({T , T }, - T T  - T T  - z*T   + x*T  ), ({T , T }, - T T  + z*T  ),
│ │ │ │ -         1   9      5 6    1 9      14      17      4   7      4 7      13
│ │ │ │ +o15 = {({T , T }, - T T  - T T  - z*T   + x*T  ), ({T , T }, - T T  + z*T  ),
│ │ │ │ +          1   9      5 6    1 9      14      17      4   7      4 7      13
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        ({T , T  }, T T  - T T   + x*T  ), ({T , T }, - T T  + z*T  ), ({T ,
│ │ │ │           1   10    4 6    1 10      20      5   9      5 9      16      3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        T }, T T  - z*T   + x*T  ), ({T , T }, - T T  - T T  - z*T   + x*T  ),
│ │ │ │         7    3 7      11      12      5   6      5 6    1 9      14      17
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ @@ -184,21 +181,24 @@
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        z*T  ), ({T , T  }, T T  - T T   - z*T   + z*T  ), ({T , T }, T T  +
│ │ │ │           17      5   10    4 9    5 10      17      19      3   8    2 6
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        T T  + T T  + y*T   - z*T  ), ({T , T }, T T  + y*T   - z*T  ), ({T ,
│ │ │ │         3 8    4 9      14      17      3   6    3 6      11      12      5
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T }, - T T  - T T   + z*T   + z*T  )}
│ │ │ │ -       7      5 7    4 10      12      20
│ │ │ │ +      T }, - T T  - T T   + z*T   + z*T  ), ({T , T }, T T  + T T  - z*T   +
│ │ │ │ +       7      5 7    4 10      12      20      3   7    4 6    3 7      11
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      y*T  ), ({T , T }, - T T  + y*T  )}
│ │ │ │ +         13      2   9      2 9      16
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  i16 : H#(H'_0)
│ │ │ │  
│ │ │ │ -o16 = 1
│ │ │ │ +o16 = -1
│ │ │ │  
│ │ │ │  o16 : S[T ..T  ]
│ │ │ │           1   99
│ │ │ │  From this we see that [T_5, T_6] sends T_37 to -1 in kk.
│ │ │ │  Another, often simpler view of the pairing is given by _b_r_a_c_k_e_t_M_a_t_r_i_x, where the
│ │ │ │  rows and columns correspond to the generators of Pi^d and Pi^e, and the entries
│ │ │ │  are the bracket products, interpreted as elements of Pi^{d+e}. Note the anti-
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/example-output/_cone_lp__Arrangement_cm__Ring__Element_rp.out
│ │ │ @@ -44,15 +44,15 @@
│ │ │  
│ │ │  o13 = {x, y, x - y, 0, - x + y, x}
│ │ │  
│ │ │  o13 : Hyperplane Arrangement 
│ │ │  
│ │ │  i14 : cA'' = trim cone(A, x)
│ │ │  
│ │ │ -o14 = {x - y, y, x}
│ │ │ +o14 = {y, x, x - y}
│ │ │  
│ │ │  o14 : Hyperplane Arrangement 
│ │ │  
│ │ │  i15 : assert isCentral cA''
│ │ │  
│ │ │  i16 : assert(# hyperplanes cA'' =!= 1 + # hyperplanes A)
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/example-output/_type__B_lp__Z__Z_cm__Ring_rp.out
│ │ │ @@ -33,16 +33,16 @@
│ │ │  o5 = {x , x  + x , x  + x , x }
│ │ │         1   1    2   1    2   2
│ │ │  
│ │ │  o5 : Hyperplane Arrangement 
│ │ │  
│ │ │  i6 : trim A3
│ │ │  
│ │ │ -o6 = {x , x , x  + x }
│ │ │ -       2   1   1    2
│ │ │ +o6 = {x  + x , x , x }
│ │ │ +       1    2   2   1
│ │ │  
│ │ │  o6 : Hyperplane Arrangement 
│ │ │  
│ │ │  i7 : ring A3
│ │ │  
│ │ │       ZZ
│ │ │  o7 = --[x ..x ]
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/html/_cone_lp__Arrangement_cm__Ring__Element_rp.html
│ │ │ @@ -167,15 +167,15 @@
│ │ │  o13 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : cA'' = trim cone(A, x)
│ │ │  
│ │ │ -o14 = {x - y, y, x}
│ │ │ +o14 = {y, x, x - y}
│ │ │  
│ │ │  o14 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : assert isCentral cA''
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -60,15 +60,15 @@
│ │ │ │  i13 : cone(A, x)
│ │ │ │  
│ │ │ │  o13 = {x, y, x - y, 0, - x + y, x}
│ │ │ │  
│ │ │ │  o13 : Hyperplane Arrangement
│ │ │ │  i14 : cA'' = trim cone(A, x)
│ │ │ │  
│ │ │ │ -o14 = {x - y, y, x}
│ │ │ │ +o14 = {y, x, x - y}
│ │ │ │  
│ │ │ │  o14 : Hyperplane Arrangement
│ │ │ │  i15 : assert isCentral cA''
│ │ │ │  i16 : assert(# hyperplanes cA'' =!= 1 + # hyperplanes A)
│ │ │ │  When the second input is a _S_y_m_b_o_l, this method creates a new ring from the
│ │ │ │  underlying ring of $A$ by adjoining the symbol as a variable and constructs the
│ │ │ │  cone in this new ring.
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/html/_type__B_lp__Z__Z_cm__Ring_rp.html
│ │ │ @@ -125,16 +125,16 @@
│ │ │  o5 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : trim A3
│ │ │  
│ │ │ -o6 = {x , x , x  + x }
│ │ │ -       2   1   1    2
│ │ │ +o6 = {x  + x , x , x }
│ │ │ +       1    2   2   1
│ │ │  
│ │ │  o6 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : ring A3
│ │ │ ├── html2text {}
│ │ │ │ @@ -51,16 +51,16 @@
│ │ │ │  
│ │ │ │  o5 = {x , x  + x , x  + x , x }
│ │ │ │         1   1    2   1    2   2
│ │ │ │  
│ │ │ │  o5 : Hyperplane Arrangement
│ │ │ │  i6 : trim A3
│ │ │ │  
│ │ │ │ -o6 = {x , x , x  + x }
│ │ │ │ -       2   1   1    2
│ │ │ │ +o6 = {x  + x , x , x }
│ │ │ │ +       1    2   2   1
│ │ │ │  
│ │ │ │  o6 : Hyperplane Arrangement
│ │ │ │  i7 : ring A3
│ │ │ │  
│ │ │ │       ZZ
│ │ │ │  o7 = --[x ..x ]
│ │ │ │        2  1   2
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Strategy_eq_gt..._rp.out
│ │ │ @@ -16,15 +16,15 @@
│ │ │  i3 : R = S/f
│ │ │  
│ │ │  o3 = R
│ │ │  
│ │ │  o3 : QuotientRing
│ │ │  
│ │ │  i4 : time R' = integralClosure R
│ │ │ - -- used 0.69883s (cpu); 0.430243s (thread); 0s (gc)
│ │ │ + -- used 0.773624s (cpu); 0.40907s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = R'
│ │ │  
│ │ │  o4 : QuotientRing
│ │ │  
│ │ │  i5 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -83,15 +83,15 @@
│ │ │  i9 : R = S/f
│ │ │  
│ │ │  o9 = R
│ │ │  
│ │ │  o9 : QuotientRing
│ │ │  
│ │ │  i10 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.788474s (cpu); 0.418463s (thread); 0s (gc)
│ │ │ + -- used 0.837092s (cpu); 0.431171s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = R'
│ │ │  
│ │ │  o10 : QuotientRing
│ │ │  
│ │ │  i11 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -150,15 +150,15 @@
│ │ │  i15 : R = S/f
│ │ │  
│ │ │  o15 = R
│ │ │  
│ │ │  o15 : QuotientRing
│ │ │  
│ │ │  i16 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.874799s (cpu); 0.497563s (thread); 0s (gc)
│ │ │ + -- used 0.841117s (cpu); 0.413169s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = R'
│ │ │  
│ │ │  o16 : QuotientRing
│ │ │  
│ │ │  i17 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -208,15 +208,15 @@
│ │ │  i20 : R = S/f
│ │ │  
│ │ │  o20 = R
│ │ │  
│ │ │  o20 : QuotientRing
│ │ │  
│ │ │  i21 : time R' = integralClosure(R, Strategy => SimplifyFractions)
│ │ │ - -- used 0.948566s (cpu); 0.526929s (thread); 0s (gc)
│ │ │ + -- used 0.937218s (cpu); 0.469773s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = R'
│ │ │  
│ │ │  o21 : QuotientRing
│ │ │  
│ │ │  i22 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -266,15 +266,15 @@
│ │ │  i25 : R = S/f
│ │ │  
│ │ │  o25 = R
│ │ │  
│ │ │  o25 : QuotientRing
│ │ │  
│ │ │  i26 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 1.74854s (cpu); 0.873262s (thread); 0s (gc)
│ │ │ + -- used 1.80255s (cpu); 0.811293s (thread); 0s (gc)
│ │ │  
│ │ │  o26 = R'
│ │ │  
│ │ │  o26 : QuotientRing
│ │ │  
│ │ │  i27 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -324,15 +324,15 @@
│ │ │  i30 : R = S/f
│ │ │  
│ │ │  o30 = R
│ │ │  
│ │ │  o30 : QuotientRing
│ │ │  
│ │ │  i31 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ - -- used 0.549469s (cpu); 0.440596s (thread); 0s (gc)
│ │ │ + -- used 0.586979s (cpu); 0.410297s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = R'
│ │ │  
│ │ │  o31 : QuotientRing
│ │ │  
│ │ │  i32 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -382,15 +382,15 @@
│ │ │  i35 : R = S/f
│ │ │  
│ │ │  o35 = R
│ │ │  
│ │ │  o35 : QuotientRing
│ │ │  
│ │ │  i36 : time R' = integralClosure R
│ │ │ - -- used 0.044023s (cpu); 0.0440201s (thread); 0s (gc)
│ │ │ + -- used 0.0528371s (cpu); 0.0528351s (thread); 0s (gc)
│ │ │  
│ │ │  o36 = R'
│ │ │  
│ │ │  o36 : QuotientRing
│ │ │  
│ │ │  i37 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -432,15 +432,15 @@
│ │ │  i40 : R = S/I
│ │ │  
│ │ │  o40 = R
│ │ │  
│ │ │  o40 : QuotientRing
│ │ │  
│ │ │  i41 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.0436521s (cpu); 0.0436527s (thread); 0s (gc)
│ │ │ + -- used 0.0515378s (cpu); 0.0515377s (thread); 0s (gc)
│ │ │  
│ │ │  o41 = R'
│ │ │  
│ │ │  o41 : QuotientRing
│ │ │  
│ │ │  i42 : icFractions R
│ │ │  
│ │ │ @@ -467,15 +467,15 @@
│ │ │  i45 : R = S/I
│ │ │  
│ │ │  o45 = R
│ │ │  
│ │ │  o45 : QuotientRing
│ │ │  
│ │ │  i46 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.0621814s (cpu); 0.0621828s (thread); 0s (gc)
│ │ │ + -- used 0.0746811s (cpu); 0.074676s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = R'
│ │ │  
│ │ │  o46 : QuotientRing
│ │ │  
│ │ │  i47 : icFractions R
│ │ │  
│ │ │ @@ -501,15 +501,15 @@
│ │ │  i50 : R = S/I
│ │ │  
│ │ │  o50 = R
│ │ │  
│ │ │  o50 : QuotientRing
│ │ │  
│ │ │  i51 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 0.0426811s (cpu); 0.0426768s (thread); 0s (gc)
│ │ │ + -- used 0.0578376s (cpu); 0.0578344s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = R'
│ │ │  
│ │ │  o51 : QuotientRing
│ │ │  
│ │ │  i52 : icFractions R
│ │ │  
│ │ │ @@ -536,15 +536,15 @@
│ │ │  i55 : R = S/I
│ │ │  
│ │ │  o55 = R
│ │ │  
│ │ │  o55 : QuotientRing
│ │ │  
│ │ │  i56 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ - -- used 0.0565581s (cpu); 0.0565591s (thread); 0s (gc)
│ │ │ + -- used 0.0707296s (cpu); 0.0707291s (thread); 0s (gc)
│ │ │  
│ │ │  o56 = R'
│ │ │  
│ │ │  o56 : QuotientRing
│ │ │  
│ │ │  i57 : icFractions R
│ │ │  
│ │ │ @@ -632,15 +632,15 @@
│ │ │  i66 : R = S/I
│ │ │  
│ │ │  o66 = R
│ │ │  
│ │ │  o66 : QuotientRing
│ │ │  
│ │ │  i67 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.0601817s (cpu); 0.0601808s (thread); 0s (gc)
│ │ │ + -- used 0.192702s (cpu); 0.101797s (thread); 0s (gc)
│ │ │  
│ │ │  o67 = R'
│ │ │  
│ │ │  o67 : QuotientRing
│ │ │  
│ │ │  i68 : icFractions R
│ │ │  
│ │ │ @@ -721,15 +721,15 @@
│ │ │  i77 : R = S/I
│ │ │  
│ │ │  o77 = R
│ │ │  
│ │ │  o77 : QuotientRing
│ │ │  
│ │ │  i78 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.399687s (cpu); 0.347773s (thread); 0s (gc)
│ │ │ + -- used 0.460915s (cpu); 0.384805s (thread); 0s (gc)
│ │ │  
│ │ │  o78 = R'
│ │ │  
│ │ │  o78 : QuotientRing
│ │ │  
│ │ │  i79 : icFractions R
│ │ │  
│ │ │ @@ -749,15 +749,15 @@
│ │ │  i81 : R = S/sub(I,S)
│ │ │  
│ │ │  o81 = R
│ │ │  
│ │ │  o81 : QuotientRing
│ │ │  
│ │ │  i82 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.511397s (cpu); 0.373479s (thread); 0s (gc)
│ │ │ + -- used 0.543551s (cpu); 0.397206s (thread); 0s (gc)
│ │ │  
│ │ │  o82 = R'
│ │ │  
│ │ │  o82 : QuotientRing
│ │ │  
│ │ │  i83 : icFractions R
│ │ │  
│ │ │ @@ -777,20 +777,20 @@
│ │ │  i85 : R = S/sub(I,S)
│ │ │  
│ │ │  o85 = R
│ │ │  
│ │ │  o85 : QuotientRing
│ │ │  
│ │ │  i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1)
│ │ │ - [jacobian time .000568686 sec #minors 4]
│ │ │ + [jacobian time .000603284 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .205692 sec  #fractions 6]
│ │ │ - [step 1:   time .231823 sec  #fractions 6]
│ │ │ - -- used 0.441545s (cpu); 0.311876s (thread); 0s (gc)
│ │ │ + [step 0:   time .213744 sec  #fractions 6]
│ │ │ + [step 1:   time .259694 sec  #fractions 6]
│ │ │ + -- used 0.477789s (cpu); 0.323215s (thread); 0s (gc)
│ │ │  
│ │ │  o86 = R'
│ │ │  
│ │ │  o86 : QuotientRing
│ │ │  
│ │ │  i87 : icFractions R
│ │ │  
│ │ │ @@ -810,20 +810,20 @@
│ │ │  i89 : R = S/sub(I,S)
│ │ │  
│ │ │  o89 = R
│ │ │  
│ │ │  o89 : QuotientRing
│ │ │  
│ │ │  i90 : time R' = integralClosure (R, Strategy => Vasconcelos, Verbosity => 1)
│ │ │ - [jacobian time .000531076 sec #minors 4]
│ │ │ + [jacobian time .00057524 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .0900187 sec  #fractions 6]
│ │ │ - [step 1:   time .361551 sec  #fractions 6]
│ │ │ - -- used 0.455508s (cpu); 0.332486s (thread); 0s (gc)
│ │ │ + [step 0:   time .105492 sec  #fractions 6]
│ │ │ + [step 1:   time .461168 sec  #fractions 6]
│ │ │ + -- used 0.570765s (cpu); 0.394537s (thread); 0s (gc)
│ │ │  
│ │ │  o90 = R'
│ │ │  
│ │ │  o90 : QuotientRing
│ │ │  
│ │ │  i91 : icFractions R
│ │ │  
│ │ │ @@ -843,20 +843,20 @@
│ │ │  i93 : R = S/sub(I,S)
│ │ │  
│ │ │  o93 = R
│ │ │  
│ │ │  o93 : QuotientRing
│ │ │  
│ │ │  i94 : time R' = integralClosure (R, Strategy => {Vasconcelos, StartWithOneMinor}, Verbosity => 1)
│ │ │ - [jacobian time .000614342 sec #minors 1]
│ │ │ + [jacobian time .000766695 sec #minors 1]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .115349 sec  #fractions 6]
│ │ │ - [step 1:   time .476597 sec  #fractions 6]
│ │ │ - -- used 0.595576s (cpu); 0.43448s (thread); 0s (gc)
│ │ │ + [step 0:   time .136626 sec  #fractions 6]
│ │ │ + [step 1:   time .525447 sec  #fractions 6]
│ │ │ + -- used 0.666555s (cpu); 0.486827s (thread); 0s (gc)
│ │ │  
│ │ │  o94 = R'
│ │ │  
│ │ │  o94 : QuotientRing
│ │ │  
│ │ │  i95 : icFractions R
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Verbosity_eq_gt..._rp.out
│ │ │ @@ -1,50 +1,50 @@
│ │ │  -- -*- M2-comint -*- hash: 13177954069434615273
│ │ │  
│ │ │  i1 : R = QQ[x,y,z]/ideal(x^8-z^6-y^2*z^4-z^3);
│ │ │  
│ │ │  i2 : time R' = integralClosure(R, Verbosity => 2)
│ │ │ - [jacobian time .000590017 sec #minors 3]
│ │ │ + [jacobian time .000607332 sec #minors 3]
│ │ │  integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │   [step 0: 
│ │ │ -      radical (use minprimes) .00261628 seconds
│ │ │ -      idlizer1:  .00964141 seconds
│ │ │ -      idlizer2:  .00985369 seconds
│ │ │ -      minpres:   .00867646 seconds
│ │ │ -  time .0425597 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00346468 seconds
│ │ │ +      idlizer1:  .0097807 seconds
│ │ │ +      idlizer2:  .0110849 seconds
│ │ │ +      minpres:   .0106139 seconds
│ │ │ +  time .0481789 sec  #fractions 4]
│ │ │   [step 1: 
│ │ │ -      radical (use minprimes) .00239377 seconds
│ │ │ -      idlizer1:  .0111541 seconds
│ │ │ -      idlizer2:  .00992828 seconds
│ │ │ -      minpres:   .0111938 seconds
│ │ │ -  time .0454337 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00275511 seconds
│ │ │ +      idlizer1:  .014126 seconds
│ │ │ +      idlizer2:  .0125221 seconds
│ │ │ +      minpres:   .0146113 seconds
│ │ │ +  time .0571029 sec  #fractions 4]
│ │ │   [step 2: 
│ │ │ -      radical (use minprimes) .00239345 seconds
│ │ │ -      idlizer1:  .0115122 seconds
│ │ │ -      idlizer2:  .00971861 seconds
│ │ │ -      minpres:   .00890232 seconds
│ │ │ -  time .0434159 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00285978 seconds
│ │ │ +      idlizer1:  .0141329 seconds
│ │ │ +      idlizer2:  .0117474 seconds
│ │ │ +      minpres:   .0110254 seconds
│ │ │ +  time .0527172 sec  #fractions 5]
│ │ │   [step 3: 
│ │ │ -      radical (use minprimes) .00252848 seconds
│ │ │ -      idlizer1:  .118442 seconds
│ │ │ -      idlizer2:  .0133284 seconds
│ │ │ -      minpres:   .0156925 seconds
│ │ │ -  time .162067 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00297411 seconds
│ │ │ +      idlizer1:  .137512 seconds
│ │ │ +      idlizer2:  .0150726 seconds
│ │ │ +      minpres:   .0185573 seconds
│ │ │ +  time .188397 sec  #fractions 5]
│ │ │   [step 4: 
│ │ │ -      radical (use minprimes) .00282827 seconds
│ │ │ -      idlizer1:  .00924351 seconds
│ │ │ -      idlizer2:  .0162379 seconds
│ │ │ -      minpres:   .0120262 seconds
│ │ │ -  time .0536223 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00285856 seconds
│ │ │ +      idlizer1:  .0112043 seconds
│ │ │ +      idlizer2:  .0182633 seconds
│ │ │ +      minpres:   .0190476 seconds
│ │ │ +  time .0671292 sec  #fractions 5]
│ │ │   [step 5: 
│ │ │ -      radical (use minprimes) .00232149 seconds
│ │ │ -      idlizer1:  .00789534 seconds
│ │ │ -  time .0169089 sec  #fractions 5]
│ │ │ - -- used 0.368261s (cpu); 0.301585s (thread); 0s (gc)
│ │ │ +      radical (use minprimes) .00283719 seconds
│ │ │ +      idlizer1:  .0103612 seconds
│ │ │ +  time .0217275 sec  #fractions 5]
│ │ │ + -- used 0.439793s (cpu); 0.360658s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = R'
│ │ │  
│ │ │  o2 : QuotientRing
│ │ │  
│ │ │  i3 : trim ideal R'
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp__Ideal_cm__Ring__Element_cm__Z__Z_rp.out
│ │ │ @@ -13,26 +13,26 @@
│ │ │  
│ │ │                  2      2    2        2   2 2     2
│ │ │  o3 = ideal (2a*b c + 3a , 2a b*c + 3b , a b  + 3c )
│ │ │  
│ │ │  o3 : Ideal of S
│ │ │  
│ │ │  i4 : time integralClosure J
│ │ │ - -- used 1.00532s (cpu); 0.717095s (thread); 0s (gc)
│ │ │ + -- used 1.41194s (cpu); 0.808542s (thread); 0s (gc)
│ │ │  
│ │ │               2 2              2 2                2          2   2     
│ │ │  o4 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │       ------------------------------------------------------------------------
│ │ │             2   3               2 2     2   5
│ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ │  
│ │ │  o4 : Ideal of S
│ │ │  
│ │ │  i5 : time integralClosure(J, Strategy=>{RadicalCodim1})
│ │ │ - -- used 0.637826s (cpu); 0.4847s (thread); 0s (gc)
│ │ │ + -- used 0.952755s (cpu); 0.556928s (thread); 0s (gc)
│ │ │  
│ │ │               2 2              2 2                2          2   2     
│ │ │  o5 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │       ------------------------------------------------------------------------
│ │ │             2   3               2 2     2   5
│ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp..._cm__Strategy_eq_gt..._rp.html
│ │ │ @@ -99,15 +99,15 @@
│ │ │  
│ │ │  o3 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time R' = integralClosure R
│ │ │ - -- used 0.69883s (cpu); 0.430243s (thread); 0s (gc)
│ │ │ + -- used 0.773624s (cpu); 0.40907s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = R'
│ │ │  
│ │ │  o4 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -186,15 +186,15 @@
│ │ │  
│ │ │  o9 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.788474s (cpu); 0.418463s (thread); 0s (gc)
│ │ │ + -- used 0.837092s (cpu); 0.431171s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = R'
│ │ │  
│ │ │  o10 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -273,15 +273,15 @@
│ │ │  
│ │ │  o15 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i16 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.874799s (cpu); 0.497563s (thread); 0s (gc)
│ │ │ + -- used 0.841117s (cpu); 0.413169s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = R'
│ │ │  
│ │ │  o16 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -348,15 +348,15 @@
│ │ │  
│ │ │  o20 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i21 : time R' = integralClosure(R, Strategy => SimplifyFractions)
│ │ │ - -- used 0.948566s (cpu); 0.526929s (thread); 0s (gc)
│ │ │ + -- used 0.937218s (cpu); 0.469773s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = R'
│ │ │  
│ │ │  o21 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -423,15 +423,15 @@
│ │ │  
│ │ │  o25 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i26 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 1.74854s (cpu); 0.873262s (thread); 0s (gc)
│ │ │ + -- used 1.80255s (cpu); 0.811293s (thread); 0s (gc)
│ │ │  
│ │ │  o26 = R'
│ │ │  
│ │ │  o26 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -498,15 +498,15 @@
│ │ │  
│ │ │  o30 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i31 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ - -- used 0.549469s (cpu); 0.440596s (thread); 0s (gc)
│ │ │ + -- used 0.586979s (cpu); 0.410297s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = R'
│ │ │  
│ │ │  o31 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -573,15 +573,15 @@
│ │ │  
│ │ │  o35 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i36 : time R' = integralClosure R
│ │ │ - -- used 0.044023s (cpu); 0.0440201s (thread); 0s (gc)
│ │ │ + -- used 0.0528371s (cpu); 0.0528351s (thread); 0s (gc)
│ │ │  
│ │ │  o36 = R'
│ │ │  
│ │ │  o36 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -643,15 +643,15 @@
│ │ │  
│ │ │  o40 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i41 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.0436521s (cpu); 0.0436527s (thread); 0s (gc)
│ │ │ + -- used 0.0515378s (cpu); 0.0515377s (thread); 0s (gc)
│ │ │  
│ │ │  o41 = R'
│ │ │  
│ │ │  o41 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -695,15 +695,15 @@
│ │ │  
│ │ │  o45 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i46 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.0621814s (cpu); 0.0621828s (thread); 0s (gc)
│ │ │ + -- used 0.0746811s (cpu); 0.074676s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = R'
│ │ │  
│ │ │  o46 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -746,15 +746,15 @@
│ │ │  
│ │ │  o50 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i51 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 0.0426811s (cpu); 0.0426768s (thread); 0s (gc)
│ │ │ + -- used 0.0578376s (cpu); 0.0578344s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = R'
│ │ │  
│ │ │  o51 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -798,15 +798,15 @@
│ │ │  
│ │ │  o55 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i56 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ - -- used 0.0565581s (cpu); 0.0565591s (thread); 0s (gc)
│ │ │ + -- used 0.0707296s (cpu); 0.0707291s (thread); 0s (gc)
│ │ │  
│ │ │  o56 = R'
│ │ │  
│ │ │  o56 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -932,15 +932,15 @@
│ │ │  
│ │ │  o66 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i67 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.0601817s (cpu); 0.0601808s (thread); 0s (gc)
│ │ │ + -- used 0.192702s (cpu); 0.101797s (thread); 0s (gc)
│ │ │  
│ │ │  o67 = R'
│ │ │  
│ │ │  o67 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -1056,15 +1056,15 @@
│ │ │  
│ │ │  o77 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i78 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.399687s (cpu); 0.347773s (thread); 0s (gc)
│ │ │ + -- used 0.460915s (cpu); 0.384805s (thread); 0s (gc)
│ │ │  
│ │ │  o78 = R'
│ │ │  
│ │ │  o78 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -1098,15 +1098,15 @@
│ │ │  
│ │ │  o81 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i82 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.511397s (cpu); 0.373479s (thread); 0s (gc)
│ │ │ + -- used 0.543551s (cpu); 0.397206s (thread); 0s (gc)
│ │ │  
│ │ │  o82 = R'
│ │ │  
│ │ │  o82 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -1140,20 +1140,20 @@
│ │ │  
│ │ │  o85 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1)
│ │ │ - [jacobian time .000568686 sec #minors 4]
│ │ │ + [jacobian time .000603284 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .205692 sec  #fractions 6]
│ │ │ - [step 1:   time .231823 sec  #fractions 6]
│ │ │ - -- used 0.441545s (cpu); 0.311876s (thread); 0s (gc)
│ │ │ + [step 0:   time .213744 sec  #fractions 6]
│ │ │ + [step 1:   time .259694 sec  #fractions 6]
│ │ │ + -- used 0.477789s (cpu); 0.323215s (thread); 0s (gc)
│ │ │  
│ │ │  o86 = R'
│ │ │  
│ │ │  o86 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -1187,20 +1187,20 @@
│ │ │  
│ │ │  o89 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i90 : time R' = integralClosure (R, Strategy => Vasconcelos, Verbosity => 1)
│ │ │ - [jacobian time .000531076 sec #minors 4]
│ │ │ + [jacobian time .00057524 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .0900187 sec  #fractions 6]
│ │ │ - [step 1:   time .361551 sec  #fractions 6]
│ │ │ - -- used 0.455508s (cpu); 0.332486s (thread); 0s (gc)
│ │ │ + [step 0:   time .105492 sec  #fractions 6]
│ │ │ + [step 1:   time .461168 sec  #fractions 6]
│ │ │ + -- used 0.570765s (cpu); 0.394537s (thread); 0s (gc)
│ │ │  
│ │ │  o90 = R'
│ │ │  
│ │ │  o90 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -1237,20 +1237,20 @@
│ │ │  
│ │ │  o93 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i94 : time R' = integralClosure (R, Strategy => {Vasconcelos, StartWithOneMinor}, Verbosity => 1)
│ │ │ - [jacobian time .000614342 sec #minors 1]
│ │ │ + [jacobian time .000766695 sec #minors 1]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .115349 sec  #fractions 6]
│ │ │ - [step 1:   time .476597 sec  #fractions 6]
│ │ │ - -- used 0.595576s (cpu); 0.43448s (thread); 0s (gc)
│ │ │ + [step 0:   time .136626 sec  #fractions 6]
│ │ │ + [step 1:   time .525447 sec  #fractions 6]
│ │ │ + -- used 0.666555s (cpu); 0.486827s (thread); 0s (gc)
│ │ │  
│ │ │  o94 = R'
│ │ │  
│ │ │  o94 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,15 +48,15 @@
│ │ │ │  o2 : Ideal of S
│ │ │ │  i3 : R = S/f
│ │ │ │  
│ │ │ │  o3 = R
│ │ │ │  
│ │ │ │  o3 : QuotientRing
│ │ │ │  i4 : time R' = integralClosure R
│ │ │ │ - -- used 0.69883s (cpu); 0.430243s (thread); 0s (gc)
│ │ │ │ + -- used 0.773624s (cpu); 0.40907s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = R'
│ │ │ │  
│ │ │ │  o4 : QuotientRing
│ │ │ │  i5 : netList (ideal R')_*
│ │ │ │  
│ │ │ │       +------------------------------------------------------------------------+
│ │ │ │ @@ -109,15 +109,15 @@
│ │ │ │  o8 : Ideal of S
│ │ │ │  i9 : R = S/f
│ │ │ │  
│ │ │ │  o9 = R
│ │ │ │  
│ │ │ │  o9 : QuotientRing
│ │ │ │  i10 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ │ - -- used 0.788474s (cpu); 0.418463s (thread); 0s (gc)
│ │ │ │ + -- used 0.837092s (cpu); 0.431171s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = R'
│ │ │ │  
│ │ │ │  o10 : QuotientRing
│ │ │ │  i11 : netList (ideal R')_*
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ @@ -199,15 +199,15 @@
│ │ │ │  o14 : Ideal of S
│ │ │ │  i15 : R = S/f
│ │ │ │  
│ │ │ │  o15 = R
│ │ │ │  
│ │ │ │  o15 : QuotientRing
│ │ │ │  i16 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ │ - -- used 0.874799s (cpu); 0.497563s (thread); 0s (gc)
│ │ │ │ + -- used 0.841117s (cpu); 0.413169s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = R'
│ │ │ │  
│ │ │ │  o16 : QuotientRing
│ │ │ │  i17 : netList (ideal R')_*
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ @@ -281,15 +281,15 @@
│ │ │ │  o19 : Ideal of S
│ │ │ │  i20 : R = S/f
│ │ │ │  
│ │ │ │  o20 = R
│ │ │ │  
│ │ │ │  o20 : QuotientRing
│ │ │ │  i21 : time R' = integralClosure(R, Strategy => SimplifyFractions)
│ │ │ │ - -- used 0.948566s (cpu); 0.526929s (thread); 0s (gc)
│ │ │ │ + -- used 0.937218s (cpu); 0.469773s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = R'
│ │ │ │  
│ │ │ │  o21 : QuotientRing
│ │ │ │  i22 : netList (ideal R')_*
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ @@ -363,15 +363,15 @@
│ │ │ │  o24 : Ideal of S
│ │ │ │  i25 : R = S/f
│ │ │ │  
│ │ │ │  o25 = R
│ │ │ │  
│ │ │ │  o25 : QuotientRing
│ │ │ │  i26 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ │ - -- used 1.74854s (cpu); 0.873262s (thread); 0s (gc)
│ │ │ │ + -- used 1.80255s (cpu); 0.811293s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o26 = R'
│ │ │ │  
│ │ │ │  o26 : QuotientRing
│ │ │ │  i27 : netList (ideal R')_*
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ @@ -445,15 +445,15 @@
│ │ │ │  o29 : Ideal of S
│ │ │ │  i30 : R = S/f
│ │ │ │  
│ │ │ │  o30 = R
│ │ │ │  
│ │ │ │  o30 : QuotientRing
│ │ │ │  i31 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ │ - -- used 0.549469s (cpu); 0.440596s (thread); 0s (gc)
│ │ │ │ + -- used 0.586979s (cpu); 0.410297s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o31 = R'
│ │ │ │  
│ │ │ │  o31 : QuotientRing
│ │ │ │  i32 : netList (ideal R')_*
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ @@ -527,15 +527,15 @@
│ │ │ │  o34 : Ideal of S
│ │ │ │  i35 : R = S/f
│ │ │ │  
│ │ │ │  o35 = R
│ │ │ │  
│ │ │ │  o35 : QuotientRing
│ │ │ │  i36 : time R' = integralClosure R
│ │ │ │ - -- used 0.044023s (cpu); 0.0440201s (thread); 0s (gc)
│ │ │ │ + -- used 0.0528371s (cpu); 0.0528351s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o36 = R'
│ │ │ │  
│ │ │ │  o36 : QuotientRing
│ │ │ │  i37 : netList (ideal R')_*
│ │ │ │  
│ │ │ │        +-----------+
│ │ │ │ @@ -573,15 +573,15 @@
│ │ │ │  o39 : Ideal of S
│ │ │ │  i40 : R = S/I
│ │ │ │  
│ │ │ │  o40 = R
│ │ │ │  
│ │ │ │  o40 : QuotientRing
│ │ │ │  i41 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ │ - -- used 0.0436521s (cpu); 0.0436527s (thread); 0s (gc)
│ │ │ │ + -- used 0.0515378s (cpu); 0.0515377s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o41 = R'
│ │ │ │  
│ │ │ │  o41 : QuotientRing
│ │ │ │  i42 : icFractions R
│ │ │ │  
│ │ │ │          2
│ │ │ │ @@ -603,15 +603,15 @@
│ │ │ │  o44 : Ideal of S
│ │ │ │  i45 : R = S/I
│ │ │ │  
│ │ │ │  o45 = R
│ │ │ │  
│ │ │ │  o45 : QuotientRing
│ │ │ │  i46 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ │ - -- used 0.0621814s (cpu); 0.0621828s (thread); 0s (gc)
│ │ │ │ + -- used 0.0746811s (cpu); 0.074676s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o46 = R'
│ │ │ │  
│ │ │ │  o46 : QuotientRing
│ │ │ │  i47 : icFractions R
│ │ │ │  
│ │ │ │         b*d
│ │ │ │ @@ -632,15 +632,15 @@
│ │ │ │  o49 : Ideal of S
│ │ │ │  i50 : R = S/I
│ │ │ │  
│ │ │ │  o50 = R
│ │ │ │  
│ │ │ │  o50 : QuotientRing
│ │ │ │  i51 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ │ - -- used 0.0426811s (cpu); 0.0426768s (thread); 0s (gc)
│ │ │ │ + -- used 0.0578376s (cpu); 0.0578344s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o51 = R'
│ │ │ │  
│ │ │ │  o51 : QuotientRing
│ │ │ │  i52 : icFractions R
│ │ │ │  
│ │ │ │          2
│ │ │ │ @@ -662,15 +662,15 @@
│ │ │ │  o54 : Ideal of S
│ │ │ │  i55 : R = S/I
│ │ │ │  
│ │ │ │  o55 = R
│ │ │ │  
│ │ │ │  o55 : QuotientRing
│ │ │ │  i56 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ │ - -- used 0.0565581s (cpu); 0.0565591s (thread); 0s (gc)
│ │ │ │ + -- used 0.0707296s (cpu); 0.0707291s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o56 = R'
│ │ │ │  
│ │ │ │  o56 : QuotientRing
│ │ │ │  i57 : icFractions R
│ │ │ │  
│ │ │ │         b*d
│ │ │ │ @@ -754,15 +754,15 @@
│ │ │ │  o65 : BettiTally
│ │ │ │  i66 : R = S/I
│ │ │ │  
│ │ │ │  o66 = R
│ │ │ │  
│ │ │ │  o66 : QuotientRing
│ │ │ │  i67 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ │ - -- used 0.0601817s (cpu); 0.0601808s (thread); 0s (gc)
│ │ │ │ + -- used 0.192702s (cpu); 0.101797s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o67 = R'
│ │ │ │  
│ │ │ │  o67 : QuotientRing
│ │ │ │  i68 : icFractions R
│ │ │ │  
│ │ │ │          2    2
│ │ │ │ @@ -838,15 +838,15 @@
│ │ │ │  o76 : BettiTally
│ │ │ │  i77 : R = S/I
│ │ │ │  
│ │ │ │  o77 = R
│ │ │ │  
│ │ │ │  o77 : QuotientRing
│ │ │ │  i78 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ │ - -- used 0.399687s (cpu); 0.347773s (thread); 0s (gc)
│ │ │ │ + -- used 0.460915s (cpu); 0.384805s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o78 = R'
│ │ │ │  
│ │ │ │  o78 : QuotientRing
│ │ │ │  i79 : icFractions R
│ │ │ │  
│ │ │ │          2    2   2     3      2
│ │ │ │ @@ -862,15 +862,15 @@
│ │ │ │  o80 : PolynomialRing
│ │ │ │  i81 : R = S/sub(I,S)
│ │ │ │  
│ │ │ │  o81 = R
│ │ │ │  
│ │ │ │  o81 : QuotientRing
│ │ │ │  i82 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ │ - -- used 0.511397s (cpu); 0.373479s (thread); 0s (gc)
│ │ │ │ + -- used 0.543551s (cpu); 0.397206s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o82 = R'
│ │ │ │  
│ │ │ │  o82 : QuotientRing
│ │ │ │  i83 : icFractions R
│ │ │ │  
│ │ │ │          2    2   2     3      2
│ │ │ │ @@ -886,20 +886,20 @@
│ │ │ │  o84 : PolynomialRing
│ │ │ │  i85 : R = S/sub(I,S)
│ │ │ │  
│ │ │ │  o85 = R
│ │ │ │  
│ │ │ │  o85 : QuotientRing
│ │ │ │  i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1)
│ │ │ │ - [jacobian time .000568686 sec #minors 4]
│ │ │ │ + [jacobian time .000603284 sec #minors 4]
│ │ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │ │  
│ │ │ │ - [step 0:   time .205692 sec  #fractions 6]
│ │ │ │ - [step 1:   time .231823 sec  #fractions 6]
│ │ │ │ - -- used 0.441545s (cpu); 0.311876s (thread); 0s (gc)
│ │ │ │ + [step 0:   time .213744 sec  #fractions 6]
│ │ │ │ + [step 1:   time .259694 sec  #fractions 6]
│ │ │ │ + -- used 0.477789s (cpu); 0.323215s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o86 = R'
│ │ │ │  
│ │ │ │  o86 : QuotientRing
│ │ │ │  i87 : icFractions R
│ │ │ │  
│ │ │ │          2    2   2     3      2
│ │ │ │ @@ -915,20 +915,20 @@
│ │ │ │  o88 : PolynomialRing
│ │ │ │  i89 : R = S/sub(I,S)
│ │ │ │  
│ │ │ │  o89 = R
│ │ │ │  
│ │ │ │  o89 : QuotientRing
│ │ │ │  i90 : time R' = integralClosure (R, Strategy => Vasconcelos, Verbosity => 1)
│ │ │ │ - [jacobian time .000531076 sec #minors 4]
│ │ │ │ + [jacobian time .00057524 sec #minors 4]
│ │ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │ │  
│ │ │ │ - [step 0:   time .0900187 sec  #fractions 6]
│ │ │ │ - [step 1:   time .361551 sec  #fractions 6]
│ │ │ │ - -- used 0.455508s (cpu); 0.332486s (thread); 0s (gc)
│ │ │ │ + [step 0:   time .105492 sec  #fractions 6]
│ │ │ │ + [step 1:   time .461168 sec  #fractions 6]
│ │ │ │ + -- used 0.570765s (cpu); 0.394537s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o90 = R'
│ │ │ │  
│ │ │ │  o90 : QuotientRing
│ │ │ │  i91 : icFractions R
│ │ │ │  
│ │ │ │          2    2   2     3      2
│ │ │ │ @@ -947,20 +947,20 @@
│ │ │ │  i93 : R = S/sub(I,S)
│ │ │ │  
│ │ │ │  o93 = R
│ │ │ │  
│ │ │ │  o93 : QuotientRing
│ │ │ │  i94 : time R' = integralClosure (R, Strategy => {Vasconcelos,
│ │ │ │  StartWithOneMinor}, Verbosity => 1)
│ │ │ │ - [jacobian time .000614342 sec #minors 1]
│ │ │ │ + [jacobian time .000766695 sec #minors 1]
│ │ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │ │  
│ │ │ │ - [step 0:   time .115349 sec  #fractions 6]
│ │ │ │ - [step 1:   time .476597 sec  #fractions 6]
│ │ │ │ - -- used 0.595576s (cpu); 0.43448s (thread); 0s (gc)
│ │ │ │ + [step 0:   time .136626 sec  #fractions 6]
│ │ │ │ + [step 1:   time .525447 sec  #fractions 6]
│ │ │ │ + -- used 0.666555s (cpu); 0.486827s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o94 = R'
│ │ │ │  
│ │ │ │  o94 : QuotientRing
│ │ │ │  i95 : icFractions R
│ │ │ │  
│ │ │ │           2     2          2   2     3      2
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp..._cm__Verbosity_eq_gt..._rp.html
│ │ │ @@ -71,52 +71,52 @@
│ │ │              
│ │ │                
    i1 : R = QQ[x,y,z]/ideal(x^8-z^6-y^2*z^4-z^3);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time R' = integralClosure(R, Verbosity => 2)
│ │ │ - [jacobian time .000590017 sec #minors 3]
│ │ │ + [jacobian time .000607332 sec #minors 3]
│ │ │  integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │   [step 0: 
│ │ │ -      radical (use minprimes) .00261628 seconds
│ │ │ -      idlizer1:  .00964141 seconds
│ │ │ -      idlizer2:  .00985369 seconds
│ │ │ -      minpres:   .00867646 seconds
│ │ │ -  time .0425597 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00346468 seconds
│ │ │ +      idlizer1:  .0097807 seconds
│ │ │ +      idlizer2:  .0110849 seconds
│ │ │ +      minpres:   .0106139 seconds
│ │ │ +  time .0481789 sec  #fractions 4]
│ │ │   [step 1: 
│ │ │ -      radical (use minprimes) .00239377 seconds
│ │ │ -      idlizer1:  .0111541 seconds
│ │ │ -      idlizer2:  .00992828 seconds
│ │ │ -      minpres:   .0111938 seconds
│ │ │ -  time .0454337 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00275511 seconds
│ │ │ +      idlizer1:  .014126 seconds
│ │ │ +      idlizer2:  .0125221 seconds
│ │ │ +      minpres:   .0146113 seconds
│ │ │ +  time .0571029 sec  #fractions 4]
│ │ │   [step 2: 
│ │ │ -      radical (use minprimes) .00239345 seconds
│ │ │ -      idlizer1:  .0115122 seconds
│ │ │ -      idlizer2:  .00971861 seconds
│ │ │ -      minpres:   .00890232 seconds
│ │ │ -  time .0434159 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00285978 seconds
│ │ │ +      idlizer1:  .0141329 seconds
│ │ │ +      idlizer2:  .0117474 seconds
│ │ │ +      minpres:   .0110254 seconds
│ │ │ +  time .0527172 sec  #fractions 5]
│ │ │   [step 3: 
│ │ │ -      radical (use minprimes) .00252848 seconds
│ │ │ -      idlizer1:  .118442 seconds
│ │ │ -      idlizer2:  .0133284 seconds
│ │ │ -      minpres:   .0156925 seconds
│ │ │ -  time .162067 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00297411 seconds
│ │ │ +      idlizer1:  .137512 seconds
│ │ │ +      idlizer2:  .0150726 seconds
│ │ │ +      minpres:   .0185573 seconds
│ │ │ +  time .188397 sec  #fractions 5]
│ │ │   [step 4: 
│ │ │ -      radical (use minprimes) .00282827 seconds
│ │ │ -      idlizer1:  .00924351 seconds
│ │ │ -      idlizer2:  .0162379 seconds
│ │ │ -      minpres:   .0120262 seconds
│ │ │ -  time .0536223 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00285856 seconds
│ │ │ +      idlizer1:  .0112043 seconds
│ │ │ +      idlizer2:  .0182633 seconds
│ │ │ +      minpres:   .0190476 seconds
│ │ │ +  time .0671292 sec  #fractions 5]
│ │ │   [step 5: 
│ │ │ -      radical (use minprimes) .00232149 seconds
│ │ │ -      idlizer1:  .00789534 seconds
│ │ │ -  time .0169089 sec  #fractions 5]
│ │ │ - -- used 0.368261s (cpu); 0.301585s (thread); 0s (gc)
│ │ │ +      radical (use minprimes) .00283719 seconds
│ │ │ +      idlizer1:  .0103612 seconds
│ │ │ +  time .0217275 sec  #fractions 5]
│ │ │ + -- used 0.439793s (cpu); 0.360658s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = R'
│ │ │  
│ │ │  o2 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,52 +12,52 @@
│ │ │ │              displayed. A value of 0 means: keep quiet.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  When the computation takes a considerable time, this function can be used to
│ │ │ │  decide if it will ever finish, or to get a feel for what is happening during
│ │ │ │  the computation.
│ │ │ │  i1 : R = QQ[x,y,z]/ideal(x^8-z^6-y^2*z^4-z^3);
│ │ │ │  i2 : time R' = integralClosure(R, Verbosity => 2)
│ │ │ │ - [jacobian time .000590017 sec #minors 3]
│ │ │ │ + [jacobian time .000607332 sec #minors 3]
│ │ │ │  integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2
│ │ │ │  
│ │ │ │   [step 0:
│ │ │ │ -      radical (use minprimes) .00261628 seconds
│ │ │ │ -      idlizer1:  .00964141 seconds
│ │ │ │ -      idlizer2:  .00985369 seconds
│ │ │ │ -      minpres:   .00867646 seconds
│ │ │ │ -  time .0425597 sec  #fractions 4]
│ │ │ │ +      radical (use minprimes) .00346468 seconds
│ │ │ │ +      idlizer1:  .0097807 seconds
│ │ │ │ +      idlizer2:  .0110849 seconds
│ │ │ │ +      minpres:   .0106139 seconds
│ │ │ │ +  time .0481789 sec  #fractions 4]
│ │ │ │   [step 1:
│ │ │ │ -      radical (use minprimes) .00239377 seconds
│ │ │ │ -      idlizer1:  .0111541 seconds
│ │ │ │ -      idlizer2:  .00992828 seconds
│ │ │ │ -      minpres:   .0111938 seconds
│ │ │ │ -  time .0454337 sec  #fractions 4]
│ │ │ │ +      radical (use minprimes) .00275511 seconds
│ │ │ │ +      idlizer1:  .014126 seconds
│ │ │ │ +      idlizer2:  .0125221 seconds
│ │ │ │ +      minpres:   .0146113 seconds
│ │ │ │ +  time .0571029 sec  #fractions 4]
│ │ │ │   [step 2:
│ │ │ │ -      radical (use minprimes) .00239345 seconds
│ │ │ │ -      idlizer1:  .0115122 seconds
│ │ │ │ -      idlizer2:  .00971861 seconds
│ │ │ │ -      minpres:   .00890232 seconds
│ │ │ │ -  time .0434159 sec  #fractions 5]
│ │ │ │ +      radical (use minprimes) .00285978 seconds
│ │ │ │ +      idlizer1:  .0141329 seconds
│ │ │ │ +      idlizer2:  .0117474 seconds
│ │ │ │ +      minpres:   .0110254 seconds
│ │ │ │ +  time .0527172 sec  #fractions 5]
│ │ │ │   [step 3:
│ │ │ │ -      radical (use minprimes) .00252848 seconds
│ │ │ │ -      idlizer1:  .118442 seconds
│ │ │ │ -      idlizer2:  .0133284 seconds
│ │ │ │ -      minpres:   .0156925 seconds
│ │ │ │ -  time .162067 sec  #fractions 5]
│ │ │ │ +      radical (use minprimes) .00297411 seconds
│ │ │ │ +      idlizer1:  .137512 seconds
│ │ │ │ +      idlizer2:  .0150726 seconds
│ │ │ │ +      minpres:   .0185573 seconds
│ │ │ │ +  time .188397 sec  #fractions 5]
│ │ │ │   [step 4:
│ │ │ │ -      radical (use minprimes) .00282827 seconds
│ │ │ │ -      idlizer1:  .00924351 seconds
│ │ │ │ -      idlizer2:  .0162379 seconds
│ │ │ │ -      minpres:   .0120262 seconds
│ │ │ │ -  time .0536223 sec  #fractions 5]
│ │ │ │ +      radical (use minprimes) .00285856 seconds
│ │ │ │ +      idlizer1:  .0112043 seconds
│ │ │ │ +      idlizer2:  .0182633 seconds
│ │ │ │ +      minpres:   .0190476 seconds
│ │ │ │ +  time .0671292 sec  #fractions 5]
│ │ │ │   [step 5:
│ │ │ │ -      radical (use minprimes) .00232149 seconds
│ │ │ │ -      idlizer1:  .00789534 seconds
│ │ │ │ -  time .0169089 sec  #fractions 5]
│ │ │ │ - -- used 0.368261s (cpu); 0.301585s (thread); 0s (gc)
│ │ │ │ +      radical (use minprimes) .00283719 seconds
│ │ │ │ +      idlizer1:  .0103612 seconds
│ │ │ │ +  time .0217275 sec  #fractions 5]
│ │ │ │ + -- used 0.439793s (cpu); 0.360658s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = R'
│ │ │ │  
│ │ │ │  o2 : QuotientRing
│ │ │ │  i3 : trim ideal R'
│ │ │ │  
│ │ │ │                       3   2                     2 2    4           4
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp__Ideal_cm__Ring__Element_cm__Z__Z_rp.html
│ │ │ @@ -109,29 +109,29 @@
│ │ │  
│ │ │  o3 : Ideal of S
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time integralClosure J
│ │ │ - -- used 1.00532s (cpu); 0.717095s (thread); 0s (gc)
│ │ │ + -- used 1.41194s (cpu); 0.808542s (thread); 0s (gc)
│ │ │  
│ │ │               2 2              2 2                2          2   2     
│ │ │  o4 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │       ------------------------------------------------------------------------
│ │ │             2   3               2 2     2   5
│ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ │  
│ │ │  o4 : Ideal of S
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time integralClosure(J, Strategy=>{RadicalCodim1})
│ │ │ - -- used 0.637826s (cpu); 0.4847s (thread); 0s (gc)
│ │ │ + -- used 0.952755s (cpu); 0.556928s (thread); 0s (gc)
│ │ │  
│ │ │               2 2              2 2                2          2   2     
│ │ │  o5 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │       ------------------------------------------------------------------------
│ │ │             2   3               2 2     2   5
│ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ │ ├── html2text {}
│ │ │ │ @@ -46,25 +46,25 @@
│ │ │ │  i3 : J = ideal jacobian ideal F
│ │ │ │  
│ │ │ │                  2      2    2        2   2 2     2
│ │ │ │  o3 = ideal (2a*b c + 3a , 2a b*c + 3b , a b  + 3c )
│ │ │ │  
│ │ │ │  o3 : Ideal of S
│ │ │ │  i4 : time integralClosure J
│ │ │ │ - -- used 1.00532s (cpu); 0.717095s (thread); 0s (gc)
│ │ │ │ + -- used 1.41194s (cpu); 0.808542s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               2 2              2 2                2          2   2
│ │ │ │  o4 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │             2   3               2 2     2   5
│ │ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ │ │  
│ │ │ │  o4 : Ideal of S
│ │ │ │  i5 : time integralClosure(J, Strategy=>{RadicalCodim1})
│ │ │ │ - -- used 0.637826s (cpu); 0.4847s (thread); 0s (gc)
│ │ │ │ + -- used 0.952755s (cpu); 0.556928s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               2 2              2 2                2          2   2
│ │ │ │  o5 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │             2   3               2 2     2   5
│ │ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_equivariant__Hilbert.out
│ │ │ @@ -25,15 +25,15 @@
│ │ │  o3 : DiagonalAction
│ │ │  
│ │ │  i4 : T.cache.?equivariantHilbert
│ │ │  
│ │ │  o4 = false
│ │ │  
│ │ │  i5 : elapsedTime equivariantHilbertSeries(T, Order => 5)
│ │ │ - -- .00274702s elapsed
│ │ │ + -- .0034381s elapsed
│ │ │  
│ │ │                    -1    -1       2 2              -2    -1 -1    -2  2  
│ │ │  o5 = 1 + (z z  + z   + z  )T + (z z  + z  + z  + z   + z  z   + z  )T  +
│ │ │             0 1    1     0        0 1    0    1    1     0  1     0      
│ │ │       ------------------------------------------------------------------------
│ │ │         3 3    2        2      -1        -3    -1      -1 -2    -2 -1    -3  3
│ │ │       (z z  + z z  + z z  + z z   + 1 + z   + z  z  + z  z   + z  z   + z  )T 
│ │ │ @@ -51,10 +51,10 @@
│ │ │           0   1
│ │ │  
│ │ │  i6 : T.cache.?equivariantHilbert
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : elapsedTime equivariantHilbertSeries(T, Order => 5);
│ │ │ - -- .000427798s elapsed
│ │ │ + -- .000586542s elapsed
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_hsop_spalgorithms.out
│ │ │ @@ -23,23 +23,23 @@
│ │ │  o3 = QQ[x..z] <- {| 0 -1 0  |, | 0 -1 0 |}
│ │ │                    | 1 0  0  |  | 1 0  0 |
│ │ │                    | 0 0  -1 |  | 0 0  1 |
│ │ │  
│ │ │  o3 : FiniteGroupAction
│ │ │  
│ │ │  i4 : time P1=primaryInvariants C4xC2
│ │ │ - -- used 0.797659s (cpu); 0.519949s (thread); 0s (gc)
│ │ │ + -- used 0.943442s (cpu); 0.638119s (thread); 0s (gc)
│ │ │  
│ │ │         2   2    2   3       3
│ │ │  o4 = {z , x  + y , x y - x*y }
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ - -- used 0.671422s (cpu); 0.364464s (thread); 0s (gc)
│ │ │ + -- used 0.784327s (cpu); 0.41924s (thread); 0s (gc)
│ │ │  
│ │ │                     8                 7                   6 2  
│ │ │  o5 = {656100000000x  - 4738500000000x y + 10209037500000x y  -
│ │ │       ------------------------------------------------------------------------
│ │ │                     5 3                  4 4                 3 5  
│ │ │       1232156250000x y  - 14757374609375x y  + 1232156250000x y  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -90,23 +90,23 @@
│ │ │       ------------------------------------------------------------------------
│ │ │          2 6    8
│ │ │       90y z  + z }
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : time secondaryInvariants(P1,C4xC2)
│ │ │ - -- used 0.0219117s (cpu); 0.0219168s (thread); 0s (gc)
│ │ │ + -- used 0.0256658s (cpu); 0.0256669s (thread); 0s (gc)
│ │ │  
│ │ │            4    4
│ │ │  o6 = {1, x  + y }
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : time secondaryInvariants(P2,C4xC2)
│ │ │ - -- used 2.00343s (cpu); 1.26795s (thread); 0s (gc)
│ │ │ + -- used 2.68634s (cpu); 1.52762s (thread); 0s (gc)
│ │ │  
│ │ │            2   2    2   4   2 2    2 2   2 2   3       3   4    4   6   2 4  
│ │ │  o7 = {1, z , x  + y , z , x z  + y z , x y , x y - x*y , x  + y , z , x z  +
│ │ │       ------------------------------------------------------------------------
│ │ │        2 4   2 2 2   3   2      3 2   4 2    4 2   4 2    2 4   5       5   6
│ │ │       y z , x y z , x y*z  - x*y z , x z  + y z , x y  + x y , x y - x*y , x 
│ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp..._cm__Degree__Bound_eq_gt..._rp.out
│ │ │ @@ -14,15 +14,15 @@
│ │ │             | 1 0 0 0 |  | 1 0 0 0 |
│ │ │             | 0 0 1 0 |  | 0 1 0 0 |
│ │ │             | 0 0 0 1 |  | 0 0 1 0 |
│ │ │  
│ │ │  o3 : FiniteGroupAction
│ │ │  
│ │ │  i4 : elapsedTime invariants S4
│ │ │ - -- .779956s elapsed
│ │ │ + -- .599156s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │ @@ -32,15 +32,15 @@
│ │ │  
│ │ │  i5 : elapsedTime invariants(S4,DegreeBound=>4)
│ │ │  
│ │ │  Warning: stopping condition not met!
│ │ │  Output may not generate the entire ring of invariants.
│ │ │  Increase value of DegreeBound.
│ │ │  
│ │ │ - -- .524625s elapsed
│ │ │ + -- .480397s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o5 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp..._cm__Use__Linear__Algebra_eq_gt..._rp.out
│ │ │ @@ -14,28 +14,28 @@
│ │ │             | 1 0 0 0 |  | 1 0 0 0 |
│ │ │             | 0 0 1 0 |  | 0 1 0 0 |
│ │ │             | 0 0 0 1 |  | 0 0 1 0 |
│ │ │  
│ │ │  o3 : FiniteGroupAction
│ │ │  
│ │ │  i4 : elapsedTime invariants S4
│ │ │ - -- .677194s elapsed
│ │ │ + -- .570454s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │        4
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : elapsedTime invariants(S4,UseLinearAlgebra=>true)
│ │ │ - -- .104964s elapsed
│ │ │ + -- .0721791s elapsed
│ │ │  
│ │ │  o5 = {x  + x  + x  + x , x x  + x x  + x x  + x x  + x x  + x x , x x x  +
│ │ │         1    2    3    4   1 2    1 3    2 3    1 4    2 4    3 4   1 2 3  
│ │ │       ------------------------------------------------------------------------
│ │ │       x x x  + x x x  + x x x , x x x x }
│ │ │        1 2 4    1 3 4    2 3 4   1 2 3 4
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_equivariant__Hilbert.html
│ │ │ @@ -92,15 +92,15 @@
│ │ │  
│ │ │  o4 = false
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime equivariantHilbertSeries(T, Order => 5)
│ │ │ - -- .00274702s elapsed
│ │ │ + -- .0034381s elapsed
│ │ │  
│ │ │                    -1    -1       2 2              -2    -1 -1    -2  2  
│ │ │  o5 = 1 + (z z  + z   + z  )T + (z z  + z  + z  + z   + z  z   + z  )T  +
│ │ │             0 1    1     0        0 1    0    1    1     0  1     0      
│ │ │       ------------------------------------------------------------------------
│ │ │         3 3    2        2      -1        -3    -1      -1 -2    -2 -1    -3  3
│ │ │       (z z  + z z  + z z  + z z   + 1 + z   + z  z  + z  z   + z  z   + z  )T 
│ │ │ @@ -124,15 +124,15 @@
│ │ │  
│ │ │  o6 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : elapsedTime equivariantHilbertSeries(T, Order => 5);
│ │ │ - -- .000427798s elapsed
    
│ │ │ + -- .000586542s elapsed
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    
│ │ │            
    For the programmer
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,15 +30,15 @@
│ │ │ │       | 0  -1 1 |
│ │ │ │  
│ │ │ │  o3 : DiagonalAction
│ │ │ │  i4 : T.cache.?equivariantHilbert
│ │ │ │  
│ │ │ │  o4 = false
│ │ │ │  i5 : elapsedTime equivariantHilbertSeries(T, Order => 5)
│ │ │ │ - -- .00274702s elapsed
│ │ │ │ + -- .0034381s elapsed
│ │ │ │  
│ │ │ │                    -1    -1       2 2              -2    -1 -1    -2  2
│ │ │ │  o5 = 1 + (z z  + z   + z  )T + (z z  + z  + z  + z   + z  z   + z  )T  +
│ │ │ │             0 1    1     0        0 1    0    1    1     0  1     0
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │         3 3    2        2      -1        -3    -1      -1 -2    -2 -1    -3  3
│ │ │ │       (z z  + z z  + z z  + z z   + 1 + z   + z  z  + z  z   + z  z   + z  )T
│ │ │ │ @@ -54,13 +54,13 @@
│ │ │ │  
│ │ │ │  o5 : ZZ[z ..z ][T]
│ │ │ │           0   1
│ │ │ │  i6 : T.cache.?equivariantHilbert
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  i7 : elapsedTime equivariantHilbertSeries(T, Order => 5);
│ │ │ │ - -- .000427798s elapsed
│ │ │ │ + -- .000586542s elapsed
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _e_q_u_i_v_a_r_i_a_n_t_H_i_l_b_e_r_t is a _s_y_m_b_o_l.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/InvariantRing/AbelianGroupsDoc.m2:185:0.
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_hsop_spalgorithms.html
│ │ │ @@ -92,26 +92,26 @@
│ │ │            
│ │ │          
│ │ │          
    The two algorithms used in primaryInvariants are timed. One sees that the Dade algorithm is faster, however the primary invariants output are all of degree 8 and have ugly coefficients.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │          
    
│ │ │                
    i4 : time P1=primaryInvariants C4xC2
│ │ │ - -- used 0.797659s (cpu); 0.519949s (thread); 0s (gc)
│ │ │ + -- used 0.943442s (cpu); 0.638119s (thread); 0s (gc)
│ │ │  
│ │ │         2   2    2   3       3
│ │ │  o4 = {z , x  + y , x y - x*y }
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │                
    i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ - -- used 0.671422s (cpu); 0.364464s (thread); 0s (gc)
│ │ │ + -- used 0.784327s (cpu); 0.41924s (thread); 0s (gc)
│ │ │  
│ │ │                     8                 7                   6 2  
│ │ │  o5 = {656100000000x  - 4738500000000x y + 10209037500000x y  -
│ │ │       ------------------------------------------------------------------------
│ │ │                     5 3                  4 4                 3 5  
│ │ │       1232156250000x y  - 14757374609375x y  + 1232156250000x y  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -168,26 +168,26 @@
│ │ │            
    
│ │ │          
    The extra work done by the default algorithm to ensure an optimal hsop is rewarded by needing to calculate a smaller collection of corresponding secondary invariants.  In fact, it has proved quicker overall to calculate the invariant ring based on the optimal algorithm rather than the Dade algorithm.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i6 : time secondaryInvariants(P1,C4xC2)
│ │ │ - -- used 0.0219117s (cpu); 0.0219168s (thread); 0s (gc)
│ │ │ + -- used 0.0256658s (cpu); 0.0256669s (thread); 0s (gc)
│ │ │  
│ │ │            4    4
│ │ │  o6 = {1, x  + y }
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : time secondaryInvariants(P2,C4xC2)
│ │ │ - -- used 2.00343s (cpu); 1.26795s (thread); 0s (gc)
│ │ │ + -- used 2.68634s (cpu); 1.52762s (thread); 0s (gc)
│ │ │  
│ │ │            2   2    2   4   2 2    2 2   2 2   3       3   4    4   6   2 4  
│ │ │  o7 = {1, z , x  + y , z , x z  + y z , x y , x y - x*y , x  + y , z , x z  +
│ │ │       ------------------------------------------------------------------------
│ │ │        2 4   2 2 2   3   2      3 2   4 2    4 2   4 2    2 4   5       5   6
│ │ │       y z , x y z , x y*z  - x*y z , x z  + y z , x y  + x y , x y - x*y , x 
│ │ │       ------------------------------------------------------------------------
│ │ │ ├── html2text {}
│ │ │ │ @@ -69,22 +69,22 @@
│ │ │ │                    | 0 0  -1 |  | 0 0  1 |
│ │ │ │  
│ │ │ │  o3 : FiniteGroupAction
│ │ │ │  The two algorithms used in _p_r_i_m_a_r_y_I_n_v_a_r_i_a_n_t_s are timed. One sees that the Dade
│ │ │ │  algorithm is faster, however the primary invariants output are all of degree 8
│ │ │ │  and have ugly coefficients.
│ │ │ │  i4 : time P1=primaryInvariants C4xC2
│ │ │ │ - -- used 0.797659s (cpu); 0.519949s (thread); 0s (gc)
│ │ │ │ + -- used 0.943442s (cpu); 0.638119s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         2   2    2   3       3
│ │ │ │  o4 = {z , x  + y , x y - x*y }
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ │ - -- used 0.671422s (cpu); 0.364464s (thread); 0s (gc)
│ │ │ │ + -- used 0.784327s (cpu); 0.41924s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                     8                 7                   6 2
│ │ │ │  o5 = {656100000000x  - 4738500000000x y + 10209037500000x y  -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                     5 3                  4 4                 3 5
│ │ │ │       1232156250000x y  - 14757374609375x y  + 1232156250000x y  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -138,22 +138,22 @@
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  The extra work done by the default algorithm to ensure an optimal hsop is
│ │ │ │  rewarded by needing to calculate a smaller collection of corresponding
│ │ │ │  secondary invariants. In fact, it has proved quicker overall to calculate the
│ │ │ │  invariant ring based on the optimal algorithm rather than the Dade algorithm.
│ │ │ │  i6 : time secondaryInvariants(P1,C4xC2)
│ │ │ │ - -- used 0.0219117s (cpu); 0.0219168s (thread); 0s (gc)
│ │ │ │ + -- used 0.0256658s (cpu); 0.0256669s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            4    4
│ │ │ │  o6 = {1, x  + y }
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : time secondaryInvariants(P2,C4xC2)
│ │ │ │ - -- used 2.00343s (cpu); 1.26795s (thread); 0s (gc)
│ │ │ │ + -- used 2.68634s (cpu); 1.52762s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            2   2    2   4   2 2    2 2   2 2   3       3   4    4   6   2 4
│ │ │ │  o7 = {1, z , x  + y , z , x z  + y z , x y , x y - x*y , x  + y , z , x z  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        2 4   2 2 2   3   2      3 2   4 2    4 2   4 2    2 4   5       5   6
│ │ │ │       y z , x y z , x y*z  - x*y z , x z  + y z , x y  + x y , x y - x*y , x
│ │ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp..._cm__Degree__Bound_eq_gt..._rp.html
│ │ │ @@ -96,15 +96,15 @@
│ │ │  
│ │ │  o3 : FiniteGroupAction
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime invariants S4
│ │ │ - -- .779956s elapsed
│ │ │ + -- .599156s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │ @@ -117,15 +117,15 @@
│ │ │              
    		
│ │ │                
    i5 : elapsedTime invariants(S4,DegreeBound=>4)
│ │ │  
│ │ │  Warning: stopping condition not met!
│ │ │  Output may not generate the entire ring of invariants.
│ │ │  Increase value of DegreeBound.
│ │ │  
│ │ │ - -- .524625s elapsed
│ │ │ + -- .480397s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o5 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,15 +33,15 @@
│ │ │ │  o3 = R <- {| 0 1 0 0 |, | 0 0 0 1 |}
│ │ │ │             | 1 0 0 0 |  | 1 0 0 0 |
│ │ │ │             | 0 0 1 0 |  | 0 1 0 0 |
│ │ │ │             | 0 0 0 1 |  | 0 0 1 0 |
│ │ │ │  
│ │ │ │  o3 : FiniteGroupAction
│ │ │ │  i4 : elapsedTime invariants S4
│ │ │ │ - -- .779956s elapsed
│ │ │ │ + -- .599156s elapsed
│ │ │ │  
│ │ │ │                            2    2    2    2   3    3    3    3   4    4    4
│ │ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        4
│ │ │ │       x }
│ │ │ │ @@ -50,15 +50,15 @@
│ │ │ │  o4 : List
│ │ │ │  i5 : elapsedTime invariants(S4,DegreeBound=>4)
│ │ │ │  
│ │ │ │  Warning: stopping condition not met!
│ │ │ │  Output may not generate the entire ring of invariants.
│ │ │ │  Increase value of DegreeBound.
│ │ │ │  
│ │ │ │ - -- .524625s elapsed
│ │ │ │ + -- .480397s elapsed
│ │ │ │  
│ │ │ │                            2    2    2    2   3    3    3    3   4    4    4
│ │ │ │  o5 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        4
│ │ │ │       x }
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp..._cm__Use__Linear__Algebra_eq_gt..._rp.html
│ │ │ @@ -96,15 +96,15 @@
│ │ │  
│ │ │  o3 : FiniteGroupAction
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime invariants S4
│ │ │ - -- .677194s elapsed
│ │ │ + -- .570454s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │ @@ -112,15 +112,15 @@
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime invariants(S4,UseLinearAlgebra=>true)
│ │ │ - -- .104964s elapsed
│ │ │ + -- .0721791s elapsed
│ │ │  
│ │ │  o5 = {x  + x  + x  + x , x x  + x x  + x x  + x x  + x x  + x x , x x x  +
│ │ │         1    2    3    4   1 2    1 3    2 3    1 4    2 4    3 4   1 2 3  
│ │ │       ------------------------------------------------------------------------
│ │ │       x x x  + x x x  + x x x , x x x x }
│ │ │        1 2 4    1 3 4    2 3 4   1 2 3 4
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,27 +35,27 @@
│ │ │ │  o3 = R <- {| 0 1 0 0 |, | 0 0 0 1 |}
│ │ │ │             | 1 0 0 0 |  | 1 0 0 0 |
│ │ │ │             | 0 0 1 0 |  | 0 1 0 0 |
│ │ │ │             | 0 0 0 1 |  | 0 0 1 0 |
│ │ │ │  
│ │ │ │  o3 : FiniteGroupAction
│ │ │ │  i4 : elapsedTime invariants S4
│ │ │ │ - -- .677194s elapsed
│ │ │ │ + -- .570454s elapsed
│ │ │ │  
│ │ │ │                            2    2    2    2   3    3    3    3   4    4    4
│ │ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        4
│ │ │ │       x }
│ │ │ │        4
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : elapsedTime invariants(S4,UseLinearAlgebra=>true)
│ │ │ │ - -- .104964s elapsed
│ │ │ │ + -- .0721791s elapsed
│ │ │ │  
│ │ │ │  o5 = {x  + x  + x  + x , x x  + x x  + x x  + x x  + x x  + x x , x x x  +
│ │ │ │         1    2    3    4   1 2    1 3    2 3    1 4    2 4    3 4   1 2 3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       x x x  + x x x  + x x x , x x x x }
│ │ │ │        1 2 4    1 3 4    2 3 4   1 2 3 4
│ │ ├── ./usr/share/doc/Macaulay2/Isomorphism/example-output/_is__Isomorphic.out
│ │ │ @@ -156,20 +156,20 @@
│ │ │                     {-1} | 0 0 0  0 0 0 0 0 0 0  0 0  0 0  0 0 0  0 0 0 0  0 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    x_2  x_1  x_0  |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    x_0  0    0    0    0    0    0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_2 x_1 x_0^2 |
│ │ │                     {-1} | 0 0 0  0 0 0 0 0 0 0  0 0  0 0  0 0 0  0 0 0 0  0 0 1 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0    x_0  -x_2 x_1  -x_3 x_2  0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_3 x_2 x_1^2 |
│ │ │  
│ │ │                                  40
│ │ │  o22 : S-module, subquotient of S
│ │ │  
│ │ │  i23 : elapsedTime isIsomorphic(T1, T2)
│ │ │ - -- 1.4004s elapsed
│ │ │ + -- 1.60473s elapsed
│ │ │  
│ │ │  o23 = true
│ │ │  
│ │ │  i24 : elapsedTime isomorphism(T1, T2)
│ │ │ - -- .00002087s elapsed
│ │ │ + -- .000037281s elapsed
│ │ │  
│ │ │  o24 = {-1} | 1      -3976  -13490 13495  -2886  2577   14757  -881   7677  
│ │ │        {-1} | -2527  -13566 2778   -6934  -14806 4619   -13099 6022   -10907
│ │ │        {-1} | -15420 5642   1489   1354   4591   11881  -5253  7296   -1098 
│ │ │        {-1} | 7909   -12428 -2260  -8465  12113  -6893  8411   4186   -9393 
│ │ │        {-1} | -9615  2934   10440  5015   8145   -5585  1360   3295   12851 
│ │ │        {-1} | -4881  -7984  12700  -10391 -10009 -14538 13207  262    -6500
│ │ ├── ./usr/share/doc/Macaulay2/Isomorphism/html/_is__Isomorphic.html
│ │ │ @@ -328,23 +328,23 @@
│ │ │                                  40
│ │ │  o22 : S-module, subquotient of S
    
│ │ │              
    
│ │ │                
    i23 : elapsedTime isIsomorphic(T1, T2)
│ │ │ - -- 1.4004s elapsed
│ │ │ + -- 1.60473s elapsed
│ │ │  
│ │ │  o23 = true
    
│ │ │              
    
│ │ │                
    i24 : elapsedTime isomorphism(T1, T2)
│ │ │ - -- .00002087s elapsed
│ │ │ + -- .000037281s elapsed
│ │ │  
│ │ │  o24 = {-1} | 1      -3976  -13490 13495  -2886  2577   14757  -881   7677  
│ │ │        {-1} | -2527  -13566 2778   -6934  -14806 4619   -13099 6022   -10907
│ │ │        {-1} | -15420 5642   1489   1354   4591   11881  -5253  7296   -1098 
│ │ │        {-1} | 7909   -12428 -2260  -8465  12113  -6893  8411   4186   -9393 
│ │ │        {-1} | -9615  2934   10440  5015   8145   -5585  1360   3295   12851 
│ │ │        {-1} | -4881  -7984  12700  -10391 -10009 -14538 13207  262    -6500
│ │ │ ├── html2text {}
│ │ │ │ @@ -684,19 +684,19 @@
│ │ │ │  0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0
│ │ │ │  0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0
│ │ │ │  0   0   0     x_3 x_2 x_1^2 |
│ │ │ │  
│ │ │ │                                  40
│ │ │ │  o22 : S-module, subquotient of S
│ │ │ │  i23 : elapsedTime isIsomorphic(T1, T2)
│ │ │ │ - -- 1.4004s elapsed
│ │ │ │ + -- 1.60473s elapsed
│ │ │ │  
│ │ │ │  o23 = true
│ │ │ │  i24 : elapsedTime isomorphism(T1, T2)
│ │ │ │ - -- .00002087s elapsed
│ │ │ │ + -- .000037281s elapsed
│ │ │ │  
│ │ │ │  o24 = {-1} | 1      -3976  -13490 13495  -2886  2577   14757  -881   7677
│ │ │ │        {-1} | -2527  -13566 2778   -6934  -14806 4619   -13099 6022   -10907
│ │ │ │        {-1} | -15420 5642   1489   1354   4591   11881  -5253  7296   -1098
│ │ │ │        {-1} | 7909   -12428 -2260  -8465  12113  -6893  8411   4186   -9393
│ │ │ │        {-1} | -9615  2934   10440  5015   8145   -5585  1360   3295   12851
│ │ │ │        {-1} | -4881  -7984  12700  -10391 -10009 -14538 13207  262    -6500
│ │ ├── ./usr/share/doc/Macaulay2/JSON/example-output/_from__J__S__O__N.out
│ │ │ @@ -39,19 +39,19 @@
│ │ │  
│ │ │  o8 = {1, 2, 3}
│ │ │  
│ │ │  o8 : List
│ │ │  
│ │ │  i9 : jsonFile = temporaryFileName() | ".json"
│ │ │  
│ │ │ -o9 = /tmp/M2-50412-0/0.json
│ │ │ +o9 = /tmp/M2-79402-0/0.json
│ │ │  
│ │ │  i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │  
│ │ │ -o10 = /tmp/M2-50412-0/0.json
│ │ │ +o10 = /tmp/M2-79402-0/0.json
│ │ │  
│ │ │  o10 : File
│ │ │  
│ │ │  i11 : fromJSON openIn jsonFile
│ │ │  
│ │ │  o11 = {1, 2, 3}
│ │ ├── ./usr/share/doc/Macaulay2/JSON/html/_from__J__S__O__N.html
│ │ │ @@ -167,22 +167,22 @@
│ │ │            
    The input may also be a file containing JSON data.
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i9 : jsonFile = temporaryFileName() | ".json"
│ │ │  
│ │ │ -o9 = /tmp/M2-50412-0/0.json
    
│ │ │ +o9 = /tmp/M2-79402-0/0.json
│ │ │              
    
│ │ │                
    i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │  
│ │ │ -o10 = /tmp/M2-50412-0/0.json
│ │ │ +o10 = /tmp/M2-79402-0/0.json
│ │ │  
│ │ │  o10 : File
    
│ │ │              
    
│ │ │                
    i11 : fromJSON openIn jsonFile
│ │ │ ├── html2text {}
│ │ │ │ @@ -53,18 +53,18 @@
│ │ │ │  
│ │ │ │  o8 = {1, 2, 3}
│ │ │ │  
│ │ │ │  o8 : List
│ │ │ │  The input may also be a file containing JSON data.
│ │ │ │  i9 : jsonFile = temporaryFileName() | ".json"
│ │ │ │  
│ │ │ │ -o9 = /tmp/M2-50412-0/0.json
│ │ │ │ +o9 = /tmp/M2-79402-0/0.json
│ │ │ │  i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │ │  
│ │ │ │ -o10 = /tmp/M2-50412-0/0.json
│ │ │ │ +o10 = /tmp/M2-79402-0/0.json
│ │ │ │  
│ │ │ │  o10 : File
│ │ │ │  i11 : fromJSON openIn jsonFile
│ │ │ │  
│ │ │ │  o11 = {1, 2, 3}
│ │ │ │  
│ │ │ │  o11 : List
│ │ ├── ./usr/share/doc/Macaulay2/Jets/example-output/___Example_sp1.out
│ │ │ @@ -17,24 +17,24 @@
│ │ │  o3 = ideal (y0*z0*x2 + x0*z0*y2 + x0*y0*z2 + z0*x1*y1 + y0*x1*z1 + x0*y1*z1,
│ │ │       ------------------------------------------------------------------------
│ │ │       y0*z0*x1 + x0*z0*y1 + x0*y0*z1, x0*y0*z0)
│ │ │  
│ │ │  o3 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
│ │ │  
│ │ │  i4 : elapsedTime jetsRadical(2,I)
│ │ │ - -- .00226289s elapsed
│ │ │ + -- .00268378s elapsed
│ │ │  
│ │ │  o4 = ideal (y0*z0*x2, x0*z0*y2, x0*y0*z2, z0*x1*y1, y0*x1*z1, x0*y1*z1,
│ │ │       ------------------------------------------------------------------------
│ │ │       y0*z0*x1, x0*z0*y1, x0*y0*z1, x0*y0*z0)
│ │ │  
│ │ │  o4 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
│ │ │  
│ │ │  i5 : elapsedTime radical J2I
│ │ │ - -- .297009s elapsed
│ │ │ + -- .255297s elapsed
│ │ │  
│ │ │  o5 = ideal (x0*y0*z0, x0*y0*z1, x0*z0*y1, y0*z0*x1, x0*y1*z1, y0*x1*z1,
│ │ │       ------------------------------------------------------------------------
│ │ │       z0*x1*y1, x0*y0*z2, x0*z0*y2, y0*z0*x2)
│ │ │  
│ │ │  o5 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
│ │ ├── ./usr/share/doc/Macaulay2/Jets/example-output/___Storing_sp__Computations.out
│ │ │ @@ -33,15 +33,15 @@
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : I.cache.?jet
│ │ │  
│ │ │  o7 = false
│ │ │  
│ │ │  i8 : elapsedTime jets(3,I)
│ │ │ - -- .0247993s elapsed
│ │ │ + -- .0110744s elapsed
│ │ │  
│ │ │                                                    2                 2
│ │ │  o8 = ideal (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o8 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │  
│ │ │  i9 : I.cache.?jet
│ │ │ @@ -53,23 +53,23 @@
│ │ │  o10 = CacheTable{jetsMatrix => | 2x0x3-y3+2x1x2 |}
│ │ │                                 | 2x0x2-y2+x1^2  |
│ │ │                                 | 2x0x1-y1       |
│ │ │                                 | x0^2-y0        |
│ │ │                   jetsMaxOrder => 3
│ │ │  
│ │ │  i11 : elapsedTime jets(3,I)
│ │ │ - -- .0146144s elapsed
│ │ │ + -- .00308626s elapsed
│ │ │  
│ │ │                                                     2                 2
│ │ │  o11 = ideal (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o11 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │  
│ │ │  i12 : elapsedTime jets(2,I)
│ │ │ - -- .00634335s elapsed
│ │ │ + -- .0027243s elapsed
│ │ │  
│ │ │                               2                 2
│ │ │  o12 = ideal (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o12 : Ideal of QQ[x0, y0][x1, y1][x2, y2]
│ │ │  
│ │ │  i13 : Q = R/I
│ │ │ @@ -148,15 +148,15 @@
│ │ │  o22 = true
│ │ │  
│ │ │  i23 : f.cache.?jet
│ │ │  
│ │ │  o23 = false
│ │ │  
│ │ │  i24 : elapsedTime jets(3,f)
│ │ │ - -- .0116862s elapsed
│ │ │ + -- .0150734s elapsed
│ │ │  
│ │ │                                                QQ[x0, y0][x1, y1][x2, y2][x3, y3]                                                      2                    2
│ │ │  o24 = map (QQ[t0][t1][t2][t3], ----------------------------------------------------------------, {t3, 2t0*t3 + 2t1*t2, t2, 2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │                                                                        2                 2
│ │ │                                 (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │                                                      QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │ @@ -173,15 +173,15 @@
│ │ │  o26 = CacheTable{jetsMatrix => | t3 2t0t3+2t1t2 |}
│ │ │                                 | t2 2t0t2+t1^2  |
│ │ │                                 | t1 2t0t1       |
│ │ │                                 | t0 t0^2        |
│ │ │                   jetsMaxOrder => 3
│ │ │  
│ │ │  i27 : elapsedTime jets(2,f)
│ │ │ - -- .000613795s elapsed
│ │ │ + -- .000798654s elapsed
│ │ │  
│ │ │                                     QQ[x0, y0][x1, y1][x2, y2]                          2                    2
│ │ │  o27 = map (QQ[t0][t1][t2], ------------------------------------------, {t2, 2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │                                              2                 2
│ │ │                             (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │                                           QQ[x0, y0][x1, y1][x2, y2]
│ │ ├── ./usr/share/doc/Macaulay2/Jets/html/___Example_sp1.html
│ │ │ @@ -87,27 +87,27 @@
│ │ │          
    
│ │ │            
    However, by [GS06, Theorem 3.1], the radical is always a (squarefree) monomial ideal. In fact, the proof of [GS06, Theorem 3.2] shows that the radical is generated by the individual terms in the generators of the ideal of jets. This observation provides an alternative algorithm for computing radicals of jets of monomial ideals, which can be faster than the default radical computation in Macaulay2.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,23 +27,23 @@
│ │ │ │  However, by [GS06, Theorem 3.1], the radical is always a (squarefree) monomial
│ │ │ │  ideal. In fact, the proof of [GS06, Theorem 3.2] shows that the radical is
│ │ │ │  generated by the individual terms in the generators of the ideal of jets. This
│ │ │ │  observation provides an alternative algorithm for computing radicals of jets of
│ │ │ │  monomial ideals, which can be faster than the default radical computation in
│ │ │ │  Macaulay2.
│ │ │ │  i4 : elapsedTime jetsRadical(2,I)
│ │ │ │ - -- .00226289s elapsed
│ │ │ │ + -- .00268378s elapsed
│ │ │ │  
│ │ │ │  o4 = ideal (y0*z0*x2, x0*z0*y2, x0*y0*z2, z0*x1*y1, y0*x1*z1, x0*y1*z1,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       y0*z0*x1, x0*z0*y1, x0*y0*z1, x0*y0*z0)
│ │ │ │  
│ │ │ │  o4 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
│ │ │ │  i5 : elapsedTime radical J2I
│ │ │ │ - -- .297009s elapsed
│ │ │ │ + -- .255297s elapsed
│ │ │ │  
│ │ │ │  o5 = ideal (x0*y0*z0, x0*y0*z1, x0*z0*y1, y0*z0*x1, x0*y1*z1, y0*x1*z1,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       z0*x1*y1, x0*y0*z2, x0*z0*y2, y0*z0*x2)
│ │ │ │  
│ │ │ │  o5 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
│ │ │ │  For a monomial hypersurface, [GS06, Theorem 3.2] describes the minimal primes
│ │ ├── ./usr/share/doc/Macaulay2/Jets/html/___Storing_sp__Computations.html
│ │ │ @@ -117,15 +117,15 @@
│ │ │  
│ │ │  o7 = false
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -146,26 +146,26 @@
│ │ │                                 | x0^2-y0        |
│ │ │                   jetsMaxOrder => 3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -290,15 +290,15 @@
│ │ │  
│ │ │  o23 = false
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : elapsedTime jetsRadical(2,I)
│ │ │ - -- .00226289s elapsed
│ │ │ + -- .00268378s elapsed
│ │ │  
│ │ │  o4 = ideal (y0*z0*x2, x0*z0*y2, x0*y0*z2, z0*x1*y1, y0*x1*z1, x0*y1*z1,
│ │ │       ------------------------------------------------------------------------
│ │ │       y0*z0*x1, x0*z0*y1, x0*y0*z1, x0*y0*z0)
│ │ │  
│ │ │  o4 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime radical J2I
│ │ │ - -- .297009s elapsed
│ │ │ + -- .255297s elapsed
│ │ │  
│ │ │  o5 = ideal (x0*y0*z0, x0*y0*z1, x0*z0*y1, y0*z0*x1, x0*y1*z1, y0*x1*z1,
│ │ │       ------------------------------------------------------------------------
│ │ │       z0*x1*y1, x0*y0*z2, x0*z0*y2, y0*z0*x2)
│ │ │  
│ │ │  o5 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
    
│ │ │              
    
│ │ │                
    i8 : elapsedTime jets(3,I)
│ │ │ - -- .0247993s elapsed
│ │ │ + -- .0110744s elapsed
│ │ │  
│ │ │                                                    2                 2
│ │ │  o8 = ideal (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o8 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3]
    
│ │ │              
    
│ │ │                
    i11 : elapsedTime jets(3,I)
│ │ │ - -- .0146144s elapsed
│ │ │ + -- .00308626s elapsed
│ │ │  
│ │ │                                                     2                 2
│ │ │  o11 = ideal (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o11 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3]
    
│ │ │              
    
│ │ │                
    i12 : elapsedTime jets(2,I)
│ │ │ - -- .00634335s elapsed
│ │ │ + -- .0027243s elapsed
│ │ │  
│ │ │                               2                 2
│ │ │  o12 = ideal (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o12 : Ideal of QQ[x0, y0][x1, y1][x2, y2]
    
│ │ │              
    
│ │ │                
    i24 : elapsedTime jets(3,f)
│ │ │ - -- .0116862s elapsed
│ │ │ + -- .0150734s elapsed
│ │ │  
│ │ │                                                QQ[x0, y0][x1, y1][x2, y2][x3, y3]                                                      2                    2
│ │ │  o24 = map (QQ[t0][t1][t2][t3], ----------------------------------------------------------------, {t3, 2t0*t3 + 2t1*t2, t2, 2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │                                                                        2                 2
│ │ │                                 (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │                                                      QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │ @@ -324,15 +324,15 @@
│ │ │                                 | t0 t0^2        |
│ │ │                   jetsMaxOrder => 3
    
│ │ │              
    
│ │ │                
    i27 : elapsedTime jets(2,f)
│ │ │ - -- .000613795s elapsed
│ │ │ + -- .000798654s elapsed
│ │ │  
│ │ │                                     QQ[x0, y0][x1, y1][x2, y2]                          2                    2
│ │ │  o27 = map (QQ[t0][t1][t2], ------------------------------------------, {t2, 2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │                                              2                 2
│ │ │                             (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │                                           QQ[x0, y0][x1, y1][x2, y2]
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,15 +41,15 @@
│ │ │ │  o6 = ideal(x  - y)
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : I.cache.?jet
│ │ │ │  
│ │ │ │  o7 = false
│ │ │ │  i8 : elapsedTime jets(3,I)
│ │ │ │ - -- .0247993s elapsed
│ │ │ │ + -- .0110744s elapsed
│ │ │ │  
│ │ │ │                                                    2                 2
│ │ │ │  o8 = ideal (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │ │  
│ │ │ │  o8 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │ │  i9 : I.cache.?jet
│ │ │ │  
│ │ │ │ @@ -58,22 +58,22 @@
│ │ │ │  
│ │ │ │  o10 = CacheTable{jetsMatrix => | 2x0x3-y3+2x1x2 |}
│ │ │ │                                 | 2x0x2-y2+x1^2  |
│ │ │ │                                 | 2x0x1-y1       |
│ │ │ │                                 | x0^2-y0        |
│ │ │ │                   jetsMaxOrder => 3
│ │ │ │  i11 : elapsedTime jets(3,I)
│ │ │ │ - -- .0146144s elapsed
│ │ │ │ + -- .00308626s elapsed
│ │ │ │  
│ │ │ │                                                     2                 2
│ │ │ │  o11 = ideal (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │ │  
│ │ │ │  o11 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │ │  i12 : elapsedTime jets(2,I)
│ │ │ │ - -- .00634335s elapsed
│ │ │ │ + -- .0027243s elapsed
│ │ │ │  
│ │ │ │                               2                 2
│ │ │ │  o12 = ideal (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │ │  
│ │ │ │  o12 : Ideal of QQ[x0, y0][x1, y1][x2, y2]
│ │ │ │  For quotient rings, data is stored under *.jet. Each jets order gives rise to a
│ │ │ │  different quotient that is stored separately under *.jet.jetsRing (order zero
│ │ │ │ @@ -153,15 +153,15 @@
│ │ │ │  i22 : isWellDefined f
│ │ │ │  
│ │ │ │  o22 = true
│ │ │ │  i23 : f.cache.?jet
│ │ │ │  
│ │ │ │  o23 = false
│ │ │ │  i24 : elapsedTime jets(3,f)
│ │ │ │ - -- .0116862s elapsed
│ │ │ │ + -- .0150734s elapsed
│ │ │ │  
│ │ │ │                                                QQ[x0, y0][x1, y1][x2, y2][x3,
│ │ │ │  y3]                                                      2                    2
│ │ │ │  o24 = map (QQ[t0][t1][t2][t3], ------------------------------------------------
│ │ │ │  ----------------, {t3, 2t0*t3 + 2t1*t2, t2, 2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │ │                                                                        2
│ │ │ │  2
│ │ │ │ @@ -183,15 +183,15 @@
│ │ │ │  
│ │ │ │  o26 = CacheTable{jetsMatrix => | t3 2t0t3+2t1t2 |}
│ │ │ │                                 | t2 2t0t2+t1^2  |
│ │ │ │                                 | t1 2t0t1       |
│ │ │ │                                 | t0 t0^2        |
│ │ │ │                   jetsMaxOrder => 3
│ │ │ │  i27 : elapsedTime jets(2,f)
│ │ │ │ - -- .000613795s elapsed
│ │ │ │ + -- .000798654s elapsed
│ │ │ │  
│ │ │ │                                     QQ[x0, y0][x1, y1][x2, y2]
│ │ │ │  2                    2
│ │ │ │  o27 = map (QQ[t0][t1][t2], ------------------------------------------, {t2,
│ │ │ │  2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │ │                                              2                 2
│ │ │ │                             (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_analyze__Strand.out
│ │ │ @@ -19,15 +19,15 @@
│ │ │        32003  0   5   0   5         32003  0   5   0   5          32003  0   5   0   5           32003  0   5   0   5           32003  0   5   0   5           32003  0   5   0   5           32003  0   5   0   5           32003  0   5   0   5           32003  0   5   0   5          32003  0   5   0   5
│ │ │                                                                                                                                                                                                                                                                                           
│ │ │       0                            1                             2                              3                              4                              5                              6                              7                              8                             9
│ │ │  
│ │ │  o3 : Complex
│ │ │  
│ │ │  i4 : L = analyzeStrand(F,a); #L
│ │ │ - -- .0269396s elapsed
│ │ │ + -- .0281998s elapsed
│ │ │  
│ │ │  o5 = 350
│ │ │  
│ │ │  i6 : betti F_a, betti F
│ │ │  
│ │ │                 0         0  1   2   3   4   5   6   7  8 9
│ │ │  o6 = (total: 833, total: 1 36 187 491 793 833 573 250 63 7)
│ │ │ @@ -46,19 +46,19 @@
│ │ │  o7 : Expression of class Product
│ │ │  
│ │ │  i8 : L3 = select(L,c->c%3==0); #L3
│ │ │  
│ │ │  o9 = 14
│ │ │  
│ │ │  i10 : carpetBettiTable(a,b,3)
│ │ │ - -- .0024042s elapsed
│ │ │ - -- .00650777s elapsed
│ │ │ - -- .0527064s elapsed
│ │ │ - -- .0736751s elapsed
│ │ │ - -- .016221s elapsed
│ │ │ + -- .00266166s elapsed
│ │ │ + -- .00750364s elapsed
│ │ │ + -- .0271065s elapsed
│ │ │ + -- .0421727s elapsed
│ │ │ + -- .00367907s elapsed
│ │ │  
│ │ │               0  1   2   3   4   5   6   7  8 9
│ │ │  o10 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │            0: 1  .   .   .   .   .   .   .  . .
│ │ │            1: . 36 160 315 288  14   .   .  . .
│ │ │            2: .  .   .   .  14 288 315 160 36 .
│ │ │            3: .  .   .   .   .   .   .   .  . 1
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Table.out
│ │ │ @@ -3,20 +3,20 @@
│ │ │  i1 : a=5,b=5
│ │ │  
│ │ │  o1 = (5, 5)
│ │ │  
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : elapsedTime T=carpetBettiTable(a,b,3)
│ │ │ - -- .00223679s elapsed
│ │ │ - -- .00804055s elapsed
│ │ │ - -- .0226273s elapsed
│ │ │ - -- .00963621s elapsed
│ │ │ - -- .00349231s elapsed
│ │ │ - -- .474407s elapsed
│ │ │ + -- .00281292s elapsed
│ │ │ + -- .00724451s elapsed
│ │ │ + -- .0263822s elapsed
│ │ │ + -- .0098365s elapsed
│ │ │ + -- .00401185s elapsed
│ │ │ + -- .424124s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o2 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -26,15 +26,15 @@
│ │ │  i3 : J=canonicalCarpet(a+b+1,b,Characteristic=>3);
│ │ │  
│ │ │                ZZ
│ │ │  o3 : Ideal of --[x ..x , y ..y ]
│ │ │                 3  0   5   0   5
│ │ │  
│ │ │  i4 : elapsedTime T'=minimalBetti J
│ │ │ - -- .247702s elapsed
│ │ │ + -- .204316s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o4 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -48,22 +48,22 @@
│ │ │           1: . . . . . . . . . .
│ │ │           2: . . . . . . . . . .
│ │ │           3: . . . . . . . . . .
│ │ │  
│ │ │  o5 : BettiTally
│ │ │  
│ │ │  i6 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ - -- .00450803s elapsed
│ │ │ - -- .017145s elapsed
│ │ │ - -- .132278s elapsed
│ │ │ - -- 1.14139s elapsed
│ │ │ - -- .517143s elapsed
│ │ │ - -- .0710902s elapsed
│ │ │ - -- .00652267s elapsed
│ │ │ - -- 6.47678s elapsed
│ │ │ + -- .00486174s elapsed
│ │ │ + -- .0189692s elapsed
│ │ │ + -- .106582s elapsed
│ │ │ + -- .94934s elapsed
│ │ │ + -- .412653s elapsed
│ │ │ + -- .0407239s elapsed
│ │ │ + -- .00717644s elapsed
│ │ │ + -- 5.64329s elapsed
│ │ │  
│ │ │  i7 : carpetBettiTable(h,7)
│ │ │  
│ │ │              0  1   2   3    4    5    6    7   8   9 10 11
│ │ │  o7 = total: 1 55 320 891 1408 1155 1155 1408 891 320 55  1
│ │ │           0: 1  .   .   .    .    .    .    .   .   .  .  .
│ │ │           1: . 55 320 891 1408 1155    .    .   .   .  .  .
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Tables.out
│ │ │ @@ -3,19 +3,19 @@
│ │ │  i1 : a=5,b=5
│ │ │  
│ │ │  o1 = (5, 5)
│ │ │  
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : h=carpetBettiTables(a,b)
│ │ │ - -- .0103685s elapsed
│ │ │ - -- .00662677s elapsed
│ │ │ - -- .0228435s elapsed
│ │ │ - -- .0337852s elapsed
│ │ │ - -- .00363175s elapsed
│ │ │ + -- .00288797s elapsed
│ │ │ + -- .00994783s elapsed
│ │ │ + -- .0254836s elapsed
│ │ │ + -- .0100659s elapsed
│ │ │ + -- .00418197s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o2 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -48,15 +48,15 @@
│ │ │  i4 : J=canonicalCarpet(a+b+1,b,Characteristic=>3);
│ │ │  
│ │ │                ZZ
│ │ │  o4 : Ideal of --[x ..x , y ..y ]
│ │ │                 3  0   5   0   5
│ │ │  
│ │ │  i5 : elapsedTime T'=minimalBetti J
│ │ │ - -- .248736s elapsed
│ │ │ + -- .212131s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o5 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -70,22 +70,22 @@
│ │ │           1: . . . . . . . . . .
│ │ │           2: . . . . . . . . . .
│ │ │           3: . . . . . . . . . .
│ │ │  
│ │ │  o6 : BettiTally
│ │ │  
│ │ │  i7 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ - -- .00536115s elapsed
│ │ │ - -- .0370005s elapsed
│ │ │ - -- .204983s elapsed
│ │ │ - -- 1.32933s elapsed
│ │ │ - -- .430969s elapsed
│ │ │ - -- .0511654s elapsed
│ │ │ - -- .00654992s elapsed
│ │ │ - -- 6.84149s elapsed
│ │ │ + -- .00494279s elapsed
│ │ │ + -- .0185525s elapsed
│ │ │ + -- .105768s elapsed
│ │ │ + -- .978405s elapsed
│ │ │ + -- .441955s elapsed
│ │ │ + -- .0408685s elapsed
│ │ │ + -- .00778108s elapsed
│ │ │ + -- 5.77872s elapsed
│ │ │  
│ │ │  i8 : keys h
│ │ │  
│ │ │  o8 = {0, 2, 3, 5}
│ │ │  
│ │ │  o8 : List
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Det.out
│ │ │ @@ -3,82 +3,82 @@
│ │ │  i1 : a=4,b=4
│ │ │  
│ │ │  o1 = (4, 4)
│ │ │  
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : d=carpetDet(a,b)
│ │ │ - -- .00694639s elapsed
│ │ │ - -- .0123481s elapsed
│ │ │ + -- .009874s elapsed
│ │ │ + -- .0126587s elapsed
│ │ │  (number Of blocks, 26)
│ │ │ - -- .000297184s elapsed
│ │ │ + -- .00027455s elapsed
│ │ │  1
│ │ │ - -- .000143437s elapsed
│ │ │ + -- .000185688s elapsed
│ │ │  1
│ │ │ - -- .000131295s elapsed
│ │ │ + -- .000178379s elapsed
│ │ │  1
│ │ │ - -- .000128579s elapsed
│ │ │ + -- .000177481s elapsed
│ │ │  1
│ │ │ - -- .000143327s elapsed
│ │ │ + -- .000169209s elapsed
│ │ │  2
│ │ │ - -- .000142685s elapsed
│ │ │ + -- .000169714s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000154078s elapsed
│ │ │ + -- .000187978s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000160099s elapsed
│ │ │ + -- .00023802s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000140462s elapsed
│ │ │ + -- .000167793s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .00013971s elapsed
│ │ │ + -- .000186754s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000138028s elapsed
│ │ │ + -- .000195247s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000132678s elapsed
│ │ │ + -- .000178733s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000120584s elapsed
│ │ │ + -- .000147258s elapsed
│ │ │  2
│ │ │ - -- .000122919s elapsed
│ │ │ + -- .000150579s elapsed
│ │ │  2
│ │ │ - -- .000134922s elapsed
│ │ │ + -- .000177083s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .00012317s elapsed
│ │ │ + -- .000161338s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000140812s elapsed
│ │ │ + -- .000175536s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000133859s elapsed
│ │ │ + -- .000157713s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000142265s elapsed
│ │ │ + -- .000166284s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000130654s elapsed
│ │ │ + -- .000178353s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000132116s elapsed
│ │ │ + -- .000167486s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000117308s elapsed
│ │ │ + -- .000184007s elapsed
│ │ │  2
│ │ │ - -- .000119734s elapsed
│ │ │ + -- .0001629s elapsed
│ │ │  1
│ │ │ - -- .000126405s elapsed
│ │ │ + -- .000148492s elapsed
│ │ │  1
│ │ │ - -- .000127438s elapsed
│ │ │ + -- .000184538s elapsed
│ │ │  1
│ │ │ - -- .000133549s elapsed
│ │ │ + -- .000162316s elapsed
│ │ │  1
│ │ │  
│ │ │  o2 = 3131031158784
│ │ │  
│ │ │  i3 : factor d
│ │ │  
│ │ │        32 6
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_compute__Bound.out
│ │ │ @@ -3,17 +3,17 @@
│ │ │  i1 : (a,b)=computeBound(6,4,3)
│ │ │  
│ │ │  o1 = (9, 7)
│ │ │  
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : computeBound 3
│ │ │ - -- .193447s elapsed
│ │ │ - -- .265378s elapsed
│ │ │ - -- .211761s elapsed
│ │ │ - -- .284175s elapsed
│ │ │ - -- .322238s elapsed
│ │ │ - -- .410165s elapsed
│ │ │ + -- .150832s elapsed
│ │ │ + -- .192664s elapsed
│ │ │ + -- .165033s elapsed
│ │ │ + -- .185753s elapsed
│ │ │ + -- .222014s elapsed
│ │ │ + -- .24144s elapsed
│ │ │  
│ │ │  o2 = 6
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_degenerate__K3__Betti__Tables.out
│ │ │ @@ -9,19 +9,19 @@
│ │ │  i2 : e=(-1,5)
│ │ │  
│ │ │  o2 = (-1, 5)
│ │ │  
│ │ │  o2 : Sequence
│ │ │  
│ │ │  i3 : h=degenerateK3BettiTables(a,b,e)
│ │ │ - -- .0634893s elapsed
│ │ │ - -- .00657433s elapsed
│ │ │ - -- .0254523s elapsed
│ │ │ - -- .00879327s elapsed
│ │ │ - -- .00336788s elapsed
│ │ │ + -- .0220704s elapsed
│ │ │ + -- .00774523s elapsed
│ │ │ + -- .0268594s elapsed
│ │ │ + -- .0104728s elapsed
│ │ │ + -- .00402033s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o3 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -49,15 +49,15 @@
│ │ │  i4 : keys h
│ │ │  
│ │ │  o4 = {0, 2, 3, 5}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : elapsedTime T= minimalBetti degenerateK3(a,b,e,Characteristic=>5)
│ │ │ - -- .309166s elapsed
│ │ │ + -- .318165s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o5 = total: 1 36 167 370 476 476 370 167 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 322 336 140  48   7  . .
│ │ │           2: .  .   7  48 140 336 322 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -77,19 +77,19 @@
│ │ │  i7 : e=(-1,5^2)
│ │ │  
│ │ │  o7 = (-1, 25)
│ │ │  
│ │ │  o7 : Sequence
│ │ │  
│ │ │  i8 : h=degenerateK3BettiTables(a,b,e)
│ │ │ - -- .00250595s elapsed
│ │ │ - -- .00659358s elapsed
│ │ │ - -- .0231823s elapsed
│ │ │ - -- .0102019s elapsed
│ │ │ - -- .00377573s elapsed
│ │ │ + -- .00302061s elapsed
│ │ │ + -- .00669575s elapsed
│ │ │ + -- .0258951s elapsed
│ │ │ + -- .0192743s elapsed
│ │ │ + -- .00438701s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o8 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1     }
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_resonance__Det.out
│ │ │ @@ -1,172 +1,172 @@
│ │ │  -- -*- M2-comint -*- hash: 1729182891690704738
│ │ │  
│ │ │  i1 : a=4
│ │ │  
│ │ │  o1 = 4
│ │ │  
│ │ │  i2 : (d1,d2)=resonanceDet(a)
│ │ │ - -- .0177477s elapsed
│ │ │ + -- .01833s elapsed
│ │ │  (number of blocks= , 18)
│ │ │  (size of the matrices, Tally{1 => 4})
│ │ │                               2 => 6
│ │ │                               3 => 2
│ │ │                               4 => 6
│ │ │         0 1
│ │ │  total: 1 1
│ │ │      7: 1 1
│ │ │ - -- .000051325s elapsed
│ │ │ + -- .000037929s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . 2
│ │ │ - -- .000095639s elapsed
│ │ │ + -- .000083524s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . .
│ │ │      9: . 2
│ │ │ - -- .000076743s elapsed
│ │ │ + -- .000094981s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │      7: 2 .
│ │ │      8: 1 .
│ │ │      9: . 1
│ │ │     10: . 2
│ │ │ - -- .000079969s elapsed
│ │ │ + -- .000103907s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (-3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      7: 1 .
│ │ │      8: 1 .
│ │ │      9: 2 2
│ │ │     10: . 1
│ │ │     11: . 1
│ │ │ - -- .000083625s elapsed
│ │ │ + -- .000106622s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      8: 1 .
│ │ │      9: 2 1
│ │ │     10: 1 2
│ │ │     11: . 1
│ │ │ - -- .000093434s elapsed
│ │ │ + -- .000089873s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │      9: 1 1
│ │ │ - -- .000024326s elapsed
│ │ │ + -- .000027256s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      9: 1 1
│ │ │     10: 1 1
│ │ │ - -- .000067026s elapsed
│ │ │ + -- .00006535s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .00009138s elapsed
│ │ │ + -- .000107582s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 2 1
│ │ │     11: 1 2
│ │ │     12: . 1
│ │ │ - -- .000095969s elapsed
│ │ │ + -- .000101465s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 1 .
│ │ │     11: 2 2
│ │ │     12: . 1
│ │ │     13: . 1
│ │ │ - -- .000090488s elapsed
│ │ │ + -- .000100193s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .000081492s elapsed
│ │ │ + -- .000083782s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │     10: 2 .
│ │ │     11: 1 .
│ │ │     12: . 1
│ │ │     13: . 2
│ │ │ - -- .00009612s elapsed
│ │ │ + -- .000080151s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     10: 1 1
│ │ │     11: 1 1
│ │ │ - -- .000065421s elapsed
│ │ │ + -- .0000702s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     11: 2 .
│ │ │     12: . .
│ │ │     13: . 2
│ │ │ - -- .000077685s elapsed
│ │ │ + -- .000064338s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     11: 1 1
│ │ │ - -- .000024636s elapsed
│ │ │ + -- .000025545s elapsed
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     12: 2 .
│ │ │     13: . 2
│ │ │ - -- .000076392s elapsed
│ │ │ + -- .000069319s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     13: 1 1
│ │ │ - -- .000027041s elapsed
│ │ │ + -- .000027655s elapsed
│ │ │  (e )
│ │ │    1
│ │ │  
│ │ │         6      32    32
│ │ │  o2 = (3 , (e )  (e )  )
│ │ │              1     2
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_analyze__Strand.html
│ │ │ @@ -102,15 +102,15 @@
│ │ │  
│ │ │  o3 : Complex
    
│ │ │              
    
│ │ │                
    i4 : L = analyzeStrand(F,a); #L
│ │ │ - -- .0269396s elapsed
│ │ │ + -- .0281998s elapsed
│ │ │  
│ │ │  o5 = 350
    
│ │ │              
    
│ │ │                
    i6 : betti F_a, betti F
│ │ │ @@ -141,19 +141,19 @@
│ │ │  
│ │ │  o9 = 14
    
│ │ │              
    
│ │ │                
    i10 : carpetBettiTable(a,b,3)
│ │ │ - -- .0024042s elapsed
│ │ │ - -- .00650777s elapsed
│ │ │ - -- .0527064s elapsed
│ │ │ - -- .0736751s elapsed
│ │ │ - -- .016221s elapsed
│ │ │ + -- .00266166s elapsed
│ │ │ + -- .00750364s elapsed
│ │ │ + -- .0271065s elapsed
│ │ │ + -- .0421727s elapsed
│ │ │ + -- .00367907s elapsed
│ │ │  
│ │ │               0  1   2   3   4   5   6   7  8 9
│ │ │  o10 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │            0: 1  .   .   .   .   .   .   .  . .
│ │ │            1: . 36 160 315 288  14   .   .  . .
│ │ │            2: .  .   .   .  14 288 315 160 36 .
│ │ │            3: .  .   .   .   .   .   .   .  . 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -49,15 +49,15 @@
│ │ │ │       0                            1                             2
│ │ │ │  3                              4                              5
│ │ │ │  6                              7                              8
│ │ │ │  9
│ │ │ │  
│ │ │ │  o3 : Complex
│ │ │ │  i4 : L = analyzeStrand(F,a); #L
│ │ │ │ - -- .0269396s elapsed
│ │ │ │ + -- .0281998s elapsed
│ │ │ │  
│ │ │ │  o5 = 350
│ │ │ │  i6 : betti F_a, betti F
│ │ │ │  
│ │ │ │                 0         0  1   2   3   4   5   6   7  8 9
│ │ │ │  o6 = (total: 833, total: 1 36 187 491 793 833 573 250 63 7)
│ │ │ │            6: 350      0: 1  .   .   .   .   .   .   .  . .
│ │ │ │ @@ -72,19 +72,19 @@
│ │ │ │  o7 = 2   3
│ │ │ │  
│ │ │ │  o7 : Expression of class Product
│ │ │ │  i8 : L3 = select(L,c->c%3==0); #L3
│ │ │ │  
│ │ │ │  o9 = 14
│ │ │ │  i10 : carpetBettiTable(a,b,3)
│ │ │ │ - -- .0024042s elapsed
│ │ │ │ - -- .00650777s elapsed
│ │ │ │ - -- .0527064s elapsed
│ │ │ │ - -- .0736751s elapsed
│ │ │ │ - -- .016221s elapsed
│ │ │ │ + -- .00266166s elapsed
│ │ │ │ + -- .00750364s elapsed
│ │ │ │ + -- .0271065s elapsed
│ │ │ │ + -- .0421727s elapsed
│ │ │ │ + -- .00367907s elapsed
│ │ │ │  
│ │ │ │               0  1   2   3   4   5   6   7  8 9
│ │ │ │  o10 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │ │            0: 1  .   .   .   .   .   .   .  . .
│ │ │ │            1: . 36 160 315 288  14   .   .  . .
│ │ │ │            2: .  .   .   .  14 288 315 160 36 .
│ │ │ │            3: .  .   .   .   .   .   .   .  . 1
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Table.html
│ │ │ @@ -83,20 +83,20 @@
│ │ │  
│ │ │  o1 : Sequence
    
│ │ │              
    
│ │ │                
    i2 : elapsedTime T=carpetBettiTable(a,b,3)
│ │ │ - -- .00223679s elapsed
│ │ │ - -- .00804055s elapsed
│ │ │ - -- .0226273s elapsed
│ │ │ - -- .00963621s elapsed
│ │ │ - -- .00349231s elapsed
│ │ │ - -- .474407s elapsed
│ │ │ + -- .00281292s elapsed
│ │ │ + -- .00724451s elapsed
│ │ │ + -- .0263822s elapsed
│ │ │ + -- .0098365s elapsed
│ │ │ + -- .00401185s elapsed
│ │ │ + -- .424124s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o2 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -112,15 +112,15 @@
│ │ │  o3 : Ideal of --[x ..x , y ..y ]
│ │ │                 3  0   5   0   5
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime T'=minimalBetti J
│ │ │ - -- .247702s elapsed
│ │ │ + -- .204316s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o4 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -140,22 +140,22 @@
│ │ │  
│ │ │  o5 : BettiTally
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ - -- .00450803s elapsed
│ │ │ - -- .017145s elapsed
│ │ │ - -- .132278s elapsed
│ │ │ - -- 1.14139s elapsed
│ │ │ - -- .517143s elapsed
│ │ │ - -- .0710902s elapsed
│ │ │ - -- .00652267s elapsed
│ │ │ - -- 6.47678s elapsed
    
│ │ │ + -- .00486174s elapsed
│ │ │ + -- .0189692s elapsed
│ │ │ + -- .106582s elapsed
│ │ │ + -- .94934s elapsed
│ │ │ + -- .412653s elapsed
│ │ │ + -- .0407239s elapsed
│ │ │ + -- .00717644s elapsed
│ │ │ + -- 5.64329s elapsed
│ │ │              
    
│ │ │                
    i7 : carpetBettiTable(h,7)
│ │ │  
│ │ │              0  1   2   3    4    5    6    7   8   9 10 11
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,20 +25,20 @@
│ │ │ │  resulting data allow us to compute the Betti tables for arbitrary primes.
│ │ │ │  i1 : a=5,b=5
│ │ │ │  
│ │ │ │  o1 = (5, 5)
│ │ │ │  
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : elapsedTime T=carpetBettiTable(a,b,3)
│ │ │ │ - -- .00223679s elapsed
│ │ │ │ - -- .00804055s elapsed
│ │ │ │ - -- .0226273s elapsed
│ │ │ │ - -- .00963621s elapsed
│ │ │ │ - -- .00349231s elapsed
│ │ │ │ - -- .474407s elapsed
│ │ │ │ + -- .00281292s elapsed
│ │ │ │ + -- .00724451s elapsed
│ │ │ │ + -- .0263822s elapsed
│ │ │ │ + -- .0098365s elapsed
│ │ │ │ + -- .00401185s elapsed
│ │ │ │ + -- .424124s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │ │  o2 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ │ @@ -46,15 +46,15 @@
│ │ │ │  o2 : BettiTally
│ │ │ │  i3 : J=canonicalCarpet(a+b+1,b,Characteristic=>3);
│ │ │ │  
│ │ │ │                ZZ
│ │ │ │  o3 : Ideal of --[x ..x , y ..y ]
│ │ │ │                 3  0   5   0   5
│ │ │ │  i4 : elapsedTime T'=minimalBetti J
│ │ │ │ - -- .247702s elapsed
│ │ │ │ + -- .204316s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │ │  o4 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ │ @@ -66,22 +66,22 @@
│ │ │ │  o5 = total: . . . . . . . . . .
│ │ │ │           1: . . . . . . . . . .
│ │ │ │           2: . . . . . . . . . .
│ │ │ │           3: . . . . . . . . . .
│ │ │ │  
│ │ │ │  o5 : BettiTally
│ │ │ │  i6 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ │ - -- .00450803s elapsed
│ │ │ │ - -- .017145s elapsed
│ │ │ │ - -- .132278s elapsed
│ │ │ │ - -- 1.14139s elapsed
│ │ │ │ - -- .517143s elapsed
│ │ │ │ - -- .0710902s elapsed
│ │ │ │ - -- .00652267s elapsed
│ │ │ │ - -- 6.47678s elapsed
│ │ │ │ + -- .00486174s elapsed
│ │ │ │ + -- .0189692s elapsed
│ │ │ │ + -- .106582s elapsed
│ │ │ │ + -- .94934s elapsed
│ │ │ │ + -- .412653s elapsed
│ │ │ │ + -- .0407239s elapsed
│ │ │ │ + -- .00717644s elapsed
│ │ │ │ + -- 5.64329s elapsed
│ │ │ │  i7 : carpetBettiTable(h,7)
│ │ │ │  
│ │ │ │              0  1   2   3    4    5    6    7   8   9 10 11
│ │ │ │  o7 = total: 1 55 320 891 1408 1155 1155 1408 891 320 55  1
│ │ │ │           0: 1  .   .   .    .    .    .    .   .   .  .  .
│ │ │ │           1: . 55 320 891 1408 1155    .    .   .   .  .  .
│ │ │ │           2: .  .   .   .    .    . 1155 1408 891 320 55  .
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Tables.html
│ │ │ @@ -80,19 +80,19 @@
│ │ │  
│ │ │  o1 : Sequence
    
│ │ │              
    
│ │ │                
    i2 : h=carpetBettiTables(a,b)
│ │ │ - -- .0103685s elapsed
│ │ │ - -- .00662677s elapsed
│ │ │ - -- .0228435s elapsed
│ │ │ - -- .0337852s elapsed
│ │ │ - -- .00363175s elapsed
│ │ │ + -- .00288797s elapsed
│ │ │ + -- .00994783s elapsed
│ │ │ + -- .0254836s elapsed
│ │ │ + -- .0100659s elapsed
│ │ │ + -- .00418197s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o2 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -134,15 +134,15 @@
│ │ │  o4 : Ideal of --[x ..x , y ..y ]
│ │ │                 3  0   5   0   5
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime T'=minimalBetti J
│ │ │ - -- .248736s elapsed
│ │ │ + -- .212131s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o5 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -162,22 +162,22 @@
│ │ │  
│ │ │  o6 : BettiTally
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ - -- .00536115s elapsed
│ │ │ - -- .0370005s elapsed
│ │ │ - -- .204983s elapsed
│ │ │ - -- 1.32933s elapsed
│ │ │ - -- .430969s elapsed
│ │ │ - -- .0511654s elapsed
│ │ │ - -- .00654992s elapsed
│ │ │ - -- 6.84149s elapsed
    
│ │ │ + -- .00494279s elapsed
│ │ │ + -- .0185525s elapsed
│ │ │ + -- .105768s elapsed
│ │ │ + -- .978405s elapsed
│ │ │ + -- .441955s elapsed
│ │ │ + -- .0408685s elapsed
│ │ │ + -- .00778108s elapsed
│ │ │ + -- 5.77872s elapsed
│ │ │              
    
│ │ │                
    i8 : keys h
│ │ │  
│ │ │  o8 = {0, 2, 3, 5}
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,19 +21,19 @@
│ │ │ │  resulting data allow us to compute the Betti tables for arbitrary primes.
│ │ │ │  i1 : a=5,b=5
│ │ │ │  
│ │ │ │  o1 = (5, 5)
│ │ │ │  
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : h=carpetBettiTables(a,b)
│ │ │ │ - -- .0103685s elapsed
│ │ │ │ - -- .00662677s elapsed
│ │ │ │ - -- .0228435s elapsed
│ │ │ │ - -- .0337852s elapsed
│ │ │ │ - -- .00363175s elapsed
│ │ │ │ + -- .00288797s elapsed
│ │ │ │ + -- .00994783s elapsed
│ │ │ │ + -- .0254836s elapsed
│ │ │ │ + -- .0100659s elapsed
│ │ │ │ + -- .00418197s elapsed
│ │ │ │  
│ │ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │ │  o2 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ │ @@ -63,15 +63,15 @@
│ │ │ │  o3 : BettiTally
│ │ │ │  i4 : J=canonicalCarpet(a+b+1,b,Characteristic=>3);
│ │ │ │  
│ │ │ │                ZZ
│ │ │ │  o4 : Ideal of --[x ..x , y ..y ]
│ │ │ │                 3  0   5   0   5
│ │ │ │  i5 : elapsedTime T'=minimalBetti J
│ │ │ │ - -- .248736s elapsed
│ │ │ │ + -- .212131s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │ │  o5 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ │ @@ -83,22 +83,22 @@
│ │ │ │  o6 = total: . . . . . . . . . .
│ │ │ │           1: . . . . . . . . . .
│ │ │ │           2: . . . . . . . . . .
│ │ │ │           3: . . . . . . . . . .
│ │ │ │  
│ │ │ │  o6 : BettiTally
│ │ │ │  i7 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ │ - -- .00536115s elapsed
│ │ │ │ - -- .0370005s elapsed
│ │ │ │ - -- .204983s elapsed
│ │ │ │ - -- 1.32933s elapsed
│ │ │ │ - -- .430969s elapsed
│ │ │ │ - -- .0511654s elapsed
│ │ │ │ - -- .00654992s elapsed
│ │ │ │ - -- 6.84149s elapsed
│ │ │ │ + -- .00494279s elapsed
│ │ │ │ + -- .0185525s elapsed
│ │ │ │ + -- .105768s elapsed
│ │ │ │ + -- .978405s elapsed
│ │ │ │ + -- .441955s elapsed
│ │ │ │ + -- .0408685s elapsed
│ │ │ │ + -- .00778108s elapsed
│ │ │ │ + -- 5.77872s elapsed
│ │ │ │  i8 : keys h
│ │ │ │  
│ │ │ │  o8 = {0, 2, 3, 5}
│ │ │ │  
│ │ │ │  o8 : List
│ │ │ │  i9 : carpetBettiTable(h,7)
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Det.html
│ │ │ @@ -80,82 +80,82 @@
│ │ │  
│ │ │  o1 : Sequence
    
│ │ │              
    
│ │ │                
    i2 : d=carpetDet(a,b)
│ │ │ - -- .00694639s elapsed
│ │ │ - -- .0123481s elapsed
│ │ │ + -- .009874s elapsed
│ │ │ + -- .0126587s elapsed
│ │ │  (number Of blocks, 26)
│ │ │ - -- .000297184s elapsed
│ │ │ + -- .00027455s elapsed
│ │ │  1
│ │ │ - -- .000143437s elapsed
│ │ │ + -- .000185688s elapsed
│ │ │  1
│ │ │ - -- .000131295s elapsed
│ │ │ + -- .000178379s elapsed
│ │ │  1
│ │ │ - -- .000128579s elapsed
│ │ │ + -- .000177481s elapsed
│ │ │  1
│ │ │ - -- .000143327s elapsed
│ │ │ + -- .000169209s elapsed
│ │ │  2
│ │ │ - -- .000142685s elapsed
│ │ │ + -- .000169714s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000154078s elapsed
│ │ │ + -- .000187978s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000160099s elapsed
│ │ │ + -- .00023802s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000140462s elapsed
│ │ │ + -- .000167793s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .00013971s elapsed
│ │ │ + -- .000186754s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000138028s elapsed
│ │ │ + -- .000195247s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000132678s elapsed
│ │ │ + -- .000178733s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000120584s elapsed
│ │ │ + -- .000147258s elapsed
│ │ │  2
│ │ │ - -- .000122919s elapsed
│ │ │ + -- .000150579s elapsed
│ │ │  2
│ │ │ - -- .000134922s elapsed
│ │ │ + -- .000177083s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .00012317s elapsed
│ │ │ + -- .000161338s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000140812s elapsed
│ │ │ + -- .000175536s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000133859s elapsed
│ │ │ + -- .000157713s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000142265s elapsed
│ │ │ + -- .000166284s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000130654s elapsed
│ │ │ + -- .000178353s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000132116s elapsed
│ │ │ + -- .000167486s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000117308s elapsed
│ │ │ + -- .000184007s elapsed
│ │ │  2
│ │ │ - -- .000119734s elapsed
│ │ │ + -- .0001629s elapsed
│ │ │  1
│ │ │ - -- .000126405s elapsed
│ │ │ + -- .000148492s elapsed
│ │ │  1
│ │ │ - -- .000127438s elapsed
│ │ │ + -- .000184538s elapsed
│ │ │  1
│ │ │ - -- .000133549s elapsed
│ │ │ + -- .000162316s elapsed
│ │ │  1
│ │ │  
│ │ │  o2 = 3131031158784
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,82 +19,82 @@
│ │ │ │  determinants and return their product.
│ │ │ │  i1 : a=4,b=4
│ │ │ │  
│ │ │ │  o1 = (4, 4)
│ │ │ │  
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : d=carpetDet(a,b)
│ │ │ │ - -- .00694639s elapsed
│ │ │ │ - -- .0123481s elapsed
│ │ │ │ + -- .009874s elapsed
│ │ │ │ + -- .0126587s elapsed
│ │ │ │  (number Of blocks, 26)
│ │ │ │ - -- .000297184s elapsed
│ │ │ │ + -- .00027455s elapsed
│ │ │ │  1
│ │ │ │ - -- .000143437s elapsed
│ │ │ │ + -- .000185688s elapsed
│ │ │ │  1
│ │ │ │ - -- .000131295s elapsed
│ │ │ │ + -- .000178379s elapsed
│ │ │ │  1
│ │ │ │ - -- .000128579s elapsed
│ │ │ │ + -- .000177481s elapsed
│ │ │ │  1
│ │ │ │ - -- .000143327s elapsed
│ │ │ │ + -- .000169209s elapsed
│ │ │ │  2
│ │ │ │ - -- .000142685s elapsed
│ │ │ │ + -- .000169714s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000154078s elapsed
│ │ │ │ + -- .000187978s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000160099s elapsed
│ │ │ │ + -- .00023802s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000140462s elapsed
│ │ │ │ + -- .000167793s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .00013971s elapsed
│ │ │ │ + -- .000186754s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000138028s elapsed
│ │ │ │ + -- .000195247s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000132678s elapsed
│ │ │ │ + -- .000178733s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000120584s elapsed
│ │ │ │ + -- .000147258s elapsed
│ │ │ │  2
│ │ │ │ - -- .000122919s elapsed
│ │ │ │ + -- .000150579s elapsed
│ │ │ │  2
│ │ │ │ - -- .000134922s elapsed
│ │ │ │ + -- .000177083s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .00012317s elapsed
│ │ │ │ + -- .000161338s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000140812s elapsed
│ │ │ │ + -- .000175536s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000133859s elapsed
│ │ │ │ + -- .000157713s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000142265s elapsed
│ │ │ │ + -- .000166284s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000130654s elapsed
│ │ │ │ + -- .000178353s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000132116s elapsed
│ │ │ │ + -- .000167486s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000117308s elapsed
│ │ │ │ + -- .000184007s elapsed
│ │ │ │  2
│ │ │ │ - -- .000119734s elapsed
│ │ │ │ + -- .0001629s elapsed
│ │ │ │  1
│ │ │ │ - -- .000126405s elapsed
│ │ │ │ + -- .000148492s elapsed
│ │ │ │  1
│ │ │ │ - -- .000127438s elapsed
│ │ │ │ + -- .000184538s elapsed
│ │ │ │  1
│ │ │ │ - -- .000133549s elapsed
│ │ │ │ + -- .000162316s elapsed
│ │ │ │  1
│ │ │ │  
│ │ │ │  o2 = 3131031158784
│ │ │ │  i3 : factor d
│ │ │ │  
│ │ │ │        32 6
│ │ │ │  o3 = 2  3
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_compute__Bound.html
│ │ │ @@ -85,20 +85,20 @@
│ │ │  
│ │ │  o1 : Sequence
│ │ │              
    
│ │ │                
    i2 : computeBound 3
│ │ │ - -- .193447s elapsed
│ │ │ - -- .265378s elapsed
│ │ │ - -- .211761s elapsed
│ │ │ - -- .284175s elapsed
│ │ │ - -- .322238s elapsed
│ │ │ - -- .410165s elapsed
│ │ │ + -- .150832s elapsed
│ │ │ + -- .192664s elapsed
│ │ │ + -- .165033s elapsed
│ │ │ + -- .185753s elapsed
│ │ │ + -- .222014s elapsed
│ │ │ + -- .24144s elapsed
│ │ │  
│ │ │  o2 = 6
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,20 +25,20 @@
│ │ │ │  classes mod k. We conjecture that c=k^2-k.
│ │ │ │  i1 : (a,b)=computeBound(6,4,3)
│ │ │ │  
│ │ │ │  o1 = (9, 7)
│ │ │ │  
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : computeBound 3
│ │ │ │ - -- .193447s elapsed
│ │ │ │ - -- .265378s elapsed
│ │ │ │ - -- .211761s elapsed
│ │ │ │ - -- .284175s elapsed
│ │ │ │ - -- .322238s elapsed
│ │ │ │ - -- .410165s elapsed
│ │ │ │ + -- .150832s elapsed
│ │ │ │ + -- .192664s elapsed
│ │ │ │ + -- .165033s elapsed
│ │ │ │ + -- .185753s elapsed
│ │ │ │ + -- .222014s elapsed
│ │ │ │ + -- .24144s elapsed
│ │ │ │  
│ │ │ │  o2 = 6
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_l_a_t_i_v_e_E_q_u_a_t_i_o_n_s -- compute the relative quadrics
│ │ │ │  ********** WWaayyss ttoo uussee ccoommppuutteeBBoouunndd:: **********
│ │ │ │      * computeBound(ZZ)
│ │ │ │      * computeBound(ZZ,ZZ,ZZ)
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_degenerate__K3__Betti__Tables.html
│ │ │ @@ -90,19 +90,19 @@
│ │ │  
│ │ │  o2 : Sequence
    
│ │ │              
    
│ │ │                
    i3 : h=degenerateK3BettiTables(a,b,e)
│ │ │ - -- .0634893s elapsed
│ │ │ - -- .00657433s elapsed
│ │ │ - -- .0254523s elapsed
│ │ │ - -- .00879327s elapsed
│ │ │ - -- .00336788s elapsed
│ │ │ + -- .0220704s elapsed
│ │ │ + -- .00774523s elapsed
│ │ │ + -- .0268594s elapsed
│ │ │ + -- .0104728s elapsed
│ │ │ + -- .00402033s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o3 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -136,15 +136,15 @@
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime T= minimalBetti degenerateK3(a,b,e,Characteristic=>5)
│ │ │ - -- .309166s elapsed
│ │ │ + -- .318165s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o5 = total: 1 36 167 370 476 476 370 167 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 322 336 140  48   7  . .
│ │ │           2: .  .   7  48 140 336 322 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -178,19 +178,19 @@
│ │ │  
│ │ │  o7 : Sequence
    
│ │ │              
    
│ │ │                
    i8 : h=degenerateK3BettiTables(a,b,e)
│ │ │ - -- .00250595s elapsed
│ │ │ - -- .00659358s elapsed
│ │ │ - -- .0231823s elapsed
│ │ │ - -- .0102019s elapsed
│ │ │ - -- .00377573s elapsed
│ │ │ + -- .00302061s elapsed
│ │ │ + -- .00669575s elapsed
│ │ │ + -- .0258951s elapsed
│ │ │ + -- .0192743s elapsed
│ │ │ + -- .00438701s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o8 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1     }
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,19 +27,19 @@
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : e=(-1,5)
│ │ │ │  
│ │ │ │  o2 = (-1, 5)
│ │ │ │  
│ │ │ │  o2 : Sequence
│ │ │ │  i3 : h=degenerateK3BettiTables(a,b,e)
│ │ │ │ - -- .0634893s elapsed
│ │ │ │ - -- .00657433s elapsed
│ │ │ │ - -- .0254523s elapsed
│ │ │ │ - -- .00879327s elapsed
│ │ │ │ - -- .00336788s elapsed
│ │ │ │ + -- .0220704s elapsed
│ │ │ │ + -- .00774523s elapsed
│ │ │ │ + -- .0268594s elapsed
│ │ │ │ + -- .0104728s elapsed
│ │ │ │ + -- .00402033s elapsed
│ │ │ │  
│ │ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │ │  o3 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ │ @@ -65,15 +65,15 @@
│ │ │ │  o3 : HashTable
│ │ │ │  i4 : keys h
│ │ │ │  
│ │ │ │  o4 = {0, 2, 3, 5}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : elapsedTime T= minimalBetti degenerateK3(a,b,e,Characteristic=>5)
│ │ │ │ - -- .309166s elapsed
│ │ │ │ + -- .318165s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │ │  o5 = total: 1 36 167 370 476 476 370 167 36 1
│ │ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │ │           1: . 36 160 322 336 140  48   7  . .
│ │ │ │           2: .  .   7  48 140 336 322 160 36 .
│ │ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ │ @@ -94,19 +94,19 @@
│ │ │ │  these mistakes.
│ │ │ │  i7 : e=(-1,5^2)
│ │ │ │  
│ │ │ │  o7 = (-1, 25)
│ │ │ │  
│ │ │ │  o7 : Sequence
│ │ │ │  i8 : h=degenerateK3BettiTables(a,b,e)
│ │ │ │ - -- .00250595s elapsed
│ │ │ │ - -- .00659358s elapsed
│ │ │ │ - -- .0231823s elapsed
│ │ │ │ - -- .0102019s elapsed
│ │ │ │ - -- .00377573s elapsed
│ │ │ │ + -- .00302061s elapsed
│ │ │ │ + -- .00669575s elapsed
│ │ │ │ + -- .0258951s elapsed
│ │ │ │ + -- .0192743s elapsed
│ │ │ │ + -- .00438701s elapsed
│ │ │ │  
│ │ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │ │  o8 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1     }
│ │ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_resonance__Det.html
│ │ │ @@ -78,172 +78,172 @@
│ │ │  
│ │ │  o1 = 4
    
│ │ │              
    
│ │ │                
    i2 : (d1,d2)=resonanceDet(a)
│ │ │ - -- .0177477s elapsed
│ │ │ + -- .01833s elapsed
│ │ │  (number of blocks= , 18)
│ │ │  (size of the matrices, Tally{1 => 4})
│ │ │                               2 => 6
│ │ │                               3 => 2
│ │ │                               4 => 6
│ │ │         0 1
│ │ │  total: 1 1
│ │ │      7: 1 1
│ │ │ - -- .000051325s elapsed
│ │ │ + -- .000037929s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . 2
│ │ │ - -- .000095639s elapsed
│ │ │ + -- .000083524s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . .
│ │ │      9: . 2
│ │ │ - -- .000076743s elapsed
│ │ │ + -- .000094981s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │      7: 2 .
│ │ │      8: 1 .
│ │ │      9: . 1
│ │ │     10: . 2
│ │ │ - -- .000079969s elapsed
│ │ │ + -- .000103907s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (-3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      7: 1 .
│ │ │      8: 1 .
│ │ │      9: 2 2
│ │ │     10: . 1
│ │ │     11: . 1
│ │ │ - -- .000083625s elapsed
│ │ │ + -- .000106622s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      8: 1 .
│ │ │      9: 2 1
│ │ │     10: 1 2
│ │ │     11: . 1
│ │ │ - -- .000093434s elapsed
│ │ │ + -- .000089873s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │      9: 1 1
│ │ │ - -- .000024326s elapsed
│ │ │ + -- .000027256s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      9: 1 1
│ │ │     10: 1 1
│ │ │ - -- .000067026s elapsed
│ │ │ + -- .00006535s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .00009138s elapsed
│ │ │ + -- .000107582s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 2 1
│ │ │     11: 1 2
│ │ │     12: . 1
│ │ │ - -- .000095969s elapsed
│ │ │ + -- .000101465s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 1 .
│ │ │     11: 2 2
│ │ │     12: . 1
│ │ │     13: . 1
│ │ │ - -- .000090488s elapsed
│ │ │ + -- .000100193s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .000081492s elapsed
│ │ │ + -- .000083782s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │     10: 2 .
│ │ │     11: 1 .
│ │ │     12: . 1
│ │ │     13: . 2
│ │ │ - -- .00009612s elapsed
│ │ │ + -- .000080151s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     10: 1 1
│ │ │     11: 1 1
│ │ │ - -- .000065421s elapsed
│ │ │ + -- .0000702s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     11: 2 .
│ │ │     12: . .
│ │ │     13: . 2
│ │ │ - -- .000077685s elapsed
│ │ │ + -- .000064338s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     11: 1 1
│ │ │ - -- .000024636s elapsed
│ │ │ + -- .000025545s elapsed
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     12: 2 .
│ │ │     13: . 2
│ │ │ - -- .000076392s elapsed
│ │ │ + -- .000069319s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     13: 1 1
│ │ │ - -- .000027041s elapsed
│ │ │ + -- .000027655s elapsed
│ │ │  (e )
│ │ │    1
│ │ │  
│ │ │         6      32    32
│ │ │  o2 = (3 , (e )  (e )  )
│ │ │              1     2
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,172 +19,172 @@
│ │ │ │  grading. Viewed as a resolution over QQ(e_1,e_2), this resolution is non-
│ │ │ │  minimal and carries further gradings. We decompose the crucial map of the a-th
│ │ │ │  strand into blocks, compute their determinants, and factor the product.
│ │ │ │  i1 : a=4
│ │ │ │  
│ │ │ │  o1 = 4
│ │ │ │  i2 : (d1,d2)=resonanceDet(a)
│ │ │ │ - -- .0177477s elapsed
│ │ │ │ + -- .01833s elapsed
│ │ │ │  (number of blocks= , 18)
│ │ │ │  (size of the matrices, Tally{1 => 4})
│ │ │ │                               2 => 6
│ │ │ │                               3 => 2
│ │ │ │                               4 => 6
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ │ │      7: 1 1
│ │ │ │ - -- .000051325s elapsed
│ │ │ │ + -- .000037929s elapsed
│ │ │ │  (e )(-1)
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │      7: 2 .
│ │ │ │      8: . 2
│ │ │ │ - -- .000095639s elapsed
│ │ │ │ + -- .000083524s elapsed
│ │ │ │      2
│ │ │ │  (e ) (e )(-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │      7: 2 .
│ │ │ │      8: . .
│ │ │ │      9: . 2
│ │ │ │ - -- .000076743s elapsed
│ │ │ │ + -- .000094981s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e )
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 3 3
│ │ │ │      7: 2 .
│ │ │ │      8: 1 .
│ │ │ │      9: . 1
│ │ │ │     10: . 2
│ │ │ │ - -- .000079969s elapsed
│ │ │ │ + -- .000103907s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (-3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      7: 1 .
│ │ │ │      8: 1 .
│ │ │ │      9: 2 2
│ │ │ │     10: . 1
│ │ │ │     11: . 1
│ │ │ │ - -- .000083625s elapsed
│ │ │ │ + -- .000106622s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      8: 1 .
│ │ │ │      9: 2 1
│ │ │ │     10: 1 2
│ │ │ │     11: . 1
│ │ │ │ - -- .000093434s elapsed
│ │ │ │ + -- .000089873s elapsed
│ │ │ │      2    3
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ │ │      9: 1 1
│ │ │ │ - -- .000024326s elapsed
│ │ │ │ + -- .000027256s elapsed
│ │ │ │  (e )(-1)
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │      9: 1 1
│ │ │ │     10: 1 1
│ │ │ │ - -- .000067026s elapsed
│ │ │ │ + -- .00006535s elapsed
│ │ │ │      2
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 2 1
│ │ │ │     10: 1 1
│ │ │ │     11: 1 2
│ │ │ │ - -- .00009138s elapsed
│ │ │ │ + -- .000107582s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e ) (-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 1 .
│ │ │ │     10: 2 1
│ │ │ │     11: 1 2
│ │ │ │     12: . 1
│ │ │ │ - -- .000095969s elapsed
│ │ │ │ + -- .000101465s elapsed
│ │ │ │      2    3
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 1 .
│ │ │ │     10: 1 .
│ │ │ │     11: 2 2
│ │ │ │     12: . 1
│ │ │ │     13: . 1
│ │ │ │ - -- .000090488s elapsed
│ │ │ │ + -- .000100193s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 2 1
│ │ │ │     10: 1 1
│ │ │ │     11: 1 2
│ │ │ │ - -- .000081492s elapsed
│ │ │ │ + -- .000083782s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e ) (-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 3 3
│ │ │ │     10: 2 .
│ │ │ │     11: 1 .
│ │ │ │     12: . 1
│ │ │ │     13: . 2
│ │ │ │ - -- .00009612s elapsed
│ │ │ │ + -- .000080151s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │     10: 1 1
│ │ │ │     11: 1 1
│ │ │ │ - -- .000065421s elapsed
│ │ │ │ + -- .0000702s elapsed
│ │ │ │      2
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │     11: 2 .
│ │ │ │     12: . .
│ │ │ │     13: . 2
│ │ │ │ - -- .000077685s elapsed
│ │ │ │ + -- .000064338s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e )
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ │ │     11: 1 1
│ │ │ │ - -- .000024636s elapsed
│ │ │ │ + -- .000025545s elapsed
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │     12: 2 .
│ │ │ │     13: . 2
│ │ │ │ - -- .000076392s elapsed
│ │ │ │ + -- .000069319s elapsed
│ │ │ │      2
│ │ │ │  (e ) (e )(-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ │ │     13: 1 1
│ │ │ │ - -- .000027041s elapsed
│ │ │ │ + -- .000027655s elapsed
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │  
│ │ │ │         6      32    32
│ │ │ │  o2 = (3 , (e )  (e )  )
│ │ │ │              1     2
│ │ ├── ./usr/share/doc/Macaulay2/LLLBases/example-output/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.out
│ │ │ @@ -7,55 +7,55 @@
│ │ │  
│ │ │  i2 : m = syz m1;
│ │ │  
│ │ │                50       47
│ │ │  o2 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i3 : time LLL m;
│ │ │ - -- used 0.00904435s (cpu); 0.00904001s (thread); 0s (gc)
│ │ │ + -- used 0.00980062s (cpu); 0.00979899s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o3 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i4 : time LLL(m, Strategy=>CohenEngine);
│ │ │ - -- used 0.0273886s (cpu); 0.02739s (thread); 0s (gc)
│ │ │ + -- used 0.0286954s (cpu); 0.0287021s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o4 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i5 : time LLL(m, Strategy=>CohenTopLevel);
│ │ │ - -- used 0.107453s (cpu); 0.10743s (thread); 0s (gc)
│ │ │ + -- used 0.122055s (cpu); 0.122061s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o5 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i6 : time LLL(m, Strategy=>{Givens,RealFP});
│ │ │ - -- used 0.011869s (cpu); 0.0118693s (thread); 0s (gc)
│ │ │ + -- used 0.0130634s (cpu); 0.013068s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o6 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i7 : time LLL(m, Strategy=>{Givens,RealQP});
│ │ │ - -- used 0.0480703s (cpu); 0.0480742s (thread); 0s (gc)
│ │ │ + -- used 0.0623315s (cpu); 0.0621652s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o7 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i8 : time LLL(m, Strategy=>{Givens,RealXD});
│ │ │ - -- used 0.0590818s (cpu); 0.0590819s (thread); 0s (gc)
│ │ │ + -- used 0.0654061s (cpu); 0.0654061s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o8 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i9 : time LLL(m, Strategy=>{Givens,RealRR});
│ │ │ - -- used 0.347092s (cpu); 0.347092s (thread); 0s (gc)
│ │ │ + -- used 0.344305s (cpu); 0.344311s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o9 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i10 : time LLL(m, Strategy=>{BKZ,Givens,RealQP});
│ │ │ - -- used 0.113298s (cpu); 0.113301s (thread); 0s (gc)
│ │ │ + -- used 0.155317s (cpu); 0.155322s (thread); 0s (gc)
│ │ │  
│ │ │                 50       47
│ │ │  o10 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/LLLBases/html/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.html
│ │ │ @@ -139,78 +139,78 @@
│ │ │                50       47
│ │ │  o2 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i3 : time LLL m;
│ │ │ - -- used 0.00904435s (cpu); 0.00904001s (thread); 0s (gc)
│ │ │ + -- used 0.00980062s (cpu); 0.00979899s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o3 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i4 : time LLL(m, Strategy=>CohenEngine);
│ │ │ - -- used 0.0273886s (cpu); 0.02739s (thread); 0s (gc)
│ │ │ + -- used 0.0286954s (cpu); 0.0287021s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o4 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i5 : time LLL(m, Strategy=>CohenTopLevel);
│ │ │ - -- used 0.107453s (cpu); 0.10743s (thread); 0s (gc)
│ │ │ + -- used 0.122055s (cpu); 0.122061s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o5 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i6 : time LLL(m, Strategy=>{Givens,RealFP});
│ │ │ - -- used 0.011869s (cpu); 0.0118693s (thread); 0s (gc)
│ │ │ + -- used 0.0130634s (cpu); 0.013068s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o6 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i7 : time LLL(m, Strategy=>{Givens,RealQP});
│ │ │ - -- used 0.0480703s (cpu); 0.0480742s (thread); 0s (gc)
│ │ │ + -- used 0.0623315s (cpu); 0.0621652s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o7 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i8 : time LLL(m, Strategy=>{Givens,RealXD});
│ │ │ - -- used 0.0590818s (cpu); 0.0590819s (thread); 0s (gc)
│ │ │ + -- used 0.0654061s (cpu); 0.0654061s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o8 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i9 : time LLL(m, Strategy=>{Givens,RealRR});
│ │ │ - -- used 0.347092s (cpu); 0.347092s (thread); 0s (gc)
│ │ │ + -- used 0.344305s (cpu); 0.344311s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o9 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i10 : time LLL(m, Strategy=>{BKZ,Givens,RealQP});
│ │ │ - -- used 0.113298s (cpu); 0.113301s (thread); 0s (gc)
│ │ │ + -- used 0.155317s (cpu); 0.155322s (thread); 0s (gc)
│ │ │  
│ │ │                 50       47
│ │ │  o10 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │        
│ │ │ ├── html2text {}
│ │ │ │ @@ -115,50 +115,50 @@
│ │ │ │                50       50
│ │ │ │  o1 : Matrix ZZ   <-- ZZ
│ │ │ │  i2 : m = syz m1;
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o2 : Matrix ZZ   <-- ZZ
│ │ │ │  i3 : time LLL m;
│ │ │ │ - -- used 0.00904435s (cpu); 0.00904001s (thread); 0s (gc)
│ │ │ │ + -- used 0.00980062s (cpu); 0.00979899s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o3 : Matrix ZZ   <-- ZZ
│ │ │ │  i4 : time LLL(m, Strategy=>CohenEngine);
│ │ │ │ - -- used 0.0273886s (cpu); 0.02739s (thread); 0s (gc)
│ │ │ │ + -- used 0.0286954s (cpu); 0.0287021s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o4 : Matrix ZZ   <-- ZZ
│ │ │ │  i5 : time LLL(m, Strategy=>CohenTopLevel);
│ │ │ │ - -- used 0.107453s (cpu); 0.10743s (thread); 0s (gc)
│ │ │ │ + -- used 0.122055s (cpu); 0.122061s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o5 : Matrix ZZ   <-- ZZ
│ │ │ │  i6 : time LLL(m, Strategy=>{Givens,RealFP});
│ │ │ │ - -- used 0.011869s (cpu); 0.0118693s (thread); 0s (gc)
│ │ │ │ + -- used 0.0130634s (cpu); 0.013068s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o6 : Matrix ZZ   <-- ZZ
│ │ │ │  i7 : time LLL(m, Strategy=>{Givens,RealQP});
│ │ │ │ - -- used 0.0480703s (cpu); 0.0480742s (thread); 0s (gc)
│ │ │ │ + -- used 0.0623315s (cpu); 0.0621652s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o7 : Matrix ZZ   <-- ZZ
│ │ │ │  i8 : time LLL(m, Strategy=>{Givens,RealXD});
│ │ │ │ - -- used 0.0590818s (cpu); 0.0590819s (thread); 0s (gc)
│ │ │ │ + -- used 0.0654061s (cpu); 0.0654061s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o8 : Matrix ZZ   <-- ZZ
│ │ │ │  i9 : time LLL(m, Strategy=>{Givens,RealRR});
│ │ │ │ - -- used 0.347092s (cpu); 0.347092s (thread); 0s (gc)
│ │ │ │ + -- used 0.344305s (cpu); 0.344311s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o9 : Matrix ZZ   <-- ZZ
│ │ │ │  i10 : time LLL(m, Strategy=>{BKZ,Givens,RealQP});
│ │ │ │ - -- used 0.113298s (cpu); 0.113301s (thread); 0s (gc)
│ │ │ │ + -- used 0.155317s (cpu); 0.155322s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                 50       47
│ │ │ │  o10 : Matrix ZZ   <-- ZZ
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  For most of the options, the columns do not need to be linearly independent.
│ │ │ │  The strategies CohenEngine and CohenTopLevel currently require the columns to
│ │ │ │  be linearly independent.
│ │ ├── ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_are__Isomorphic.out
│ │ │ @@ -16,14 +16,14 @@
│ │ │  
│ │ │                3       8
│ │ │  o4 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i5 : P = convexHull(M);
│ │ │  
│ │ │  i6 : time areIsomorphic(P,P);
│ │ │ - -- used 0.593992s (cpu); 0.441215s (thread); 0s (gc)
│ │ │ + -- used 1.08404s (cpu); 0.538763s (thread); 0s (gc)
│ │ │  
│ │ │  i7 : time areIsomorphic(P,P,smoothTest=>false);
│ │ │ - -- used 0.449515s (cpu); 0.292918s (thread); 0s (gc)
│ │ │ + -- used 0.968266s (cpu); 0.377071s (thread); 0s (gc)
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/LatticePolytopes/html/_are__Isomorphic.html
│ │ │ @@ -120,21 +120,21 @@
│ │ │              
    		
│ │ │                
    i5 : P = convexHull(M);
    
│ │ │              
    
│ │ │                
    i6 : time areIsomorphic(P,P);
│ │ │ - -- used 0.593992s (cpu); 0.441215s (thread); 0s (gc)
    
│ │ │ + -- used 1.08404s (cpu); 0.538763s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i7 : time areIsomorphic(P,P,smoothTest=>false);
│ │ │ - -- used 0.449515s (cpu); 0.292918s (thread); 0s (gc)
    
│ │ │ + -- used 0.968266s (cpu); 0.377071s (thread); 0s (gc)
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    
│ │ │            
    Ways to use areIsomorphic:
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,17 +35,17 @@
│ │ │ │       | 0 0 1 0 1 0 1 1 |
│ │ │ │       | 0 0 0 1 0 1 1 1 |
│ │ │ │  
│ │ │ │                3       8
│ │ │ │  o4 : Matrix ZZ  <-- ZZ
│ │ │ │  i5 : P = convexHull(M);
│ │ │ │  i6 : time areIsomorphic(P,P);
│ │ │ │ - -- used 0.593992s (cpu); 0.441215s (thread); 0s (gc)
│ │ │ │ + -- used 1.08404s (cpu); 0.538763s (thread); 0s (gc)
│ │ │ │  i7 : time areIsomorphic(P,P,smoothTest=>false);
│ │ │ │ - -- used 0.449515s (cpu); 0.292918s (thread); 0s (gc)
│ │ │ │ + -- used 0.968266s (cpu); 0.377071s (thread); 0s (gc)
│ │ │ │  ********** WWaayyss ttoo uussee aarreeIIssoommoorrpphhiicc:: **********
│ │ │ │      * areIsomorphic(Matrix,Matrix)
│ │ │ │      * areIsomorphic(Polyhedron,Polyhedron)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _a_r_e_I_s_o_m_o_r_p_h_i_c is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_find__Region.out
│ │ │ @@ -29,21 +29,21 @@
│ │ │  i5 : findRegion({{0,0},{4,4}},M,f)
│ │ │  
│ │ │  o5 = {{1, 2}, {3, 1}}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : elapsedTime findRegion({{0,0},{4,4}},M,f)
│ │ │ - -- .129699s elapsed
│ │ │ + -- .0819399s elapsed
│ │ │  
│ │ │  o6 = {{1, 2}, {3, 1}}
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : elapsedTime findRegion({{0,0},{4,4}},M,f,Inner=>{{1,2},{3,1}},Outer=>{{1,1}})
│ │ │ - -- .0121746s elapsed
│ │ │ + -- .0148293s elapsed
│ │ │  
│ │ │  o7 = {{1, 2}, {3, 1}}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_linear__Truncations__Bound.out
│ │ │ @@ -30,21 +30,21 @@
│ │ │  i5 : apply(L, d -> isLinearComplex res prune truncate(d,M))
│ │ │  
│ │ │  o5 = {true, true}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : elapsedTime linearTruncations({{2,2,2},{4,4,4}}, M)
│ │ │ - -- 4.16059s elapsed
│ │ │ + -- 3.05188s elapsed
│ │ │  
│ │ │  o6 = {{4, 3, 3}, {4, 4, 2}}
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : elapsedTime linearTruncationsBound M
│ │ │ - -- .0266295s elapsed
│ │ │ + -- .0275374s elapsed
│ │ │  
│ │ │  o7 = {{4, 3, 3}, {4, 4, 2}}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/html/_find__Region.html
│ │ │ @@ -125,25 +125,25 @@
│ │ │          
    
│ │ │            
    If some degrees d are known to satisfy f(d,M), then they can be specified using the option Inner in order to expedite the computation.  Similarly, degrees not above those given in Outer will be assumed not to satisfy f(d,M). If f takes options these can also be given to findRegion.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i6 : elapsedTime findRegion({{0,0},{4,4}},M,f)
│ │ │ - -- .129699s elapsed
│ │ │ + -- .0819399s elapsed
│ │ │  
│ │ │  o6 = {{1, 2}, {3, 1}}
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime findRegion({{0,0},{4,4}},M,f,Inner=>{{1,2},{3,1}},Outer=>{{1,1}})
│ │ │ - -- .0121746s elapsed
│ │ │ + -- .0148293s elapsed
│ │ │  
│ │ │  o7 = {{1, 2}, {3, 1}}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,22 +48,22 @@
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  If some degrees d are known to satisfy f(d,M), then they can be specified using
│ │ │ │  the option Inner in order to expedite the computation. Similarly, degrees not
│ │ │ │  above those given in Outer will be assumed not to satisfy f(d,M). If f takes
│ │ │ │  options these can also be given to findRegion.
│ │ │ │  i6 : elapsedTime findRegion({{0,0},{4,4}},M,f)
│ │ │ │ - -- .129699s elapsed
│ │ │ │ + -- .0819399s elapsed
│ │ │ │  
│ │ │ │  o6 = {{1, 2}, {3, 1}}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : elapsedTime findRegion({{0,0},{4,4}},M,f,Inner=>{{1,2},{3,1}},Outer=>{
│ │ │ │  {1,1}})
│ │ │ │ - -- .0121746s elapsed
│ │ │ │ + -- .0148293s elapsed
│ │ │ │  
│ │ │ │  o7 = {{1, 2}, {3, 1}}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  ********** CCoonnttrriibbuuttoorrss **********
│ │ │ │  Mahrud Sayrafi contributed to the code for this function.
│ │ │ │  ********** CCaavveeaatt **********
│ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/html/_linear__Truncations__Bound.html
│ │ │ @@ -123,25 +123,25 @@
│ │ │          
    
│ │ │            
    The output is a list of the minimal multidegrees $d$ such that the sum of the positive coordinates of $b-d$ is at most $i$ for all degrees $b$ appearing in the i-th step of the resolution of $M$.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i6 : elapsedTime linearTruncations({{2,2,2},{4,4,4}}, M)
│ │ │ - -- 4.16059s elapsed
│ │ │ + -- 3.05188s elapsed
│ │ │  
│ │ │  o6 = {{4, 3, 3}, {4, 4, 2}}
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime linearTruncationsBound M
│ │ │ - -- .0266295s elapsed
│ │ │ + -- .0275374s elapsed
│ │ │  
│ │ │  o7 = {{4, 3, 3}, {4, 4, 2}}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,21 +48,21 @@
│ │ │ │  o5 = {true, true}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  The output is a list of the minimal multidegrees $d$ such that the sum of the
│ │ │ │  positive coordinates of $b-d$ is at most $i$ for all degrees $b$ appearing in
│ │ │ │  the i-th step of the resolution of $M$.
│ │ │ │  i6 : elapsedTime linearTruncations({{2,2,2},{4,4,4}}, M)
│ │ │ │ - -- 4.16059s elapsed
│ │ │ │ + -- 3.05188s elapsed
│ │ │ │  
│ │ │ │  o6 = {{4, 3, 3}, {4, 4, 2}}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : elapsedTime linearTruncationsBound M
│ │ │ │ - -- .0266295s elapsed
│ │ │ │ + -- .0275374s elapsed
│ │ │ │  
│ │ │ │  o7 = {{4, 3, 3}, {4, 4, 2}}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  In general linearTruncationsBound will not find the minimal degrees where $M$
│ │ │ │  has a linear resolution but will be faster than repeatedly truncating $M$.
│ │ ├── ./usr/share/doc/Macaulay2/LocalRings/example-output/_hilbert__Samuel__Function.out
│ │ │ @@ -15,15 +15,15 @@
│ │ │  
│ │ │  o4 = cokernel | x5+y3+z3 y5+x3+z3 z5+x3+y3 |
│ │ │  
│ │ │                                1
│ │ │  o4 : RP-module, quotient of RP
│ │ │  
│ │ │  i5 : elapsedTime hilbertSamuelFunction(M, 0, 6)
│ │ │ - -- .238651s elapsed
│ │ │ + -- .199547s elapsed
│ │ │  
│ │ │  o5 = {1, 3, 6, 7, 6, 3, 1}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : oo//sum
│ │ │  
│ │ │ @@ -44,21 +44,21 @@
│ │ │  
│ │ │                2   3
│ │ │  o10 = ideal (x , y )
│ │ │  
│ │ │  o10 : Ideal of RP
│ │ │  
│ │ │  i11 : elapsedTime hilbertSamuelFunction(N, 0, 5) -- n+1 -- 0.02 seconds
│ │ │ - -- .0754774s elapsed
│ │ │ + -- .0166883s elapsed
│ │ │  
│ │ │  o11 = {1, 2, 3, 4, 5, 6}
│ │ │  
│ │ │  o11 : List
│ │ │  
│ │ │  i12 : elapsedTime hilbertSamuelFunction(q, N, 0, 5) -- 6(n+1) -- 0.32 seconds
│ │ │ - -- .381402s elapsed
│ │ │ + -- .263885s elapsed
│ │ │  
│ │ │  o12 = {6, 12, 18, 24, 30, 36}
│ │ │  
│ │ │  o12 : List
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/LocalRings/html/_hilbert__Samuel__Function.html
│ │ │ @@ -111,15 +111,15 @@
│ │ │                                1
│ │ │  o4 : RP-module, quotient of RP
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime hilbertSamuelFunction(M, 0, 6)
│ │ │ - -- .238651s elapsed
│ │ │ + -- .199547s elapsed
│ │ │  
│ │ │  o5 = {1, 3, 6, 7, 6, 3, 1}
│ │ │  
│ │ │  o5 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -163,25 +163,25 @@
│ │ │  
│ │ │  o10 : Ideal of RP
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : elapsedTime hilbertSamuelFunction(N, 0, 5) -- n+1 -- 0.02 seconds
│ │ │ - -- .0754774s elapsed
│ │ │ + -- .0166883s elapsed
│ │ │  
│ │ │  o11 = {1, 2, 3, 4, 5, 6}
│ │ │  
│ │ │  o11 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : elapsedTime hilbertSamuelFunction(q, N, 0, 5) -- 6(n+1) -- 0.32 seconds
│ │ │ - -- .381402s elapsed
│ │ │ + -- .263885s elapsed
│ │ │  
│ │ │  o12 = {6, 12, 18, 24, 30, 36}
│ │ │  
│ │ │  o12 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,15 +41,15 @@
│ │ │ │  i4 : M = RP^1/I
│ │ │ │  
│ │ │ │  o4 = cokernel | x5+y3+z3 y5+x3+z3 z5+x3+y3 |
│ │ │ │  
│ │ │ │                                1
│ │ │ │  o4 : RP-module, quotient of RP
│ │ │ │  i5 : elapsedTime hilbertSamuelFunction(M, 0, 6)
│ │ │ │ - -- .238651s elapsed
│ │ │ │ + -- .199547s elapsed
│ │ │ │  
│ │ │ │  o5 = {1, 3, 6, 7, 6, 3, 1}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : oo//sum
│ │ │ │  
│ │ │ │  o6 = 27
│ │ │ │ @@ -65,21 +65,21 @@
│ │ │ │  i10 : q = ideal"x2,y3"
│ │ │ │  
│ │ │ │                2   3
│ │ │ │  o10 = ideal (x , y )
│ │ │ │  
│ │ │ │  o10 : Ideal of RP
│ │ │ │  i11 : elapsedTime hilbertSamuelFunction(N, 0, 5) -- n+1 -- 0.02 seconds
│ │ │ │ - -- .0754774s elapsed
│ │ │ │ + -- .0166883s elapsed
│ │ │ │  
│ │ │ │  o11 = {1, 2, 3, 4, 5, 6}
│ │ │ │  
│ │ │ │  o11 : List
│ │ │ │  i12 : elapsedTime hilbertSamuelFunction(q, N, 0, 5) -- 6(n+1) -- 0.32 seconds
│ │ │ │ - -- .381402s elapsed
│ │ │ │ + -- .263885s elapsed
│ │ │ │  
│ │ │ │  o12 = {6, 12, 18, 24, 30, 36}
│ │ │ │  
│ │ │ │  o12 : List
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  Hilbert-Samuel function with respect to a parameter ideal other than the
│ │ │ │  maximal ideal can be slower.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Command.out
│ │ │ @@ -5,12 +5,12 @@
│ │ │  i2 : f
│ │ │  
│ │ │  o2 = 1073741824
│ │ │  
│ │ │  i3 : (c = Command "date";)
│ │ │  
│ │ │  i4 : c
│ │ │ -Sun Dec 14 15:26:56 UTC 2025
│ │ │ +Thu Jan  1 11:02:31 UTC 2026
│ │ │  
│ │ │  o4 = 0
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Database.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 9579076464446459296
│ │ │  
│ │ │  i1 : filename = temporaryFileName () | ".dbm"
│ │ │  
│ │ │ -o1 = /tmp/M2-11641-0/0.dbm
│ │ │ +o1 = /tmp/M2-13231-0/0.dbm
│ │ │  
│ │ │  i2 : x = openDatabaseOut filename
│ │ │  
│ │ │ -o2 = /tmp/M2-11641-0/0.dbm
│ │ │ +o2 = /tmp/M2-13231-0/0.dbm
│ │ │  
│ │ │  o2 : Database
│ │ │  
│ │ │  i3 : x#"first" = "hi there"
│ │ │  
│ │ │  o3 = hi there
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Cstats.out
│ │ │ @@ -1,19 +1,19 @@
│ │ │  -- -*- M2-comint -*- hash: 1731899428494721487
│ │ │  
│ │ │  i1 : s = GCstats()
│ │ │  
│ │ │ -o1 = HashTable{"bytesAlloc" => 42969169706        }
│ │ │ +o1 = HashTable{"bytesAlloc" => 43062247930        }
│ │ │                 "GC_free_space_divisor" => 3
│ │ │                 "GC_LARGE_ALLOC_WARN_INTERVAL" => 1
│ │ │                 "gcCpuTimeSecs" => 0
│ │ │ -               "heapSize" => 206680064
│ │ │ -               "numGCs" => 795
│ │ │ -               "numGCThreads" => 6
│ │ │ +               "heapSize" => 225296384
│ │ │ +               "numGCs" => 783
│ │ │ +               "numGCThreads" => 16
│ │ │  
│ │ │  o1 : HashTable
│ │ │  
│ │ │  i2 : s#"heapSize"
│ │ │  
│ │ │ -o2 = 206680064
│ │ │ +o2 = 225296384
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Minimal__Generators.out
│ │ │ @@ -40,20 +40,20 @@
│ │ │  o6 : PolynomialRing
│ │ │  
│ │ │  i7 : I = monomialCurveIdeal(R, {1,4,5,9});
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 : time J = truncate(8, I, MinimalGenerators => false);
│ │ │ - -- used 0.00906552s (cpu); 0.00905868s (thread); 0s (gc)
│ │ │ + -- used 0.00528296s (cpu); 0.00527776s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of R
│ │ │  
│ │ │  i9 : time K = truncate(8, I, MinimalGenerators => true);
│ │ │ - -- used 0.0794664s (cpu); 0.0794742s (thread); 0s (gc)
│ │ │ + -- used 0.0550328s (cpu); 0.0550419s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of R
│ │ │  
│ │ │  i10 : numgens J
│ │ │  
│ │ │  o10 = 1067
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___S__V__D_lp..._cm__Divide__Conquer_eq_gt..._rp.out
│ │ │ @@ -3,13 +3,13 @@
│ │ │  i1 : M = random(RR^200, RR^200);
│ │ │  
│ │ │                  200         200
│ │ │  o1 : Matrix RR      <-- RR
│ │ │                53          53
│ │ │  
│ │ │  i2 : time SVD(M);
│ │ │ - -- used 0.0253142s (cpu); 0.0253129s (thread); 0s (gc)
│ │ │ + -- used 0.0380489s (cpu); 0.0379843s (thread); 0s (gc)
│ │ │  
│ │ │  i3 : time SVD(M, DivideConquer=>true);
│ │ │ - -- used 0.0274055s (cpu); 0.0274142s (thread); 0s (gc)
│ │ │ + -- used 0.0381925s (cpu); 0.0379757s (thread); 0s (gc)
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_a_spfirst_sp__Macaulay2_spsession.out
│ │ │ @@ -351,15 +351,15 @@
│ │ │                 | b e h k n q |
│ │ │                 | c f i l o r |
│ │ │  
│ │ │                               3
│ │ │  o58 : R-module, quotient of R
│ │ │  
│ │ │  i59 : time C = resolution M
│ │ │ - -- used 0.00192778s (cpu); 0.00191945s (thread); 0s (gc)
│ │ │ + -- used 0.00195051s (cpu); 0.00194491s (thread); 0s (gc)
│ │ │  
│ │ │         3      6      15      18      6
│ │ │  o59 = R  <-- R  <-- R   <-- R   <-- R  <-- 0
│ │ │                                              
│ │ │        0      1      2       3       4      5
│ │ │  
│ │ │  o59 : ChainComplex
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_at__End__Of__File_lp__File_rp.out
│ │ │ @@ -14,10 +14,10 @@
│ │ │  
│ │ │  i4 : peek read f
│ │ │  
│ │ │  o4 = "hi there"
│ │ │  
│ │ │  i5 : atEndOfFile f
│ │ │  
│ │ │ -o5 = false
│ │ │ +o5 = true
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_benchmark.out
│ │ │ @@ -1,9 +1,9 @@
│ │ │  -- -*- M2-comint -*- hash: 1330379359420
│ │ │  
│ │ │  i1 : benchmark "sqrt 2p100000"
│ │ │  
│ │ │ -o1 = .000290697861367332
│ │ │ +o1 = .0003478640801445259
│ │ │  
│ │ │  o1 : RR (of precision 53)
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_caching_spcomputation_spresults.out
│ │ │ @@ -4,20 +4,20 @@
│ │ │  
│ │ │  i2 : R = QQ[x,y,z];
│ │ │  
│ │ │  i3 : M = coker vars R;
│ │ │  
│ │ │  i4 : elapsedTime pdim' M
│ │ │   -- computing pdim'
│ │ │ - -- .00670047s elapsed
│ │ │ + -- .00382538s elapsed
│ │ │  
│ │ │  o4 = 3
│ │ │  
│ │ │  i5 : elapsedTime pdim' M
│ │ │ - -- .000001513s elapsed
│ │ │ + -- .000002683s elapsed
│ │ │  
│ │ │  o5 = 3
│ │ │  
│ │ │  i6 : peek M.cache
│ │ │  
│ │ │  o6 = CacheTable{cache => MutableHashTable{}                                              }
│ │ │                  isHomogeneous => true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cancel__Task_lp__Task_rp.out
│ │ │ @@ -18,29 +18,29 @@
│ │ │  
│ │ │  o4 = <>
│ │ │  
│ │ │  o4 : Task
│ │ │  
│ │ │  i5 : n
│ │ │  
│ │ │ -o5 = 711206
│ │ │ +o5 = 1095015
│ │ │  
│ │ │  i6 : sleep 1
│ │ │  
│ │ │  o6 = 0
│ │ │  
│ │ │  i7 : t
│ │ │  
│ │ │  o7 = <>
│ │ │  
│ │ │  o7 : Task
│ │ │  
│ │ │  i8 : n
│ │ │  
│ │ │ -o8 = 1453533
│ │ │ +o8 = 2220814
│ │ │  
│ │ │  i9 : isReady t
│ │ │  
│ │ │  o9 = false
│ │ │  
│ │ │  i10 : cancelTask t
│ │ │  
│ │ │ @@ -53,22 +53,22 @@
│ │ │  
│ │ │  o12 = <>
│ │ │  
│ │ │  o12 : Task
│ │ │  
│ │ │  i13 : n
│ │ │  
│ │ │ -o13 = 1453746
│ │ │ +o13 = 2221001
│ │ │  
│ │ │  i14 : sleep 1
│ │ │  
│ │ │  o14 = 0
│ │ │  
│ │ │  i15 : n
│ │ │  
│ │ │ -o15 = 1453746
│ │ │ +o15 = 2221001
│ │ │  
│ │ │  i16 : isReady t
│ │ │  
│ │ │  o16 = false
│ │ │  
│ │ │  i17 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_change__Directory.out
│ │ │ @@ -1,19 +1,19 @@
│ │ │  -- -*- M2-comint -*- hash: 8535510246140175278
│ │ │  
│ │ │  i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10463-0/0
│ │ │ +o1 = /tmp/M2-10833-0/0
│ │ │  
│ │ │  i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-10463-0/0
│ │ │ +o2 = /tmp/M2-10833-0/0
│ │ │  
│ │ │  i3 : changeDirectory dir
│ │ │  
│ │ │ -o3 = /tmp/M2-10463-0/0/
│ │ │ +o3 = /tmp/M2-10833-0/0/
│ │ │  
│ │ │  i4 : currentDirectory()
│ │ │  
│ │ │ -o4 = /tmp/M2-10463-0/0/
│ │ │ +o4 = /tmp/M2-10833-0/0/
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_check.out
│ │ │ @@ -4,51 +4,51 @@
│ │ │  
│ │ │  o1 = FirstPackage
│ │ │  
│ │ │  o1 : Package
│ │ │  
│ │ │  i2 : check_1 FirstPackage
│ │ │   -- warning: reloading FirstPackage; recreate instances of types from this package
│ │ │ - -- capturing check(1, "FirstPackage")        -- .15147s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .122544s elapsed
│ │ │  
│ │ │  i3 : check FirstPackage
│ │ │ - -- capturing check(0, "FirstPackage")        -- .150181s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .150965s elapsed
│ │ │ + -- capturing check(0, "FirstPackage")        -- .116342s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .118608s elapsed
│ │ │  
│ │ │  i4 : check_1 "FirstPackage"
│ │ │ - -- capturing check(1, "FirstPackage")        -- .152579s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .118608s elapsed
│ │ │  
│ │ │  i5 : check "FirstPackage"
│ │ │ - -- capturing check(0, "FirstPackage")        -- .152053s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .151867s elapsed
│ │ │ + -- capturing check(0, "FirstPackage")        -- .120072s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .119513s elapsed
│ │ │  
│ │ │  i6 : tests(1, "FirstPackage")
│ │ │  
│ │ │  o6 = TestInput[/usr/share/Macaulay2/FirstPackage.m2:58:5-60:3]
│ │ │  
│ │ │  o6 : TestInput
│ │ │  
│ │ │  i7 : check oo
│ │ │ - -- capturing check(1, "FirstPackage")        -- .153083s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .119399s elapsed
│ │ │  
│ │ │  i8 : tests "FirstPackage"
│ │ │  
│ │ │  o8 = {0 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:54:5-56:3]}
│ │ │       {1 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:58:5-60:3]}
│ │ │  
│ │ │  o8 : NumberedVerticalList
│ │ │  
│ │ │  i9 : check oo
│ │ │ - -- capturing check(0, "FirstPackage")        -- .152703s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .150901s elapsed
│ │ │ + -- capturing check(0, "FirstPackage")        -- .118011s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .116374s elapsed
│ │ │  
│ │ │  i10 : tests "FirstPackage"
│ │ │  
│ │ │  o10 = {0 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:54:5-56:3]}
│ │ │        {1 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:58:5-60:3]}
│ │ │  
│ │ │  o10 : NumberedVerticalList
│ │ │  
│ │ │  i11 : check 1
│ │ │ - -- capturing check(1, "FirstPackage")        -- .151346s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .123542s elapsed
│ │ │  
│ │ │  i12 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_communicating_spwith_spprograms.out
│ │ │ @@ -1,25 +1,25 @@
│ │ │  -- -*- M2-comint -*- hash: 10365735446967377456
│ │ │  
│ │ │  i1 : run "uname -a"
│ │ │ -Linux sbuild 6.12.57+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.57-1 (2025-11-05) x86_64 GNU/Linux
│ │ │ +Linux sbuild 6.12.57+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.57-1 (2025-11-05) x86_64 GNU/Linux
│ │ │  
│ │ │  o1 = 0
│ │ │  
│ │ │  i2 : "!grep a" << " ba \n bc \n ad \n ef \n" << close
│ │ │   ba 
│ │ │   ad 
│ │ │  
│ │ │  o2 = !grep a
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : peek get "!uname -a"
│ │ │  
│ │ │ -o3 = "Linux sbuild 6.12.57+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ +o3 = "Linux sbuild 6.12.57+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │       6.12.57-1 (2025-11-05) x86_64 GNU/Linux\n"
│ │ │  
│ │ │  i4 : f = openInOut "!grep -E '^in'"
│ │ │  
│ │ │  o4 = !grep -E '^in'
│ │ │  
│ │ │  o4 : File
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_sp__Groebner_spbases.out
│ │ │ @@ -126,15 +126,15 @@
│ │ │  
│ │ │                  ZZ
│ │ │  o23 : Ideal of ----[x..z, w]
│ │ │                 1277
│ │ │  
│ │ │  i24 : gb I
│ │ │  
│ │ │ -   -- registering gb 5 at 0x7f67b957e540
│ │ │ +   -- registering gb 5 at 0x7fb15fc15540
│ │ │  
│ │ │     -- [gb]{2}(2)mm{3}(1)m{4}(2)om{5}(1)onumber of (nonminimal) gb elements = 4
│ │ │     -- number of monomials                = 8
│ │ │     -- #reduction steps = 2
│ │ │     -- #spairs done = 6
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ │ @@ -177,15 +177,15 @@
│ │ │  
│ │ │  i32 : f = random(R^1,R^{-3,-3,-5,-6});
│ │ │  
│ │ │                1      4
│ │ │  o32 : Matrix R  <-- R
│ │ │  
│ │ │  i33 : time betti gb f
│ │ │ - -- used 0.303879s (cpu); 0.305901s (thread); 0s (gc)
│ │ │ + -- used 0.219908s (cpu); 0.21961s (thread); 0s (gc)
│ │ │  
│ │ │               0  1
│ │ │  o33 = total: 1 53
│ │ │            0: 1  .
│ │ │            1: .  .
│ │ │            2: .  2
│ │ │            3: .  1
│ │ │ @@ -208,15 +208,15 @@
│ │ │  
│ │ │              3    5     8     9    12     14    17
│ │ │  o35 = 1 - 2T  - T  + 2T  + 2T  - T   - 2T   + T
│ │ │  
│ │ │  o35 : ZZ[T]
│ │ │  
│ │ │  i36 : time betti gb f
│ │ │ - -- used 0.00799525s (cpu); 0.00545052s (thread); 0s (gc)
│ │ │ + -- used 0.00304346s (cpu); 0.00303251s (thread); 0s (gc)
│ │ │  
│ │ │               0  1
│ │ │  o36 = total: 1 53
│ │ │            0: 1  .
│ │ │            1: .  .
│ │ │            2: .  2
│ │ │            3: .  1
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_copy__Directory_lp__String_cm__String_rp.out
│ │ │ @@ -1,76 +1,76 @@
│ │ │  -- -*- M2-comint -*- hash: 11422793294564310273
│ │ │  
│ │ │  i1 : src = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-11185-0/0/
│ │ │ +o1 = /tmp/M2-12295-0/0/
│ │ │  
│ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-11185-0/1/
│ │ │ +o2 = /tmp/M2-12295-0/1/
│ │ │  
│ │ │  i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-11185-0/0/a/
│ │ │ +o3 = /tmp/M2-12295-0/0/a/
│ │ │  
│ │ │  i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-11185-0/0/b/
│ │ │ +o4 = /tmp/M2-12295-0/0/b/
│ │ │  
│ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-11185-0/0/b/c/
│ │ │ +o5 = /tmp/M2-12295-0/0/b/c/
│ │ │  
│ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11185-0/0/a/f
│ │ │ +o6 = /tmp/M2-12295-0/0/a/f
│ │ │  
│ │ │  o6 : File
│ │ │  
│ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11185-0/0/a/g
│ │ │ +o7 = /tmp/M2-12295-0/0/a/g
│ │ │  
│ │ │  o7 : File
│ │ │  
│ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-11185-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-12295-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
│ │ │  
│ │ │  i9 : stack findFiles src
│ │ │  
│ │ │ -o9 = /tmp/M2-11185-0/0/
│ │ │ -     /tmp/M2-11185-0/0/b/
│ │ │ -     /tmp/M2-11185-0/0/b/c/
│ │ │ -     /tmp/M2-11185-0/0/b/c/g
│ │ │ -     /tmp/M2-11185-0/0/a/
│ │ │ -     /tmp/M2-11185-0/0/a/g
│ │ │ -     /tmp/M2-11185-0/0/a/f
│ │ │ +o9 = /tmp/M2-12295-0/0/
│ │ │ +     /tmp/M2-12295-0/0/a/
│ │ │ +     /tmp/M2-12295-0/0/a/g
│ │ │ +     /tmp/M2-12295-0/0/a/f
│ │ │ +     /tmp/M2-12295-0/0/b/
│ │ │ +     /tmp/M2-12295-0/0/b/c/
│ │ │ +     /tmp/M2-12295-0/0/b/c/g
│ │ │  
│ │ │  i10 : copyDirectory(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-11185-0/0/b/c/g -> /tmp/M2-11185-0/1/b/c/g
│ │ │ - -- copying: /tmp/M2-11185-0/0/a/g -> /tmp/M2-11185-0/1/a/g
│ │ │ - -- copying: /tmp/M2-11185-0/0/a/f -> /tmp/M2-11185-0/1/a/f
│ │ │ + -- copying: /tmp/M2-12295-0/0/a/g -> /tmp/M2-12295-0/1/a/g
│ │ │ + -- copying: /tmp/M2-12295-0/0/a/f -> /tmp/M2-12295-0/1/a/f
│ │ │ + -- copying: /tmp/M2-12295-0/0/b/c/g -> /tmp/M2-12295-0/1/b/c/g
│ │ │  
│ │ │  i11 : copyDirectory(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11185-0/0/b/c/g not newer than /tmp/M2-11185-0/1/b/c/g
│ │ │ - -- skipping: /tmp/M2-11185-0/0/a/g not newer than /tmp/M2-11185-0/1/a/g
│ │ │ - -- skipping: /tmp/M2-11185-0/0/a/f not newer than /tmp/M2-11185-0/1/a/f
│ │ │ + -- skipping: /tmp/M2-12295-0/0/a/g not newer than /tmp/M2-12295-0/1/a/g
│ │ │ + -- skipping: /tmp/M2-12295-0/0/a/f not newer than /tmp/M2-12295-0/1/a/f
│ │ │ + -- skipping: /tmp/M2-12295-0/0/b/c/g not newer than /tmp/M2-12295-0/1/b/c/g
│ │ │  
│ │ │  i12 : stack findFiles dst
│ │ │  
│ │ │ -o12 = /tmp/M2-11185-0/1/
│ │ │ -      /tmp/M2-11185-0/1/a/
│ │ │ -      /tmp/M2-11185-0/1/a/f
│ │ │ -      /tmp/M2-11185-0/1/a/g
│ │ │ -      /tmp/M2-11185-0/1/b/
│ │ │ -      /tmp/M2-11185-0/1/b/c/
│ │ │ -      /tmp/M2-11185-0/1/b/c/g
│ │ │ +o12 = /tmp/M2-12295-0/1/
│ │ │ +      /tmp/M2-12295-0/1/a/
│ │ │ +      /tmp/M2-12295-0/1/a/g
│ │ │ +      /tmp/M2-12295-0/1/a/f
│ │ │ +      /tmp/M2-12295-0/1/b/
│ │ │ +      /tmp/M2-12295-0/1/b/c/
│ │ │ +      /tmp/M2-12295-0/1/b/c/g
│ │ │  
│ │ │  i13 : get (dst|"b/c/g")
│ │ │  
│ │ │  o13 = ho there
│ │ │  
│ │ │  i14 : rm = d -> if isDirectory d then removeDirectory d else removeFile d
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_copy__File_lp__String_cm__String_rp.out
│ │ │ @@ -1,41 +1,41 @@
│ │ │  -- -*- M2-comint -*- hash: 11539475420155775110
│ │ │  
│ │ │  i1 : src = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10970-0/0
│ │ │ +o1 = /tmp/M2-11860-0/0
│ │ │  
│ │ │  i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-10970-0/1
│ │ │ +o2 = /tmp/M2-11860-0/1
│ │ │  
│ │ │  i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10970-0/0
│ │ │ +o3 = /tmp/M2-11860-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : copyFile(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-10970-0/0 -> /tmp/M2-10970-0/1
│ │ │ + -- copying: /tmp/M2-11860-0/0 -> /tmp/M2-11860-0/1
│ │ │  
│ │ │  i5 : get dst
│ │ │  
│ │ │  o5 = hi there
│ │ │  
│ │ │  i6 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-10970-0/0 not newer than /tmp/M2-10970-0/1
│ │ │ + -- skipping: /tmp/M2-11860-0/0 not newer than /tmp/M2-11860-0/1
│ │ │  
│ │ │  i7 : src << "ho there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-10970-0/0
│ │ │ +o7 = /tmp/M2-11860-0/0
│ │ │  
│ │ │  o7 : File
│ │ │  
│ │ │  i8 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-10970-0/0 not newer than /tmp/M2-10970-0/1
│ │ │ + -- skipping: /tmp/M2-11860-0/0 not newer than /tmp/M2-11860-0/1
│ │ │  
│ │ │  i9 : get dst
│ │ │  
│ │ │  o9 = hi there
│ │ │  
│ │ │  i10 : removeFile src
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cpu__Time.out
│ │ │ @@ -1,23 +1,23 @@
│ │ │  -- -*- M2-comint -*- hash: 15508153783232232453
│ │ │  
│ │ │  i1 : t1 = cpuTime()
│ │ │  
│ │ │ -o1 = 354.029649282
│ │ │ +o1 = 319.890774352
│ │ │  
│ │ │  o1 : RR (of precision 53)
│ │ │  
│ │ │  i2 : for i from 0 to 1000000 do 223131321321*324234324324;
│ │ │  
│ │ │  i3 : t2 = cpuTime()
│ │ │  
│ │ │ -o3 = 355.996910491
│ │ │ +o3 = 320.686724796
│ │ │  
│ │ │  o3 : RR (of precision 53)
│ │ │  
│ │ │  i4 : t2-t1
│ │ │  
│ │ │ -o4 = 1.967261209000014
│ │ │ +o4 = .7959504440000273
│ │ │  
│ │ │  o4 : RR (of precision 53)
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current__Time.out
│ │ │ @@ -1,24 +1,24 @@
│ │ │  -- -*- M2-comint -*- hash: 3660839476107967259
│ │ │  
│ │ │  i1 : currentTime()
│ │ │  
│ │ │ -o1 = 1765726091
│ │ │ +o1 = 1767265403
│ │ │  
│ │ │  i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 )
│ │ │  
│ │ │ -o2 = 55.95363237235333
│ │ │ +o2 = 56.00241122780173
│ │ │  
│ │ │  o2 : RR (of precision 53)
│ │ │  
│ │ │  i3 : 12 * (oo - floor oo)
│ │ │  
│ │ │ -o3 = 11.44358846823999
│ │ │ +o3 = .02893473362075838
│ │ │  
│ │ │  o3 : RR (of precision 53)
│ │ │  
│ │ │  i4 : run "date"
│ │ │ -Sun Dec 14 15:28:11 UTC 2025
│ │ │ +Thu Jan  1 11:03:23 UTC 2026
│ │ │  
│ │ │  o4 = 0
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elapsed__Time.out
│ │ │ @@ -1,8 +1,8 @@
│ │ │  -- -*- M2-comint -*- hash: 1330565958025
│ │ │  
│ │ │  i1 : elapsedTime sleep 1
│ │ │ - -- 1.00015s elapsed
│ │ │ + -- 1.00013s 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
│ │ │ -     -- 1.00015 seconds
│ │ │ +     -- 1.00014 seconds
│ │ │  
│ │ │  o1 : Time
│ │ │  
│ │ │  i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{1.00015, 0}
│ │ │ +o2 = Time{1.00014, 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 0.464405s (cpu); 0.277254s (thread); 0s (gc)
│ │ │ + -- used 0.163882s (cpu); 0.163881s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = | x3y9 5148txy3 108729sxy2z2 sy4z 46644741sxy3z 143sy5 6sxy4
│ │ │       ------------------------------------------------------------------------
│ │ │       563515116021sx2y3 4374txy2z3 612704350498473090tx2yz3 217458ty4z2
│ │ │       ------------------------------------------------------------------------
│ │ │       267076255345488270sy3z4 5256861933965245618410txyz6
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -85,15 +85,15 @@
│ │ │  
│ │ │                 3                   3     2               3
│ │ │  o7 = ideal (- s  - s*t + x - 1, - t  - 3t  + y - t, - s*t  + z)
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 : time G = eliminate(I,{s,t})
│ │ │ - -- used 0.422625s (cpu); 0.238689s (thread); 0s (gc)
│ │ │ + -- used 0.403807s (cpu); 0.181s (thread); 0s (gc)
│ │ │  
│ │ │              3 9     2 9     2 8      2 6 3       9    2 7         8   
│ │ │  o8 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │       ------------------------------------------------------------------------
│ │ │           7 2       2 5 3       6 3    7 3        5 4       3 6    9       7 
│ │ │       7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -154,15 +154,15 @@
│ │ │  i10 : R1 = QQ[x,y,z,s,t, Degrees=>{3,3,4,1,1}];
│ │ │  
│ │ │  i11 : I1 = substitute(I,R1);
│ │ │  
│ │ │  o11 : Ideal of R1
│ │ │  
│ │ │  i12 : time G = eliminate(I1,{s,t})
│ │ │ - -- used 0.286447s (cpu); 0.110979s (thread); 0s (gc)
│ │ │ + -- used 0.0304523s (cpu); 0.0304574s (thread); 0s (gc)
│ │ │  
│ │ │               3 9     2 6 3       3 6    9     2 8         5 4      2 7  
│ │ │  o12 = ideal(x y  - 3x y z  + 3x*y z  - z  - 6x y z - 15x*y z  + 21y z  -
│ │ │        -----------------------------------------------------------------------
│ │ │          2 9       2 5 3       6 3    7 3         2 6     3 6       7 2  
│ │ │        3x y  - 324x y z  + 6x*y z  - y z  - 405x*y z  - 3y z  + 7x*y z  -
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -228,15 +228,15 @@
│ │ │  
│ │ │                     3             3     2         3
│ │ │  o16 = map (A, B, {s  + s*t + 1, t  + 3t  + t, s*t })
│ │ │  
│ │ │  o16 : RingMap A <-- B
│ │ │  
│ │ │  i17 : time G = kernel F
│ │ │ - -- used 0.404757s (cpu); 0.227468s (thread); 0s (gc)
│ │ │ + -- used 0.100185s (cpu); 0.100191s (thread); 0s (gc)
│ │ │  
│ │ │               3 9     2 9     2 8      2 6 3       9    2 7         8   
│ │ │  o17 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │        -----------------------------------------------------------------------
│ │ │            7 2       2 5 3       6 3    7 3        5 4       3 6    9       7 
│ │ │        7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -297,23 +297,23 @@
│ │ │  i19 : use ring I
│ │ │  
│ │ │  o19 = R
│ │ │  
│ │ │  o19 : PolynomialRing
│ │ │  
│ │ │  i20 : time f1 = resultant(I_0,I_2,s)
│ │ │ - -- used 0.00191261s (cpu); 0.00191299s (thread); 0s (gc)
│ │ │ + -- used 0.00161638s (cpu); 0.00161341s (thread); 0s (gc)
│ │ │  
│ │ │           9    9      7    3
│ │ │  o20 = x*t  - t  - z*t  - z
│ │ │  
│ │ │  o20 : R
│ │ │  
│ │ │  i21 : time f2 = resultant(I_1,f1,t)
│ │ │ - -- used 0.0583203s (cpu); 0.0583314s (thread); 0s (gc)
│ │ │ + -- used 0.0355171s (cpu); 0.0355287s (thread); 0s (gc)
│ │ │  
│ │ │           3 9     2 9     2 8      2 6 3       9    2 7         8        7 2  
│ │ │  o21 = - x y  + 3x y  + 6x y z + 3x y z  - 3x*y  + x y z - 12x*y z - 7x*y z  +
│ │ │        -----------------------------------------------------------------------
│ │ │            2 5 3       6 3    7 3        5 4       3 6    9       7      8   
│ │ │        324x y z  - 6x*y z  + y z  + 15x*y z  - 3x*y z  + y  - 2x*y z + 6y z +
│ │ │        -----------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_end__Package.out
│ │ │ @@ -59,15 +59,15 @@
│ │ │                                      Version => 0.0
│ │ │               package prefix => /usr/
│ │ │               PackageIsLoaded => true
│ │ │               pkgname => Foo
│ │ │               private dictionary => Foo#"private dictionary"
│ │ │               processed documentation => MutableHashTable{}
│ │ │               raw documentation => MutableHashTable{}
│ │ │ -             source directory => /tmp/M2-10191-0/91-rundir/
│ │ │ +             source directory => /tmp/M2-10311-0/91-rundir/
│ │ │               source file => stdio
│ │ │               test inputs => MutableList{}
│ │ │  
│ │ │  i7 : dictionaryPath
│ │ │  
│ │ │  o7 = {Foo.Dictionary, Varieties.Dictionary, Isomorphism.Dictionary,
│ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Exists.out
│ │ │ @@ -1,20 +1,20 @@
│ │ │  -- -*- M2-comint -*- hash: 7475038936570224899
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10558-0/0
│ │ │ +o1 = /tmp/M2-11028-0/0
│ │ │  
│ │ │  i2 : fileExists fn
│ │ │  
│ │ │  o2 = false
│ │ │  
│ │ │  i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10558-0/0
│ │ │ +o3 = /tmp/M2-11028-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : fileExists fn
│ │ │  
│ │ │  o4 = true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Length.out
│ │ │ @@ -1,28 +1,28 @@
│ │ │  -- -*- M2-comint -*- hash: 1216695447195237994
│ │ │  
│ │ │  i1 : f = temporaryFileName() << "hi there"
│ │ │  
│ │ │ -o1 = /tmp/M2-12150-0/0
│ │ │ +o1 = /tmp/M2-14270-0/0
│ │ │  
│ │ │  o1 : File
│ │ │  
│ │ │  i2 : fileLength f
│ │ │  
│ │ │  o2 = 8
│ │ │  
│ │ │  i3 : close f
│ │ │  
│ │ │ -o3 = /tmp/M2-12150-0/0
│ │ │ +o3 = /tmp/M2-14270-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : filename = toString f
│ │ │  
│ │ │ -o4 = /tmp/M2-12150-0/0
│ │ │ +o4 = /tmp/M2-14270-0/0
│ │ │  
│ │ │  i5 : fileLength filename
│ │ │  
│ │ │  o5 = 8
│ │ │  
│ │ │  i6 : get filename
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__File_rp.out
│ │ │ @@ -1,25 +1,25 @@
│ │ │  -- -*- M2-comint -*- hash: 11202140621123993633
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11375-0/0
│ │ │ +o1 = /tmp/M2-12685-0/0
│ │ │  
│ │ │  i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-11375-0/0
│ │ │ +o2 = /tmp/M2-12685-0/0
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : fileMode f
│ │ │  
│ │ │  o3 = 420
│ │ │  
│ │ │  i4 : close f
│ │ │  
│ │ │ -o4 = /tmp/M2-11375-0/0
│ │ │ +o4 = /tmp/M2-12685-0/0
│ │ │  
│ │ │  o4 : File
│ │ │  
│ │ │  i5 : removeFile fn
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__String_rp.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 4782570202197464532
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10989-0/0
│ │ │ +o1 = /tmp/M2-11899-0/0
│ │ │  
│ │ │  i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-10989-0/0
│ │ │ +o2 = /tmp/M2-11899-0/0
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : fileMode fn
│ │ │  
│ │ │  o3 = 420
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__Z__Z_cm__File_rp.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 17473878267845575442
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10854-0/0
│ │ │ +o1 = /tmp/M2-11624-0/0
│ │ │  
│ │ │  i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-10854-0/0
│ │ │ +o2 = /tmp/M2-11624-0/0
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : m = 7 + 7*8 + 7*64
│ │ │  
│ │ │  o3 = 511
│ │ │  
│ │ │ @@ -18,15 +18,15 @@
│ │ │  
│ │ │  i5 : fileMode f
│ │ │  
│ │ │  o5 = 511
│ │ │  
│ │ │  i6 : close f
│ │ │  
│ │ │ -o6 = /tmp/M2-10854-0/0
│ │ │ +o6 = /tmp/M2-11624-0/0
│ │ │  
│ │ │  o6 : File
│ │ │  
│ │ │  i7 : fileMode fn
│ │ │  
│ │ │  o7 = 511
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__Z__Z_cm__String_rp.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 16772784390799334723
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11977-0/0
│ │ │ +o1 = /tmp/M2-13917-0/0
│ │ │  
│ │ │  i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-11977-0/0
│ │ │ +o2 = /tmp/M2-13917-0/0
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : m = fileMode fn
│ │ │  
│ │ │  o3 = 420
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Time.out
│ │ │ @@ -1,7 +1,7 @@
│ │ │  -- -*- M2-comint -*- hash: 1331310711075
│ │ │  
│ │ │  i1 : currentTime() - fileTime "."
│ │ │  
│ │ │ -o1 = 61
│ │ │ +o1 = 46
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_force__G__B_lp..._cm__Syzygy__Matrix_eq_gt..._rp.out
│ │ │ @@ -29,15 +29,15 @@
│ │ │       {4} | 0     x2-3  y3-1  |
│ │ │  
│ │ │               3      3
│ │ │  o6 : Matrix R  <-- R
│ │ │  
│ │ │  i7 : syz f
│ │ │  
│ │ │ -   -- registering gb 0 at 0x7f5498d2ce00
│ │ │ +   -- registering gb 0 at 0x7f1938812e00
│ │ │  
│ │ │     -- [gb]{2}(1)m{3}(1)m{4}(1)m{5}(1)z{6}(1)z{7}(1)znumber of (nonminimal) gb elements = 3
│ │ │     -- number of monomials                = 9
│ │ │     -- #reduction steps = 6
│ │ │     -- #spairs done = 6
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_get.out
│ │ │ @@ -10,11 +10,11 @@
│ │ │  
│ │ │  o2 = hi there
│ │ │  
│ │ │  i3 : removeFile "test-file"
│ │ │  
│ │ │  i4 : get "!date"
│ │ │  
│ │ │ -o4 = Sun Dec 14 15:27:22 UTC 2025
│ │ │ +o4 = Thu Jan  1 11:02:48 UTC 2026
│ │ │  
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_instances.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │                 defaultPrecision => 53
│ │ │                 engineDebugLevel => 0
│ │ │                 errorDepth => 0
│ │ │                 gbTrace => 0
│ │ │                 interpreterDepth => 1
│ │ │                 lineNumber => 2
│ │ │                 loadDepth => 3
│ │ │ -               maxAllowableThreads => 7
│ │ │ +               maxAllowableThreads => 17
│ │ │                 maxExponent => 1073741823
│ │ │                 minExponent => -1073741824
│ │ │                 numTBBThreads => 0
│ │ │                 o1 => 2432902008176640000
│ │ │                 oo => 2432902008176640000
│ │ │                 printingAccuracy => -1
│ │ │                 printingLeadLimit => 5
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Directory.out
│ │ │ @@ -2,19 +2,19 @@
│ │ │  
│ │ │  i1 : isDirectory "."
│ │ │  
│ │ │  o1 = true
│ │ │  
│ │ │  i2 : fn = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-10380-0/0
│ │ │ +o2 = /tmp/M2-10670-0/0
│ │ │  
│ │ │  i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10380-0/0
│ │ │ +o3 = /tmp/M2-10670-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : isDirectory fn
│ │ │  
│ │ │  o4 = false
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Pseudoprime_lp__Z__Z_rp.out
│ │ │ @@ -75,15 +75,15 @@
│ │ │  o17 = false
│ │ │  
│ │ │  i18 : isPrime(m*m*m1*m1*m2^6)
│ │ │  
│ │ │  o18 = false
│ │ │  
│ │ │  i19 : elapsedTime facs = factor(m*m1)
│ │ │ - -- 4.33674s elapsed
│ │ │ + -- 5.79079s elapsed
│ │ │  
│ │ │  o19 = 1000000000000000000000000000057*1000000000000000000010000000083
│ │ │  
│ │ │  o19 : Expression of class Product
│ │ │  
│ │ │  i20 : facs = facs//toList/toList
│ │ │  
│ │ │ @@ -97,17 +97,17 @@
│ │ │  
│ │ │  i22 : m3 = nextPrime (m^3)
│ │ │  
│ │ │  o22 = 10000000000000000000000000001710000000000000000000000000097470000000000
│ │ │        00000000000000185613
│ │ │  
│ │ │  i23 : elapsedTime isPrime m3
│ │ │ - -- .0569545s elapsed
│ │ │ + -- .0555585s elapsed
│ │ │  
│ │ │  o23 = true
│ │ │  
│ │ │  i24 : elapsedTime isPseudoprime m3
│ │ │ - -- .000114113s elapsed
│ │ │ + -- .000106514s elapsed
│ │ │  
│ │ │  o24 = true
│ │ │  
│ │ │  i25 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Regular__File.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 4782205245758053629
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12188-0/0
│ │ │ +o1 = /tmp/M2-14348-0/0
│ │ │  
│ │ │  i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-12188-0/0
│ │ │ +o2 = /tmp/M2-14348-0/0
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : isRegularFile fn
│ │ │  
│ │ │  o3 = true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_make__Directory_lp__String_rp.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 5113372159204571746
│ │ │  
│ │ │  i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10722-0/0
│ │ │ +o1 = /tmp/M2-11352-0/0
│ │ │  
│ │ │  i2 : makeDirectory (dir|"/a/b/c")
│ │ │  
│ │ │ -o2 = /tmp/M2-10722-0/0/a/b/c
│ │ │ +o2 = /tmp/M2-11352-0/0/a/b/c
│ │ │  
│ │ │  i3 : removeDirectory (dir|"/a/b/c")
│ │ │  
│ │ │  i4 : removeDirectory (dir|"/a/b")
│ │ │  
│ │ │  i5 : removeDirectory (dir|"/a")
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Allowable__Threads.out
│ │ │ @@ -1,7 +1,7 @@
│ │ │  -- -*- M2-comint -*- hash: 1331887830690
│ │ │  
│ │ │  i1 : maxAllowableThreads
│ │ │  
│ │ │ -o1 = 7
│ │ │ +o1 = 17
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_memoize.out
│ │ │ @@ -3,31 +3,31 @@
│ │ │  i1 : fib = n -> if n <= 1 then 1 else fib(n-1) + fib(n-2)
│ │ │  
│ │ │  o1 = fib
│ │ │  
│ │ │  o1 : FunctionClosure
│ │ │  
│ │ │  i2 : time fib 28
│ │ │ - -- used 1.26333s (cpu); 0.726987s (thread); 0s (gc)
│ │ │ + -- used 0.754762s (cpu); 0.577694s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 514229
│ │ │  
│ │ │  i3 : fib = memoize fib
│ │ │  
│ │ │  o3 = fib
│ │ │  
│ │ │  o3 : FunctionClosure
│ │ │  
│ │ │  i4 : time fib 28
│ │ │ - -- used 7.5542e-05s (cpu); 7.4801e-05s (thread); 0s (gc)
│ │ │ + -- used 8.2955e-05s (cpu); 8.1779e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 514229
│ │ │  
│ │ │  i5 : time fib 28
│ │ │ - -- used 3.987e-06s (cpu); 3.627e-06s (thread); 0s (gc)
│ │ │ + -- used 3.696e-06s (cpu); 3.447e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 514229
│ │ │  
│ │ │  i6 : fib = memoize( n -> fib(n-1) + fib(n-2) )
│ │ │  
│ │ │  o6 = fib
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_methods.out
│ │ │ @@ -17,20 +17,20 @@
│ │ │       {12 => (poincare, BettiTally)                                }
│ │ │       {13 => (hilbertPolynomial, ZZ, BettiTally)                   }
│ │ │       {14 => (degree, BettiTally)                                  }
│ │ │       {15 => (hilbertSeries, ZZ, BettiTally)                       }
│ │ │       {16 => (pdim, BettiTally)                                    }
│ │ │       {17 => (regularity, BettiTally)                              }
│ │ │       {18 => (mathML, BettiTally)                                  }
│ │ │ -     {19 => (codim, BettiTally)                                   }
│ │ │ +     {19 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │       {20 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ -     {21 => (dual, BettiTally)                                    }
│ │ │ -     {22 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ -     {23 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ -     {24 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ +     {21 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ +     {22 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ +     {23 => (codim, BettiTally)                                   }
│ │ │ +     {24 => (dual, BettiTally)                                    }
│ │ │       {25 => (^, Ring, BettiTally)                                 }
│ │ │  
│ │ │  o1 : NumberedVerticalList
│ │ │  
│ │ │  i2 : methods resolution
│ │ │  
│ │ │  o2 = {0 => (resolution, Ideal) }
│ │ │ @@ -60,20 +60,20 @@
│ │ │       {1 => (++, Module, GradedModule)}
│ │ │       {2 => (++, Module, Module)      }
│ │ │  
│ │ │  o4 : NumberedVerticalList
│ │ │  
│ │ │  i5 : methods( Matrix, Matrix )
│ │ │  
│ │ │ -o5 = {0 => (contract, Matrix, Matrix)                            }
│ │ │ -     {1 => (diff, Matrix, Matrix)                                }
│ │ │ -     {2 => (diff', Matrix, Matrix)                               }
│ │ │ -     {3 => (-, Matrix, Matrix)                                   }
│ │ │ +o5 = {0 => (+, Matrix, Matrix)                                   }
│ │ │ +     {1 => (-, Matrix, Matrix)                                   }
│ │ │ +     {2 => (contract, Matrix, Matrix)                            }
│ │ │ +     {3 => (diff, Matrix, Matrix)                                }
│ │ │       {4 => (contract', Matrix, Matrix)                           }
│ │ │ -     {5 => (+, Matrix, Matrix)                                   }
│ │ │ +     {5 => (diff', Matrix, Matrix)                               }
│ │ │       {6 => (markedGB, Matrix, Matrix)                            }
│ │ │       {7 => (Hom, Matrix, Matrix)                                 }
│ │ │       {8 => (==, Matrix, Matrix)                                  }
│ │ │       {9 => (*, Matrix, Matrix)                                   }
│ │ │       {10 => (|, Matrix, Matrix)                                  }
│ │ │       {11 => (||, Matrix, Matrix)                                 }
│ │ │       {12 => (subquotient, Matrix, Matrix)                        }
│ │ │ @@ -88,18 +88,18 @@
│ │ │       {21 => (quotient, Matrix, Matrix)                           }
│ │ │       {22 => (quotient', Matrix, Matrix)                          }
│ │ │       {23 => (remainder', Matrix, Matrix)                         }
│ │ │       {24 => (%, Matrix, Matrix)                                  }
│ │ │       {25 => (remainder, Matrix, Matrix)                          }
│ │ │       {26 => (pushout, Matrix, Matrix)                            }
│ │ │       {27 => (solve, Matrix, Matrix)                              }
│ │ │ -     {28 => (pullback, Matrix, Matrix)                           }
│ │ │ +     {28 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │       {29 => (intersect, Matrix, Matrix)                          }
│ │ │ -     {30 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ -     {31 => (tensor, Matrix, Matrix)                             }
│ │ │ +     {30 => (tensor, Matrix, Matrix)                             }
│ │ │ +     {31 => (pullback, Matrix, Matrix)                           }
│ │ │       {32 => (substitute, Matrix, Matrix)                         }
│ │ │       {33 => (yonedaProduct, Matrix, Matrix)                      }
│ │ │       {34 => (isShortExactSequence, Matrix, Matrix)               }
│ │ │       {35 => (horseshoeResolution, Matrix, Matrix)                }
│ │ │       {36 => (connectingExtMap, Module, Matrix, Matrix)           }
│ │ │       {37 => (connectingExtMap, Matrix, Matrix, Module)           }
│ │ │       {38 => (connectingTorMap, Module, Matrix, Matrix)           }
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Betti.out
│ │ │ @@ -9,15 +9,15 @@
│ │ │  i2 : S = ring I
│ │ │  
│ │ │  o2 = S
│ │ │  
│ │ │  o2 : PolynomialRing
│ │ │  
│ │ │  i3 : elapsedTime C = minimalBetti I
│ │ │ - -- 1.82886s elapsed
│ │ │ + -- 2.45521s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o3 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -26,44 +26,44 @@
│ │ │  o3 : BettiTally
│ │ │  
│ │ │  i4 : I = ideal I_*;
│ │ │  
│ │ │  o4 : Ideal of S
│ │ │  
│ │ │  i5 : elapsedTime C = minimalBetti(I, DegreeLimit=>2)
│ │ │ - -- .745147s elapsed
│ │ │ + -- .925445s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7
│ │ │  o5 = total: 1 35 140 385 819 1080 735 196
│ │ │           0: 1  .   .   .   .    .   .   .
│ │ │           1: . 35 140 189  84    .   .   .
│ │ │           2: .  .   . 196 735 1080 735 196
│ │ │  
│ │ │  o5 : BettiTally
│ │ │  
│ │ │  i6 : I = ideal I_*;
│ │ │  
│ │ │  o6 : Ideal of S
│ │ │  
│ │ │  i7 : elapsedTime C = minimalBetti(I, DegreeLimit=>1, LengthLimit=>5)
│ │ │ - -- .0317322s elapsed
│ │ │ + -- .0362305s elapsed
│ │ │  
│ │ │              0  1   2   3  4
│ │ │  o7 = total: 1 35 140 189 84
│ │ │           0: 1  .   .   .  .
│ │ │           1: . 35 140 189 84
│ │ │  
│ │ │  o7 : BettiTally
│ │ │  
│ │ │  i8 : I = ideal I_*;
│ │ │  
│ │ │  o8 : Ideal of S
│ │ │  
│ │ │  i9 : elapsedTime C = minimalBetti(I, LengthLimit=>5)
│ │ │ - -- 1.20016s elapsed
│ │ │ + -- 3.28835s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5
│ │ │  o9 = total: 1 35 140 385 819 1080
│ │ │           0: 1  .   .   .   .    .
│ │ │           1: . 35 140 189  84    .
│ │ │           2: .  .   . 196 735 1080
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_mkdir.out
│ │ │ @@ -1,22 +1,22 @@
│ │ │  -- -*- M2-comint -*- hash: 15555226809509933135
│ │ │  
│ │ │  i1 : p = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-10741-0/0/
│ │ │ +o1 = /tmp/M2-11391-0/0/
│ │ │  
│ │ │  i2 : mkdir p
│ │ │  
│ │ │  i3 : isDirectory p
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 : (fn = p | "foo") << "hi there" << close
│ │ │  
│ │ │ -o4 = /tmp/M2-10741-0/0/foo
│ │ │ +o4 = /tmp/M2-11391-0/0/foo
│ │ │  
│ │ │  o4 : File
│ │ │  
│ │ │  i5 : get fn
│ │ │  
│ │ │  o5 = hi there
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_move__File_lp__String_cm__String_rp.out
│ │ │ @@ -1,31 +1,31 @@
│ │ │  -- -*- M2-comint -*- hash: 4857944042471093218
│ │ │  
│ │ │  i1 : src = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10615-0/0
│ │ │ +o1 = /tmp/M2-11145-0/0
│ │ │  
│ │ │  i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-10615-0/1
│ │ │ +o2 = /tmp/M2-11145-0/1
│ │ │  
│ │ │  i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10615-0/0
│ │ │ +o3 = /tmp/M2-11145-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : moveFile(src,dst,Verbose=>true)
│ │ │ ---moving: /tmp/M2-10615-0/0 -> /tmp/M2-10615-0/1
│ │ │ +--moving: /tmp/M2-11145-0/0 -> /tmp/M2-11145-0/1
│ │ │  
│ │ │  i5 : get dst
│ │ │  
│ │ │  o5 = hi there
│ │ │  
│ │ │  i6 : bak = moveFile(dst,Verbose=>true)
│ │ │ ---backup file created: /tmp/M2-10615-0/1.bak
│ │ │ +--backup file created: /tmp/M2-11145-0/1.bak
│ │ │  
│ │ │ -o6 = /tmp/M2-10615-0/1.bak
│ │ │ +o6 = /tmp/M2-11145-0/1.bak
│ │ │  
│ │ │  i7 : removeFile bak
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_nanosleep.out
│ │ │ @@ -1,8 +1,8 @@
│ │ │  -- -*- M2-comint -*- hash: 1331114612441
│ │ │  
│ │ │  i1 : elapsedTime nanosleep 500000000
│ │ │ - -- .500135s elapsed
│ │ │ + -- .500115s 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);
│ │ │ - -- .640674s elapsed
│ │ │ + -- .681356s elapsed
│ │ │  
│ │ │  i4 : elapsedTime parallelApply(1..100, n -> sort L);
│ │ │ - -- .307919s elapsed
│ │ │ + -- .182068s elapsed
│ │ │  
│ │ │  i5 : allowableThreads
│ │ │  
│ │ │  o5 = 5
│ │ │  
│ │ │  i6 : allowableThreads = maxAllowableThreads
│ │ │  
│ │ │ -o6 = 7
│ │ │ +o6 = 17
│ │ │  
│ │ │  i7 : R = QQ[x,y,z];
│ │ │  
│ │ │  i8 : I = ideal(x^2 + 2*y^2 - y - 2*z, x^2 - 8*y^2 + 10*z - 1, x^2 - 7*y*z)
│ │ │  
│ │ │               2     2            2     2             2
│ │ │  o8 = ideal (x  + 2y  - y - 2z, x  - 8y  + 10z - 1, x  - 7y*z)
│ │ │ @@ -41,15 +41,15 @@
│ │ │  
│ │ │  o10 = <>
│ │ │  
│ │ │  o10 : Task
│ │ │  
│ │ │  i11 : t
│ │ │  
│ │ │ -o11 = <>
│ │ │ +o11 = <>
│ │ │  
│ │ │  o11 : Task
│ │ │  
│ │ │  i12 : isReady t
│ │ │  
│ │ │  o12 = false
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallelism_spin_spengine_spcomputations.out
│ │ │ @@ -67,15 +67,15 @@
│ │ │  i3 : S = ring I
│ │ │  
│ │ │  o3 = S
│ │ │  
│ │ │  o3 : PolynomialRing
│ │ │  
│ │ │  i4 : elapsedTime minimalBetti I
│ │ │ - -- 2.07143s elapsed
│ │ │ + -- 2.1915s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o4 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -84,15 +84,15 @@
│ │ │  o4 : BettiTally
│ │ │  
│ │ │  i5 : I = ideal I_*;
│ │ │  
│ │ │  o5 : Ideal of S
│ │ │  
│ │ │  i6 : elapsedTime minimalBetti(I, ParallelizeByDegree => true)
│ │ │ - -- 1.82839s elapsed
│ │ │ + -- 2.25423s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o6 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -105,15 +105,15 @@
│ │ │  o7 : Ideal of S
│ │ │  
│ │ │  i8 : numTBBThreads = 1
│ │ │  
│ │ │  o8 = 1
│ │ │  
│ │ │  i9 : elapsedTime minimalBetti(I)
│ │ │ - -- 1.75494s elapsed
│ │ │ + -- 2.17331s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o9 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -132,15 +132,15 @@
│ │ │  o11 = 0
│ │ │  
│ │ │  i12 : I = ideal I_*;
│ │ │  
│ │ │  o12 : Ideal of S
│ │ │  
│ │ │  i13 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ - -- 2.23687s elapsed
│ │ │ + -- 3.78121s elapsed
│ │ │  
│ │ │         1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o13 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S
│ │ │                                                                                                    
│ │ │        0      1       2        3        4         5         6         7         8        9        10
│ │ │  
│ │ │  o13 : Complex
│ │ │ @@ -150,15 +150,15 @@
│ │ │  o14 = 1
│ │ │  
│ │ │  i15 : I = ideal I_*;
│ │ │  
│ │ │  o15 : Ideal of S
│ │ │  
│ │ │  i16 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ - -- 2.55496s elapsed
│ │ │ + -- 2.48371s elapsed
│ │ │  
│ │ │         1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o16 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S
│ │ │                                                                                                    
│ │ │        0      1       2        3        4         5         6         7         8        9        10
│ │ │  
│ │ │  o16 : Complex
│ │ │ @@ -174,43 +174,43 @@
│ │ │  o18 : PolynomialRing
│ │ │  
│ │ │  i19 : I = ideal random(S^1, S^{4:-5});
│ │ │  
│ │ │  o19 : Ideal of S
│ │ │  
│ │ │  i20 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 5.136s elapsed
│ │ │ + -- 3.695s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o20 : Matrix S  <-- S
│ │ │  
│ │ │  i21 : numTBBThreads = 1
│ │ │  
│ │ │  o21 = 1
│ │ │  
│ │ │  i22 : I = ideal I_*;
│ │ │  
│ │ │  o22 : Ideal of S
│ │ │  
│ │ │  i23 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 7.09108s elapsed
│ │ │ + -- 8.58327s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o23 : Matrix S  <-- S
│ │ │  
│ │ │  i24 : numTBBThreads = 10
│ │ │  
│ │ │  o24 = 10
│ │ │  
│ │ │  i25 : I = ideal I_*;
│ │ │  
│ │ │  o25 : Ideal of S
│ │ │  
│ │ │  i26 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 4.22028s elapsed
│ │ │ + -- 3.71049s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o26 : Matrix S  <-- S
│ │ │  
│ │ │  i27 : needsPackage "AssociativeAlgebras"
│ │ │  
│ │ │  o27 = AssociativeAlgebras
│ │ │ @@ -233,15 +233,15 @@
│ │ │  o30 = ideal (5a  + 2b*c + 3c*b, 3a*c + 5b  + 2c*a, 2a*b + 3b*a + 5c )
│ │ │  
│ │ │                  ZZ
│ │ │  o30 : Ideal of ---<|a, b, c|>
│ │ │                 101
│ │ │  
│ │ │  i31 : elapsedTime NCGB(I, 22);
│ │ │ - -- 1.006s elapsed
│ │ │ + -- 1.04567s elapsed
│ │ │  
│ │ │                 ZZ            1       ZZ            148
│ │ │  o31 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │                101                   101
│ │ │  
│ │ │  i32 : I = ideal I_*
│ │ │  
│ │ │ @@ -253,14 +253,14 @@
│ │ │                 101
│ │ │  
│ │ │  i33 : numTBBThreads = 1
│ │ │  
│ │ │  o33 = 1
│ │ │  
│ │ │  i34 : elapsedTime NCGB(I, 22);
│ │ │ - -- 1.19796s elapsed
│ │ │ + -- 1.55933s elapsed
│ │ │  
│ │ │                 ZZ            1       ZZ            148
│ │ │  o34 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │                101                   101
│ │ │  
│ │ │  i35 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_poincare.out
│ │ │ @@ -146,65 +146,65 @@
│ │ │  o26 : ZZ[T]
│ │ │  
│ │ │  i27 : gbTrace = 3
│ │ │  
│ │ │  o27 = 3
│ │ │  
│ │ │  i28 : time poincare I
│ │ │ - -- used 0.00277298s (cpu); 2.153e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00166338s (cpu); 1.2346e-05s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o28 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o28 : ZZ[T]
│ │ │  
│ │ │  i29 : time gens gb I;
│ │ │  
│ │ │ -   -- registering gb 19 at 0x7f3fe9e7e540
│ │ │ +   -- registering gb 19 at 0x7fb20a72f380
│ │ │  
│ │ │     -- [gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(2,6)mm{7}(1,4)m{8}(0,2)number of (nonminimal) gb elements = 11
│ │ │     -- number of monomials                = 4186
│ │ │     -- #reduction steps = 38
│ │ │     -- #spairs done = 11
│ │ │     -- ncalls = 10
│ │ │     -- nloop = 29
│ │ │     -- nsaved = 0
│ │ │ -   --  -- used 0.0251999s (cpu); 0.0250353s (thread); 0s (gc)
│ │ │ +   --  -- used 0.0143391s (cpu); 0.0143519s (thread); 0s (gc)
│ │ │  
│ │ │                1      11
│ │ │  o29 : Matrix R  <-- R
│ │ │  
│ │ │  i30 : R = QQ[a..d];
│ │ │  
│ │ │  i31 : I = ideal random(R^1, R^{3:-3});
│ │ │  
│ │ │ -   -- registering gb 20 at 0x7f3fe9e7e380
│ │ │ +   -- registering gb 20 at 0x7fb20a72f000
│ │ │  
│ │ │     -- [gb]number of (nonminimal) gb elements = 0
│ │ │     -- number of monomials                = 0
│ │ │     -- #reduction steps = 0
│ │ │     -- #spairs done = 0
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ │     -- nsaved = 0
│ │ │     -- 
│ │ │  o31 : Ideal of R
│ │ │  
│ │ │  i32 : time p = poincare I
│ │ │  
│ │ │ -   -- registering gb 21 at 0x7f3fe9e7e000
│ │ │ +   -- registering gb 21 at 0x7fb207cf1c40
│ │ │  
│ │ │     -- [gb]{3}(3)mmm{4}(2)mm{5}(3)mmm{6}(6)mmoooo{7}(4)mooo{8}(2)oonumber of (nonminimal) gb elements = 11
│ │ │     -- number of monomials                = 267
│ │ │     -- #reduction steps = 236
│ │ │     -- #spairs done = 30
│ │ │     -- ncalls = 10
│ │ │     -- nloop = 20
│ │ │     -- nsaved = 0
│ │ │ -   --  -- used 0.00799718s (cpu); 0.00944086s (thread); 0s (gc)
│ │ │ +   --  -- used 0.00776975s (cpu); 0.00544307s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o32 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o32 : ZZ[T]
│ │ │  
│ │ │  i33 : S = QQ[a..d, MonomialOrder => Eliminate 2]
│ │ │ @@ -254,27 +254,27 @@
│ │ │  
│ │ │  i36 : gbTrace = 3
│ │ │  
│ │ │  o36 = 3
│ │ │  
│ │ │  i37 : time gens gb J;
│ │ │  
│ │ │ -   -- registering gb 22 at 0x7f3fe9ab3e00
│ │ │ +   -- registering gb 22 at 0x7fb207cf1a80
│ │ │  
│ │ │     -- [gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(3,7)mmm{7}(3,8)mmm{8}(3,9)mmm{9}(3,9)m
│ │ │     -- mm{10}(2,8)mm{11}(1,5)m{12}(1,3)m{13}(1,3)m{14}(1,3)m{15}(1,3)m{16}(1,3)m
│ │ │     -- {17}(1,3)m{18}(1,3)m{19}(1,3)m{20}(1,3)m{21}(1,3)m{22}(1,3)m{23}(1,3)m{24}(1,3)m
│ │ │     -- {25}(1,3)m{26}(1,3)m{27}(1,3)m{28}(0,2)number of (nonminimal) gb elements = 39
│ │ │     -- number of monomials                = 1051
│ │ │     -- #reduction steps = 284
│ │ │     -- #spairs done = 53
│ │ │     -- ncalls = 46
│ │ │     -- nloop = 54
│ │ │     -- nsaved = 0
│ │ │ -   --  -- used 0.0840427s (cpu); 0.0832054s (thread); 0s (gc)
│ │ │ +   --  -- used 0.0519744s (cpu); 0.0506s (thread); 0s (gc)
│ │ │  
│ │ │                1      39
│ │ │  o37 : Matrix S  <-- S
│ │ │  
│ │ │  i38 : selectInSubring(1, gens gb J)
│ │ │  
│ │ │  o38 = | 188529931266160087758259645374082357642621166724936033369975727480205
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_printing_spto_spa_spfile.out
│ │ │ @@ -12,19 +12,19 @@
│ │ │  
│ │ │  o2 = stdio
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : fn = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-10932-0/0
│ │ │ +o3 = /tmp/M2-11782-0/0
│ │ │  
│ │ │  i4 : fn << "hi there" << endl << close
│ │ │  
│ │ │ -o4 = /tmp/M2-10932-0/0
│ │ │ +o4 = /tmp/M2-11782-0/0
│ │ │  
│ │ │  o4 : File
│ │ │  
│ │ │  i5 : get fn
│ │ │  
│ │ │  o5 = hi there
│ │ │  
│ │ │ @@ -49,27 +49,27 @@
│ │ │  x   + 10x  + 45x  + 120x  + 210x  + 252x  + 210x  + 120x  + 45x  + 10x + 1
│ │ │  o8 = stdio
│ │ │  
│ │ │  o8 : File
│ │ │  
│ │ │  i9 : fn << f << close
│ │ │  
│ │ │ -o9 = /tmp/M2-10932-0/0
│ │ │ +o9 = /tmp/M2-11782-0/0
│ │ │  
│ │ │  o9 : File
│ │ │  
│ │ │  i10 : get fn
│ │ │  
│ │ │  o10 =  10      9      8       7       6       5       4       3      2
│ │ │        x   + 10x  + 45x  + 120x  + 210x  + 252x  + 210x  + 120x  + 45x  + 10x
│ │ │        + 1
│ │ │  
│ │ │  i11 : fn << toExternalString f << close
│ │ │  
│ │ │ -o11 = /tmp/M2-10932-0/0
│ │ │ +o11 = /tmp/M2-11782-0/0
│ │ │  
│ │ │  o11 : File
│ │ │  
│ │ │  i12 : get fn
│ │ │  
│ │ │  o12 = x^10+10*x^9+45*x^8+120*x^7+210*x^6+252*x^5+210*x^4+120*x^3+45*x^2+10*x+
│ │ │        1
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_process__I__D.out
│ │ │ @@ -1,7 +1,7 @@
│ │ │  -- -*- M2-comint -*- hash: 1330513630563
│ │ │  
│ │ │  i1 : processID()
│ │ │  
│ │ │ -o1 = 10191
│ │ │ +o1 = 10311
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_profile.out
│ │ │ @@ -9,35 +9,35 @@
│ │ │  
│ │ │                4       5
│ │ │  o1 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i2 : profileSummary
│ │ │  
│ │ │  o2 = #run  %time   position                         
│ │ │ -     1     94.52   ../../m2/matrix1.m2:279:4-282:58 
│ │ │ -     1     92.12   ../../m2/matrix1.m2:281:22-281:43
│ │ │ -     1     44.16   ../../m2/matrix1.m2:193:25-193:52
│ │ │ -     1     30.59   ../../m2/matrix1.m2:114:5-156:72 
│ │ │ -     1     29.47   ../../m2/matrix1.m2:140:10-155:16
│ │ │ -     1     23.83   ../../m2/matrix1.m2:181:4-181:42 
│ │ │ -     1     22.54   ../../m2/set.m2:127:5-127:61     
│ │ │ -     1     20.86   ../../m2/matrix1.m2:45:10-49:22  
│ │ │ -     1     3.30    ../../m2/matrix1.m2:112:5-112:29 
│ │ │ -     1     2.34    ../../m2/matrix1.m2:141:13-141:78
│ │ │ -     1     2.18    ../../m2/matrix1.m2:96:5-109:11  
│ │ │ -     1     1.42    ../../m2/matrix1.m2:281:7-281:16 
│ │ │ -     1     1.29    ../../m2/matrix1.m2:147:20-147:64
│ │ │ -     1     1.29    ../../m2/matrix1.m2:111:5-111:91 
│ │ │ -     1     1.27    ../../m2/matrix1.m2:276:4-277:73 
│ │ │ -     1     1.02    ../../m2/matrix1.m2:98:10-98:46  
│ │ │ -     1     .97     ../../m2/matrix1.m2:182:4-184:74 
│ │ │ -     1     .81     ../../m2/modules.m2:279:4-279:52 
│ │ │ -     20    .64     ../../m2/matrix1.m2:191:14-192:67
│ │ │ -     20    .47     ../../m2/matrix1.m2:47:43-47:71  
│ │ │ -     1     .0038s  elapsed total                    
│ │ │ +     1     93.35   ../../m2/matrix1.m2:279:4-282:58 
│ │ │ +     1     90.65   ../../m2/matrix1.m2:281:22-281:43
│ │ │ +     1     43.01   ../../m2/matrix1.m2:193:25-193:52
│ │ │ +     1     29.94   ../../m2/matrix1.m2:114:5-156:72 
│ │ │ +     1     28.72   ../../m2/matrix1.m2:140:10-155:16
│ │ │ +     1     23.46   ../../m2/matrix1.m2:181:4-181:42 
│ │ │ +     1     22.12   ../../m2/set.m2:127:5-127:61     
│ │ │ +     1     20.2    ../../m2/matrix1.m2:45:10-49:22  
│ │ │ +     1     3.26    ../../m2/matrix1.m2:112:5-112:29 
│ │ │ +     1     2.8     ../../m2/matrix1.m2:141:13-141:78
│ │ │ +     1     2.16    ../../m2/matrix1.m2:96:5-109:11  
│ │ │ +     1     1.69    ../../m2/matrix1.m2:281:7-281:16 
│ │ │ +     1     1.37    ../../m2/matrix1.m2:276:4-277:73 
│ │ │ +     1     1.26    ../../m2/matrix1.m2:147:20-147:64
│ │ │ +     1     1.14    ../../m2/matrix1.m2:111:5-111:91 
│ │ │ +     1     1.10    ../../m2/matrix1.m2:182:4-184:74 
│ │ │ +     1     1.08    ../../m2/matrix1.m2:98:10-98:46  
│ │ │ +     20    .95     ../../m2/matrix1.m2:191:14-192:67
│ │ │ +     20    .67     ../../m2/matrix1.m2:47:43-47:71  
│ │ │ +     20    .60     ../../m2/matrix1.m2:190:17-190:29
│ │ │ +     1     .0035s  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 0.173474s (cpu); 0.138239s (thread); 0s (gc)
│ │ │ + -- used 0.247677s (cpu); 0.101492s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = ideal (x  - 53x , x  + 8x , x  - 4x )
│ │ │               2      3   1     3   0     3
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : R=kk[x_0..x_5];
│ │ │ @@ -33,15 +33,15 @@
│ │ │  i9 : codim I, degree I
│ │ │  
│ │ │  o9 = (3, 10)
│ │ │  
│ │ │  o9 : Sequence
│ │ │  
│ │ │  i10 : time randomKRationalPoint(I)
│ │ │ - -- used 0.663337s (cpu); 0.35806s (thread); 0s (gc)
│ │ │ + -- used 0.767075s (cpu); 0.303818s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = ideal (x  - 27x , x  - 16x , x  - 9x , x  + 44x , x  - 52x )
│ │ │                4      5   3      5   2     5   1      5   0      5
│ │ │  
│ │ │  o10 : Ideal of R
│ │ │  
│ │ │  i11 : p=10007,kk=ZZ/p,R=kk[x_0..x_2]
│ │ │ @@ -58,12 +58,12 @@
│ │ │  
│ │ │  i14 : I=ideal random(n,R);
│ │ │  
│ │ │  o14 : Ideal of R
│ │ │  
│ │ │  i15 : time (#select(apply(100,i->(degs=apply(decompose(I+ideal random(1,R)),c->degree c);
│ │ │                       #select(degs,d->d==1))),f->f>0))
│ │ │ - -- used 3.65398s (cpu); 1.92733s (thread); 0s (gc)
│ │ │ + -- used 4.37816s (cpu); 1.88301s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 58
│ │ │  
│ │ │  i16 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_read__Directory.out
│ │ │ @@ -1,26 +1,26 @@
│ │ │  -- -*- M2-comint -*- hash: 20910736704070514
│ │ │  
│ │ │  i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11565-0/0
│ │ │ +o1 = /tmp/M2-13075-0/0
│ │ │  
│ │ │  i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-11565-0/0
│ │ │ +o2 = /tmp/M2-13075-0/0
│ │ │  
│ │ │  i3 : (fn = dir | "/" | "foo") << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11565-0/0/foo
│ │ │ +o3 = /tmp/M2-13075-0/0/foo
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : readDirectory dir
│ │ │  
│ │ │ -o4 = {., .., foo}
│ │ │ +o4 = {.., ., foo}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : removeFile fn
│ │ │  
│ │ │  i6 : removeDirectory dir
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_reading_spfiles.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 13513555104200944796
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11107-0/0
│ │ │ +o1 = /tmp/M2-12137-0/0
│ │ │  
│ │ │  i2 : fn << "z^6+3*x*z^4+6*y*z^4+3*x^2*z^2+12*x*y*z^2+12*y^2*z^2+x^3+6*x^2*y+12*x*y^2+8*y^3" << endl << close
│ │ │  
│ │ │ -o2 = /tmp/M2-11107-0/0
│ │ │ +o2 = /tmp/M2-12137-0/0
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : get fn
│ │ │  
│ │ │  o3 = z^6+3*x*z^4+6*y*z^4+3*x^2*z^2+12*x*y*z^2+12*y^2*z^2+x^3+6*x^2*y+12*x*y^2
│ │ │       +8*y^3
│ │ │ @@ -38,15 +38,15 @@
│ │ │  
│ │ │  o6 : Expression of class Product
│ │ │  
│ │ │  i7 : fn << "sample = 2^100
│ │ │       print sample
│ │ │       " << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11107-0/0
│ │ │ +o7 = /tmp/M2-12137-0/0
│ │ │  
│ │ │  o7 : File
│ │ │  
│ │ │  i8 : get fn
│ │ │  
│ │ │  o8 = sample = 2^100
│ │ │       print sample
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_readlink.out
│ │ │ @@ -1,12 +1,12 @@
│ │ │  -- -*- M2-comint -*- hash: 4408639611478781130
│ │ │  
│ │ │  i1 : p = temporaryFileName ()
│ │ │  
│ │ │ -o1 = /tmp/M2-11806-0/0
│ │ │ +o1 = /tmp/M2-13556-0/0
│ │ │  
│ │ │  i2 : symlinkFile ("foo", p)
│ │ │  
│ │ │  i3 : readlink p
│ │ │  
│ │ │  o3 = foo
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_realpath.out
│ │ │ @@ -1,39 +1,39 @@
│ │ │  -- -*- M2-comint -*- hash: 324072347213224656
│ │ │  
│ │ │  i1 : realpath "."
│ │ │  
│ │ │ -o1 = /tmp/M2-10191-0/86-rundir/
│ │ │ +o1 = /tmp/M2-10311-0/86-rundir/
│ │ │  
│ │ │  i2 : p = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11825-0/0
│ │ │ +o2 = /tmp/M2-13595-0/0
│ │ │  
│ │ │  i3 : q = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-11825-0/1
│ │ │ +o3 = /tmp/M2-13595-0/1
│ │ │  
│ │ │  i4 : symlinkFile(p,q)
│ │ │  
│ │ │  i5 : p << close
│ │ │  
│ │ │ -o5 = /tmp/M2-11825-0/0
│ │ │ +o5 = /tmp/M2-13595-0/0
│ │ │  
│ │ │  o5 : File
│ │ │  
│ │ │  i6 : readlink q
│ │ │  
│ │ │ -o6 = /tmp/M2-11825-0/0
│ │ │ +o6 = /tmp/M2-13595-0/0
│ │ │  
│ │ │  i7 : realpath q
│ │ │  
│ │ │ -o7 = /tmp/M2-11825-0/0
│ │ │ +o7 = /tmp/M2-13595-0/0
│ │ │  
│ │ │  i8 : removeFile p
│ │ │  
│ │ │  i9 : removeFile q
│ │ │  
│ │ │  i10 : realpath ""
│ │ │  
│ │ │ -o10 = /tmp/M2-10191-0/86-rundir/
│ │ │ +o10 = /tmp/M2-10311-0/86-rundir/
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_register__Finalizer.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 1729384374372662693
│ │ │  
│ │ │  i1 : for i from 1 to 9 do (x := 0 .. 10000 ; registerFinalizer(x, "-- finalizing sequence #"|i|" --"))
│ │ │  
│ │ │  i2 : collectGarbage() 
│ │ │  --finalization: (1)[3]: -- finalizing sequence #4 --
│ │ │ ---finalization: (2)[7]: -- finalizing sequence #8 --
│ │ │ ---finalization: (3)[4]: -- finalizing sequence #5 --
│ │ │ ---finalization: (4)[1]: -- finalizing sequence #2 --
│ │ │ ---finalization: (5)[6]: -- finalizing sequence #7 --
│ │ │ ---finalization: (6)[5]: -- finalizing sequence #6 --
│ │ │ ---finalization: (7)[8]: -- finalizing sequence #9 --
│ │ │ ---finalization: (8)[2]: -- finalizing sequence #3 --
│ │ │ ---finalization: (9)[0]: -- finalizing sequence #1 --
│ │ │ +--finalization: (2)[6]: -- finalizing sequence #7 --
│ │ │ +--finalization: (3)[1]: -- finalizing sequence #2 --
│ │ │ +--finalization: (4)[5]: -- finalizing sequence #6 --
│ │ │ +--finalization: (5)[8]: -- finalizing sequence #9 --
│ │ │ +--finalization: (6)[2]: -- finalizing sequence #3 --
│ │ │ +--finalization: (7)[0]: -- finalizing sequence #1 --
│ │ │ +--finalization: (8)[4]: -- finalizing sequence #5 --
│ │ │ +--finalization: (9)[7]: -- finalizing sequence #8 --
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_remove__Directory.out
│ │ │ @@ -1,19 +1,19 @@
│ │ │  -- -*- M2-comint -*- hash: 8532980310097060089
│ │ │  
│ │ │  i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10779-0/0
│ │ │ +o1 = /tmp/M2-11469-0/0
│ │ │  
│ │ │  i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-10779-0/0
│ │ │ +o2 = /tmp/M2-11469-0/0
│ │ │  
│ │ │  i3 : readDirectory dir
│ │ │  
│ │ │ -o3 = {., ..}
│ │ │ +o3 = {.., .}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : removeDirectory dir
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_root__Path.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1731420232148149387
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10283-0/0
│ │ │ +o1 = /tmp/M2-10473-0/0
│ │ │  
│ │ │  i2 : rootPath | fn
│ │ │  
│ │ │ -o2 = /tmp/M2-10283-0/0
│ │ │ +o2 = /tmp/M2-10473-0/0
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_root__U__R__I.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1731420231525572968
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11508-0/0
│ │ │ +o1 = /tmp/M2-12958-0/0
│ │ │  
│ │ │  i2 : rootURI | fn
│ │ │  
│ │ │ -o2 = file:///tmp/M2-11508-0/0
│ │ │ +o2 = file:///tmp/M2-12958-0/0
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_saving_sppolynomials_spand_spmatrices_spin_spfiles.out
│ │ │ @@ -25,19 +25,19 @@
│ │ │  o4 = image | x2 x2-y2 xyz7 |
│ │ │  
│ │ │                               1
│ │ │  o4 : R-module, submodule of R
│ │ │  
│ │ │  i5 : f = temporaryFileName()
│ │ │  
│ │ │ -o5 = /tmp/M2-11356-0/0
│ │ │ +o5 = /tmp/M2-12646-0/0
│ │ │  
│ │ │  i6 : f << toString (p,m,M) << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11356-0/0
│ │ │ +o6 = /tmp/M2-12646-0/0
│ │ │  
│ │ │  o6 : File
│ │ │  
│ │ │  i7 : get f
│ │ │  
│ │ │  o7 = (x^3-3*x^2*y+3*x*y^2-y^3-3*x^2+6*x*y-3*y^2+3*x-3*y-1,matrix {{x^2,
│ │ │       x^2-y^2, x*y*z^7}},image matrix {{x^2, x^2-y^2, x*y*z^7}})
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_serial__Number.out
│ │ │ @@ -1,15 +1,15 @@
│ │ │  -- -*- M2-comint -*- hash: 5271760183816554957
│ │ │  
│ │ │  i1 : serialNumber asdf
│ │ │  
│ │ │ -o1 = 1426273
│ │ │ +o1 = 1526273
│ │ │  
│ │ │  i2 : serialNumber foo
│ │ │  
│ │ │ -o2 = 1426275
│ │ │ +o2 = 1526275
│ │ │  
│ │ │  i3 : serialNumber ZZ
│ │ │  
│ │ │  o3 = 1000050
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_solve.out
│ │ │ @@ -189,18 +189,18 @@
│ │ │  o25 = 40
│ │ │  
│ │ │  i26 : A = mutableMatrix(CC_53, N, N); fillMatrix A;
│ │ │  
│ │ │  i28 : B = mutableMatrix(CC_53, N, 2); fillMatrix B;
│ │ │  
│ │ │  i30 : time X = solve(A,B);
│ │ │ - -- used 0.000227156s (cpu); 0.000219682s (thread); 0s (gc)
│ │ │ + -- used 0.000227852s (cpu); 0.00021655s (thread); 0s (gc)
│ │ │  
│ │ │  i31 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.000163036s (cpu); 0.000163226s (thread); 0s (gc)
│ │ │ + -- used 0.000112839s (cpu); 0.000112646s (thread); 0s (gc)
│ │ │  
│ │ │  i32 : norm(A*X-B)
│ │ │  
│ │ │  o32 = 5.111850690840453e-15
│ │ │  
│ │ │  o32 : RR (of precision 53)
│ │ │  
│ │ │ @@ -209,18 +209,18 @@
│ │ │  o33 = 100
│ │ │  
│ │ │  i34 : A = mutableMatrix(CC_100, N, N); fillMatrix A;
│ │ │  
│ │ │  i36 : B = mutableMatrix(CC_100, N, 2); fillMatrix B;
│ │ │  
│ │ │  i38 : time X = solve(A,B);
│ │ │ - -- used 0.482514s (cpu); 0.3008s (thread); 0s (gc)
│ │ │ + -- used 0.135732s (cpu); 0.13574s (thread); 0s (gc)
│ │ │  
│ │ │  i39 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.237275s (cpu); 0.23721s (thread); 0s (gc)
│ │ │ + -- used 0.138289s (cpu); 0.138302s (thread); 0s (gc)
│ │ │  
│ │ │  i40 : norm(A*X-B)
│ │ │  
│ │ │  o40 = 1.491578274689709814082355885932e-28
│ │ │  
│ │ │  o40 : RR (of precision 100)
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_symlink__Directory_lp__String_cm__String_rp.out
│ │ │ @@ -1,60 +1,60 @@
│ │ │  -- -*- M2-comint -*- hash: 2989513528213557691
│ │ │  
│ │ │  i1 : src = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-11147-0/0/
│ │ │ +o1 = /tmp/M2-12217-0/0/
│ │ │  
│ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-11147-0/1/
│ │ │ +o2 = /tmp/M2-12217-0/1/
│ │ │  
│ │ │  i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-11147-0/0/a/
│ │ │ +o3 = /tmp/M2-12217-0/0/a/
│ │ │  
│ │ │  i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-11147-0/0/b/
│ │ │ +o4 = /tmp/M2-12217-0/0/b/
│ │ │  
│ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-11147-0/0/b/c/
│ │ │ +o5 = /tmp/M2-12217-0/0/b/c/
│ │ │  
│ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11147-0/0/a/f
│ │ │ +o6 = /tmp/M2-12217-0/0/a/f
│ │ │  
│ │ │  o6 : File
│ │ │  
│ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11147-0/0/a/g
│ │ │ +o7 = /tmp/M2-12217-0/0/a/g
│ │ │  
│ │ │  o7 : File
│ │ │  
│ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-11147-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-12217-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
│ │ │  
│ │ │  i9 : symlinkDirectory(src,dst,Verbose=>true)
│ │ │ ---symlinking: ../../../0/b/c/g -> /tmp/M2-11147-0/1/b/c/g
│ │ │ ---symlinking: ../../0/a/g -> /tmp/M2-11147-0/1/a/g
│ │ │ ---symlinking: ../../0/a/f -> /tmp/M2-11147-0/1/a/f
│ │ │ +--symlinking: ../../0/a/g -> /tmp/M2-12217-0/1/a/g
│ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-12217-0/1/a/f
│ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-12217-0/1/b/c/g
│ │ │  
│ │ │  i10 : get (dst|"b/c/g")
│ │ │  
│ │ │  o10 = ho there
│ │ │  
│ │ │  i11 : symlinkDirectory(src,dst,Verbose=>true,Undo=>true)
│ │ │ ---unsymlinking: ../../../0/b/c/g -> /tmp/M2-11147-0/1/b/c/g
│ │ │ ---unsymlinking: ../../0/a/g -> /tmp/M2-11147-0/1/a/g
│ │ │ ---unsymlinking: ../../0/a/f -> /tmp/M2-11147-0/1/a/f
│ │ │ +--unsymlinking: ../../0/a/g -> /tmp/M2-12217-0/1/a/g
│ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-12217-0/1/a/f
│ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-12217-0/1/b/c/g
│ │ │  
│ │ │  i12 : rm = d -> if isDirectory d then removeDirectory d else removeFile d
│ │ │  
│ │ │  o12 = rm
│ │ │  
│ │ │  o12 : FunctionClosure
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_symlink__File.out
│ │ │ @@ -1,12 +1,12 @@
│ │ │  -- -*- M2-comint -*- hash: 9343844672940306595
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11204-0/0
│ │ │ +o1 = /tmp/M2-12334-0/0
│ │ │  
│ │ │  i2 : symlinkFile("qwert", fn)
│ │ │  
│ │ │  i3 : fileExists fn
│ │ │  
│ │ │  o3 = false
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_temporary__File__Name.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1731926531291302106
│ │ │  
│ │ │  i1 : temporaryFileName () | ".tex"
│ │ │  
│ │ │ -o1 = /tmp/M2-12169-0/0.tex
│ │ │ +o1 = /tmp/M2-14309-0/0.tex
│ │ │  
│ │ │  i2 : temporaryFileName () | ".html"
│ │ │  
│ │ │ -o2 = /tmp/M2-12169-0/1.html
│ │ │ +o2 = /tmp/M2-14309-0/1.html
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_time.out
│ │ │ @@ -1,8 +1,8 @@
│ │ │  -- -*- M2-comint -*- hash: 1332435500723
│ │ │  
│ │ │  i1 : time 3^30
│ │ │ - -- used 2.18e-05s (cpu); 1.1101e-05s (thread); 0s (gc)
│ │ │ + -- used 1.9043e-05s (cpu); 5.659e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = 205891132094649
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_timing.out
│ │ │ @@ -1,14 +1,14 @@
│ │ │  -- -*- M2-comint -*- hash: 1730988300469098603
│ │ │  
│ │ │  i1 : timing 3^30
│ │ │  
│ │ │  o1 = 205891132094649
│ │ │ -     -- .000018144 seconds
│ │ │ +     -- .000019906 seconds
│ │ │  
│ │ │  o1 : Time
│ │ │  
│ │ │  i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{.000018144, 205891132094649}
│ │ │ +o2 = Time{.000019906, 205891132094649}
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_version.out
│ │ │ @@ -34,15 +34,15 @@
│ │ │                 "memtailor version" => 1.1
│ │ │                 "mpfi version" => 1.5.4
│ │ │                 "mpfr version" => 4.2.2
│ │ │                 "mpsolve version" => 3.2.2
│ │ │                 "mysql version" => not present
│ │ │                 "normaliz version" => 3.11.0
│ │ │                 "ntl version" => 11.5.1
│ │ │ -               "operating system release" => 6.12.57+deb13-amd64
│ │ │ +               "operating system release" => 6.12.57+deb13-cloud-amd64
│ │ │                 "operating system" => Linux
│ │ │                 "packages" => Style FirstPackage Macaulay2Doc Parsing Classic Browse Benchmark Text SimpleDoc PackageTemplate Saturation PrimaryDecomposition FourierMotzkin Dmodules WeylAlgebras HolonomicSystems BernsteinSato ConnectionMatrices Depth Elimination GenericInitialIdeal IntegralClosure HyperplaneArrangements LexIdeals Markov NoetherNormalization Points ReesAlgebra Regularity SchurRings SymmetricPolynomials SchurFunctors SimplicialComplexes LLLBases TangentCone ChainComplexExtras Varieties Schubert2 PushForward LocalRings PruneComplex BoijSoederberg BGG Bruns InvolutiveBases ConwayPolynomials EdgeIdeals FourTiTwo StatePolytope Polyhedra Truncations Polymake gfanInterface PieriMaps Normaliz Posets XML OpenMath SCSCP RationalPoints MapleInterface ConvexInterface SRdeformations NumericalAlgebraicGeometry BeginningMacaulay2 FormalGroupLaws Graphics WeylGroups HodgeIntegrals Cyclotomic Binomials Kronecker Nauty ToricVectorBundles ModuleDeformations PHCpack SimplicialDecomposability BooleanGB AdjointIdeal Parametrization Serialization NAGtypes NormalToricVarieties DGAlgebras Graphs GraphicalModels BIBasis KustinMiller Units NautyGraphs VersalDeformations CharacteristicClasses RandomIdeals RandomObjects RandomPlaneCurves RandomSpaceCurves RandomGenus14Curves RandomCanonicalCurves RandomCurves TensorComplexes MonomialAlgebras QthPower EliminationMatrices EllipticIntegrals Triplets CompleteIntersectionResolutions EagonResolution MCMApproximations MultiplierIdeals InvariantRing QuillenSuslin EnumerationCurves Book3264Examples WeilDivisors EllipticCurves HighestWeights MinimalPrimes Bertini TorAlgebra Permanents BinomialEdgeIdeals TateOnProducts LatticePolytopes FiniteFittingIdeals HigherCIOperators LieAlgebraRepresentations ConformalBlocks M0nbar AnalyzeSheafOnP1 MultiplierIdealsDim2 RunExternalM2 NumericalSchubertCalculus ToricTopology Cremona Resultants VectorFields SLPexpressions Miura ResidualIntersections Visualize EquivariantGB ExampleSystems RationalMaps FastMinors RandomPoints SwitchingFields SpectralSequences SectionRing OldPolyhedra OldToricVectorBundles K3Carpets ChainComplexOperations NumericalCertification PhylogeneticTrees MonodromySolver ReactionNetworks PackageCitations NumericSolutions GradedLieAlgebras InverseSystems Pullback EngineTests SVDComplexes RandomComplexes CohomCalg Topcom Triangulations ReflexivePolytopesDB AbstractToricVarieties TestIdeals FrobeniusThresholds NonPrincipalTestIdeals Seminormalization AlgebraicSplines TriangularSets Chordal Tropical SymbolicPowers Complexes OldChainComplexes GroebnerWalk RandomMonomialIdeals Matroids NumericalImplicitization NonminimalComplexes CoincidentRootLoci RelativeCanonicalResolution RandomCurvesOverVerySmallFiniteFields StronglyStableIdeals SLnEquivariantMatrices CorrespondenceScrolls NCAlgebra SpaceCurves ExteriorIdeals ToricInvariants SegreClasses SemidefiniteProgramming SumsOfSquares MultiGradedRationalMap AssociativeAlgebras VirtualResolutions Quasidegrees DiffAlg DeterminantalRepresentations FGLM SpechtModule SchurComplexes SimplicialPosets SlackIdeals PositivityToricBundles SparseResultants DecomposableSparseSystems MixedMultiplicity PencilsOfQuadrics ThreadedGB AdjunctionForSurfaces VectorGraphics GKMVarieties MonomialIntegerPrograms NoetherianOperators Hadamard StatGraphs GraphicalModelsMLE EigenSolver MultiplicitySequence ResolutionsOfStanleyReisnerRings NumericalLinearAlgebra ResLengthThree MonomialOrbits MultiprojectiveVarieties SpecialFanoFourfolds RationalPoints2 SuperLinearAlgebra SubalgebraBases AInfinity LinearTruncations ThinSincereQuivers Python BettiCharacters Jets FunctionFieldDesingularization HomotopyLieAlgebra TSpreadIdeals RealRoots ExteriorModules K3Surfaces GroebnerStrata QuaternaryQuartics CotangentSchubert OnlineLookup MergeTeX Probability Isomorphism CodingTheory WhitneyStratifications JSON ForeignFunctions GeometricDecomposability PseudomonomialPrimaryDecomposition PolyominoIdeals MatchingFields CellularResolutions SagbiGbDetection A1BrouwerDegrees QuadraticIdealExamplesByRoos TerraciniLoci MatrixSchubert RInterface OIGroebnerBases PlaneCurveLinearSeries Valuations SchurVeronese VNumber TropicalToric MultigradedBGG AbstractSimplicialComplexes MultigradedImplicitization Msolve Permutations SCMAlgebras NumericalSemigroups ExteriorExtensions Oscillators IncidenceCorrespondenceCohomology ToricHigherDirectImages Brackets IntegerProgramming GameTheory AllMarkovBases Tableaux CpMackeyFunctors JSONRPC MatrixFactorizations PathSignatures
│ │ │                 "pointer size" => 8
│ │ │                 "python version" => 3.13.11
│ │ │                 "readline version" => 8.3
│ │ │                 "scscp version" => not present
│ │ │                 "tbb version" => 2022.1
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Command.html
│ │ │ @@ -84,15 +84,15 @@
│ │ │              
│ │ │                
    i3 : (c = Command "date";)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : c
│ │ │ -Sun Dec 14 15:26:56 UTC 2025
│ │ │ +Thu Jan  1 11:02:31 UTC 2026
│ │ │  
│ │ │  o4 = 0
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,15 +19,15 @@
│ │ │ │  in a file), then it gets executed with empty argument list.
│ │ │ │  i1 : (f = Command ( () -> 2^30 );)
│ │ │ │  i2 : f
│ │ │ │  
│ │ │ │  o2 = 1073741824
│ │ │ │  i3 : (c = Command "date";)
│ │ │ │  i4 : c
│ │ │ │ -Sun Dec 14 15:26:56 UTC 2025
│ │ │ │ +Thu Jan  1 11:02:31 UTC 2026
│ │ │ │  
│ │ │ │  o4 = 0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_u_n -- run an external command
│ │ │ │      * _A_f_t_e_r_E_v_a_l -- top level method applied after evaluation
│ │ │ │  ********** MMeetthhooddss tthhaatt uussee aa ccoommmmaanndd:: **********
│ │ │ │      * code(Command) -- see _c_o_d_e -- display source code
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Database.html
│ │ │ @@ -52,22 +52,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │  A database file is just like a hash table, except both the keys and values have to be strings.  In this example we create a database file, store a few entries, remove an entry with remove, close the file, and then remove the file.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : filename = temporaryFileName () | ".dbm"
│ │ │  
│ │ │ -o1 = /tmp/M2-11641-0/0.dbm
    
│ │ │ +o1 = /tmp/M2-13231-0/0.dbm
│ │ │              
    
│ │ │                
    i2 : x = openDatabaseOut filename
│ │ │  
│ │ │ -o2 = /tmp/M2-11641-0/0.dbm
│ │ │ +o2 = /tmp/M2-13231-0/0.dbm
│ │ │  
│ │ │  o2 : Database
    
│ │ │              
    
│ │ │                
    i3 : x#"first" = "hi there"
│ │ │ ├── html2text {}
│ │ │ │ @@ -7,18 +7,18 @@
│ │ │ │  ************ DDaattaabbaassee ---- tthhee ccllaassss ooff aallll ddaattaabbaassee ffiilleess ************
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  A database file is just like a hash table, except both the keys and values have
│ │ │ │  to be strings. In this example we create a database file, store a few entries,
│ │ │ │  remove an entry with _r_e_m_o_v_e, close the file, and then remove the file.
│ │ │ │  i1 : filename = temporaryFileName () | ".dbm"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11641-0/0.dbm
│ │ │ │ +o1 = /tmp/M2-13231-0/0.dbm
│ │ │ │  i2 : x = openDatabaseOut filename
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11641-0/0.dbm
│ │ │ │ +o2 = /tmp/M2-13231-0/0.dbm
│ │ │ │  
│ │ │ │  o2 : Database
│ │ │ │  i3 : x#"first" = "hi there"
│ │ │ │  
│ │ │ │  o3 = hi there
│ │ │ │  i4 : x#"first"
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___G__Cstats.html
│ │ │ @@ -53,33 +53,33 @@
│ │ │          
    Description
    
│ │ │          
    Macaulay2 uses the Hans Boehm garbage collector to reclaim unused memory.  The function GCstats provides information about its status, such as the total number of bytes allocated, the current heap size, the number of garbage collections done, the number of threads used in each collection, the total cpu time spent in garbage collection, etc.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : s = GCstats()
│ │ │  
│ │ │ -o1 = HashTable{"bytesAlloc" => 42969169706        }
│ │ │ +o1 = HashTable{"bytesAlloc" => 43062247930        }
│ │ │                 "GC_free_space_divisor" => 3
│ │ │                 "GC_LARGE_ALLOC_WARN_INTERVAL" => 1
│ │ │                 "gcCpuTimeSecs" => 0
│ │ │ -               "heapSize" => 206680064
│ │ │ -               "numGCs" => 795
│ │ │ -               "numGCThreads" => 6
│ │ │ +               "heapSize" => 225296384
│ │ │ +               "numGCs" => 783
│ │ │ +               "numGCThreads" => 16
│ │ │  
│ │ │  o1 : HashTable
    
│ │ │              
    
│ │ │          
    The value returned is a hash table, from which individual bits of information can be easily extracted, as follows.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i2 : s#"heapSize"
│ │ │  
│ │ │ -o2 = 206680064
    
│ │ │ +o2 = 225296384
│ │ │              
    
│ │ │          
    Any entries whose keys are all upper case give the values of environment variables affecting the operation of the garbage collector that have been specified by the user.
    
│ │ │          
    For further information about the individual items in the table, we refer the user to the source code and documentation of the garbage collector.
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,28 +9,28 @@
│ │ │ │  Macaulay2 uses the Hans Boehm _g_a_r_b_a_g_e_ _c_o_l_l_e_c_t_o_r to reclaim unused memory. The
│ │ │ │  function GCstats provides information about its status, such as the total
│ │ │ │  number of bytes allocated, the current heap size, the number of garbage
│ │ │ │  collections done, the number of threads used in each collection, the total cpu
│ │ │ │  time spent in garbage collection, etc.
│ │ │ │  i1 : s = GCstats()
│ │ │ │  
│ │ │ │ -o1 = HashTable{"bytesAlloc" => 42969169706        }
│ │ │ │ +o1 = HashTable{"bytesAlloc" => 43062247930        }
│ │ │ │                 "GC_free_space_divisor" => 3
│ │ │ │                 "GC_LARGE_ALLOC_WARN_INTERVAL" => 1
│ │ │ │                 "gcCpuTimeSecs" => 0
│ │ │ │ -               "heapSize" => 206680064
│ │ │ │ -               "numGCs" => 795
│ │ │ │ -               "numGCThreads" => 6
│ │ │ │ +               "heapSize" => 225296384
│ │ │ │ +               "numGCs" => 783
│ │ │ │ +               "numGCThreads" => 16
│ │ │ │  
│ │ │ │  o1 : HashTable
│ │ │ │  The value returned is a hash table, from which individual bits of information
│ │ │ │  can be easily extracted, as follows.
│ │ │ │  i2 : s#"heapSize"
│ │ │ │  
│ │ │ │ -o2 = 206680064
│ │ │ │ +o2 = 225296384
│ │ │ │  Any entries whose keys are all upper case give the values of environment
│ │ │ │  variables affecting the operation of the garbage collector that have been
│ │ │ │  specified by the user.
│ │ │ │  For further information about the individual items in the table, we refer the
│ │ │ │  user to the source code and documentation of the garbage collector.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _G_C_ _g_a_r_b_a_g_e_ _c_o_l_l_e_c_t_o_r
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Minimal__Generators.html
│ │ │ @@ -128,23 +128,23 @@
│ │ │  
│ │ │  o7 : Ideal of R
    
│ │ │              
    
│ │ │                
    i8 : time J = truncate(8, I, MinimalGenerators => false);
│ │ │ - -- used 0.00906552s (cpu); 0.00905868s (thread); 0s (gc)
│ │ │ + -- used 0.00528296s (cpu); 0.00527776s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of R
    
│ │ │              
    
│ │ │                
    i9 : time K = truncate(8, I, MinimalGenerators => true);
│ │ │ - -- used 0.0794664s (cpu); 0.0794742s (thread); 0s (gc)
│ │ │ + -- used 0.0550328s (cpu); 0.0550419s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of R
    
│ │ │              
    
│ │ │                
    i10 : numgens J
│ │ │ ├── html2text {}
│ │ │ │ @@ -46,19 +46,19 @@
│ │ │ │  o6 = R
│ │ │ │  
│ │ │ │  o6 : PolynomialRing
│ │ │ │  i7 : I = monomialCurveIdeal(R, {1,4,5,9});
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  i8 : time J = truncate(8, I, MinimalGenerators => false);
│ │ │ │ - -- used 0.00906552s (cpu); 0.00905868s (thread); 0s (gc)
│ │ │ │ + -- used 0.00528296s (cpu); 0.00527776s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 : Ideal of R
│ │ │ │  i9 : time K = truncate(8, I, MinimalGenerators => true);
│ │ │ │ - -- used 0.0794664s (cpu); 0.0794742s (thread); 0s (gc)
│ │ │ │ + -- used 0.0550328s (cpu); 0.0550419s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 : Ideal of R
│ │ │ │  i10 : numgens J
│ │ │ │  
│ │ │ │  o10 = 1067
│ │ │ │  i11 : numgens K
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___S__V__D_lp..._cm__Divide__Conquer_eq_gt..._rp.html
│ │ │ @@ -68,21 +68,21 @@
│ │ │  o1 : Matrix RR      <-- RR
│ │ │                53          53
    
│ │ │              
    
│ │ │                
    i2 : time SVD(M);
│ │ │ - -- used 0.0253142s (cpu); 0.0253129s (thread); 0s (gc)
    
│ │ │ + -- used 0.0380489s (cpu); 0.0379843s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i3 : time SVD(M, DivideConquer=>true);
│ │ │ - -- used 0.0274055s (cpu); 0.0274142s (thread); 0s (gc)
    
│ │ │ + -- used 0.0381925s (cpu); 0.0379757s (thread); 0s (gc)
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    
│ │ │            
    Functions with optional argument named DivideConquer:
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,17 +11,17 @@
│ │ │ │  For large matrices, this algorithm is often much faster.
│ │ │ │  i1 : M = random(RR^200, RR^200);
│ │ │ │  
│ │ │ │                  200         200
│ │ │ │  o1 : Matrix RR      <-- RR
│ │ │ │                53          53
│ │ │ │  i2 : time SVD(M);
│ │ │ │ - -- used 0.0253142s (cpu); 0.0253129s (thread); 0s (gc)
│ │ │ │ + -- used 0.0380489s (cpu); 0.0379843s (thread); 0s (gc)
│ │ │ │  i3 : time SVD(M, DivideConquer=>true);
│ │ │ │ - -- used 0.0274055s (cpu); 0.0274142s (thread); 0s (gc)
│ │ │ │ + -- used 0.0381925s (cpu); 0.0379757s (thread); 0s (gc)
│ │ │ │  ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd DDiivviiddeeCCoonnqquueerr:: **********
│ │ │ │      * _S_V_D_(_._._._,_D_i_v_i_d_e_C_o_n_q_u_e_r_=_>_._._._) -- whether to use the LAPACK divide and
│ │ │ │        conquer SVD algorithm
│ │ │ │  ********** FFuurrtthheerr iinnffoorrmmaattiioonn **********
│ │ │ │      * Default value: _t_r_u_e
│ │ │ │      * Function: _S_V_D -- singular value decomposition of a matrix
│ │ │ │      * Option key: _D_i_v_i_d_e_C_o_n_q_u_e_r -- an optional argument
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_a_spfirst_sp__Macaulay2_spsession.html
│ │ │ @@ -826,15 +826,15 @@
│ │ │          
    
│ │ │            
    We may use resolution to produce a projective resolution of it, and time to report the time required.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i59 : time C = resolution M
│ │ │ - -- used 0.00192778s (cpu); 0.00191945s (thread); 0s (gc)
│ │ │ + -- used 0.00195051s (cpu); 0.00194491s (thread); 0s (gc)
│ │ │  
│ │ │         3      6      15      18      6
│ │ │  o59 = R  <-- R  <-- R   <-- R   <-- R  <-- 0
│ │ │                                              
│ │ │        0      1      2       3       4      5
│ │ │  
│ │ │  o59 : ChainComplex
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -390,15 +390,15 @@
│ │ │ │                 | c f i l o r |
│ │ │ │  
│ │ │ │                               3
│ │ │ │  o58 : R-module, quotient of R
│ │ │ │  We may use _r_e_s_o_l_u_t_i_o_n to produce a projective resolution of it, and _t_i_m_e to
│ │ │ │  report the time required.
│ │ │ │  i59 : time C = resolution M
│ │ │ │ - -- used 0.00192778s (cpu); 0.00191945s (thread); 0s (gc)
│ │ │ │ + -- used 0.00195051s (cpu); 0.00194491s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         3      6      15      18      6
│ │ │ │  o59 = R  <-- R  <-- R   <-- R   <-- R  <-- 0
│ │ │ │  
│ │ │ │        0      1      2       3       4      5
│ │ │ │  
│ │ │ │  o59 : ChainComplex
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_at__End__Of__File_lp__File_rp.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  o4 = "hi there"
│ │ │              
    
│ │ │                
    i5 : atEndOfFile f
│ │ │  
│ │ │ -o5 = false
    
│ │ │ +o5 = true
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    
│ │ │            
    Ways to use this method:
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,13 +23,13 @@
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : peek read f
│ │ │ │  
│ │ │ │  o4 = "hi there"
│ │ │ │  i5 : atEndOfFile f
│ │ │ │  
│ │ │ │ -o5 = false
│ │ │ │ +o5 = true
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _a_t_E_n_d_O_f_F_i_l_e_(_F_i_l_e_) -- test for end of file
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_files.m2:374:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_benchmark.html
│ │ │ @@ -68,15 +68,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │  Produces an accurate timing for the code contained in the string s.  The value returned is the number of seconds.        
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : benchmark "sqrt 2p100000"
│ │ │  
│ │ │ -o1 = .000290697861367332
│ │ │ +o1 = .0003478640801445259
│ │ │  
│ │ │  o1 : RR (of precision 53)
    
│ │ │              
    
│ │ │  The snippet of code provided will be run enough times to register meaningfully on the clock, and the garbage collector will be called beforehand.      
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,15 +12,15 @@
│ │ │ │            o a _r_e_a_l_ _n_u_m_b_e_r, the number of seconds it takes to evaluate the code
│ │ │ │              in s
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Produces an accurate timing for the code contained in the string s. The value
│ │ │ │  returned is the number of seconds.
│ │ │ │  i1 : benchmark "sqrt 2p100000"
│ │ │ │  
│ │ │ │ -o1 = .000290697861367332
│ │ │ │ +o1 = .0003478640801445259
│ │ │ │  
│ │ │ │  o1 : RR (of precision 53)
│ │ │ │  The snippet of code provided will be run enough times to register meaningfully
│ │ │ │  on the clock, and the garbage collector will be called beforehand.
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _b_e_n_c_h_m_a_r_k is a _f_u_n_c_t_i_o_n_ _c_l_o_s_u_r_e.
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_caching_spcomputation_spresults.html
│ │ │ @@ -69,23 +69,23 @@
│ │ │                
    i3 : M = coker vars R;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : elapsedTime pdim' M
│ │ │   -- computing pdim'
│ │ │ - -- .00670047s elapsed
│ │ │ + -- .00382538s elapsed
│ │ │  
│ │ │  o4 = 3
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime pdim' M
│ │ │ - -- .000001513s elapsed
│ │ │ + -- .000002683s elapsed
│ │ │  
│ │ │  o5 = 3
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : peek M.cache
│ │ │ ├── html2text {}
│ │ │ │ @@ -8,19 +8,19 @@
│ │ │ │  Here is a simple example of caching a computation in a _C_a_c_h_e_T_a_b_l_e, using the
│ │ │ │  augmented null coalescing operator _?_?_=.
│ │ │ │  i1 : pdim' = M -> M.cache.pdim' ??= ( printerr "computing pdim'"; pdim M );
│ │ │ │  i2 : R = QQ[x,y,z];
│ │ │ │  i3 : M = coker vars R;
│ │ │ │  i4 : elapsedTime pdim' M
│ │ │ │   -- computing pdim'
│ │ │ │ - -- .00670047s elapsed
│ │ │ │ + -- .00382538s elapsed
│ │ │ │  
│ │ │ │  o4 = 3
│ │ │ │  i5 : elapsedTime pdim' M
│ │ │ │ - -- .000001513s elapsed
│ │ │ │ + -- .000002683s elapsed
│ │ │ │  
│ │ │ │  o5 = 3
│ │ │ │  i6 : peek M.cache
│ │ │ │  
│ │ │ │  o6 = CacheTable{cache => MutableHashTable{}
│ │ │ │  }
│ │ │ │                  isHomogeneous => true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cancel__Task_lp__Task_rp.html
│ │ │ @@ -104,15 +104,15 @@
│ │ │  o4 : Task
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : n
│ │ │  
│ │ │ -o5 = 711206
    
│ │ │ +o5 = 1095015
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : sleep 1
│ │ │  
│ │ │  o6 = 0
    
│ │ │ @@ -127,15 +127,15 @@
│ │ │  o7 : Task
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : n
│ │ │  
│ │ │ -o8 = 1453533
    
│ │ │ +o8 = 2220814
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : isReady t
│ │ │  
│ │ │  o9 = false
    
│ │ │ @@ -163,29 +163,29 @@
│ │ │  o12 : Task
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : n
│ │ │  
│ │ │ -o13 = 1453746
    
│ │ │ +o13 = 2221001
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : sleep 1
│ │ │  
│ │ │  o14 = 0
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : n
│ │ │  
│ │ │ -o15 = 1453746
    
│ │ │ +o15 = 2221001
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i16 : isReady t
│ │ │  
│ │ │  o16 = false
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,26 +28,26 @@
│ │ │ │  i4 : t
│ │ │ │  
│ │ │ │  o4 = <>
│ │ │ │  
│ │ │ │  o4 : Task
│ │ │ │  i5 : n
│ │ │ │  
│ │ │ │ -o5 = 711206
│ │ │ │ +o5 = 1095015
│ │ │ │  i6 : sleep 1
│ │ │ │  
│ │ │ │  o6 = 0
│ │ │ │  i7 : t
│ │ │ │  
│ │ │ │  o7 = <>
│ │ │ │  
│ │ │ │  o7 : Task
│ │ │ │  i8 : n
│ │ │ │  
│ │ │ │ -o8 = 1453533
│ │ │ │ +o8 = 2220814
│ │ │ │  i9 : isReady t
│ │ │ │  
│ │ │ │  o9 = false
│ │ │ │  i10 : cancelTask t
│ │ │ │  i11 : sleep 2
│ │ │ │  stdio:2:25:(3):[1]: error: interrupted
│ │ │ │  
│ │ │ │ @@ -55,21 +55,21 @@
│ │ │ │  i12 : t
│ │ │ │  
│ │ │ │  o12 = <>
│ │ │ │  
│ │ │ │  o12 : Task
│ │ │ │  i13 : n
│ │ │ │  
│ │ │ │ -o13 = 1453746
│ │ │ │ +o13 = 2221001
│ │ │ │  i14 : sleep 1
│ │ │ │  
│ │ │ │  o14 = 0
│ │ │ │  i15 : n
│ │ │ │  
│ │ │ │ -o15 = 1453746
│ │ │ │ +o15 = 2221001
│ │ │ │  i16 : isReady t
│ │ │ │  
│ │ │ │  o16 = false
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _c_a_n_c_e_l_T_a_s_k_(_T_a_s_k_) -- stop a task
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_change__Directory.html
│ │ │ @@ -71,36 +71,36 @@
│ │ │            
    Change the current working directory to dir.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10463-0/0
    
│ │ │ +o1 = /tmp/M2-10833-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-10463-0/0
    
│ │ │ +o2 = /tmp/M2-10833-0/0
│ │ │              
    
│ │ │                
    i3 : changeDirectory dir
│ │ │  
│ │ │ -o3 = /tmp/M2-10463-0/0/
    
│ │ │ +o3 = /tmp/M2-10833-0/0/
│ │ │              
    
│ │ │                
    i4 : currentDirectory()
│ │ │  
│ │ │ -o4 = /tmp/M2-10463-0/0/
    
│ │ │ +o4 = /tmp/M2-10833-0/0/
│ │ │              
    
│ │ │          
    
│ │ │            
    If dir is omitted, then the current working directory is changed to the user's home directory.
    
│ │ │          
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,24 +11,24 @@
│ │ │ │            o dir, a _s_t_r_i_n_g,
│ │ │ │      * Outputs:
│ │ │ │            o a _s_t_r_i_n_g, the new working directory;
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Change the current working directory to dir.
│ │ │ │  i1 : dir = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10463-0/0
│ │ │ │ +o1 = /tmp/M2-10833-0/0
│ │ │ │  i2 : makeDirectory dir
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10463-0/0
│ │ │ │ +o2 = /tmp/M2-10833-0/0
│ │ │ │  i3 : changeDirectory dir
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10463-0/0/
│ │ │ │ +o3 = /tmp/M2-10833-0/0/
│ │ │ │  i4 : currentDirectory()
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-10463-0/0/
│ │ │ │ +o4 = /tmp/M2-10833-0/0/
│ │ │ │  If dir is omitted, then the current working directory is changed to the user's
│ │ │ │  home directory.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_u_r_r_e_n_t_D_i_r_e_c_t_o_r_y -- current working directory
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _c_h_a_n_g_e_D_i_r_e_c_t_o_r_y is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_check.html
│ │ │ @@ -95,40 +95,40 @@
│ │ │  o1 : Package
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : check_1 FirstPackage
│ │ │   -- warning: reloading FirstPackage; recreate instances of types from this package
│ │ │ - -- capturing check(1, "FirstPackage")        -- .15147s elapsed
    
│ │ │ + -- capturing check(1, "FirstPackage")        -- .122544s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : check FirstPackage
│ │ │ - -- capturing check(0, "FirstPackage")        -- .150181s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .150965s elapsed
    
│ │ │ + -- capturing check(0, "FirstPackage")        -- .116342s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .118608s elapsed
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    Alternatively, if the package is installed somewhere accessible, one can do the following.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : check_1 "FirstPackage"
│ │ │ - -- capturing check(1, "FirstPackage")        -- .152579s elapsed
    
│ │ │ + -- capturing check(1, "FirstPackage")        -- .118608s elapsed
│ │ │              
    
│ │ │                
    i5 : check "FirstPackage"
│ │ │ - -- capturing check(0, "FirstPackage")        -- .152053s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .151867s elapsed
    
│ │ │ + -- capturing check(0, "FirstPackage")        -- .120072s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .119513s elapsed
│ │ │              
    
│ │ │          
    
│ │ │            
    A TestInput object (or a list of such objects) can also be run directly.
    
│ │ │          
    
│ │ │          
│ │ │ @@ -140,15 +140,15 @@
│ │ │  
│ │ │  o6 : TestInput
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : check oo
│ │ │ - -- capturing check(1, "FirstPackage")        -- .153083s elapsed
    
│ │ │ + -- capturing check(1, "FirstPackage")        -- .119399s elapsed
│ │ │              
    
│ │ │                
    i8 : tests "FirstPackage"
│ │ │  
│ │ │  o8 = {0 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:54:5-56:3]}
│ │ │ @@ -156,16 +156,16 @@
│ │ │  
│ │ │  o8 : NumberedVerticalList
    
│ │ │              
    
│ │ │                
    i9 : check oo
│ │ │ - -- capturing check(0, "FirstPackage")        -- .152703s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .150901s elapsed
    
│ │ │ + -- capturing check(0, "FirstPackage")        -- .118011s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .116374s elapsed
│ │ │              
    
│ │ │          
    
│ │ │            
    If only an integer is passed as an argument, then the test with that index from the last call to tests is run.
    
│ │ │          
    
│ │ │          
│ │ │ @@ -178,15 +178,15 @@
│ │ │  
│ │ │  o10 : NumberedVerticalList
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i11 : check 1
│ │ │ - -- capturing check(1, "FirstPackage")        -- .151346s elapsed
    
│ │ │ + -- capturing check(1, "FirstPackage")        -- .123542s elapsed
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    Caveat
    
│ │ │          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -42,52 +42,52 @@
│ │ │ │  
│ │ │ │  o1 = FirstPackage
│ │ │ │  
│ │ │ │  o1 : Package
│ │ │ │  i2 : check_1 FirstPackage
│ │ │ │   -- warning: reloading FirstPackage; recreate instances of types from this
│ │ │ │  package
│ │ │ │ - -- capturing check(1, "FirstPackage")        -- .15147s elapsed
│ │ │ │ + -- capturing check(1, "FirstPackage")        -- .122544s elapsed
│ │ │ │  i3 : check FirstPackage
│ │ │ │ - -- capturing check(0, "FirstPackage")        -- .150181s elapsed
│ │ │ │ - -- capturing check(1, "FirstPackage")        -- .150965s elapsed
│ │ │ │ + -- capturing check(0, "FirstPackage")        -- .116342s elapsed
│ │ │ │ + -- capturing check(1, "FirstPackage")        -- .118608s elapsed
│ │ │ │  Alternatively, if the package is installed somewhere accessible, one can do the
│ │ │ │  following.
│ │ │ │  i4 : check_1 "FirstPackage"
│ │ │ │ - -- capturing check(1, "FirstPackage")        -- .152579s elapsed
│ │ │ │ + -- capturing check(1, "FirstPackage")        -- .118608s elapsed
│ │ │ │  i5 : check "FirstPackage"
│ │ │ │ - -- capturing check(0, "FirstPackage")        -- .152053s elapsed
│ │ │ │ - -- capturing check(1, "FirstPackage")        -- .151867s elapsed
│ │ │ │ + -- capturing check(0, "FirstPackage")        -- .120072s elapsed
│ │ │ │ + -- capturing check(1, "FirstPackage")        -- .119513s elapsed
│ │ │ │  A _T_e_s_t_I_n_p_u_t object (or a list of such objects) can also be run directly.
│ │ │ │  i6 : tests(1, "FirstPackage")
│ │ │ │  
│ │ │ │  o6 = TestInput[/usr/share/Macaulay2/FirstPackage.m2:58:5-60:3]
│ │ │ │  
│ │ │ │  o6 : TestInput
│ │ │ │  i7 : check oo
│ │ │ │ - -- capturing check(1, "FirstPackage")        -- .153083s elapsed
│ │ │ │ + -- capturing check(1, "FirstPackage")        -- .119399s elapsed
│ │ │ │  i8 : tests "FirstPackage"
│ │ │ │  
│ │ │ │  o8 = {0 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:54:5-56:3]}
│ │ │ │       {1 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:58:5-60:3]}
│ │ │ │  
│ │ │ │  o8 : NumberedVerticalList
│ │ │ │  i9 : check oo
│ │ │ │ - -- capturing check(0, "FirstPackage")        -- .152703s elapsed
│ │ │ │ - -- capturing check(1, "FirstPackage")        -- .150901s elapsed
│ │ │ │ + -- capturing check(0, "FirstPackage")        -- .118011s elapsed
│ │ │ │ + -- capturing check(1, "FirstPackage")        -- .116374s elapsed
│ │ │ │  If only an integer is passed as an argument, then the test with that index from
│ │ │ │  the last call to _t_e_s_t_s is run.
│ │ │ │  i10 : tests "FirstPackage"
│ │ │ │  
│ │ │ │  o10 = {0 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:54:5-56:3]}
│ │ │ │        {1 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:58:5-60:3]}
│ │ │ │  
│ │ │ │  o10 : NumberedVerticalList
│ │ │ │  i11 : check 1
│ │ │ │ - -- capturing check(1, "FirstPackage")        -- .151346s elapsed
│ │ │ │ + -- capturing check(1, "FirstPackage")        -- .123542s elapsed
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  Currently, if the package was only partially loaded because the documentation
│ │ │ │  was obtainable from a database (see _b_e_g_i_n_D_o_c_u_m_e_n_t_a_t_i_o_n), then the package will
│ │ │ │  be reloaded, this time completely, to ensure that all tests are considered;
│ │ │ │  this may affect user objects of types declared by the package, as they may be
│ │ │ │  not usable by the new instance of the package. In a future version, either the
│ │ │ │  tests and the documentation will both be cached, or neither will.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_communicating_spwith_spprograms.html
│ │ │ @@ -50,15 +50,15 @@
│ │ │      
    
│ │ │        
    communicating with programs
    
│ │ │        
    
│ │ │  The most naive way to interact with another program is simply to run it, let it communicate directly with the user, and wait for it to finish.  This is done with the run command.        
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : run "uname -a"
│ │ │ -Linux sbuild 6.12.57+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.57-1 (2025-11-05) x86_64 GNU/Linux
│ │ │ +Linux sbuild 6.12.57+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.57-1 (2025-11-05) x86_64 GNU/Linux
│ │ │  
│ │ │  o1 = 0
    
│ │ │              
    
│ │ │  To run a program and provide it with input, one way is use the operator <<, with a file name whose first character is an exclamation point; the rest of the file name will be taken as the command to run, as in the following example.        
│ │ │            
│ │ │ @@ -74,15 +74,15 @@
│ │ │            
│ │ │          
    
│ │ │  More often, one wants to write Macaulay2 code to obtain and manipulate the output from the other program.  If the program requires no input data, then we can use get with a file name whose first character is an exclamation point.  In the following example, we also peek at the string to see whether it includes a newline character.        
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i3 : peek get "!uname -a"
│ │ │  
│ │ │ -o3 = "Linux sbuild 6.12.57+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ +o3 = "Linux sbuild 6.12.57+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │       6.12.57-1 (2025-11-05) x86_64 GNU/Linux\n"
    
│ │ │              
    
│ │ │  Bidirectional communication with a program is also possible.  We use openInOut to create a file that serves as a bidirectional connection to a program.  That file is called an input output file.  In this example we open a connection to the unix utility grep and use it to locate the symbol names in Macaulay2 that begin with in.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,15 +33,15 @@
│ │ │ │  take(N,-2));
│ │ │ │  
│ │ │ │  o4 : ProjectiveVariety, curve in PP^8
│ │ │ │  i5 : ? X
│ │ │ │  
│ │ │ │  o5 = curve in PP^8 cut out by 17 hypersurfaces of degrees 1^2 2^15
│ │ │ │  i6 : time f = X ===> Y;
│ │ │ │ - -- used 3.0434s (cpu); 1.80018s (thread); 0s (gc)
│ │ │ │ + -- used 4.13499s (cpu); 2.08126s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 : MultirationalMap (automorphism of PP^8)
│ │ │ │  i7 : f X
│ │ │ │  
│ │ │ │  o7 = Y
│ │ │ │  
│ │ │ │  o7 : ProjectiveVariety, curve in PP^8
│ │ │ │ @@ -53,15 +53,15 @@
│ │ │ │  i9 : V = random({{2},{1}},X);
│ │ │ │  
│ │ │ │  o9 : ProjectiveVariety, 6-dimensional subvariety of PP^8
│ │ │ │  i10 : W = random({{2},{1}},Y);
│ │ │ │  
│ │ │ │  o10 : ProjectiveVariety, 6-dimensional subvariety of PP^8
│ │ │ │  i11 : time g = V ===> W;
│ │ │ │ - -- used 3.14065s (cpu); 1.91214s (thread); 0s (gc)
│ │ │ │ + -- used 3.69451s (cpu); 2.10155s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 : MultirationalMap (automorphism of PP^8)
│ │ │ │  i12 : g||W
│ │ │ │  
│ │ │ │  o12 = multi-rational map consisting of one single rational map
│ │ │ │        source variety: 6-dimensional subvariety of PP^8 cut out by 2
│ │ │ │  hypersurfaces of degrees 1^1 2^1
│ │ │ │ @@ -144,15 +144,15 @@
│ │ │ │  i15 : Z = projectiveVariety pfaffians(4,A);
│ │ │ │  
│ │ │ │  o15 : ProjectiveVariety, 6-dimensional subvariety of PP^9
│ │ │ │  i16 : ? Z
│ │ │ │  
│ │ │ │  o16 = 6-dimensional subvariety of PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │ │  i17 : time h = Z ===> GG_K(1,4)
│ │ │ │ - -- used 8.07742s (cpu); 4.95873s (thread); 0s (gc)
│ │ │ │ + -- used 6.52083s (cpu); 4.57396s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 = h
│ │ │ │  
│ │ │ │  o17 : MultirationalMap (isomorphism from PP^9 to PP^9)
│ │ │ │  i18 : h || GG_K(1,4)
│ │ │ │  
│ │ │ │  o18 = multi-rational map consisting of one single rational map
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp^_st_st_sp__Multiprojective__Variety.html
│ │ │ @@ -89,15 +89,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, 4-dimensional subvariety of PP^2 x PP^4
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,24 +25,24 @@
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^7 to PP^7 x PP^7)
│ │ │ │  i4 : Z = source Phi;
│ │ │ │  
│ │ │ │  o4 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^7
│ │ │ │  i5 : time Phi Z;
│ │ │ │ - -- used 0.0951343s (cpu); 0.0963862s (thread); 0s (gc)
│ │ │ │ + -- used 0.130263s (cpu); 0.111376s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^7
│ │ │ │  i6 : dim oo, degree oo, degrees oo
│ │ │ │  
│ │ │ │  o6 = (4, 80, {({0, 2}, 5), ({1, 1}, 33), ({2, 0}, 5)})
│ │ │ │  
│ │ │ │  o6 : Sequence
│ │ │ │  i7 : time Phi (point Z + point Z + point Z)
│ │ │ │ - -- used 2.05743s (cpu); 1.42394s (thread); 0s (gc)
│ │ │ │ + -- used 2.51637s (cpu); 1.40924s (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 6.12.57+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.57-1 (2025-
│ │ │ │ -11-05) x86_64 GNU/Linux
│ │ │ │ +Linux sbuild 6.12.57+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.57-1
│ │ │ │ +(2025-11-05) x86_64 GNU/Linux
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  To run a program and provide it with input, one way is use the operator _<_<,
│ │ │ │  with a file name whose first character is an exclamation point; the rest of the
│ │ │ │  file name will be taken as the command to run, as in the following example.
│ │ │ │  i2 : "!grep a" << " ba \n bc \n ad \n ef \n" << close
│ │ │ │   ba
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │  More often, one wants to write Macaulay2 code to obtain and manipulate the
│ │ │ │  output from the other program. If the program requires no input data, then we
│ │ │ │  can use _g_e_t with a file name whose first character is an exclamation point. In
│ │ │ │  the following example, we also peek at the string to see whether it includes a
│ │ │ │  newline character.
│ │ │ │  i3 : peek get "!uname -a"
│ │ │ │  
│ │ │ │ -o3 = "Linux sbuild 6.12.57+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ │ +o3 = "Linux sbuild 6.12.57+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ │       6.12.57-1 (2025-11-05) x86_64 GNU/Linux\n"
│ │ │ │  Bidirectional communication with a program is also possible. We use _o_p_e_n_I_n_O_u_t
│ │ │ │  to create a file that serves as a bidirectional connection to a program. That
│ │ │ │  file is called an input output file. In this example we open a connection to
│ │ │ │  the unix utility grep and use it to locate the symbol names in Macaulay2 that
│ │ │ │  begin with in.
│ │ │ │  i4 : f = openInOut "!grep -E '^in'"
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_computing_sp__Groebner_spbases.html
│ │ │ @@ -269,15 +269,15 @@
│ │ │                 1277
│ │ │              
    
│ │ │                
    i24 : gb I
│ │ │  
│ │ │ -   -- registering gb 5 at 0x7f67b957e540
│ │ │ +   -- registering gb 5 at 0x7fb15fc15540
│ │ │  
│ │ │     -- [gb]{2}(2)mm{3}(1)m{4}(2)om{5}(1)onumber of (nonminimal) gb elements = 4
│ │ │     -- number of monomials                = 8
│ │ │     -- #reduction steps = 2
│ │ │     -- #spairs done = 6
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ │ @@ -373,15 +373,15 @@
│ │ │                1      4
│ │ │  o32 : Matrix R  <-- R
    
│ │ │              
    
│ │ │                
    i33 : time betti gb f
│ │ │ - -- used 0.303879s (cpu); 0.305901s (thread); 0s (gc)
│ │ │ + -- used 0.219908s (cpu); 0.21961s (thread); 0s (gc)
│ │ │  
│ │ │               0  1
│ │ │  o33 = total: 1 53
│ │ │            0: 1  .
│ │ │            1: .  .
│ │ │            2: .  2
│ │ │            3: .  1
│ │ │ @@ -417,15 +417,15 @@
│ │ │  
│ │ │  o35 : ZZ[T]
    
│ │ │              
    
│ │ │                
    i36 : time betti gb f
│ │ │ - -- used 0.00799525s (cpu); 0.00545052s (thread); 0s (gc)
│ │ │ + -- used 0.00304346s (cpu); 0.00303251s (thread); 0s (gc)
│ │ │  
│ │ │               0  1
│ │ │  o36 = total: 1 53
│ │ │            0: 1  .
│ │ │            1: .  .
│ │ │            2: .  2
│ │ │            3: .  1
│ │ │ ├── html2text {}
│ │ │ │ @@ -140,15 +140,15 @@
│ │ │ │  o23 = ideal (x*y - z , y  - w )
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o23 : Ideal of ----[x..z, w]
│ │ │ │                 1277
│ │ │ │  i24 : gb I
│ │ │ │  
│ │ │ │ -   -- registering gb 5 at 0x7f67b957e540
│ │ │ │ +   -- registering gb 5 at 0x7fb15fc15540
│ │ │ │  
│ │ │ │     -- [gb]{2}(2)mm{3}(1)m{4}(2)om{5}(1)onumber of (nonminimal) gb elements = 4
│ │ │ │     -- number of monomials                = 8
│ │ │ │     -- #reduction steps = 2
│ │ │ │     -- #spairs done = 6
│ │ │ │     -- ncalls = 0
│ │ │ │     -- nloop = 0
│ │ │ │ @@ -213,15 +213,15 @@
│ │ │ │  
│ │ │ │  o31 : ZZ[T]
│ │ │ │  i32 : f = random(R^1,R^{-3,-3,-5,-6});
│ │ │ │  
│ │ │ │                1      4
│ │ │ │  o32 : Matrix R  <-- R
│ │ │ │  i33 : time betti gb f
│ │ │ │ - -- used 0.303879s (cpu); 0.305901s (thread); 0s (gc)
│ │ │ │ + -- used 0.219908s (cpu); 0.21961s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o33 = total: 1 53
│ │ │ │            0: 1  .
│ │ │ │            1: .  .
│ │ │ │            2: .  2
│ │ │ │            3: .  1
│ │ │ │ @@ -245,15 +245,15 @@
│ │ │ │  i35 : poincare cokernel f = (1-T^3)*(1-T^3)*(1-T^5)*(1-T^6) -- cache poincare
│ │ │ │  
│ │ │ │              3    5     8     9    12     14    17
│ │ │ │  o35 = 1 - 2T  - T  + 2T  + 2T  - T   - 2T   + T
│ │ │ │  
│ │ │ │  o35 : ZZ[T]
│ │ │ │  i36 : time betti gb f
│ │ │ │ - -- used 0.00799525s (cpu); 0.00545052s (thread); 0s (gc)
│ │ │ │ + -- used 0.00304346s (cpu); 0.00303251s (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/M2-11185-0/0/
    
│ │ │ +o1 = /tmp/M2-12295-0/0/
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-11185-0/1/
    
│ │ │ +o2 = /tmp/M2-12295-0/1/
│ │ │              
    
│ │ │                
    i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-11185-0/0/a/
    
│ │ │ +o3 = /tmp/M2-12295-0/0/a/
│ │ │              
    
│ │ │                
    i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-11185-0/0/b/
    
│ │ │ +o4 = /tmp/M2-12295-0/0/b/
│ │ │              
    
│ │ │                
    i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-11185-0/0/b/c/
    
│ │ │ +o5 = /tmp/M2-12295-0/0/b/c/
│ │ │              
    
│ │ │                
    i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11185-0/0/a/f
│ │ │ +o6 = /tmp/M2-12295-0/0/a/f
│ │ │  
│ │ │  o6 : File
    
│ │ │              
    
│ │ │                
    i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11185-0/0/a/g
│ │ │ +o7 = /tmp/M2-12295-0/0/a/g
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │                
    i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-11185-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-12295-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
    
│ │ │              
    
│ │ │                
    i9 : stack findFiles src
│ │ │  
│ │ │ -o9 = /tmp/M2-11185-0/0/
│ │ │ -     /tmp/M2-11185-0/0/b/
│ │ │ -     /tmp/M2-11185-0/0/b/c/
│ │ │ -     /tmp/M2-11185-0/0/b/c/g
│ │ │ -     /tmp/M2-11185-0/0/a/
│ │ │ -     /tmp/M2-11185-0/0/a/g
│ │ │ -     /tmp/M2-11185-0/0/a/f
    
│ │ │ +o9 = /tmp/M2-12295-0/0/
│ │ │ +     /tmp/M2-12295-0/0/a/
│ │ │ +     /tmp/M2-12295-0/0/a/g
│ │ │ +     /tmp/M2-12295-0/0/a/f
│ │ │ +     /tmp/M2-12295-0/0/b/
│ │ │ +     /tmp/M2-12295-0/0/b/c/
│ │ │ +     /tmp/M2-12295-0/0/b/c/g
│ │ │              
    
│ │ │                
    i10 : copyDirectory(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-11185-0/0/b/c/g -> /tmp/M2-11185-0/1/b/c/g
│ │ │ - -- copying: /tmp/M2-11185-0/0/a/g -> /tmp/M2-11185-0/1/a/g
│ │ │ - -- copying: /tmp/M2-11185-0/0/a/f -> /tmp/M2-11185-0/1/a/f
    
│ │ │ + -- copying: /tmp/M2-12295-0/0/a/g -> /tmp/M2-12295-0/1/a/g
│ │ │ + -- copying: /tmp/M2-12295-0/0/a/f -> /tmp/M2-12295-0/1/a/f
│ │ │ + -- copying: /tmp/M2-12295-0/0/b/c/g -> /tmp/M2-12295-0/1/b/c/g
│ │ │              
    
│ │ │                
    i11 : copyDirectory(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11185-0/0/b/c/g not newer than /tmp/M2-11185-0/1/b/c/g
│ │ │ - -- skipping: /tmp/M2-11185-0/0/a/g not newer than /tmp/M2-11185-0/1/a/g
│ │ │ - -- skipping: /tmp/M2-11185-0/0/a/f not newer than /tmp/M2-11185-0/1/a/f
    
│ │ │ + -- skipping: /tmp/M2-12295-0/0/a/g not newer than /tmp/M2-12295-0/1/a/g
│ │ │ + -- skipping: /tmp/M2-12295-0/0/a/f not newer than /tmp/M2-12295-0/1/a/f
│ │ │ + -- skipping: /tmp/M2-12295-0/0/b/c/g not newer than /tmp/M2-12295-0/1/b/c/g
│ │ │              
    
│ │ │                
    i12 : stack findFiles dst
│ │ │  
│ │ │ -o12 = /tmp/M2-11185-0/1/
│ │ │ -      /tmp/M2-11185-0/1/a/
│ │ │ -      /tmp/M2-11185-0/1/a/f
│ │ │ -      /tmp/M2-11185-0/1/a/g
│ │ │ -      /tmp/M2-11185-0/1/b/
│ │ │ -      /tmp/M2-11185-0/1/b/c/
│ │ │ -      /tmp/M2-11185-0/1/b/c/g
    
│ │ │ +o12 = /tmp/M2-12295-0/1/
│ │ │ +      /tmp/M2-12295-0/1/a/
│ │ │ +      /tmp/M2-12295-0/1/a/g
│ │ │ +      /tmp/M2-12295-0/1/a/f
│ │ │ +      /tmp/M2-12295-0/1/b/
│ │ │ +      /tmp/M2-12295-0/1/b/c/
│ │ │ +      /tmp/M2-12295-0/1/b/c/g
│ │ │              
    
│ │ │                
    i13 : get (dst|"b/c/g")
│ │ │  
│ │ │  o13 = ho there
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,68 +25,68 @@
│ │ │ │              individual file operations
│ │ │ │      * Consequences:
│ │ │ │            o a copy of the directory tree rooted at src is created, rooted at
│ │ │ │              dst
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : src = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11185-0/0/
│ │ │ │ +o1 = /tmp/M2-12295-0/0/
│ │ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11185-0/1/
│ │ │ │ +o2 = /tmp/M2-12295-0/1/
│ │ │ │  i3 : makeDirectory (src|"a/")
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11185-0/0/a/
│ │ │ │ +o3 = /tmp/M2-12295-0/0/a/
│ │ │ │  i4 : makeDirectory (src|"b/")
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11185-0/0/b/
│ │ │ │ +o4 = /tmp/M2-12295-0/0/b/
│ │ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-11185-0/0/b/c/
│ │ │ │ +o5 = /tmp/M2-12295-0/0/b/c/
│ │ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11185-0/0/a/f
│ │ │ │ +o6 = /tmp/M2-12295-0/0/a/f
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11185-0/0/a/g
│ │ │ │ +o7 = /tmp/M2-12295-0/0/a/g
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │ │  
│ │ │ │ -o8 = /tmp/M2-11185-0/0/b/c/g
│ │ │ │ +o8 = /tmp/M2-12295-0/0/b/c/g
│ │ │ │  
│ │ │ │  o8 : File
│ │ │ │  i9 : stack findFiles src
│ │ │ │  
│ │ │ │ -o9 = /tmp/M2-11185-0/0/
│ │ │ │ -     /tmp/M2-11185-0/0/b/
│ │ │ │ -     /tmp/M2-11185-0/0/b/c/
│ │ │ │ -     /tmp/M2-11185-0/0/b/c/g
│ │ │ │ -     /tmp/M2-11185-0/0/a/
│ │ │ │ -     /tmp/M2-11185-0/0/a/g
│ │ │ │ -     /tmp/M2-11185-0/0/a/f
│ │ │ │ +o9 = /tmp/M2-12295-0/0/
│ │ │ │ +     /tmp/M2-12295-0/0/a/
│ │ │ │ +     /tmp/M2-12295-0/0/a/g
│ │ │ │ +     /tmp/M2-12295-0/0/a/f
│ │ │ │ +     /tmp/M2-12295-0/0/b/
│ │ │ │ +     /tmp/M2-12295-0/0/b/c/
│ │ │ │ +     /tmp/M2-12295-0/0/b/c/g
│ │ │ │  i10 : copyDirectory(src,dst,Verbose=>true)
│ │ │ │ - -- copying: /tmp/M2-11185-0/0/b/c/g -> /tmp/M2-11185-0/1/b/c/g
│ │ │ │ - -- copying: /tmp/M2-11185-0/0/a/g -> /tmp/M2-11185-0/1/a/g
│ │ │ │ - -- copying: /tmp/M2-11185-0/0/a/f -> /tmp/M2-11185-0/1/a/f
│ │ │ │ + -- copying: /tmp/M2-12295-0/0/a/g -> /tmp/M2-12295-0/1/a/g
│ │ │ │ + -- copying: /tmp/M2-12295-0/0/a/f -> /tmp/M2-12295-0/1/a/f
│ │ │ │ + -- copying: /tmp/M2-12295-0/0/b/c/g -> /tmp/M2-12295-0/1/b/c/g
│ │ │ │  i11 : copyDirectory(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ │ - -- skipping: /tmp/M2-11185-0/0/b/c/g not newer than /tmp/M2-11185-0/1/b/c/g
│ │ │ │ - -- skipping: /tmp/M2-11185-0/0/a/g not newer than /tmp/M2-11185-0/1/a/g
│ │ │ │ - -- skipping: /tmp/M2-11185-0/0/a/f not newer than /tmp/M2-11185-0/1/a/f
│ │ │ │ + -- skipping: /tmp/M2-12295-0/0/a/g not newer than /tmp/M2-12295-0/1/a/g
│ │ │ │ + -- skipping: /tmp/M2-12295-0/0/a/f not newer than /tmp/M2-12295-0/1/a/f
│ │ │ │ + -- skipping: /tmp/M2-12295-0/0/b/c/g not newer than /tmp/M2-12295-0/1/b/c/g
│ │ │ │  i12 : stack findFiles dst
│ │ │ │  
│ │ │ │ -o12 = /tmp/M2-11185-0/1/
│ │ │ │ -      /tmp/M2-11185-0/1/a/
│ │ │ │ -      /tmp/M2-11185-0/1/a/f
│ │ │ │ -      /tmp/M2-11185-0/1/a/g
│ │ │ │ -      /tmp/M2-11185-0/1/b/
│ │ │ │ -      /tmp/M2-11185-0/1/b/c/
│ │ │ │ -      /tmp/M2-11185-0/1/b/c/g
│ │ │ │ +o12 = /tmp/M2-12295-0/1/
│ │ │ │ +      /tmp/M2-12295-0/1/a/
│ │ │ │ +      /tmp/M2-12295-0/1/a/g
│ │ │ │ +      /tmp/M2-12295-0/1/a/f
│ │ │ │ +      /tmp/M2-12295-0/1/b/
│ │ │ │ +      /tmp/M2-12295-0/1/b/c/
│ │ │ │ +      /tmp/M2-12295-0/1/b/c/g
│ │ │ │  i13 : get (dst|"b/c/g")
│ │ │ │  
│ │ │ │  o13 = ho there
│ │ │ │  Now we remove the files and directories we created.
│ │ │ │  i14 : rm = d -> if isDirectory d then removeDirectory d else removeFile d
│ │ │ │  
│ │ │ │  o14 = rm
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_copy__File_lp__String_cm__String_rp.html
│ │ │ @@ -78,65 +78,65 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -31,16 +31,16 @@
│ │ │ │  o1 : PolynomialRing
│ │ │ │  i2 : I = ideal(x, y, z)
│ │ │ │  
│ │ │ │  o2 = ideal (x, y, z)
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : msolveGB(I, Verbosity => 2, Threads => 6)
│ │ │ │ - -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-53632-0/0-in.ms -o /tmp/
│ │ │ │ -M2-53632-0/0-out.ms
│ │ │ │ + -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-84650-0/0-in.ms -o /tmp/
│ │ │ │ +M2-84650-0/0-out.ms
│ │ │ │  
│ │ │ │  --------------- INPUT DATA ---------------
│ │ │ │  #variables                       3
│ │ │ │  #equations                       3
│ │ │ │  #invalid equations               0
│ │ │ │  field characteristic             0
│ │ │ │  homogeneous input?               1
│ │ │ │ @@ -51,15 +51,15 @@
│ │ │ │  initial hash table size     131072 (2^17)
│ │ │ │  max pair selection             ALL
│ │ │ │  reduce gb                        1
│ │ │ │  #threads                         6
│ │ │ │  info level                       2
│ │ │ │  generate pbm files               0
│ │ │ │  ------------------------------------------
│ │ │ │ -Initial prime = 1196244169
│ │ │ │ +Initial prime = 1266683059
│ │ │ │  
│ │ │ │  Legend for f4 information
│ │ │ │  --------------------------------------------------------
│ │ │ │  deg       current degree of pairs selected in this round
│ │ │ │  sel       number of pairs selected in this round
│ │ │ │  pairs     total number of pairs in pair list
│ │ │ │  mat       matrix dimensions (# rows x # columns)
│ │ │ │ @@ -73,26 +73,26 @@
│ │ │ │  deg     sel   pairs        mat          density            new data
│ │ │ │  time(rd) in sec (real|cpu)
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------------
│ │ │ │  reduce final basis        3 x 3          33.33%        3 new       0 zero
│ │ │ │ -0.02 | 0.07
│ │ │ │ +0.00 | 0.00
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------------
│ │ │ │  
│ │ │ │  ---------------- TIMINGS ----------------
│ │ │ │ -overall(elapsed)        0.06 sec
│ │ │ │ -overall(cpu)            0.17 sec
│ │ │ │ +overall(elapsed)        0.00 sec
│ │ │ │ +overall(cpu)            0.00 sec
│ │ │ │  select                  0.00 sec   0.0%
│ │ │ │ -symbolic prep.          0.00 sec   0.0%
│ │ │ │ -update                  0.03 sec  57.3%
│ │ │ │ -convert                 0.02 sec  42.4%
│ │ │ │ -linear algebra          0.00 sec   0.0%
│ │ │ │ +symbolic prep.          0.00 sec   0.3%
│ │ │ │ +update                  0.00 sec  73.3%
│ │ │ │ +convert                 0.00 sec   7.6%
│ │ │ │ +linear algebra          0.00 sec   1.0%
│ │ │ │  reduce gb               0.00 sec   0.0%
│ │ │ │  -----------------------------------------
│ │ │ │  
│ │ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │ │  size of basis                     3
│ │ │ │  #terms in basis                   3
│ │ │ │  #pairs reduced                    0
│ │ │ │ @@ -106,18 +106,18 @@
│ │ │ │  -----------------------------------------
│ │ │ │  
│ │ │ │  
│ │ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │ │  [3]
│ │ │ │  #polynomials to lift              3
│ │ │ │  -----------------------------------------
│ │ │ │ -New prime = 1107170621
│ │ │ │ +New prime = 1184773069
│ │ │ │  
│ │ │ │  ---------------- TIMINGS ----------------
│ │ │ │ -multi-mod overall(elapsed)      0.02 sec
│ │ │ │ +multi-mod overall(elapsed)      0.00 sec
│ │ │ │  learning phase                  0.00 Gops/sec
│ │ │ │  application phase               0.00 Gops/sec
│ │ │ │  -----------------------------------------
│ │ │ │  
│ │ │ │  multi-modular steps
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------------
│ │ │ │ @@ -136,15 +136,15 @@
│ │ │ │  CRT     (elapsed)               0.00 sec
│ │ │ │  ratrecon(elapsed)               0.00 sec
│ │ │ │  -----------------------------------------
│ │ │ │  
│ │ │ │  
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----
│ │ │ │ -msolve overall time           0.19 sec (elapsed) /  0.49 sec (cpu)
│ │ │ │ +msolve overall time           0.01 sec (elapsed) /  0.04 sec (cpu)
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----
│ │ │ │  
│ │ │ │  o3 = | z y x |
│ │ │ │  
│ │ │ │               1      3
│ │ │ │  o3 : Matrix R  <-- R
│ │ ├── ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_components__Of__Kernel.out
│ │ │ @@ -23,19 +23,19 @@
│ │ │  o4 : RingMap S <-- R
│ │ │  
│ │ │  i5 : peek componentsOfKernel(2, F)
│ │ │  warning: computation begun over finite field. resulting polynomials may not lie in the ideal
│ │ │  computing total degree: 1
│ │ │  number of monomials = 6
│ │ │  number of distinct multidegrees = 6
│ │ │ - -- .00186857s elapsed
│ │ │ + -- .00246071s elapsed
│ │ │  computing total degree: 2
│ │ │  number of monomials = 21
│ │ │  number of distinct multidegrees = 18
│ │ │ - -- .00833047s elapsed
│ │ │ + -- .0106961s elapsed
│ │ │  
│ │ │  o5 = MutableHashTable{{0, 1, 0, 0, 1} => {}                   }
│ │ │                        {0, 1, 0, 1, 0} => {}
│ │ │                        {0, 1, 1, 0, 0} => {}
│ │ │                        {0, 2, 0, 0, 2} => {}
│ │ │                        {0, 2, 0, 1, 1} => {}
│ │ │                        {0, 2, 0, 2, 0} => {}
│ │ ├── ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_components__Of__Kernel.html
│ │ │ @@ -117,19 +117,19 @@
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,23 +20,23 @@
│ │ │ │  o1 : PolynomialRing
│ │ │ │  i2 : I = ideal"xy,yz,zx"
│ │ │ │  
│ │ │ │  o2 = ideal (x*y, y*z, x*z)
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : elapsedTime jMult I
│ │ │ │ - -- .0234547s elapsed
│ │ │ │ + -- .027638s elapsed
│ │ │ │  
│ │ │ │  o3 = 2
│ │ │ │  i4 : elapsedTime monjMult I
│ │ │ │ - -- .107713s elapsed
│ │ │ │ + -- .0842469s elapsed
│ │ │ │  
│ │ │ │  o4 = 2
│ │ │ │  i5 : elapsedTime multiplicitySequence I
│ │ │ │ - -- .181349s elapsed
│ │ │ │ + -- .140034s 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/M2-10970-0/0
    
│ │ │ +o1 = /tmp/M2-11860-0/0
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-10970-0/1
    
│ │ │ +o2 = /tmp/M2-11860-0/1
│ │ │              
    
│ │ │                
    i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10970-0/0
│ │ │ +o3 = /tmp/M2-11860-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : copyFile(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-10970-0/0 -> /tmp/M2-10970-0/1
    
│ │ │ + -- copying: /tmp/M2-11860-0/0 -> /tmp/M2-11860-0/1
│ │ │              
    
│ │ │                
    i5 : get dst
│ │ │  
│ │ │  o5 = hi there
    
│ │ │              
    
│ │ │                
    i6 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-10970-0/0 not newer than /tmp/M2-10970-0/1
    
│ │ │ + -- skipping: /tmp/M2-11860-0/0 not newer than /tmp/M2-11860-0/1
│ │ │              
    
│ │ │                
    i7 : src << "ho there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-10970-0/0
│ │ │ +o7 = /tmp/M2-11860-0/0
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │                
    i8 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-10970-0/0 not newer than /tmp/M2-10970-0/1
    
│ │ │ + -- skipping: /tmp/M2-11860-0/0 not newer than /tmp/M2-11860-0/1
│ │ │              
    
│ │ │                
    i9 : get dst
│ │ │  
│ │ │  o9 = hi there
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,37 +18,37 @@
│ │ │ │            o Verbose => a _B_o_o_l_e_a_n_ _v_a_l_u_e, default value false, whether to report
│ │ │ │              individual file operations
│ │ │ │      * Consequences:
│ │ │ │            o the file may be copied
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : src = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10970-0/0
│ │ │ │ +o1 = /tmp/M2-11860-0/0
│ │ │ │  i2 : dst = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10970-0/1
│ │ │ │ +o2 = /tmp/M2-11860-0/1
│ │ │ │  i3 : src << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10970-0/0
│ │ │ │ +o3 = /tmp/M2-11860-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : copyFile(src,dst,Verbose=>true)
│ │ │ │ - -- copying: /tmp/M2-10970-0/0 -> /tmp/M2-10970-0/1
│ │ │ │ + -- copying: /tmp/M2-11860-0/0 -> /tmp/M2-11860-0/1
│ │ │ │  i5 : get dst
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ │ - -- skipping: /tmp/M2-10970-0/0 not newer than /tmp/M2-10970-0/1
│ │ │ │ + -- skipping: /tmp/M2-11860-0/0 not newer than /tmp/M2-11860-0/1
│ │ │ │  i7 : src << "ho there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-10970-0/0
│ │ │ │ +o7 = /tmp/M2-11860-0/0
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ │ - -- skipping: /tmp/M2-10970-0/0 not newer than /tmp/M2-10970-0/1
│ │ │ │ + -- skipping: /tmp/M2-11860-0/0 not newer than /tmp/M2-11860-0/1
│ │ │ │  i9 : get dst
│ │ │ │  
│ │ │ │  o9 = hi there
│ │ │ │  i10 : removeFile src
│ │ │ │  i11 : removeFile dst
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_o_p_y_D_i_r_e_c_t_o_r_y
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cpu__Time.html
│ │ │ @@ -64,38 +64,38 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : t1 = cpuTime()
│ │ │  
│ │ │ -o1 = 354.029649282
│ │ │ +o1 = 319.890774352
│ │ │  
│ │ │  o1 : RR (of precision 53)
    
│ │ │              
    
│ │ │                
    i2 : for i from 0 to 1000000 do 223131321321*324234324324;
    
│ │ │              
    
│ │ │                
    i3 : t2 = cpuTime()
│ │ │  
│ │ │ -o3 = 355.996910491
│ │ │ +o3 = 320.686724796
│ │ │  
│ │ │  o3 : RR (of precision 53)
    
│ │ │              
    
│ │ │                
    i4 : t2-t1
│ │ │  
│ │ │ -o4 = 1.967261209000014
│ │ │ +o4 = .7959504440000273
│ │ │  
│ │ │  o4 : RR (of precision 53)
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,26 +9,26 @@
│ │ │ │              cpuTime()
│ │ │ │      * Outputs:
│ │ │ │            o a _r_e_a_l_ _n_u_m_b_e_r, the number of seconds of cpu time used since the
│ │ │ │              program was started
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : t1 = cpuTime()
│ │ │ │  
│ │ │ │ -o1 = 354.029649282
│ │ │ │ +o1 = 319.890774352
│ │ │ │  
│ │ │ │  o1 : RR (of precision 53)
│ │ │ │  i2 : for i from 0 to 1000000 do 223131321321*324234324324;
│ │ │ │  i3 : t2 = cpuTime()
│ │ │ │  
│ │ │ │ -o3 = 355.996910491
│ │ │ │ +o3 = 320.686724796
│ │ │ │  
│ │ │ │  o3 : RR (of precision 53)
│ │ │ │  i4 : t2-t1
│ │ │ │  
│ │ │ │ -o4 = 1.967261209000014
│ │ │ │ +o4 = .7959504440000273
│ │ │ │  
│ │ │ │  o4 : RR (of precision 53)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_i_m_e -- time a computation
│ │ │ │      * _t_i_m_i_n_g -- time a computation
│ │ │ │      * _c_u_r_r_e_n_t_T_i_m_e -- get the current time
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_current__Time.html
│ │ │ @@ -64,48 +64,48 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : currentTime()
│ │ │  
│ │ │ -o1 = 1765726091
    
│ │ │ +o1 = 1767265403
│ │ │              
    
│ │ │          
    We can compute, roughly, how many years ago the epoch began as follows.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 )
│ │ │  
│ │ │ -o2 = 55.95363237235333
│ │ │ +o2 = 56.00241122780173
│ │ │  
│ │ │  o2 : RR (of precision 53)
    
│ │ │              
    
│ │ │          
    We can also compute how many months account for the fractional part of that number.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i3 : 12 * (oo - floor oo)
│ │ │  
│ │ │ -o3 = 11.44358846823999
│ │ │ +o3 = .02893473362075838
│ │ │  
│ │ │  o3 : RR (of precision 53)
    
│ │ │              
    
│ │ │          
    Compare that to the current date, available from a standard Unix command.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : run "date"
│ │ │ -Sun Dec 14 15:28:11 UTC 2025
│ │ │ +Thu Jan  1 11:03:23 UTC 2026
│ │ │  
│ │ │  o4 = 0
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,31 +9,31 @@
│ │ │ │              currentTime()
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the current time, in seconds since 00:00:00 1970-01-01
│ │ │ │              UTC, the beginning of the epoch
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : currentTime()
│ │ │ │  
│ │ │ │ -o1 = 1765726091
│ │ │ │ +o1 = 1767265403
│ │ │ │  We can compute, roughly, how many years ago the epoch began as follows.
│ │ │ │  i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 )
│ │ │ │  
│ │ │ │ -o2 = 55.95363237235333
│ │ │ │ +o2 = 56.00241122780173
│ │ │ │  
│ │ │ │  o2 : RR (of precision 53)
│ │ │ │  We can also compute how many months account for the fractional part of that
│ │ │ │  number.
│ │ │ │  i3 : 12 * (oo - floor oo)
│ │ │ │  
│ │ │ │ -o3 = 11.44358846823999
│ │ │ │ +o3 = .02893473362075838
│ │ │ │  
│ │ │ │  o3 : RR (of precision 53)
│ │ │ │  Compare that to the current date, available from a standard Unix command.
│ │ │ │  i4 : run "date"
│ │ │ │ -Sun Dec 14 15:28:11 UTC 2025
│ │ │ │ +Thu Jan  1 11:03:23 UTC 2026
│ │ │ │  
│ │ │ │  o4 = 0
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _c_u_r_r_e_n_t_T_i_m_e is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_system.m2:1849:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_elapsed__Time.html
│ │ │ @@ -59,15 +59,15 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │  elapsedTime e evaluates e, prints the amount of time elapsed, and returns the value of e.        
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : elapsedTime sleep 1
│ │ │ - -- 1.00015s elapsed
│ │ │ + -- 1.00013s elapsed
│ │ │  
│ │ │  o1 = 0
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -7,15 +7,15 @@
│ │ │ │  ************ eellaappsseeddTTiimmee ---- ttiimmee aa ccoommppuuttaattiioonn iinncclluuddiinngg ttiimmee eellaappsseedd ************
│ │ │ │      *   Usage:
│ │ │ │              elapsedTime e
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  elapsedTime e evaluates e, prints the amount of time elapsed, and returns the
│ │ │ │  value of e.
│ │ │ │  i1 : elapsedTime sleep 1
│ │ │ │ - -- 1.00015s elapsed
│ │ │ │ + -- 1.00013s 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
│ │ │ -     -- 1.00015 seconds
│ │ │ +     -- 1.00014 seconds
│ │ │  
│ │ │  o1 : Time
    
│ │ │              
    
│ │ │                
    i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{1.00015, 0}
    
│ │ │ +o2 = Time{1.00014, 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
│ │ │ │ -     -- 1.00015 seconds
│ │ │ │ +     -- 1.00014 seconds
│ │ │ │  
│ │ │ │  o1 : Time
│ │ │ │  i2 : peek oo
│ │ │ │  
│ │ │ │ -o2 = Time{1.00015, 0}
│ │ │ │ +o2 = Time{1.00014, 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 0.464405s (cpu); 0.277254s (thread); 0s (gc)
│ │ │ + -- used 0.163882s (cpu); 0.163881s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = | x3y9 5148txy3 108729sxy2z2 sy4z 46644741sxy3z 143sy5 6sxy4
│ │ │       ------------------------------------------------------------------------
│ │ │       563515116021sx2y3 4374txy2z3 612704350498473090tx2yz3 217458ty4z2
│ │ │       ------------------------------------------------------------------------
│ │ │       267076255345488270sy3z4 5256861933965245618410txyz6
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -162,15 +162,15 @@
│ │ │  
│ │ │  o7 : Ideal of R
    
│ │ │              
    
│ │ │                
    i8 : time G = eliminate(I,{s,t})
│ │ │ - -- used 0.422625s (cpu); 0.238689s (thread); 0s (gc)
│ │ │ + -- used 0.403807s (cpu); 0.181s (thread); 0s (gc)
│ │ │  
│ │ │              3 9     2 9     2 8      2 6 3       9    2 7         8   
│ │ │  o8 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │       ------------------------------------------------------------------------
│ │ │           7 2       2 5 3       6 3    7 3        5 4       3 6    9       7 
│ │ │       7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -245,15 +245,15 @@
│ │ │  
│ │ │  o11 : Ideal of R1
    
│ │ │              
    
│ │ │                
    i12 : time G = eliminate(I1,{s,t})
│ │ │ - -- used 0.286447s (cpu); 0.110979s (thread); 0s (gc)
│ │ │ + -- used 0.0304523s (cpu); 0.0304574s (thread); 0s (gc)
│ │ │  
│ │ │               3 9     2 6 3       3 6    9     2 8         5 4      2 7  
│ │ │  o12 = ideal(x y  - 3x y z  + 3x*y z  - z  - 6x y z - 15x*y z  + 21y z  -
│ │ │        -----------------------------------------------------------------------
│ │ │          2 9       2 5 3       6 3    7 3         2 6     3 6       7 2  
│ │ │        3x y  - 324x y z  + 6x*y z  - y z  - 405x*y z  - 3y z  + 7x*y z  -
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -337,15 +337,15 @@
│ │ │  
│ │ │  o16 : RingMap A <-- B
    
│ │ │              
    
│ │ │                
    i17 : time G = kernel F
│ │ │ - -- used 0.404757s (cpu); 0.227468s (thread); 0s (gc)
│ │ │ + -- used 0.100185s (cpu); 0.100191s (thread); 0s (gc)
│ │ │  
│ │ │               3 9     2 9     2 8      2 6 3       9    2 7         8   
│ │ │  o17 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │        -----------------------------------------------------------------------
│ │ │            7 2       2 5 3       6 3    7 3        5 4       3 6    9       7 
│ │ │        7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -418,26 +418,26 @@
│ │ │  
│ │ │  o19 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i20 : time f1 = resultant(I_0,I_2,s)
│ │ │ - -- used 0.00191261s (cpu); 0.00191299s (thread); 0s (gc)
│ │ │ + -- used 0.00161638s (cpu); 0.00161341s (thread); 0s (gc)
│ │ │  
│ │ │           9    9      7    3
│ │ │  o20 = x*t  - t  - z*t  - z
│ │ │  
│ │ │  o20 : R
    
│ │ │              
    
│ │ │                
    i21 : time f2 = resultant(I_1,f1,t)
│ │ │ - -- used 0.0583203s (cpu); 0.0583314s (thread); 0s (gc)
│ │ │ + -- used 0.0355171s (cpu); 0.0355287s (thread); 0s (gc)
│ │ │  
│ │ │           3 9     2 9     2 8      2 6 3       9    2 7         8        7 2  
│ │ │  o21 = - x y  + 3x y  + 6x y z + 3x y z  - 3x*y  + x y z - 12x*y z - 7x*y z  +
│ │ │        -----------------------------------------------------------------------
│ │ │            2 5 3       6 3    7 3        5 4       3 6    9       7      8   
│ │ │        324x y z  - 6x*y z  + y z  + 15x*y z  - 3x*y z  + y  - 2x*y z + 6y z +
│ │ │        -----------------------------------------------------------------------
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,15 +13,15 @@
│ │ │ │  i2 : I = ideal(x-s^3-s*t-1, y-t^3-3*t^2-t, z-s*t^3)
│ │ │ │  
│ │ │ │                 3                   3     2               3
│ │ │ │  o2 = ideal (- s  - s*t + x - 1, - t  - 3t  - t + y, - s*t  + z)
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : time leadTerm gens gb I
│ │ │ │ - -- used 0.464405s (cpu); 0.277254s (thread); 0s (gc)
│ │ │ │ + -- used 0.163882s (cpu); 0.163881s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = | x3y9 5148txy3 108729sxy2z2 sy4z 46644741sxy3z 143sy5 6sxy4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       563515116021sx2y3 4374txy2z3 612704350498473090tx2yz3 217458ty4z2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       267076255345488270sy3z4 5256861933965245618410txyz6
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -89,15 +89,15 @@
│ │ │ │  i7 : I = ideal(x-s^3-s*t-1, y-t^3-3*t^2-t, z-s*t^3)
│ │ │ │  
│ │ │ │                 3                   3     2               3
│ │ │ │  o7 = ideal (- s  - s*t + x - 1, - t  - 3t  + y - t, - s*t  + z)
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  i8 : time G = eliminate(I,{s,t})
│ │ │ │ - -- used 0.422625s (cpu); 0.238689s (thread); 0s (gc)
│ │ │ │ + -- used 0.403807s (cpu); 0.181s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              3 9     2 9     2 8      2 6 3       9    2 7         8
│ │ │ │  o8 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │           7 2       2 5 3       6 3    7 3        5 4       3 6    9       7
│ │ │ │       7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -156,15 +156,15 @@
│ │ │ │  Sometimes giving the variables different degrees will speed up the
│ │ │ │  computations. Here, we set the degrees of x, y, and z to be the total degrees.
│ │ │ │  i10 : R1 = QQ[x,y,z,s,t, Degrees=>{3,3,4,1,1}];
│ │ │ │  i11 : I1 = substitute(I,R1);
│ │ │ │  
│ │ │ │  o11 : Ideal of R1
│ │ │ │  i12 : time G = eliminate(I1,{s,t})
│ │ │ │ - -- used 0.286447s (cpu); 0.110979s (thread); 0s (gc)
│ │ │ │ + -- used 0.0304523s (cpu); 0.0304574s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               3 9     2 6 3       3 6    9     2 8         5 4      2 7
│ │ │ │  o12 = ideal(x y  - 3x y z  + 3x*y z  - z  - 6x y z - 15x*y z  + 21y z  -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │          2 9       2 5 3       6 3    7 3         2 6     3 6       7 2
│ │ │ │        3x y  - 324x y z  + 6x*y z  - y z  - 405x*y z  - 3y z  + 7x*y z  -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ @@ -227,15 +227,15 @@
│ │ │ │  i16 : F = map(A,B,{s^3+s*t+1, t^3+3*t^2+t, s*t^3})
│ │ │ │  
│ │ │ │                     3             3     2         3
│ │ │ │  o16 = map (A, B, {s  + s*t + 1, t  + 3t  + t, s*t })
│ │ │ │  
│ │ │ │  o16 : RingMap A <-- B
│ │ │ │  i17 : time G = kernel F
│ │ │ │ - -- used 0.404757s (cpu); 0.227468s (thread); 0s (gc)
│ │ │ │ + -- used 0.100185s (cpu); 0.100191s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               3 9     2 9     2 8      2 6 3       9    2 7         8
│ │ │ │  o17 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │            7 2       2 5 3       6 3    7 3        5 4       3 6    9       7
│ │ │ │        7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ @@ -296,22 +296,22 @@
│ │ │ │  involve the variables s and t.
│ │ │ │  i19 : use ring I
│ │ │ │  
│ │ │ │  o19 = R
│ │ │ │  
│ │ │ │  o19 : PolynomialRing
│ │ │ │  i20 : time f1 = resultant(I_0,I_2,s)
│ │ │ │ - -- used 0.00191261s (cpu); 0.00191299s (thread); 0s (gc)
│ │ │ │ + -- used 0.00161638s (cpu); 0.00161341s (thread); 0s (gc)
│ │ │ │  
│ │ │ │           9    9      7    3
│ │ │ │  o20 = x*t  - t  - z*t  - z
│ │ │ │  
│ │ │ │  o20 : R
│ │ │ │  i21 : time f2 = resultant(I_1,f1,t)
│ │ │ │ - -- used 0.0583203s (cpu); 0.0583314s (thread); 0s (gc)
│ │ │ │ + -- used 0.0355171s (cpu); 0.0355287s (thread); 0s (gc)
│ │ │ │  
│ │ │ │           3 9     2 9     2 8      2 6 3       9    2 7         8        7 2
│ │ │ │  o21 = - x y  + 3x y  + 6x y z + 3x y z  - 3x*y  + x y z - 12x*y z - 7x*y z  +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │            2 5 3       6 3    7 3        5 4       3 6    9       7      8
│ │ │ │        324x y z  - 6x*y z  + y z  + 15x*y z  - 3x*y z  + y  - 2x*y z + 6y z +
│ │ │ │        -----------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_end__Package.html
│ │ │ @@ -154,15 +154,15 @@
│ │ │                                      Version => 0.0
│ │ │               package prefix => /usr/
│ │ │               PackageIsLoaded => true
│ │ │               pkgname => Foo
│ │ │               private dictionary => Foo#"private dictionary"
│ │ │               processed documentation => MutableHashTable{}
│ │ │               raw documentation => MutableHashTable{}
│ │ │ -             source directory => /tmp/M2-10191-0/91-rundir/
│ │ │ +             source directory => /tmp/M2-10311-0/91-rundir/
│ │ │               source file => stdio
│ │ │               test inputs => MutableList{}
    
│ │ │              
    
│ │ │                
    i7 : dictionaryPath
│ │ │ ├── html2text {}
│ │ │ │ @@ -77,15 +77,15 @@
│ │ │ │                                      Version => 0.0
│ │ │ │               package prefix => /usr/
│ │ │ │               PackageIsLoaded => true
│ │ │ │               pkgname => Foo
│ │ │ │               private dictionary => Foo#"private dictionary"
│ │ │ │               processed documentation => MutableHashTable{}
│ │ │ │               raw documentation => MutableHashTable{}
│ │ │ │ -             source directory => /tmp/M2-10191-0/91-rundir/
│ │ │ │ +             source directory => /tmp/M2-10311-0/91-rundir/
│ │ │ │               source file => stdio
│ │ │ │               test inputs => MutableList{}
│ │ │ │  i7 : dictionaryPath
│ │ │ │  
│ │ │ │  o7 = {Foo.Dictionary, Varieties.Dictionary, Isomorphism.Dictionary,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       Truncations.Dictionary, Polyhedra.Dictionary, Saturation.Dictionary,
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Exists.html
│ │ │ @@ -68,29 +68,29 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -510,27 +510,27 @@
│ │ │  o36 = 3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10558-0/0
    
│ │ │ +o1 = /tmp/M2-11028-0/0
│ │ │              
    
│ │ │                
    i2 : fileExists fn
│ │ │  
│ │ │  o2 = false
    
│ │ │              
    
│ │ │                
    i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10558-0/0
│ │ │ +o3 = /tmp/M2-11028-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : fileExists fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,21 +10,21 @@
│ │ │ │      * Inputs:
│ │ │ │            o fn, a _s_t_r_i_n_g
│ │ │ │      * Outputs:
│ │ │ │            o a _B_o_o_l_e_a_n_ _v_a_l_u_e, whether a file with the filename or path fn exists
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10558-0/0
│ │ │ │ +o1 = /tmp/M2-11028-0/0
│ │ │ │  i2 : fileExists fn
│ │ │ │  
│ │ │ │  o2 = false
│ │ │ │  i3 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10558-0/0
│ │ │ │ +o3 = /tmp/M2-11028-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : fileExists fn
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  i5 : removeFile fn
│ │ │ │  If fn refers to a symbolic link, then whether the file exists is determined by
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Length.html
│ │ │ @@ -69,15 +69,15 @@
│ │ │          
    Description
    
│ │ │          
    The length of an open output file is determined from the internal count of the number of bytes written so far.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : f = temporaryFileName() << "hi there"
│ │ │  
│ │ │ -o1 = /tmp/M2-12150-0/0
│ │ │ +o1 = /tmp/M2-14270-0/0
│ │ │  
│ │ │  o1 : File
    
│ │ │              
    
│ │ │                
    i2 : fileLength f
│ │ │ @@ -85,24 +85,24 @@
│ │ │  o2 = 8
    
│ │ │              
    
│ │ │                
    i3 : close f
│ │ │  
│ │ │ -o3 = /tmp/M2-12150-0/0
│ │ │ +o3 = /tmp/M2-14270-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : filename = toString f
│ │ │  
│ │ │ -o4 = /tmp/M2-12150-0/0
    
│ │ │ +o4 = /tmp/M2-14270-0/0
│ │ │              
    
│ │ │                
    i5 : fileLength filename
│ │ │  
│ │ │  o5 = 8
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,28 +12,28 @@
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the length of the file f or the file whose name is f
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  The length of an open output file is determined from the internal count of the
│ │ │ │  number of bytes written so far.
│ │ │ │  i1 : f = temporaryFileName() << "hi there"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12150-0/0
│ │ │ │ +o1 = /tmp/M2-14270-0/0
│ │ │ │  
│ │ │ │  o1 : File
│ │ │ │  i2 : fileLength f
│ │ │ │  
│ │ │ │  o2 = 8
│ │ │ │  i3 : close f
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-12150-0/0
│ │ │ │ +o3 = /tmp/M2-14270-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : filename = toString f
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-12150-0/0
│ │ │ │ +o4 = /tmp/M2-14270-0/0
│ │ │ │  i5 : fileLength filename
│ │ │ │  
│ │ │ │  o5 = 8
│ │ │ │  i6 : get filename
│ │ │ │  
│ │ │ │  o6 = hi there
│ │ │ │  i7 : length oo
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Mode_lp__File_rp.html
│ │ │ @@ -69,22 +69,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -425,15 +425,15 @@
│ │ │  
│ │ │  o33 = 1
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11375-0/0
    
│ │ │ +o1 = /tmp/M2-12685-0/0
│ │ │              
    
│ │ │                
    i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-11375-0/0
│ │ │ +o2 = /tmp/M2-12685-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : fileMode f
│ │ │ @@ -92,15 +92,15 @@
│ │ │  o3 = 420
    
│ │ │              
    
│ │ │                
    i4 : close f
│ │ │  
│ │ │ -o4 = /tmp/M2-11375-0/0
│ │ │ +o4 = /tmp/M2-12685-0/0
│ │ │  
│ │ │  o4 : File
    
│ │ │              
    
│ │ │                
    i5 : removeFile fn
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,26 +11,26 @@
│ │ │ │      * Inputs:
│ │ │ │            o f, a _f_i_l_e
│ │ │ │      * Outputs:
│ │ │ │            o the mode of the open file f
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11375-0/0
│ │ │ │ +o1 = /tmp/M2-12685-0/0
│ │ │ │  i2 : f = fn << "hi there"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11375-0/0
│ │ │ │ +o2 = /tmp/M2-12685-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : fileMode f
│ │ │ │  
│ │ │ │  o3 = 420
│ │ │ │  i4 : close f
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11375-0/0
│ │ │ │ +o4 = /tmp/M2-12685-0/0
│ │ │ │  
│ │ │ │  o4 : File
│ │ │ │  i5 : removeFile fn
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _f_i_l_e_M_o_d_e_(_F_i_l_e_) -- get file mode
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Mode_lp__String_rp.html
│ │ │ @@ -69,22 +69,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │              
│ │ │              
│ │ │                
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10989-0/0
    
│ │ │ +o1 = /tmp/M2-11899-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-10989-0/0
│ │ │ +o2 = /tmp/M2-11899-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : fileMode fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,18 +11,18 @@
│ │ │ │      * Inputs:
│ │ │ │            o fn, a _s_t_r_i_n_g
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the mode of the file located at the filename or path fn
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10989-0/0
│ │ │ │ +o1 = /tmp/M2-11899-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10989-0/0
│ │ │ │ +o2 = /tmp/M2-11899-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : fileMode fn
│ │ │ │  
│ │ │ │  o3 = 420
│ │ │ │  i4 : removeFile fn
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Mode_lp__Z__Z_cm__File_rp.html
│ │ │ @@ -73,22 +73,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -245,23 +245,23 @@
│ │ │  o22 = 10000000000000000000000000001710000000000000000000000000097470000000000
│ │ │        00000000000000185613
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10854-0/0
    
│ │ │ +o1 = /tmp/M2-11624-0/0
│ │ │              
    
│ │ │                
    i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-10854-0/0
│ │ │ +o2 = /tmp/M2-11624-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : m = 7 + 7*8 + 7*64
│ │ │ @@ -108,15 +108,15 @@
│ │ │  o5 = 511
    
│ │ │              
    
│ │ │                
    i6 : close f
│ │ │  
│ │ │ -o6 = /tmp/M2-10854-0/0
│ │ │ +o6 = /tmp/M2-11624-0/0
│ │ │  
│ │ │  o6 : File
    
│ │ │              
    
│ │ │                
    i7 : fileMode fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,30 +12,30 @@
│ │ │ │            o mo, an _i_n_t_e_g_e_r
│ │ │ │            o f, a _f_i_l_e
│ │ │ │      * Consequences:
│ │ │ │            o the mode of the open file f is set to mo
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10854-0/0
│ │ │ │ +o1 = /tmp/M2-11624-0/0
│ │ │ │  i2 : f = fn << "hi there"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10854-0/0
│ │ │ │ +o2 = /tmp/M2-11624-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : m = 7 + 7*8 + 7*64
│ │ │ │  
│ │ │ │  o3 = 511
│ │ │ │  i4 : fileMode(m,f)
│ │ │ │  i5 : fileMode f
│ │ │ │  
│ │ │ │  o5 = 511
│ │ │ │  i6 : close f
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-10854-0/0
│ │ │ │ +o6 = /tmp/M2-11624-0/0
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : fileMode fn
│ │ │ │  
│ │ │ │  o7 = 511
│ │ │ │  i8 : removeFile fn
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Mode_lp__Z__Z_cm__String_rp.html
│ │ │ @@ -73,22 +73,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11977-0/0
    
│ │ │ +o1 = /tmp/M2-13917-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-11977-0/0
│ │ │ +o2 = /tmp/M2-13917-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : m = fileMode fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,18 +13,18 @@
│ │ │ │            o fn, a _s_t_r_i_n_g
│ │ │ │      * Consequences:
│ │ │ │            o the mode of the file located at the filename or path fn is set to
│ │ │ │              mo
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11977-0/0
│ │ │ │ +o1 = /tmp/M2-13917-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11977-0/0
│ │ │ │ +o2 = /tmp/M2-13917-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : m = fileMode fn
│ │ │ │  
│ │ │ │  o3 = 420
│ │ │ │  i4 : fileMode(m|7,fn)
│ │ │ │  i5 : fileMode fn
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Time.html
│ │ │ @@ -76,15 +76,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │  The value is the number of seconds since 00:00:00 1970-01-01 UTC, the beginning of the epoch, so the number of seconds ago a file or directory was modified may be found by using the following code.        
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : currentTime() - fileTime "."
│ │ │  
│ │ │ -o1 = 61
    
│ │ │ +o1 = 46
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,15 +18,15 @@
│ │ │ │              returns null if no error occurs
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  The value is the number of seconds since 00:00:00 1970-01-01 UTC, the beginning
│ │ │ │  of the epoch, so the number of seconds ago a file or directory was modified may
│ │ │ │  be found by using the following code.
│ │ │ │  i1 : currentTime() - fileTime "."
│ │ │ │  
│ │ │ │ -o1 = 61
│ │ │ │ +o1 = 46
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_u_r_r_e_n_t_T_i_m_e -- get the current time
│ │ │ │      * _f_i_l_e_ _m_a_n_i_p_u_l_a_t_i_o_n -- Unix file manipulation functions
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _f_i_l_e_T_i_m_e is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_force__G__B_lp..._cm__Syzygy__Matrix_eq_gt..._rp.html
│ │ │ @@ -120,15 +120,15 @@
│ │ │  o6 : Matrix R  <-- R
    
│ │ │              
    
│ │ │                
    i7 : syz f
│ │ │  
│ │ │ -   -- registering gb 0 at 0x7f5498d2ce00
│ │ │ +   -- registering gb 0 at 0x7f1938812e00
│ │ │  
│ │ │     -- [gb]{2}(1)m{3}(1)m{4}(1)m{5}(1)z{6}(1)z{7}(1)znumber of (nonminimal) gb elements = 3
│ │ │     -- number of monomials                = 9
│ │ │     -- #reduction steps = 6
│ │ │     -- #spairs done = 6
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ │ ├── html2text {}
│ │ │ │ @@ -37,15 +37,15 @@
│ │ │ │       {3} | x2-3  0     -z4+2 |
│ │ │ │       {4} | 0     x2-3  y3-1  |
│ │ │ │  
│ │ │ │               3      3
│ │ │ │  o6 : Matrix R  <-- R
│ │ │ │  i7 : syz f
│ │ │ │  
│ │ │ │ -   -- registering gb 0 at 0x7f5498d2ce00
│ │ │ │ +   -- registering gb 0 at 0x7f1938812e00
│ │ │ │  
│ │ │ │     -- [gb]{2}(1)m{3}(1)m{4}(1)m{5}(1)z{6}(1)z{7}(1)znumber of (nonminimal) gb
│ │ │ │  elements = 3
│ │ │ │     -- number of monomials                = 9
│ │ │ │     -- #reduction steps = 6
│ │ │ │     -- #spairs done = 6
│ │ │ │     -- ncalls = 0
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_get.html
│ │ │ @@ -96,15 +96,15 @@
│ │ │                
    i3 : removeFile "test-file"
    
│ │ │              
    
│ │ │                
    i4 : get "!date"
│ │ │  
│ │ │ -o4 = Sun Dec 14 15:27:22 UTC 2025
    
│ │ │ +o4 = Thu Jan  1 11:02:48 UTC 2026
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,15 +25,15 @@
│ │ │ │  o1 : File
│ │ │ │  i2 : get "test-file"
│ │ │ │  
│ │ │ │  o2 = hi there
│ │ │ │  i3 : removeFile "test-file"
│ │ │ │  i4 : get "!date"
│ │ │ │  
│ │ │ │ -o4 = Sun Dec 14 15:27:22 UTC 2025
│ │ │ │ +o4 = Thu Jan  1 11:02:48 UTC 2026
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_a_d -- read from a file
│ │ │ │      * _r_e_m_o_v_e_F_i_l_e -- remove a file
│ │ │ │      * _c_l_o_s_e -- close a file
│ │ │ │      * _F_i_l_e_ _<_<_ _T_h_i_n_g -- print to a file
│ │ │ │  ********** WWaayyss ttoo uussee ggeett:: **********
│ │ │ │      * get(File)
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_instances.html
│ │ │ @@ -84,15 +84,15 @@
│ │ │                 defaultPrecision => 53
│ │ │                 engineDebugLevel => 0
│ │ │                 errorDepth => 0
│ │ │                 gbTrace => 0
│ │ │                 interpreterDepth => 1
│ │ │                 lineNumber => 2
│ │ │                 loadDepth => 3
│ │ │ -               maxAllowableThreads => 7
│ │ │ +               maxAllowableThreads => 17
│ │ │                 maxExponent => 1073741823
│ │ │                 minExponent => -1073741824
│ │ │                 numTBBThreads => 0
│ │ │                 o1 => 2432902008176640000
│ │ │                 oo => 2432902008176640000
│ │ │                 printingAccuracy => -1
│ │ │                 printingLeadLimit => 5
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,15 +23,15 @@
│ │ │ │                 defaultPrecision => 53
│ │ │ │                 engineDebugLevel => 0
│ │ │ │                 errorDepth => 0
│ │ │ │                 gbTrace => 0
│ │ │ │                 interpreterDepth => 1
│ │ │ │                 lineNumber => 2
│ │ │ │                 loadDepth => 3
│ │ │ │ -               maxAllowableThreads => 7
│ │ │ │ +               maxAllowableThreads => 17
│ │ │ │                 maxExponent => 1073741823
│ │ │ │                 minExponent => -1073741824
│ │ │ │                 numTBBThreads => 0
│ │ │ │                 o1 => 2432902008176640000
│ │ │ │                 oo => 2432902008176640000
│ │ │ │                 printingAccuracy => -1
│ │ │ │                 printingLeadLimit => 5
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Directory.html
│ │ │ @@ -75,22 +75,22 @@
│ │ │  o1 = true
    
│ │ │              
    
│ │ │                
    i2 : fn = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-10380-0/0
    
│ │ │ +o2 = /tmp/M2-10670-0/0
│ │ │              
    
│ │ │                
    i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10380-0/0
│ │ │ +o3 = /tmp/M2-10670-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : isDirectory fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,18 +13,18 @@
│ │ │ │            o a _B_o_o_l_e_a_n_ _v_a_l_u_e, whether fn is the path to a directory
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : isDirectory "."
│ │ │ │  
│ │ │ │  o1 = true
│ │ │ │  i2 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10380-0/0
│ │ │ │ +o2 = /tmp/M2-10670-0/0
│ │ │ │  i3 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10380-0/0
│ │ │ │ +o3 = /tmp/M2-10670-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : isDirectory fn
│ │ │ │  
│ │ │ │  o4 = false
│ │ │ │  i5 : removeFile fn
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Pseudoprime_lp__Z__Z_rp.html
│ │ │ @@ -211,15 +211,15 @@
│ │ │  
│ │ │  o18 = false
    
│ │ │              
    
│ │ │                
    i19 : elapsedTime facs = factor(m*m1)
│ │ │ - -- 4.33674s elapsed
│ │ │ + -- 5.79079s elapsed
│ │ │  
│ │ │  o19 = 1000000000000000000000000000057*1000000000000000000010000000083
│ │ │  
│ │ │  o19 : Expression of class Product
    
│ │ │              
    
│ │ │                
    i23 : elapsedTime isPrime m3
│ │ │ - -- .0569545s elapsed
│ │ │ + -- .0555585s elapsed
│ │ │  
│ │ │  o23 = true
    
│ │ │              
    
│ │ │                
    i24 : elapsedTime isPseudoprime m3
│ │ │ - -- .000114113s elapsed
│ │ │ + -- .000106514s elapsed
│ │ │  
│ │ │  o24 = true
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -80,15 +80,15 @@
│ │ │ │  i17 : isPrime (m*m1)
│ │ │ │  
│ │ │ │  o17 = false
│ │ │ │  i18 : isPrime(m*m*m1*m1*m2^6)
│ │ │ │  
│ │ │ │  o18 = false
│ │ │ │  i19 : elapsedTime facs = factor(m*m1)
│ │ │ │ - -- 4.33674s elapsed
│ │ │ │ + -- 5.79079s elapsed
│ │ │ │  
│ │ │ │  o19 = 1000000000000000000000000000057*1000000000000000000010000000083
│ │ │ │  
│ │ │ │  o19 : Expression of class Product
│ │ │ │  i20 : facs = facs//toList/toList
│ │ │ │  
│ │ │ │  o20 = {{1000000000000000000000000000057, 1},
│ │ │ │ @@ -98,19 +98,19 @@
│ │ │ │  o20 : List
│ │ │ │  i21 : assert(set facs === set {{m,1}, {m1,1}})
│ │ │ │  i22 : m3 = nextPrime (m^3)
│ │ │ │  
│ │ │ │  o22 = 10000000000000000000000000001710000000000000000000000000097470000000000
│ │ │ │        00000000000000185613
│ │ │ │  i23 : elapsedTime isPrime m3
│ │ │ │ - -- .0569545s elapsed
│ │ │ │ + -- .0555585s elapsed
│ │ │ │  
│ │ │ │  o23 = true
│ │ │ │  i24 : elapsedTime isPseudoprime m3
│ │ │ │ - -- .000114113s elapsed
│ │ │ │ + -- .000106514s elapsed
│ │ │ │  
│ │ │ │  o24 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_P_r_i_m_e_(_Z_Z_) -- whether a integer or polynomial is prime
│ │ │ │      * _f_a_c_t_o_r_(_Z_Z_) -- factor a ring element
│ │ │ │      * _n_e_x_t_P_r_i_m_e_(_N_u_m_b_e_r_) -- compute the smallest prime greater than or equal to
│ │ │ │        a given number
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Regular__File.html
│ │ │ @@ -68,22 +68,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │  In UNIX, a regular file is one that is not special in some way.  Special files include symbolic links and directories.  A regular file is a sequence of bytes stored permanently in a file system.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12188-0/0
    
│ │ │ +o1 = /tmp/M2-14348-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-12188-0/0
│ │ │ +o2 = /tmp/M2-14348-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : isRegularFile fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,18 +13,18 @@
│ │ │ │            o a _B_o_o_l_e_a_n_ _v_a_l_u_e, whether fn is the path to a regular file
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  In UNIX, a regular file is one that is not special in some way. Special files
│ │ │ │  include symbolic links and directories. A regular file is a sequence of bytes
│ │ │ │  stored permanently in a file system.
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12188-0/0
│ │ │ │ +o1 = /tmp/M2-14348-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12188-0/0
│ │ │ │ +o2 = /tmp/M2-14348-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : isRegularFile fn
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : removeFile fn
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_make__Directory_lp__String_rp.html
│ │ │ @@ -76,22 +76,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,18 +13,18 @@
│ │ │ │      * Outputs:
│ │ │ │            o a _s_t_r_i_n_g, the name of the newly made directory
│ │ │ │      * Consequences:
│ │ │ │            o the directory is made, with as many new path components as needed
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : dir = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10722-0/0
│ │ │ │ +o1 = /tmp/M2-11352-0/0
│ │ │ │  i2 : makeDirectory (dir|"/a/b/c")
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10722-0/0/a/b/c
│ │ │ │ +o2 = /tmp/M2-11352-0/0/a/b/c
│ │ │ │  i3 : removeDirectory (dir|"/a/b/c")
│ │ │ │  i4 : removeDirectory (dir|"/a/b")
│ │ │ │  i5 : removeDirectory (dir|"/a")
│ │ │ │  A filename starting with ~/ will have the tilde replaced by the home directory.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_k_d_i_r
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_max__Allowable__Threads.html
│ │ │ @@ -64,15 +64,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    
│ │ │                
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10722-0/0
    
│ │ │ +o1 = /tmp/M2-11352-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory (dir|"/a/b/c")
│ │ │  
│ │ │ -o2 = /tmp/M2-10722-0/0/a/b/c
    
│ │ │ +o2 = /tmp/M2-11352-0/0/a/b/c
│ │ │              
    
│ │ │                
    i3 : removeDirectory (dir|"/a/b/c")
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : maxAllowableThreads
│ │ │  
│ │ │ -o1 = 7
    
│ │ │ +o1 = 17
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,15 +9,15 @@
│ │ │ │      *   Usage:
│ │ │ │              maxAllowableThreads
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the maximum number to which _a_l_l_o_w_a_b_l_e_T_h_r_e_a_d_s can be set
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : maxAllowableThreads
│ │ │ │  
│ │ │ │ -o1 = 7
│ │ │ │ +o1 = 17
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _p_a_r_a_l_l_e_l_ _p_r_o_g_r_a_m_m_i_n_g_ _w_i_t_h_ _t_h_r_e_a_d_s_ _a_n_d_ _t_a_s_k_s
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _m_a_x_A_l_l_o_w_a_b_l_e_T_h_r_e_a_d_s is an _i_n_t_e_g_e_r.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_threads.m2:498:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_memoize.html
│ │ │ @@ -61,15 +61,15 @@
│ │ │  
│ │ │  o1 : FunctionClosure
    
│ │ │              
    
│ │ │                
    i2 : time fib 28
│ │ │ - -- used 1.26333s (cpu); 0.726987s (thread); 0s (gc)
│ │ │ + -- used 0.754762s (cpu); 0.577694s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 514229
    
│ │ │              
    
│ │ │                
    i3 : fib = memoize fib
│ │ │ @@ -78,23 +78,23 @@
│ │ │  
│ │ │  o3 : FunctionClosure
    
│ │ │              
    
│ │ │                
    i4 : time fib 28
│ │ │ - -- used 7.5542e-05s (cpu); 7.4801e-05s (thread); 0s (gc)
│ │ │ + -- used 8.2955e-05s (cpu); 8.1779e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 514229
    
│ │ │              
    
│ │ │                
    i5 : time fib 28
│ │ │ - -- used 3.987e-06s (cpu); 3.627e-06s (thread); 0s (gc)
│ │ │ + -- used 3.696e-06s (cpu); 3.447e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 514229
    
│ │ │              
    
│ │ │          
    An optional second argument to memoize provides a list of initial values, each of the form x => v, where v is the value to be provided for the argument x.
    
│ │ │          
    Alternatively, values can be provided after defining the memoized function using the syntax f x = v. A slightly more efficient implementation of the above would be
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,28 +11,28 @@
│ │ │ │  arguments are presented.
│ │ │ │  i1 : fib = n -> if n <= 1 then 1 else fib(n-1) + fib(n-2)
│ │ │ │  
│ │ │ │  o1 = fib
│ │ │ │  
│ │ │ │  o1 : FunctionClosure
│ │ │ │  i2 : time fib 28
│ │ │ │ - -- used 1.26333s (cpu); 0.726987s (thread); 0s (gc)
│ │ │ │ + -- used 0.754762s (cpu); 0.577694s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = 514229
│ │ │ │  i3 : fib = memoize fib
│ │ │ │  
│ │ │ │  o3 = fib
│ │ │ │  
│ │ │ │  o3 : FunctionClosure
│ │ │ │  i4 : time fib 28
│ │ │ │ - -- used 7.5542e-05s (cpu); 7.4801e-05s (thread); 0s (gc)
│ │ │ │ + -- used 8.2955e-05s (cpu); 8.1779e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 514229
│ │ │ │  i5 : time fib 28
│ │ │ │ - -- used 3.987e-06s (cpu); 3.627e-06s (thread); 0s (gc)
│ │ │ │ + -- used 3.696e-06s (cpu); 3.447e-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
│ │ │ @@ -89,20 +89,20 @@
│ │ │       {12 => (poincare, BettiTally)                                }
│ │ │       {13 => (hilbertPolynomial, ZZ, BettiTally)                   }
│ │ │       {14 => (degree, BettiTally)                                  }
│ │ │       {15 => (hilbertSeries, ZZ, BettiTally)                       }
│ │ │       {16 => (pdim, BettiTally)                                    }
│ │ │       {17 => (regularity, BettiTally)                              }
│ │ │       {18 => (mathML, BettiTally)                                  }
│ │ │ -     {19 => (codim, BettiTally)                                   }
│ │ │ +     {19 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │       {20 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ -     {21 => (dual, BettiTally)                                    }
│ │ │ -     {22 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ -     {23 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ -     {24 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ +     {21 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ +     {22 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ +     {23 => (codim, BettiTally)                                   }
│ │ │ +     {24 => (dual, BettiTally)                                    }
│ │ │       {25 => (^, Ring, BettiTally)                                 }
│ │ │  
│ │ │  o1 : NumberedVerticalList
    
│ │ │                
    
│ │ │ @@ -188,20 +188,20 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                  
    i5 : methods( Matrix, Matrix )
│ │ │  
│ │ │ -o5 = {0 => (contract, Matrix, Matrix)                            }
│ │ │ -     {1 => (diff, Matrix, Matrix)                                }
│ │ │ -     {2 => (diff', Matrix, Matrix)                               }
│ │ │ -     {3 => (-, Matrix, Matrix)                                   }
│ │ │ +o5 = {0 => (+, Matrix, Matrix)                                   }
│ │ │ +     {1 => (-, Matrix, Matrix)                                   }
│ │ │ +     {2 => (contract, Matrix, Matrix)                            }
│ │ │ +     {3 => (diff, Matrix, Matrix)                                }
│ │ │       {4 => (contract', Matrix, Matrix)                           }
│ │ │ -     {5 => (+, Matrix, Matrix)                                   }
│ │ │ +     {5 => (diff', Matrix, Matrix)                               }
│ │ │       {6 => (markedGB, Matrix, Matrix)                            }
│ │ │       {7 => (Hom, Matrix, Matrix)                                 }
│ │ │       {8 => (==, Matrix, Matrix)                                  }
│ │ │       {9 => (*, Matrix, Matrix)                                   }
│ │ │       {10 => (|, Matrix, Matrix)                                  }
│ │ │       {11 => (||, Matrix, Matrix)                                 }
│ │ │       {12 => (subquotient, Matrix, Matrix)                        }
│ │ │ @@ -216,18 +216,18 @@
│ │ │       {21 => (quotient, Matrix, Matrix)                           }
│ │ │       {22 => (quotient', Matrix, Matrix)                          }
│ │ │       {23 => (remainder', Matrix, Matrix)                         }
│ │ │       {24 => (%, Matrix, Matrix)                                  }
│ │ │       {25 => (remainder, Matrix, Matrix)                          }
│ │ │       {26 => (pushout, Matrix, Matrix)                            }
│ │ │       {27 => (solve, Matrix, Matrix)                              }
│ │ │ -     {28 => (pullback, Matrix, Matrix)                           }
│ │ │ +     {28 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │       {29 => (intersect, Matrix, Matrix)                          }
│ │ │ -     {30 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ -     {31 => (tensor, Matrix, Matrix)                             }
│ │ │ +     {30 => (tensor, Matrix, Matrix)                             }
│ │ │ +     {31 => (pullback, Matrix, Matrix)                           }
│ │ │       {32 => (substitute, Matrix, Matrix)                         }
│ │ │       {33 => (yonedaProduct, Matrix, Matrix)                      }
│ │ │       {34 => (isShortExactSequence, Matrix, Matrix)               }
│ │ │       {35 => (horseshoeResolution, Matrix, Matrix)                }
│ │ │       {36 => (connectingExtMap, Module, Matrix, Matrix)           }
│ │ │       {37 => (connectingExtMap, Matrix, Matrix, Module)           }
│ │ │       {38 => (connectingTorMap, Module, Matrix, Matrix)           }
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,20 +29,20 @@
│ │ │ │       {12 => (poincare, BettiTally)                                }
│ │ │ │       {13 => (hilbertPolynomial, ZZ, BettiTally)                   }
│ │ │ │       {14 => (degree, BettiTally)                                  }
│ │ │ │       {15 => (hilbertSeries, ZZ, BettiTally)                       }
│ │ │ │       {16 => (pdim, BettiTally)                                    }
│ │ │ │       {17 => (regularity, BettiTally)                              }
│ │ │ │       {18 => (mathML, BettiTally)                                  }
│ │ │ │ -     {19 => (codim, BettiTally)                                   }
│ │ │ │ +     {19 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ │       {20 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ │ -     {21 => (dual, BettiTally)                                    }
│ │ │ │ -     {22 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ │ -     {23 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ │ -     {24 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ │ +     {21 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ │ +     {22 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ │ +     {23 => (codim, BettiTally)                                   }
│ │ │ │ +     {24 => (dual, BettiTally)                                    }
│ │ │ │       {25 => (^, Ring, BettiTally)                                 }
│ │ │ │  
│ │ │ │  o1 : NumberedVerticalList
│ │ │ │  i2 : methods resolution
│ │ │ │  
│ │ │ │  o2 = {0 => (resolution, Ideal) }
│ │ │ │       {1 => (resolution, Module)}
│ │ │ │ @@ -85,20 +85,20 @@
│ │ │ │      * Inputs:
│ │ │ │            o X, a _t_y_p_e
│ │ │ │            o Y, a _t_y_p_e
│ │ │ │      * Outputs:
│ │ │ │            o a _v_e_r_t_i_c_a_l_ _l_i_s_t of those methods associated with
│ │ │ │  i5 : methods( Matrix, Matrix )
│ │ │ │  
│ │ │ │ -o5 = {0 => (contract, Matrix, Matrix)                            }
│ │ │ │ -     {1 => (diff, Matrix, Matrix)                                }
│ │ │ │ -     {2 => (diff', Matrix, Matrix)                               }
│ │ │ │ -     {3 => (-, Matrix, Matrix)                                   }
│ │ │ │ +o5 = {0 => (+, Matrix, Matrix)                                   }
│ │ │ │ +     {1 => (-, Matrix, Matrix)                                   }
│ │ │ │ +     {2 => (contract, Matrix, Matrix)                            }
│ │ │ │ +     {3 => (diff, Matrix, Matrix)                                }
│ │ │ │       {4 => (contract', Matrix, Matrix)                           }
│ │ │ │ -     {5 => (+, Matrix, Matrix)                                   }
│ │ │ │ +     {5 => (diff', Matrix, Matrix)                               }
│ │ │ │       {6 => (markedGB, Matrix, Matrix)                            }
│ │ │ │       {7 => (Hom, Matrix, Matrix)                                 }
│ │ │ │       {8 => (==, Matrix, Matrix)                                  }
│ │ │ │       {9 => (*, Matrix, Matrix)                                   }
│ │ │ │       {10 => (|, Matrix, Matrix)                                  }
│ │ │ │       {11 => (||, Matrix, Matrix)                                 }
│ │ │ │       {12 => (subquotient, Matrix, Matrix)                        }
│ │ │ │ @@ -113,18 +113,18 @@
│ │ │ │       {21 => (quotient, Matrix, Matrix)                           }
│ │ │ │       {22 => (quotient', Matrix, Matrix)                          }
│ │ │ │       {23 => (remainder', Matrix, Matrix)                         }
│ │ │ │       {24 => (%, Matrix, Matrix)                                  }
│ │ │ │       {25 => (remainder, Matrix, Matrix)                          }
│ │ │ │       {26 => (pushout, Matrix, Matrix)                            }
│ │ │ │       {27 => (solve, Matrix, Matrix)                              }
│ │ │ │ -     {28 => (pullback, Matrix, Matrix)                           }
│ │ │ │ +     {28 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ │       {29 => (intersect, Matrix, Matrix)                          }
│ │ │ │ -     {30 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ │ -     {31 => (tensor, Matrix, Matrix)                             }
│ │ │ │ +     {30 => (tensor, Matrix, Matrix)                             }
│ │ │ │ +     {31 => (pullback, Matrix, Matrix)                           }
│ │ │ │       {32 => (substitute, Matrix, Matrix)                         }
│ │ │ │       {33 => (yonedaProduct, Matrix, Matrix)                      }
│ │ │ │       {34 => (isShortExactSequence, Matrix, Matrix)               }
│ │ │ │       {35 => (horseshoeResolution, Matrix, Matrix)                }
│ │ │ │       {36 => (connectingExtMap, Module, Matrix, Matrix)           }
│ │ │ │       {37 => (connectingExtMap, Matrix, Matrix, Module)           }
│ │ │ │       {38 => (connectingTorMap, Module, Matrix, Matrix)           }
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_minimal__Betti.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o2 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime C = minimalBetti I
│ │ │ - -- 1.82886s elapsed
│ │ │ + -- 2.45521s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o3 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -125,15 +125,15 @@
│ │ │  
│ │ │  o4 : Ideal of S
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime C = minimalBetti(I, DegreeLimit=>2)
│ │ │ - -- .745147s elapsed
│ │ │ + -- .925445s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7
│ │ │  o5 = total: 1 35 140 385 819 1080 735 196
│ │ │           0: 1  .   .   .   .    .   .   .
│ │ │           1: . 35 140 189  84    .   .   .
│ │ │           2: .  .   . 196 735 1080 735 196
│ │ │  
│ │ │ @@ -146,15 +146,15 @@
│ │ │  
│ │ │  o6 : Ideal of S
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime C = minimalBetti(I, DegreeLimit=>1, LengthLimit=>5)
│ │ │ - -- .0317322s elapsed
│ │ │ + -- .0362305s elapsed
│ │ │  
│ │ │              0  1   2   3  4
│ │ │  o7 = total: 1 35 140 189 84
│ │ │           0: 1  .   .   .  .
│ │ │           1: . 35 140 189 84
│ │ │  
│ │ │  o7 : BettiTally
    
│ │ │ @@ -166,15 +166,15 @@
│ │ │  
│ │ │  o8 : Ideal of S
│ │ │              
    
│ │ │                
    i9 : elapsedTime C = minimalBetti(I, LengthLimit=>5)
│ │ │ - -- 1.20016s elapsed
│ │ │ + -- 3.28835s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5
│ │ │  o9 = total: 1 35 140 385 819 1080
│ │ │           0: 1  .   .   .   .    .
│ │ │           1: . 35 140 189  84    .
│ │ │           2: .  .   . 196 735 1080
│ │ │ ├── html2text {}
│ │ │ │ @@ -43,15 +43,15 @@
│ │ │ │  0,5   1,5   2,5   3,5   4,5   0,6   1,6   2,6   3,6   4,6   5,6
│ │ │ │  i2 : S = ring I
│ │ │ │  
│ │ │ │  o2 = S
│ │ │ │  
│ │ │ │  o2 : PolynomialRing
│ │ │ │  i3 : elapsedTime C = minimalBetti I
│ │ │ │ - -- 1.82886s elapsed
│ │ │ │ + -- 2.45521s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │ │  o3 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ │ @@ -60,40 +60,40 @@
│ │ │ │  o3 : BettiTally
│ │ │ │  One can compute smaller parts of the Betti table, by using _D_e_g_r_e_e_L_i_m_i_t and/or
│ │ │ │  _L_e_n_g_t_h_L_i_m_i_t.
│ │ │ │  i4 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o4 : Ideal of S
│ │ │ │  i5 : elapsedTime C = minimalBetti(I, DegreeLimit=>2)
│ │ │ │ - -- .745147s elapsed
│ │ │ │ + -- .925445s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5   6   7
│ │ │ │  o5 = total: 1 35 140 385 819 1080 735 196
│ │ │ │           0: 1  .   .   .   .    .   .   .
│ │ │ │           1: . 35 140 189  84    .   .   .
│ │ │ │           2: .  .   . 196 735 1080 735 196
│ │ │ │  
│ │ │ │  o5 : BettiTally
│ │ │ │  i6 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o6 : Ideal of S
│ │ │ │  i7 : elapsedTime C = minimalBetti(I, DegreeLimit=>1, LengthLimit=>5)
│ │ │ │ - -- .0317322s elapsed
│ │ │ │ + -- .0362305s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3  4
│ │ │ │  o7 = total: 1 35 140 189 84
│ │ │ │           0: 1  .   .   .  .
│ │ │ │           1: . 35 140 189 84
│ │ │ │  
│ │ │ │  o7 : BettiTally
│ │ │ │  i8 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o8 : Ideal of S
│ │ │ │  i9 : elapsedTime C = minimalBetti(I, LengthLimit=>5)
│ │ │ │ - -- 1.20016s elapsed
│ │ │ │ + -- 3.28835s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5
│ │ │ │  o9 = total: 1 35 140 385 819 1080
│ │ │ │           0: 1  .   .   .   .    .
│ │ │ │           1: . 35 140 189  84    .
│ │ │ │           2: .  .   . 196 735 1080
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_mkdir.html
│ │ │ @@ -72,15 +72,15 @@
│ │ │          
    Description
    
│ │ │          
    Only one directory will be made, so the components of the path p other than the last must already exist.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -91,15 +91,15 @@
│ │ │  o3 = true
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : p = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-10741-0/0/
    
│ │ │ +o1 = /tmp/M2-11391-0/0/
│ │ │              
    
│ │ │                
    i2 : mkdir p
    
│ │ │              
    
│ │ │                
    i4 : (fn = p | "foo") << "hi there" << close
│ │ │  
│ │ │ -o4 = /tmp/M2-10741-0/0/foo
│ │ │ +o4 = /tmp/M2-11391-0/0/foo
│ │ │  
│ │ │  o4 : File
    
│ │ │              
    
│ │ │                
    i5 : get fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,22 +12,22 @@
│ │ │ │      * Consequences:
│ │ │ │            o a directory will be created at the path p
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Only one directory will be made, so the components of the path p other than the
│ │ │ │  last must already exist.
│ │ │ │  i1 : p = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10741-0/0/
│ │ │ │ +o1 = /tmp/M2-11391-0/0/
│ │ │ │  i2 : mkdir p
│ │ │ │  i3 : isDirectory p
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : (fn = p | "foo") << "hi there" << close
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-10741-0/0/foo
│ │ │ │ +o4 = /tmp/M2-11391-0/0/foo
│ │ │ │  
│ │ │ │  o4 : File
│ │ │ │  i5 : get fn
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : removeFile fn
│ │ │ │  i7 : removeDirectory p
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_move__File_lp__String_cm__String_rp.html
│ │ │ @@ -81,52 +81,52 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,32 +20,32 @@
│ │ │ │            o the name of the backup file if one was created, or _n_u_l_l
│ │ │ │      * Consequences:
│ │ │ │            o the file will be moved by creating a new link to the file and
│ │ │ │              removing the old one
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : src = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10615-0/0
│ │ │ │ +o1 = /tmp/M2-11145-0/0
│ │ │ │  i2 : dst = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10615-0/1
│ │ │ │ +o2 = /tmp/M2-11145-0/1
│ │ │ │  i3 : src << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10615-0/0
│ │ │ │ +o3 = /tmp/M2-11145-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : moveFile(src,dst,Verbose=>true)
│ │ │ │ ---moving: /tmp/M2-10615-0/0 -> /tmp/M2-10615-0/1
│ │ │ │ +--moving: /tmp/M2-11145-0/0 -> /tmp/M2-11145-0/1
│ │ │ │  i5 : get dst
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : bak = moveFile(dst,Verbose=>true)
│ │ │ │ ---backup file created: /tmp/M2-10615-0/1.bak
│ │ │ │ +--backup file created: /tmp/M2-11145-0/1.bak
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-10615-0/1.bak
│ │ │ │ +o6 = /tmp/M2-11145-0/1.bak
│ │ │ │  i7 : removeFile bak
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_o_p_y_F_i_l_e
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * moveFile(String)
│ │ │ │      * _m_o_v_e_F_i_l_e_(_S_t_r_i_n_g_,_S_t_r_i_n_g_)
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_nanosleep.html
│ │ │ @@ -51,15 +51,15 @@
│ │ │        
    nanosleep -- sleep for a given number of nanoseconds
    
│ │ │        
    
│ │ │          
    Description
    
│ │ │  nanosleep n -- sleeps for n nanoseconds.        
    
│ │ │                
    i1 : src = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10615-0/0
    
│ │ │ +o1 = /tmp/M2-11145-0/0
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-10615-0/1
    
│ │ │ +o2 = /tmp/M2-11145-0/1
│ │ │              
    
│ │ │                
    i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10615-0/0
│ │ │ +o3 = /tmp/M2-11145-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : moveFile(src,dst,Verbose=>true)
│ │ │ ---moving: /tmp/M2-10615-0/0 -> /tmp/M2-10615-0/1
    
│ │ │ +--moving: /tmp/M2-11145-0/0 -> /tmp/M2-11145-0/1
│ │ │              
    
│ │ │                
    i5 : get dst
│ │ │  
│ │ │  o5 = hi there
    
│ │ │              
    
│ │ │                
    i6 : bak = moveFile(dst,Verbose=>true)
│ │ │ ---backup file created: /tmp/M2-10615-0/1.bak
│ │ │ +--backup file created: /tmp/M2-11145-0/1.bak
│ │ │  
│ │ │ -o6 = /tmp/M2-10615-0/1.bak
    
│ │ │ +o6 = /tmp/M2-11145-0/1.bak
│ │ │              
    
│ │ │                
    i7 : removeFile bak
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : elapsedTime nanosleep 500000000
│ │ │ - -- .500135s elapsed
│ │ │ + -- .500115s elapsed
│ │ │  
│ │ │  o1 = 0
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -4,15 +4,15 @@
│ │ │ │  [q                   ]
│ │ │ │  _n_e_x_t | _p_r_e_v_i_o_u_s | _f_o_r_w_a_r_d | _b_a_c_k_w_a_r_d | _u_p | _i_n_d_e_x | _t_o_c
│ │ │ │  ===============================================================================
│ │ │ │  ************ nnaannoosslleeeepp ---- sslleeeepp ffoorr aa ggiivveenn nnuummbbeerr ooff nnaannoosseeccoonnddss ************
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  nanosleep n -- sleeps for n nanoseconds.
│ │ │ │  i1 : elapsedTime nanosleep 500000000
│ │ │ │ - -- .500135s elapsed
│ │ │ │ + -- .500115s 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);
│ │ │ - -- .640674s elapsed
    
│ │ │ + -- .681356s elapsed
│ │ │              
    
│ │ │                
    i4 : elapsedTime parallelApply(1..100, n -> sort L);
│ │ │ - -- .307919s elapsed
    
│ │ │ + -- .182068s 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.
    
│ │ │          
    
│ │ │          
│ │ │ @@ -150,15 +150,15 @@
│ │ │            
    Note that schedule returns a task, not the result of the computation, which will be accessible only after the task has completed the computation.
    
│ │ │          
│ │ │          
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i11 : t
│ │ │  
│ │ │ -o11 = <<task, running>>
│ │ │ +o11 = <<task, created>>
│ │ │  
│ │ │  o11 : Task
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Use isReady to check whether the result is available yet.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,17 +17,17 @@
│ │ │ │  big computation. If the list is long, it will be split into chunks for each
│ │ │ │  core, reducing the overhead. But the speedup is still limited by the different
│ │ │ │  threads competing for memory, including cpu caches; it is like running
│ │ │ │  Macaulay2 on a computer that is running other big programs at the same time. We
│ │ │ │  can see this using _e_l_a_p_s_e_d_T_i_m_e.
│ │ │ │  i2 : L = random toList (1..10000);
│ │ │ │  i3 : elapsedTime         apply(1..100, n -> sort L);
│ │ │ │ - -- .640674s elapsed
│ │ │ │ + -- .681356s elapsed
│ │ │ │  i4 : elapsedTime parallelApply(1..100, n -> sort L);
│ │ │ │ - -- .307919s elapsed
│ │ │ │ + -- .182068s elapsed
│ │ │ │  You will have to try it on your examples to see how much they speed up.
│ │ │ │  Warning: Threads computing in parallel can give wrong answers if their code is
│ │ │ │  not "thread safe", meaning they make modifications to memory without ensuring
│ │ │ │  the modifications get safely communicated to other threads. (Thread safety can
│ │ │ │  slow computations some.) Currently, modifications to Macaulay2 variables and
│ │ │ │  mutable hash tables are thread safe, but not changes inside mutable lists.
│ │ │ │  Also, access to external libraries such as singular, etc., may not currently be
│ │ │ │ @@ -39,15 +39,15 @@
│ │ │ │  _a_l_l_o_w_a_b_l_e_T_h_r_e_a_d_s, and may be examined and changed as follows. (_a_l_l_o_w_a_b_l_e_T_h_r_e_a_d_s
│ │ │ │  is temporarily increased if necessary inside _p_a_r_a_l_l_e_l_A_p_p_l_y.)
│ │ │ │  i5 : allowableThreads
│ │ │ │  
│ │ │ │  o5 = 5
│ │ │ │  i6 : allowableThreads = maxAllowableThreads
│ │ │ │  
│ │ │ │ -o6 = 7
│ │ │ │ +o6 = 17
│ │ │ │  To run a function in another thread use _s_c_h_e_d_u_l_e, as in the following example.
│ │ │ │  i7 : R = QQ[x,y,z];
│ │ │ │  i8 : I = ideal(x^2 + 2*y^2 - y - 2*z, x^2 - 8*y^2 + 10*z - 1, x^2 - 7*y*z)
│ │ │ │  
│ │ │ │               2     2            2     2             2
│ │ │ │  o8 = ideal (x  + 2y  - y - 2z, x  - 8y  + 10z - 1, x  - 7y*z)
│ │ │ │  
│ │ │ │ @@ -62,15 +62,15 @@
│ │ │ │  o10 = <>
│ │ │ │  
│ │ │ │  o10 : Task
│ │ │ │  Note that _s_c_h_e_d_u_l_e returns a task, not the result of the computation, which
│ │ │ │  will be accessible only after the task has completed the computation.
│ │ │ │  i11 : t
│ │ │ │  
│ │ │ │ -o11 = <>
│ │ │ │ +o11 = <>
│ │ │ │  
│ │ │ │  o11 : Task
│ │ │ │  Use _i_s_R_e_a_d_y to check whether the result is available yet.
│ │ │ │  i12 : isReady t
│ │ │ │  
│ │ │ │  o12 = false
│ │ │ │  To wait for the result and then retrieve it, use _t_a_s_k_R_e_s_u_l_t.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_parallelism_spin_spengine_spcomputations.html
│ │ │ @@ -137,15 +137,15 @@
│ │ │  
│ │ │  o3 : PolynomialRing
│ │ │              
    
│ │ │                
    i4 : elapsedTime minimalBetti I
│ │ │ - -- 2.07143s elapsed
│ │ │ + -- 2.1915s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o4 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -160,15 +160,15 @@
│ │ │  
│ │ │  o5 : Ideal of S
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime minimalBetti(I, ParallelizeByDegree => true)
│ │ │ - -- 1.82839s elapsed
│ │ │ + -- 2.25423s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o6 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -190,15 +190,15 @@
│ │ │  
│ │ │  o8 = 1
    
│ │ │              
    
│ │ │                
    i9 : elapsedTime minimalBetti(I)
│ │ │ - -- 1.75494s elapsed
│ │ │ + -- 2.17331s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o9 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -231,15 +231,15 @@
│ │ │  
│ │ │  o12 : Ideal of S
    
│ │ │              
    
│ │ │                
    i13 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ - -- 2.23687s elapsed
│ │ │ + -- 3.78121s elapsed
│ │ │  
│ │ │         1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o13 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S
│ │ │                                                                                                    
│ │ │        0      1       2        3        4         5         6         7         8        9        10
│ │ │  
│ │ │  o13 : Complex
    
│ │ │ @@ -258,15 +258,15 @@
│ │ │  
│ │ │  o15 : Ideal of S
│ │ │              
    
│ │ │                
    i16 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ - -- 2.55496s elapsed
│ │ │ + -- 2.48371s elapsed
│ │ │  
│ │ │         1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o16 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S
│ │ │                                                                                                    
│ │ │        0      1       2        3        4         5         6         7         8        9        10
│ │ │  
│ │ │  o16 : Complex
    
│ │ │ @@ -299,15 +299,15 @@
│ │ │  
│ │ │  o19 : Ideal of S
│ │ │              
    
│ │ │                
    i20 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 5.136s elapsed
│ │ │ + -- 3.695s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o20 : Matrix S  <-- S
    
│ │ │              
    
│ │ │ @@ -322,15 +322,15 @@
│ │ │  
│ │ │  o22 : Ideal of S
│ │ │              
    
│ │ │                
    i23 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 7.09108s elapsed
│ │ │ + -- 8.58327s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o23 : Matrix S  <-- S
    
│ │ │              
    
│ │ │ @@ -345,15 +345,15 @@
│ │ │  
│ │ │  o25 : Ideal of S
│ │ │              
    
│ │ │                
    i26 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 4.22028s elapsed
│ │ │ + -- 3.71049s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o26 : Matrix S  <-- S
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -396,15 +396,15 @@
│ │ │  o30 : Ideal of ---<|a, b, c|>
│ │ │                 101
│ │ │              
    
│ │ │                
    i31 : elapsedTime NCGB(I, 22);
│ │ │ - -- 1.006s elapsed
│ │ │ + -- 1.04567s elapsed
│ │ │  
│ │ │                 ZZ            1       ZZ            148
│ │ │  o31 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │                101                   101
    
│ │ │              
    
│ │ │                
    i34 : elapsedTime NCGB(I, 22);
│ │ │ - -- 1.19796s elapsed
│ │ │ + -- 1.55933s elapsed
│ │ │  
│ │ │                 ZZ            1       ZZ            148
│ │ │  o34 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │                101                   101
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -93,30 +93,30 @@
│ │ │ │  0,5   1,5   2,5   3,5   4,5   0,6   1,6   2,6   3,6   4,6   5,6
│ │ │ │  i3 : S = ring I
│ │ │ │  
│ │ │ │  o3 = S
│ │ │ │  
│ │ │ │  o3 : PolynomialRing
│ │ │ │  i4 : elapsedTime minimalBetti I
│ │ │ │ - -- 2.07143s elapsed
│ │ │ │ + -- 2.1915s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │ │  o4 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ │           4: .  .   .   .   .    .   .   .   .  .  1
│ │ │ │  
│ │ │ │  o4 : BettiTally
│ │ │ │  i5 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o5 : Ideal of S
│ │ │ │  i6 : elapsedTime minimalBetti(I, ParallelizeByDegree => true)
│ │ │ │ - -- 1.82839s elapsed
│ │ │ │ + -- 2.25423s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │ │  o6 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ │ @@ -126,15 +126,15 @@
│ │ │ │  i7 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o7 : Ideal of S
│ │ │ │  i8 : numTBBThreads = 1
│ │ │ │  
│ │ │ │  o8 = 1
│ │ │ │  i9 : elapsedTime minimalBetti(I)
│ │ │ │ - -- 1.75494s elapsed
│ │ │ │ + -- 2.17331s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │ │  o9 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ │ @@ -149,15 +149,15 @@
│ │ │ │  i11 : numTBBThreads = 0
│ │ │ │  
│ │ │ │  o11 = 0
│ │ │ │  i12 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o12 : Ideal of S
│ │ │ │  i13 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ │ - -- 2.23687s elapsed
│ │ │ │ + -- 3.78121s elapsed
│ │ │ │  
│ │ │ │         1      35      241      841      1781      2464      2294      1432
│ │ │ │  576      135      14
│ │ │ │  o13 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-
│ │ │ │  - S    <-- S    <-- S
│ │ │ │  
│ │ │ │  
│ │ │ │ @@ -168,15 +168,15 @@
│ │ │ │  i14 : numTBBThreads = 1
│ │ │ │  
│ │ │ │  o14 = 1
│ │ │ │  i15 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o15 : Ideal of S
│ │ │ │  i16 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ │ - -- 2.55496s elapsed
│ │ │ │ + -- 2.48371s elapsed
│ │ │ │  
│ │ │ │         1      35      241      841      1781      2464      2294      1432
│ │ │ │  576      135      14
│ │ │ │  o16 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-
│ │ │ │  - S    <-- S    <-- S
│ │ │ │  
│ │ │ │  
│ │ │ │ @@ -195,37 +195,37 @@
│ │ │ │  o18 = S
│ │ │ │  
│ │ │ │  o18 : PolynomialRing
│ │ │ │  i19 : I = ideal random(S^1, S^{4:-5});
│ │ │ │  
│ │ │ │  o19 : Ideal of S
│ │ │ │  i20 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ │ - -- 5.136s elapsed
│ │ │ │ + -- 3.695s elapsed
│ │ │ │  
│ │ │ │                1      108
│ │ │ │  o20 : Matrix S  <-- S
│ │ │ │  i21 : numTBBThreads = 1
│ │ │ │  
│ │ │ │  o21 = 1
│ │ │ │  i22 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o22 : Ideal of S
│ │ │ │  i23 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ │ - -- 7.09108s elapsed
│ │ │ │ + -- 8.58327s elapsed
│ │ │ │  
│ │ │ │                1      108
│ │ │ │  o23 : Matrix S  <-- S
│ │ │ │  i24 : numTBBThreads = 10
│ │ │ │  
│ │ │ │  o24 = 10
│ │ │ │  i25 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o25 : Ideal of S
│ │ │ │  i26 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ │ - -- 4.22028s elapsed
│ │ │ │ + -- 3.71049s elapsed
│ │ │ │  
│ │ │ │                1      108
│ │ │ │  o26 : Matrix S  <-- S
│ │ │ │  For Gröbner basis computation in associative algebras, ParallelizeByDegree is
│ │ │ │  not relevant. In this case, use numTBBThreads to control the amount of
│ │ │ │  parallelism.
│ │ │ │  i27 : needsPackage "AssociativeAlgebras"
│ │ │ │ @@ -246,15 +246,15 @@
│ │ │ │                 2                         2                         2
│ │ │ │  o30 = ideal (5a  + 2b*c + 3c*b, 3a*c + 5b  + 2c*a, 2a*b + 3b*a + 5c )
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o30 : Ideal of ---<|a, b, c|>
│ │ │ │                 101
│ │ │ │  i31 : elapsedTime NCGB(I, 22);
│ │ │ │ - -- 1.006s elapsed
│ │ │ │ + -- 1.04567s elapsed
│ │ │ │  
│ │ │ │                 ZZ            1       ZZ            148
│ │ │ │  o31 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │ │                101                   101
│ │ │ │  i32 : I = ideal I_*
│ │ │ │  
│ │ │ │                 2                         2                         2
│ │ │ │ @@ -263,15 +263,15 @@
│ │ │ │                  ZZ
│ │ │ │  o32 : Ideal of ---<|a, b, c|>
│ │ │ │                 101
│ │ │ │  i33 : numTBBThreads = 1
│ │ │ │  
│ │ │ │  o33 = 1
│ │ │ │  i34 : elapsedTime NCGB(I, 22);
│ │ │ │ - -- 1.19796s elapsed
│ │ │ │ + -- 1.55933s elapsed
│ │ │ │  
│ │ │ │                 ZZ            1       ZZ            148
│ │ │ │  o34 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │ │                101                   101
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_i_n_i_m_a_l_B_e_t_t_i -- minimal betti numbers of (the minimal free resolution of)
│ │ │ │        a homogeneous ideal or module
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_poincare.html
│ │ │ @@ -370,36 +370,36 @@
│ │ │  
│ │ │  o27 = 3
│ │ │              
    
│ │ │                
    i28 : time poincare I
│ │ │ - -- used 0.00277298s (cpu); 2.153e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00166338s (cpu); 1.2346e-05s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o28 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o28 : ZZ[T]
    
│ │ │              
    
│ │ │                
    i29 : time gens gb I;
│ │ │  
│ │ │ -   -- registering gb 19 at 0x7f3fe9e7e540
│ │ │ +   -- registering gb 19 at 0x7fb20a72f380
│ │ │  
│ │ │     -- [gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(2,6)mm{7}(1,4)m{8}(0,2)number of (nonminimal) gb elements = 11
│ │ │     -- number of monomials                = 4186
│ │ │     -- #reduction steps = 38
│ │ │     -- #spairs done = 11
│ │ │     -- ncalls = 10
│ │ │     -- nloop = 29
│ │ │     -- nsaved = 0
│ │ │ -   --  -- used 0.0251999s (cpu); 0.0250353s (thread); 0s (gc)
│ │ │ +   --  -- used 0.0143391s (cpu); 0.0143519s (thread); 0s (gc)
│ │ │  
│ │ │                1      11
│ │ │  o29 : Matrix R  <-- R
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -411,15 +411,15 @@
│ │ │                
    i30 : R = QQ[a..d];
    
│ │ │              
    
│ │ │                
    i31 : I = ideal random(R^1, R^{3:-3});
│ │ │  
│ │ │ -   -- registering gb 20 at 0x7f3fe9e7e380
│ │ │ +   -- registering gb 20 at 0x7fb20a72f000
│ │ │  
│ │ │     -- [gb]number of (nonminimal) gb elements = 0
│ │ │     -- number of monomials                = 0
│ │ │     -- #reduction steps = 0
│ │ │     -- #spairs done = 0
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ │ @@ -428,24 +428,24 @@
│ │ │  o31 : Ideal of R
    
│ │ │              
    
│ │ │                
    i32 : time p = poincare I
│ │ │  
│ │ │ -   -- registering gb 21 at 0x7f3fe9e7e000
│ │ │ +   -- registering gb 21 at 0x7fb207cf1c40
│ │ │  
│ │ │     -- [gb]{3}(3)mmm{4}(2)mm{5}(3)mmm{6}(6)mmoooo{7}(4)mooo{8}(2)oonumber of (nonminimal) gb elements = 11
│ │ │     -- number of monomials                = 267
│ │ │     -- #reduction steps = 236
│ │ │     -- #spairs done = 30
│ │ │     -- ncalls = 10
│ │ │     -- nloop = 20
│ │ │     -- nsaved = 0
│ │ │ -   --  -- used 0.00799718s (cpu); 0.00944086s (thread); 0s (gc)
│ │ │ +   --  -- used 0.00776975s (cpu); 0.00544307s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o32 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o32 : ZZ[T]
    
│ │ │              
    
│ │ │                
    i37 : time gens gb J;
│ │ │  
│ │ │ -   -- registering gb 22 at 0x7f3fe9ab3e00
│ │ │ +   -- registering gb 22 at 0x7fb207cf1a80
│ │ │  
│ │ │     -- [gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(3,7)mmm{7}(3,8)mmm{8}(3,9)mmm{9}(3,9)m
│ │ │     -- mm{10}(2,8)mm{11}(1,5)m{12}(1,3)m{13}(1,3)m{14}(1,3)m{15}(1,3)m{16}(1,3)m
│ │ │     -- {17}(1,3)m{18}(1,3)m{19}(1,3)m{20}(1,3)m{21}(1,3)m{22}(1,3)m{23}(1,3)m{24}(1,3)m
│ │ │     -- {25}(1,3)m{26}(1,3)m{27}(1,3)m{28}(0,2)number of (nonminimal) gb elements = 39
│ │ │     -- number of monomials                = 1051
│ │ │     -- #reduction steps = 284
│ │ │     -- #spairs done = 53
│ │ │     -- ncalls = 46
│ │ │     -- nloop = 54
│ │ │     -- nsaved = 0
│ │ │ -   --  -- used 0.0840427s (cpu); 0.0832054s (thread); 0s (gc)
│ │ │ +   --  -- used 0.0519744s (cpu); 0.0506s (thread); 0s (gc)
│ │ │  
│ │ │                1      39
│ │ │  o37 : Matrix S  <-- S
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -177,66 +177,66 @@
│ │ │ │  o26 = 1 - 3T  + 3T  - T
│ │ │ │  
│ │ │ │  o26 : ZZ[T]
│ │ │ │  i27 : gbTrace = 3
│ │ │ │  
│ │ │ │  o27 = 3
│ │ │ │  i28 : time poincare I
│ │ │ │ - -- used 0.00277298s (cpu); 2.153e-05s (thread); 0s (gc)
│ │ │ │ + -- used 0.00166338s (cpu); 1.2346e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              3     6    9
│ │ │ │  o28 = 1 - 3T  + 3T  - T
│ │ │ │  
│ │ │ │  o28 : ZZ[T]
│ │ │ │  i29 : time gens gb I;
│ │ │ │  
│ │ │ │ -   -- registering gb 19 at 0x7f3fe9e7e540
│ │ │ │ +   -- registering gb 19 at 0x7fb20a72f380
│ │ │ │  
│ │ │ │     -- [gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(2,6)mm{7}(1,4)m{8}(0,2)number of
│ │ │ │  (nonminimal) gb elements = 11
│ │ │ │     -- number of monomials                = 4186
│ │ │ │     -- #reduction steps = 38
│ │ │ │     -- #spairs done = 11
│ │ │ │     -- ncalls = 10
│ │ │ │     -- nloop = 29
│ │ │ │     -- nsaved = 0
│ │ │ │ -   --  -- used 0.0251999s (cpu); 0.0250353s (thread); 0s (gc)
│ │ │ │ +   --  -- used 0.0143391s (cpu); 0.0143519s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                1      11
│ │ │ │  o29 : Matrix R  <-- R
│ │ │ │  In this case, the savings is minimal, but often it can be dramatic. Another
│ │ │ │  important situation is to compute a Gröbner basis using a different monomial
│ │ │ │  order.
│ │ │ │  i30 : R = QQ[a..d];
│ │ │ │  i31 : I = ideal random(R^1, R^{3:-3});
│ │ │ │  
│ │ │ │ -   -- registering gb 20 at 0x7f3fe9e7e380
│ │ │ │ +   -- registering gb 20 at 0x7fb20a72f000
│ │ │ │  
│ │ │ │     -- [gb]number of (nonminimal) gb elements = 0
│ │ │ │     -- number of monomials                = 0
│ │ │ │     -- #reduction steps = 0
│ │ │ │     -- #spairs done = 0
│ │ │ │     -- ncalls = 0
│ │ │ │     -- nloop = 0
│ │ │ │     -- nsaved = 0
│ │ │ │     --
│ │ │ │  o31 : Ideal of R
│ │ │ │  i32 : time p = poincare I
│ │ │ │  
│ │ │ │ -   -- registering gb 21 at 0x7f3fe9e7e000
│ │ │ │ +   -- registering gb 21 at 0x7fb207cf1c40
│ │ │ │  
│ │ │ │     -- [gb]{3}(3)mmm{4}(2)mm{5}(3)mmm{6}(6)mmoooo{7}(4)mooo{8}(2)oonumber of
│ │ │ │  (nonminimal) gb elements = 11
│ │ │ │     -- number of monomials                = 267
│ │ │ │     -- #reduction steps = 236
│ │ │ │     -- #spairs done = 30
│ │ │ │     -- ncalls = 10
│ │ │ │     -- nloop = 20
│ │ │ │     -- nsaved = 0
│ │ │ │ -   --  -- used 0.00799718s (cpu); 0.00944086s (thread); 0s (gc)
│ │ │ │ +   --  -- used 0.00776975s (cpu); 0.00544307s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              3     6    9
│ │ │ │  o32 = 1 - 3T  + 3T  - T
│ │ │ │  
│ │ │ │  o32 : ZZ[T]
│ │ │ │  i33 : S = QQ[a..d, MonomialOrder => Eliminate 2]
│ │ │ │  
│ │ │ │ @@ -281,30 +281,30 @@
│ │ │ │  
│ │ │ │  o35 : ZZ[T]
│ │ │ │  i36 : gbTrace = 3
│ │ │ │  
│ │ │ │  o36 = 3
│ │ │ │  i37 : time gens gb J;
│ │ │ │  
│ │ │ │ -   -- registering gb 22 at 0x7f3fe9ab3e00
│ │ │ │ +   -- registering gb 22 at 0x7fb207cf1a80
│ │ │ │  
│ │ │ │     -- [gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(3,7)mmm{7}(3,8)mmm{8}(3,9)mmm{9}
│ │ │ │  (3,9)m
│ │ │ │     -- mm{10}(2,8)mm{11}(1,5)m{12}(1,3)m{13}(1,3)m{14}(1,3)m{15}(1,3)m{16}(1,3)m
│ │ │ │     -- {17}(1,3)m{18}(1,3)m{19}(1,3)m{20}(1,3)m{21}(1,3)m{22}(1,3)m{23}(1,3)m
│ │ │ │  {24}(1,3)m
│ │ │ │     -- {25}(1,3)m{26}(1,3)m{27}(1,3)m{28}(0,2)number of (nonminimal) gb elements
│ │ │ │  = 39
│ │ │ │     -- number of monomials                = 1051
│ │ │ │     -- #reduction steps = 284
│ │ │ │     -- #spairs done = 53
│ │ │ │     -- ncalls = 46
│ │ │ │     -- nloop = 54
│ │ │ │     -- nsaved = 0
│ │ │ │ -   --  -- used 0.0840427s (cpu); 0.0832054s (thread); 0s (gc)
│ │ │ │ +   --  -- used 0.0519744s (cpu); 0.0506s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                1      39
│ │ │ │  o37 : Matrix S  <-- S
│ │ │ │  i38 : selectInSubring(1, gens gb J)
│ │ │ │  
│ │ │ │  o38 = | 188529931266160087758259645374082357642621166724936033369975727480205
│ │ │ │        -----------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_printing_spto_spa_spfile.html
│ │ │ @@ -97,22 +97,22 @@
│ │ │  o2 : File
│ │ │              
    
│ │ │                
    i3 : fn = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-10932-0/0
    
│ │ │ +o3 = /tmp/M2-11782-0/0
│ │ │              
    
│ │ │                
    i4 : fn << "hi there" << endl << close
│ │ │  
│ │ │ -o4 = /tmp/M2-10932-0/0
│ │ │ +o4 = /tmp/M2-11782-0/0
│ │ │  
│ │ │  o4 : File
    
│ │ │              
    
│ │ │                
    i5 : get fn
│ │ │ @@ -151,15 +151,15 @@
│ │ │  o8 : File
    
│ │ │              
    
│ │ │                
    i9 : fn << f << close
│ │ │  
│ │ │ -o9 = /tmp/M2-10932-0/0
│ │ │ +o9 = /tmp/M2-11782-0/0
│ │ │  
│ │ │  o9 : File
    
│ │ │              
    
│ │ │                
    i10 : get fn
│ │ │ @@ -169,15 +169,15 @@
│ │ │        + 1
    
│ │ │              
    
│ │ │                
    i11 : fn << toExternalString f << close
│ │ │  
│ │ │ -o11 = /tmp/M2-10932-0/0
│ │ │ +o11 = /tmp/M2-11782-0/0
│ │ │  
│ │ │  o11 : File
    
│ │ │              
    
│ │ │                
    i12 : get fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -36,18 +36,18 @@
│ │ │ │  -- ho there --
│ │ │ │  
│ │ │ │  o2 = stdio
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10932-0/0
│ │ │ │ +o3 = /tmp/M2-11782-0/0
│ │ │ │  i4 : fn << "hi there" << endl << close
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-10932-0/0
│ │ │ │ +o4 = /tmp/M2-11782-0/0
│ │ │ │  
│ │ │ │  o4 : File
│ │ │ │  i5 : get fn
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : R = QQ[x]
│ │ │ │  
│ │ │ │ @@ -66,25 +66,25 @@
│ │ │ │   10      9      8       7       6       5       4       3      2
│ │ │ │  x   + 10x  + 45x  + 120x  + 210x  + 252x  + 210x  + 120x  + 45x  + 10x + 1
│ │ │ │  o8 = stdio
│ │ │ │  
│ │ │ │  o8 : File
│ │ │ │  i9 : fn << f << close
│ │ │ │  
│ │ │ │ -o9 = /tmp/M2-10932-0/0
│ │ │ │ +o9 = /tmp/M2-11782-0/0
│ │ │ │  
│ │ │ │  o9 : File
│ │ │ │  i10 : get fn
│ │ │ │  
│ │ │ │  o10 =  10      9      8       7       6       5       4       3      2
│ │ │ │        x   + 10x  + 45x  + 120x  + 210x  + 252x  + 210x  + 120x  + 45x  + 10x
│ │ │ │        + 1
│ │ │ │  i11 : fn << toExternalString f << close
│ │ │ │  
│ │ │ │ -o11 = /tmp/M2-10932-0/0
│ │ │ │ +o11 = /tmp/M2-11782-0/0
│ │ │ │  
│ │ │ │  o11 : File
│ │ │ │  i12 : get fn
│ │ │ │  
│ │ │ │  o12 = x^10+10*x^9+45*x^8+120*x^7+210*x^6+252*x^5+210*x^4+120*x^3+45*x^2+10*x+
│ │ │ │        1
│ │ │ │  i13 : value get fn
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_process__I__D.html
│ │ │ @@ -64,15 +64,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : processID()
│ │ │  
│ │ │ -o1 = 10191
    
│ │ │ +o1 = 10311
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -8,15 +8,15 @@
│ │ │ │      *   Usage:
│ │ │ │              processID()
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the process identifier of the current Macaulay2 process
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : processID()
│ │ │ │  
│ │ │ │ -o1 = 10191
│ │ │ │ +o1 = 10311
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _g_r_o_u_p_I_D -- the process group identifier
│ │ │ │      * _s_e_t_G_r_o_u_p_I_D -- set the process group identifier
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _p_r_o_c_e_s_s_I_D is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_profile.html
│ │ │ @@ -91,35 +91,35 @@
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -131,15 +131,15 @@
│ │ │  
│ │ │  o9 : Sequence
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -173,15 +173,15 @@
│ │ │  o14 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i2 : profileSummary
│ │ │  
│ │ │  o2 = #run  %time   position                         
│ │ │ -     1     94.52   ../../m2/matrix1.m2:279:4-282:58 
│ │ │ -     1     92.12   ../../m2/matrix1.m2:281:22-281:43
│ │ │ -     1     44.16   ../../m2/matrix1.m2:193:25-193:52
│ │ │ -     1     30.59   ../../m2/matrix1.m2:114:5-156:72 
│ │ │ -     1     29.47   ../../m2/matrix1.m2:140:10-155:16
│ │ │ -     1     23.83   ../../m2/matrix1.m2:181:4-181:42 
│ │ │ -     1     22.54   ../../m2/set.m2:127:5-127:61     
│ │ │ -     1     20.86   ../../m2/matrix1.m2:45:10-49:22  
│ │ │ -     1     3.30    ../../m2/matrix1.m2:112:5-112:29 
│ │ │ -     1     2.34    ../../m2/matrix1.m2:141:13-141:78
│ │ │ -     1     2.18    ../../m2/matrix1.m2:96:5-109:11  
│ │ │ -     1     1.42    ../../m2/matrix1.m2:281:7-281:16 
│ │ │ -     1     1.29    ../../m2/matrix1.m2:147:20-147:64
│ │ │ -     1     1.29    ../../m2/matrix1.m2:111:5-111:91 
│ │ │ -     1     1.27    ../../m2/matrix1.m2:276:4-277:73 
│ │ │ -     1     1.02    ../../m2/matrix1.m2:98:10-98:46  
│ │ │ -     1     .97     ../../m2/matrix1.m2:182:4-184:74 
│ │ │ -     1     .81     ../../m2/modules.m2:279:4-279:52 
│ │ │ -     20    .64     ../../m2/matrix1.m2:191:14-192:67
│ │ │ -     20    .47     ../../m2/matrix1.m2:47:43-47:71  
│ │ │ -     1     .0038s  elapsed total                    
    
│ │ │ +     1     93.35   ../../m2/matrix1.m2:279:4-282:58 
│ │ │ +     1     90.65   ../../m2/matrix1.m2:281:22-281:43
│ │ │ +     1     43.01   ../../m2/matrix1.m2:193:25-193:52
│ │ │ +     1     29.94   ../../m2/matrix1.m2:114:5-156:72 
│ │ │ +     1     28.72   ../../m2/matrix1.m2:140:10-155:16
│ │ │ +     1     23.46   ../../m2/matrix1.m2:181:4-181:42 
│ │ │ +     1     22.12   ../../m2/set.m2:127:5-127:61     
│ │ │ +     1     20.2    ../../m2/matrix1.m2:45:10-49:22  
│ │ │ +     1     3.26    ../../m2/matrix1.m2:112:5-112:29 
│ │ │ +     1     2.8     ../../m2/matrix1.m2:141:13-141:78
│ │ │ +     1     2.16    ../../m2/matrix1.m2:96:5-109:11  
│ │ │ +     1     1.69    ../../m2/matrix1.m2:281:7-281:16 
│ │ │ +     1     1.37    ../../m2/matrix1.m2:276:4-277:73 
│ │ │ +     1     1.26    ../../m2/matrix1.m2:147:20-147:64
│ │ │ +     1     1.14    ../../m2/matrix1.m2:111:5-111:91 
│ │ │ +     1     1.10    ../../m2/matrix1.m2:182:4-184:74 
│ │ │ +     1     1.08    ../../m2/matrix1.m2:98:10-98:46  
│ │ │ +     20    .95     ../../m2/matrix1.m2:191:14-192:67
│ │ │ +     20    .67     ../../m2/matrix1.m2:47:43-47:71  
│ │ │ +     20    .60     ../../m2/matrix1.m2:190:17-190:29
│ │ │ +     1     .0035s  elapsed total                    
│ │ │              
    
│ │ │                
    i3 : coverageSummary
│ │ │  
│ │ │  o3 = covered lines:
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,35 +25,35 @@
│ │ │ │                4       5
│ │ │ │  o1 : Matrix ZZ  <-- ZZ
│ │ │ │  Afterwards, running profileSummary and coverageSummary produces easy to read
│ │ │ │  tables summarizing the accumulated data so far in different ways.
│ │ │ │  i2 : profileSummary
│ │ │ │  
│ │ │ │  o2 = #run  %time   position
│ │ │ │ -     1     94.52   ../../m2/matrix1.m2:279:4-282:58
│ │ │ │ -     1     92.12   ../../m2/matrix1.m2:281:22-281:43
│ │ │ │ -     1     44.16   ../../m2/matrix1.m2:193:25-193:52
│ │ │ │ -     1     30.59   ../../m2/matrix1.m2:114:5-156:72
│ │ │ │ -     1     29.47   ../../m2/matrix1.m2:140:10-155:16
│ │ │ │ -     1     23.83   ../../m2/matrix1.m2:181:4-181:42
│ │ │ │ -     1     22.54   ../../m2/set.m2:127:5-127:61
│ │ │ │ -     1     20.86   ../../m2/matrix1.m2:45:10-49:22
│ │ │ │ -     1     3.30    ../../m2/matrix1.m2:112:5-112:29
│ │ │ │ -     1     2.34    ../../m2/matrix1.m2:141:13-141:78
│ │ │ │ -     1     2.18    ../../m2/matrix1.m2:96:5-109:11
│ │ │ │ -     1     1.42    ../../m2/matrix1.m2:281:7-281:16
│ │ │ │ -     1     1.29    ../../m2/matrix1.m2:147:20-147:64
│ │ │ │ -     1     1.29    ../../m2/matrix1.m2:111:5-111:91
│ │ │ │ -     1     1.27    ../../m2/matrix1.m2:276:4-277:73
│ │ │ │ -     1     1.02    ../../m2/matrix1.m2:98:10-98:46
│ │ │ │ -     1     .97     ../../m2/matrix1.m2:182:4-184:74
│ │ │ │ -     1     .81     ../../m2/modules.m2:279:4-279:52
│ │ │ │ -     20    .64     ../../m2/matrix1.m2:191:14-192:67
│ │ │ │ -     20    .47     ../../m2/matrix1.m2:47:43-47:71
│ │ │ │ -     1     .0038s  elapsed total
│ │ │ │ +     1     93.35   ../../m2/matrix1.m2:279:4-282:58
│ │ │ │ +     1     90.65   ../../m2/matrix1.m2:281:22-281:43
│ │ │ │ +     1     43.01   ../../m2/matrix1.m2:193:25-193:52
│ │ │ │ +     1     29.94   ../../m2/matrix1.m2:114:5-156:72
│ │ │ │ +     1     28.72   ../../m2/matrix1.m2:140:10-155:16
│ │ │ │ +     1     23.46   ../../m2/matrix1.m2:181:4-181:42
│ │ │ │ +     1     22.12   ../../m2/set.m2:127:5-127:61
│ │ │ │ +     1     20.2    ../../m2/matrix1.m2:45:10-49:22
│ │ │ │ +     1     3.26    ../../m2/matrix1.m2:112:5-112:29
│ │ │ │ +     1     2.8     ../../m2/matrix1.m2:141:13-141:78
│ │ │ │ +     1     2.16    ../../m2/matrix1.m2:96:5-109:11
│ │ │ │ +     1     1.69    ../../m2/matrix1.m2:281:7-281:16
│ │ │ │ +     1     1.37    ../../m2/matrix1.m2:276:4-277:73
│ │ │ │ +     1     1.26    ../../m2/matrix1.m2:147:20-147:64
│ │ │ │ +     1     1.14    ../../m2/matrix1.m2:111:5-111:91
│ │ │ │ +     1     1.10    ../../m2/matrix1.m2:182:4-184:74
│ │ │ │ +     1     1.08    ../../m2/matrix1.m2:98:10-98:46
│ │ │ │ +     20    .95     ../../m2/matrix1.m2:191:14-192:67
│ │ │ │ +     20    .67     ../../m2/matrix1.m2:47:43-47:71
│ │ │ │ +     20    .60     ../../m2/matrix1.m2:190:17-190:29
│ │ │ │ +     1     .0035s  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 0.173474s (cpu); 0.138239s (thread); 0s (gc)
│ │ │ + -- used 0.247677s (cpu); 0.101492s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = ideal (x  - 53x , x  + 8x , x  - 4x )
│ │ │               2      3   1     3   0     3
│ │ │  
│ │ │  o6 : Ideal of R
    
│ │ │              
    
│ │ │                
    i10 : time randomKRationalPoint(I)
│ │ │ - -- used 0.663337s (cpu); 0.35806s (thread); 0s (gc)
│ │ │ + -- used 0.767075s (cpu); 0.303818s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = ideal (x  - 27x , x  - 16x , x  - 9x , x  + 44x , x  - 52x )
│ │ │                4      5   3      5   2     5   1      5   0      5
│ │ │  
│ │ │  o10 : Ideal of R
    
│ │ │              
    
│ │ │                
    i15 : time (#select(apply(100,i->(degs=apply(decompose(I+ideal random(1,R)),c->degree c);
│ │ │                       #select(degs,d->d==1))),f->f>0))
│ │ │ - -- used 3.65398s (cpu); 1.92733s (thread); 0s (gc)
│ │ │ + -- used 4.37816s (cpu); 1.88301s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 58
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : codim I, degree I
│ │ │ │  
│ │ │ │  o5 = (2, 10)
│ │ │ │  
│ │ │ │  o5 : Sequence
│ │ │ │  i6 : time randomKRationalPoint(I)
│ │ │ │ - -- used 0.173474s (cpu); 0.138239s (thread); 0s (gc)
│ │ │ │ + -- used 0.247677s (cpu); 0.101492s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = ideal (x  - 53x , x  + 8x , x  - 4x )
│ │ │ │               2      3   1     3   0     3
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : R=kk[x_0..x_5];
│ │ │ │  i8 : I=minors(3,random(R^5,R^{3:-1}));
│ │ │ │ @@ -45,15 +45,15 @@
│ │ │ │  o8 : Ideal of R
│ │ │ │  i9 : codim I, degree I
│ │ │ │  
│ │ │ │  o9 = (3, 10)
│ │ │ │  
│ │ │ │  o9 : Sequence
│ │ │ │  i10 : time randomKRationalPoint(I)
│ │ │ │ - -- used 0.663337s (cpu); 0.35806s (thread); 0s (gc)
│ │ │ │ + -- used 0.767075s (cpu); 0.303818s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = ideal (x  - 27x , x  - 16x , x  - 9x , x  + 44x , x  - 52x )
│ │ │ │                4      5   3      5   2     5   1      5   0      5
│ │ │ │  
│ │ │ │  o10 : Ideal of R
│ │ │ │  The claim that $63 \%$ of the intersections contain a K-rational point can be
│ │ │ │  experimentally tested:
│ │ │ │ @@ -69,15 +69,15 @@
│ │ │ │  o13 : RR (of precision 53)
│ │ │ │  i14 : I=ideal random(n,R);
│ │ │ │  
│ │ │ │  o14 : Ideal of R
│ │ │ │  i15 : time (#select(apply(100,i->(degs=apply(decompose(I+ideal random(1,R)),c-
│ │ │ │  >degree c);
│ │ │ │                       #select(degs,d->d==1))),f->f>0))
│ │ │ │ - -- used 3.65398s (cpu); 1.92733s (thread); 0s (gc)
│ │ │ │ + -- used 4.37816s (cpu); 1.88301s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 58
│ │ │ │  ********** WWaayyss ttoo uussee rraannddoommKKRRaattiioonnaallPPooiinntt:: **********
│ │ │ │      * randomKRationalPoint(Ideal)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _r_a_n_d_o_m_K_R_a_t_i_o_n_a_l_P_o_i_n_t is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_read__Directory.html
│ │ │ @@ -68,38 +68,38 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11565-0/0
    
│ │ │ +o1 = /tmp/M2-13075-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-11565-0/0
    
│ │ │ +o2 = /tmp/M2-13075-0/0
│ │ │              
    
│ │ │                
    i3 : (fn = dir | "/" | "foo") << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11565-0/0/foo
│ │ │ +o3 = /tmp/M2-13075-0/0/foo
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : readDirectory dir
│ │ │  
│ │ │ -o4 = {., .., foo}
│ │ │ +o4 = {.., ., foo}
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │                
    i5 : removeFile fn
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,26 +10,26 @@
│ │ │ │      * Inputs:
│ │ │ │            o dir, a _s_t_r_i_n_g, a filename or path to a directory
│ │ │ │      * Outputs:
│ │ │ │            o a _l_i_s_t, the list of filenames stored in the directory
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : dir = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11565-0/0
│ │ │ │ +o1 = /tmp/M2-13075-0/0
│ │ │ │  i2 : makeDirectory dir
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11565-0/0
│ │ │ │ +o2 = /tmp/M2-13075-0/0
│ │ │ │  i3 : (fn = dir | "/" | "foo") << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11565-0/0/foo
│ │ │ │ +o3 = /tmp/M2-13075-0/0/foo
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : readDirectory dir
│ │ │ │  
│ │ │ │ -o4 = {., .., foo}
│ │ │ │ +o4 = {.., ., foo}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : removeFile fn
│ │ │ │  i6 : removeDirectory dir
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_m_o_v_e_D_i_r_e_c_t_o_r_y -- remove a directory
│ │ │ │      * _r_e_m_o_v_e_F_i_l_e -- remove a file
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_reading_spfiles.html
│ │ │ @@ -52,22 +52,22 @@
│ │ │        
    
│ │ │  Sometimes a file will contain a single expression whose value you wish to have access to.  For example, it might be a polynomial produced by another program.  The function get can be used to obtain the entire contents of a file as a single string.  We illustrate this here with a file whose name is expression.        
    
│ │ │  First we create the file by writing the desired text to it.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11107-0/0
    
│ │ │ +o1 = /tmp/M2-12137-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "z^6+3*x*z^4+6*y*z^4+3*x^2*z^2+12*x*y*z^2+12*y^2*z^2+x^3+6*x^2*y+12*x*y^2+8*y^3" << endl << close
│ │ │  
│ │ │ -o2 = /tmp/M2-11107-0/0
│ │ │ +o2 = /tmp/M2-12137-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │  Now we get the contents of the file, as a single string.        
│ │ │            
│ │ │ @@ -116,15 +116,15 @@
│ │ │  Often a file will contain code written in the Macaulay2 language. Let's create such a file.        
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : fn << "sample = 2^100
│ │ │       print sample
│ │ │       " << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11107-0/0
│ │ │ +o7 = /tmp/M2-12137-0/0
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │  Now verify that it contains the desired text with get.        
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -8,20 +8,20 @@
│ │ │ │  Sometimes a file will contain a single expression whose value you wish to have
│ │ │ │  access to. For example, it might be a polynomial produced by another program.
│ │ │ │  The function _g_e_t can be used to obtain the entire contents of a file as a
│ │ │ │  single string. We illustrate this here with a file whose name is expression.
│ │ │ │  First we create the file by writing the desired text to it.
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11107-0/0
│ │ │ │ +o1 = /tmp/M2-12137-0/0
│ │ │ │  i2 : fn <<
│ │ │ │  "z^6+3*x*z^4+6*y*z^4+3*x^2*z^2+12*x*y*z^2+12*y^2*z^2+x^3+6*x^2*y+12*x*y^2+8*y^3"
│ │ │ │  << endl << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11107-0/0
│ │ │ │ +o2 = /tmp/M2-12137-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  Now we get the contents of the file, as a single string.
│ │ │ │  i3 : get fn
│ │ │ │  
│ │ │ │  o3 = z^6+3*x*z^4+6*y*z^4+3*x^2*z^2+12*x*y*z^2+12*y^2*z^2+x^3+6*x^2*y+12*x*y^2
│ │ │ │       +8*y^3
│ │ │ │ @@ -51,15 +51,15 @@
│ │ │ │  o6 : Expression of class Product
│ │ │ │  Often a file will contain code written in the Macaulay2 language. Let's create
│ │ │ │  such a file.
│ │ │ │  i7 : fn << "sample = 2^100
│ │ │ │       print sample
│ │ │ │       " << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11107-0/0
│ │ │ │ +o7 = /tmp/M2-12137-0/0
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  Now verify that it contains the desired text with _g_e_t.
│ │ │ │  i8 : get fn
│ │ │ │  
│ │ │ │  o8 = sample = 2^100
│ │ │ │       print sample
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_readlink.html
│ │ │ @@ -68,15 +68,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,15 +11,15 @@
│ │ │ │            o fn, a _s_t_r_i_n_g, a filename or path to a file
│ │ │ │      * Outputs:
│ │ │ │            o a _s_t_r_i_n_g, the resolved path to a symbolic link, or null if the file
│ │ │ │              was not a symbolic link.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : p = temporaryFileName ()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11806-0/0
│ │ │ │ +o1 = /tmp/M2-13556-0/0
│ │ │ │  i2 : symlinkFile ("foo", p)
│ │ │ │  i3 : readlink p
│ │ │ │  
│ │ │ │  o3 = foo
│ │ │ │  i4 : removeFile p
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_a_l_p_a_t_h -- convert a filename to one passing through no symbolic links
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_realpath.html
│ │ │ @@ -68,57 +68,57 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    
│ │ │                
    i1 : p = temporaryFileName ()
│ │ │  
│ │ │ -o1 = /tmp/M2-11806-0/0
    
│ │ │ +o1 = /tmp/M2-13556-0/0
│ │ │              
    
│ │ │                
    i2 : symlinkFile ("foo", p)
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -130,15 +130,15 @@
│ │ │          
    
│ │ │                
    i1 : realpath "."
│ │ │  
│ │ │ -o1 = /tmp/M2-10191-0/86-rundir/
    
│ │ │ +o1 = /tmp/M2-10311-0/86-rundir/
│ │ │              
    
│ │ │                
    i2 : p = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11825-0/0
    
│ │ │ +o2 = /tmp/M2-13595-0/0
│ │ │              
    
│ │ │                
    i3 : q = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-11825-0/1
    
│ │ │ +o3 = /tmp/M2-13595-0/1
│ │ │              
    
│ │ │                
    i4 : symlinkFile(p,q)
    
│ │ │              
    
│ │ │                
    i5 : p << close
│ │ │  
│ │ │ -o5 = /tmp/M2-11825-0/0
│ │ │ +o5 = /tmp/M2-13595-0/0
│ │ │  
│ │ │  o5 : File
    
│ │ │              
    
│ │ │                
    i6 : readlink q
│ │ │  
│ │ │ -o6 = /tmp/M2-11825-0/0
    
│ │ │ +o6 = /tmp/M2-13595-0/0
│ │ │              
    
│ │ │                
    i7 : realpath q
│ │ │  
│ │ │ -o7 = /tmp/M2-11825-0/0
    
│ │ │ +o7 = /tmp/M2-13595-0/0
│ │ │              
    
│ │ │                
    i8 : removeFile p
    
│ │ │              
    
│ │ │          
    The empty string is interpreted as a reference to the current directory.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : realpath ""
│ │ │  
│ │ │ -o10 = /tmp/M2-10191-0/86-rundir/
    
│ │ │ +o10 = /tmp/M2-10311-0/86-rundir/
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    Caveat
    
│ │ │  Every component of the path must exist in the file system and be accessible to the user. Terminal slashes will be dropped.  Warning: under most operating systems, the value returned is an absolute path (one starting at the root of the file system), but under Solaris, this system call may, in certain circumstances, return a relative path when given a relative path.      
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,39 +12,39 @@
│ │ │ │            o fn, a _s_t_r_i_n_g, a filename, or path to a file
│ │ │ │      * Outputs:
│ │ │ │            o a _s_t_r_i_n_g, a pathname to fn passing through no symbolic links, and
│ │ │ │              ending with a slash if fn refers to a directory
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : realpath "."
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10191-0/86-rundir/
│ │ │ │ +o1 = /tmp/M2-10311-0/86-rundir/
│ │ │ │  i2 : p = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11825-0/0
│ │ │ │ +o2 = /tmp/M2-13595-0/0
│ │ │ │  i3 : q = temporaryFileName()
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11825-0/1
│ │ │ │ +o3 = /tmp/M2-13595-0/1
│ │ │ │  i4 : symlinkFile(p,q)
│ │ │ │  i5 : p << close
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-11825-0/0
│ │ │ │ +o5 = /tmp/M2-13595-0/0
│ │ │ │  
│ │ │ │  o5 : File
│ │ │ │  i6 : readlink q
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11825-0/0
│ │ │ │ +o6 = /tmp/M2-13595-0/0
│ │ │ │  i7 : realpath q
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11825-0/0
│ │ │ │ +o7 = /tmp/M2-13595-0/0
│ │ │ │  i8 : removeFile p
│ │ │ │  i9 : removeFile q
│ │ │ │  The empty string is interpreted as a reference to the current directory.
│ │ │ │  i10 : realpath ""
│ │ │ │  
│ │ │ │ -o10 = /tmp/M2-10191-0/86-rundir/
│ │ │ │ +o10 = /tmp/M2-10311-0/86-rundir/
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  Every component of the path must exist in the file system and be accessible to
│ │ │ │  the user. Terminal slashes will be dropped. Warning: under most operating
│ │ │ │  systems, the value returned is an absolute path (one starting at the root of
│ │ │ │  the file system), but under Solaris, this system call may, in certain
│ │ │ │  circumstances, return a relative path when given a relative path.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_register__Finalizer.html
│ │ │ @@ -77,22 +77,22 @@
│ │ │                
    i1 : for i from 1 to 9 do (x := 0 .. 10000 ; registerFinalizer(x, "-- finalizing sequence #"|i|" --"))
    
│ │ │              
    
│ │ │                
    i2 : collectGarbage() 
│ │ │  --finalization: (1)[3]: -- finalizing sequence #4 --
│ │ │ ---finalization: (2)[7]: -- finalizing sequence #8 --
│ │ │ ---finalization: (3)[4]: -- finalizing sequence #5 --
│ │ │ ---finalization: (4)[1]: -- finalizing sequence #2 --
│ │ │ ---finalization: (5)[6]: -- finalizing sequence #7 --
│ │ │ ---finalization: (6)[5]: -- finalizing sequence #6 --
│ │ │ ---finalization: (7)[8]: -- finalizing sequence #9 --
│ │ │ ---finalization: (8)[2]: -- finalizing sequence #3 --
│ │ │ ---finalization: (9)[0]: -- finalizing sequence #1 --
    
│ │ │ +--finalization: (2)[6]: -- finalizing sequence #7 --
│ │ │ +--finalization: (3)[1]: -- finalizing sequence #2 --
│ │ │ +--finalization: (4)[5]: -- finalizing sequence #6 --
│ │ │ +--finalization: (5)[8]: -- finalizing sequence #9 --
│ │ │ +--finalization: (6)[2]: -- finalizing sequence #3 --
│ │ │ +--finalization: (7)[0]: -- finalizing sequence #1 --
│ │ │ +--finalization: (8)[4]: -- finalizing sequence #5 --
│ │ │ +--finalization: (9)[7]: -- finalizing sequence #8 --
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    Caveat
    
│ │ │  This function should mainly be used for debugging.  Having a large number of finalizers might degrade the performance of the program.  Moreover, registering two or more objects that are members of a circular chain of pointers for finalization will result in a memory leak, with none of the objects in the chain being freed, even if nothing else points to any of them.      
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,22 +15,22 @@
│ │ │ │            o A finalizer is registered with the garbage collector to print a
│ │ │ │              string when that object is collected as garbage
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : for i from 1 to 9 do (x := 0 .. 10000 ; registerFinalizer(x, "-
│ │ │ │  - finalizing sequence #"|i|" --"))
│ │ │ │  i2 : collectGarbage()
│ │ │ │  --finalization: (1)[3]: -- finalizing sequence #4 --
│ │ │ │ ---finalization: (2)[7]: -- finalizing sequence #8 --
│ │ │ │ ---finalization: (3)[4]: -- finalizing sequence #5 --
│ │ │ │ ---finalization: (4)[1]: -- finalizing sequence #2 --
│ │ │ │ ---finalization: (5)[6]: -- finalizing sequence #7 --
│ │ │ │ ---finalization: (6)[5]: -- finalizing sequence #6 --
│ │ │ │ ---finalization: (7)[8]: -- finalizing sequence #9 --
│ │ │ │ ---finalization: (8)[2]: -- finalizing sequence #3 --
│ │ │ │ ---finalization: (9)[0]: -- finalizing sequence #1 --
│ │ │ │ +--finalization: (2)[6]: -- finalizing sequence #7 --
│ │ │ │ +--finalization: (3)[1]: -- finalizing sequence #2 --
│ │ │ │ +--finalization: (4)[5]: -- finalizing sequence #6 --
│ │ │ │ +--finalization: (5)[8]: -- finalizing sequence #9 --
│ │ │ │ +--finalization: (6)[2]: -- finalizing sequence #3 --
│ │ │ │ +--finalization: (7)[0]: -- finalizing sequence #1 --
│ │ │ │ +--finalization: (8)[4]: -- finalizing sequence #5 --
│ │ │ │ +--finalization: (9)[7]: -- finalizing sequence #8 --
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  This function should mainly be used for debugging. Having a large number of
│ │ │ │  finalizers might degrade the performance of the program. Moreover, registering
│ │ │ │  two or more objects that are members of a circular chain of pointers for
│ │ │ │  finalization will result in a memory leak, with none of the objects in the
│ │ │ │  chain being freed, even if nothing else points to any of them.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_remove__Directory.html
│ │ │ @@ -71,29 +71,29 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10779-0/0
    
│ │ │ +o1 = /tmp/M2-11469-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-10779-0/0
    
│ │ │ +o2 = /tmp/M2-11469-0/0
│ │ │              
    
│ │ │                
    i3 : readDirectory dir
│ │ │  
│ │ │ -o3 = {., ..}
│ │ │ +o3 = {.., .}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
    
│ │ │                
    i4 : removeDirectory dir
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,21 +10,21 @@
│ │ │ │      * Inputs:
│ │ │ │            o dir, a _s_t_r_i_n_g, a filename or path to a directory
│ │ │ │      * Consequences:
│ │ │ │            o the directory is removed
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : dir = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10779-0/0
│ │ │ │ +o1 = /tmp/M2-11469-0/0
│ │ │ │  i2 : makeDirectory dir
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10779-0/0
│ │ │ │ +o2 = /tmp/M2-11469-0/0
│ │ │ │  i3 : readDirectory dir
│ │ │ │  
│ │ │ │ -o3 = {., ..}
│ │ │ │ +o3 = {.., .}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : removeDirectory dir
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_a_d_D_i_r_e_c_t_o_r_y -- read the contents of a directory
│ │ │ │      * _m_a_k_e_D_i_r_e_c_t_o_r_y -- make a directory
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_root__Path.html
│ │ │ @@ -65,22 +65,22 @@
│ │ │          
    Description
    
│ │ │          
    This string may be concatenated with an absolute path to get one understandable by external programs.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10283-0/0
    
│ │ │ +o1 = /tmp/M2-10473-0/0
│ │ │              
    
│ │ │                
    i2 : rootPath | fn
│ │ │  
│ │ │ -o2 = /tmp/M2-10283-0/0
    
│ │ │ +o2 = /tmp/M2-10473-0/0
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,18 +11,18 @@
│ │ │ │            o a _s_t_r_i_n_g, the path, as seen by external programs, to the root of
│ │ │ │              the file system seen by Macaulay2
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This string may be concatenated with an absolute path to get one understandable
│ │ │ │  by external programs.
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10283-0/0
│ │ │ │ +o1 = /tmp/M2-10473-0/0
│ │ │ │  i2 : rootPath | fn
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10283-0/0
│ │ │ │ +o2 = /tmp/M2-10473-0/0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_o_o_t_U_R_I
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _r_o_o_t_P_a_t_h is a _s_t_r_i_n_g.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_system.m2:2025:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_root__U__R__I.html
│ │ │ @@ -65,22 +65,22 @@
│ │ │          
      Description
      
│ │ │          
      This string may be concatenated with an absolute path to get one understandable by an external browser.
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11508-0/0
      
│ │ │ +o1 = /tmp/M2-12958-0/0
│ │ │              
      
│ │ │                
      i2 : rootURI | fn
│ │ │  
│ │ │ -o2 = file:///tmp/M2-11508-0/0
      
│ │ │ +o2 = file:///tmp/M2-12958-0/0
│ │ │              
      
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,18 +11,18 @@
│ │ │ │            o a _s_t_r_i_n_g, the path, as seen by an external browser, to the root of
│ │ │ │              the file system seen by Macaulay2
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This string may be concatenated with an absolute path to get one understandable
│ │ │ │  by an external browser.
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11508-0/0
│ │ │ │ +o1 = /tmp/M2-12958-0/0
│ │ │ │  i2 : rootURI | fn
│ │ │ │  
│ │ │ │ -o2 = file:///tmp/M2-11508-0/0
│ │ │ │ +o2 = file:///tmp/M2-12958-0/0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_o_o_t_P_a_t_h
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _r_o_o_t_U_R_I is a _s_t_r_i_n_g.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_system.m2:2041:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_saving_sppolynomials_spand_spmatrices_spin_spfiles.html
│ │ │ @@ -90,22 +90,22 @@
│ │ │  o4 : R-module, submodule of R
│ │ │              
    
│ │ │                
    i5 : f = temporaryFileName()
│ │ │  
│ │ │ -o5 = /tmp/M2-11356-0/0
    
│ │ │ +o5 = /tmp/M2-12646-0/0
│ │ │              
    
│ │ │                
    i6 : f << toString (p,m,M) << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11356-0/0
│ │ │ +o6 = /tmp/M2-12646-0/0
│ │ │  
│ │ │  o6 : File
    
│ │ │              
    
│ │ │                
    i7 : get f
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,18 +28,18 @@
│ │ │ │  
│ │ │ │  o4 = image | x2 x2-y2 xyz7 |
│ │ │ │  
│ │ │ │                               1
│ │ │ │  o4 : R-module, submodule of R
│ │ │ │  i5 : f = temporaryFileName()
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-11356-0/0
│ │ │ │ +o5 = /tmp/M2-12646-0/0
│ │ │ │  i6 : f << toString (p,m,M) << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11356-0/0
│ │ │ │ +o6 = /tmp/M2-12646-0/0
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : get f
│ │ │ │  
│ │ │ │  o7 = (x^3-3*x^2*y+3*x*y^2-y^3-3*x^2+6*x*y-3*y^2+3*x-3*y-1,matrix {{x^2,
│ │ │ │       x^2-y^2, x*y*z^7}},image matrix {{x^2, x^2-y^2, x*y*z^7}})
│ │ │ │  i8 : (p',m',M') = value get f
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_serial__Number.html
│ │ │ @@ -68,22 +68,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : serialNumber asdf
│ │ │  
│ │ │ -o1 = 1426273
    
│ │ │ +o1 = 1526273
│ │ │              
    
│ │ │                
    i2 : serialNumber foo
│ │ │  
│ │ │ -o2 = 1426275
    
│ │ │ +o2 = 1526275
│ │ │              
    
│ │ │                
    i3 : serialNumber ZZ
│ │ │  
│ │ │  o3 = 1000050
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,18 +10,18 @@
│ │ │ │      * Inputs:
│ │ │ │            o x
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the serial number of x
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : serialNumber asdf
│ │ │ │  
│ │ │ │ -o1 = 1426273
│ │ │ │ +o1 = 1526273
│ │ │ │  i2 : serialNumber foo
│ │ │ │  
│ │ │ │ -o2 = 1426275
│ │ │ │ +o2 = 1526275
│ │ │ │  i3 : serialNumber ZZ
│ │ │ │  
│ │ │ │  o3 = 1000050
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _s_e_r_i_a_l_N_u_m_b_e_r is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_solve.html
│ │ │ @@ -366,21 +366,21 @@
│ │ │                  		
│ │ │                
    i28 : B = mutableMatrix(CC_53, N, 2); fillMatrix B;
    
│ │ │              
    
│ │ │                
    i30 : time X = solve(A,B);
│ │ │ - -- used 0.000227156s (cpu); 0.000219682s (thread); 0s (gc)
    
│ │ │ + -- used 0.000227852s (cpu); 0.00021655s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i31 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.000163036s (cpu); 0.000163226s (thread); 0s (gc)
    
│ │ │ + -- used 0.000112839s (cpu); 0.000112646s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i32 : norm(A*X-B)
│ │ │  
│ │ │  o32 = 5.111850690840453e-15
│ │ │ @@ -411,21 +411,21 @@
│ │ │              
    		
│ │ │                
    i36 : B = mutableMatrix(CC_100, N, 2); fillMatrix B;
    
│ │ │              
    
│ │ │                
    i38 : time X = solve(A,B);
│ │ │ - -- used 0.482514s (cpu); 0.3008s (thread); 0s (gc)
    
│ │ │ + -- used 0.135732s (cpu); 0.13574s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i39 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.237275s (cpu); 0.23721s (thread); 0s (gc)
    
│ │ │ + -- used 0.138289s (cpu); 0.138302s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i40 : norm(A*X-B)
│ │ │  
│ │ │  o40 = 1.491578274689709814082355885932e-28
│ │ │ ├── html2text {}
│ │ │ │ @@ -192,33 +192,33 @@
│ │ │ │  i24 : printingPrecision = 4;
│ │ │ │  i25 : N = 40
│ │ │ │  
│ │ │ │  o25 = 40
│ │ │ │  i26 : A = mutableMatrix(CC_53, N, N); fillMatrix A;
│ │ │ │  i28 : B = mutableMatrix(CC_53, N, 2); fillMatrix B;
│ │ │ │  i30 : time X = solve(A,B);
│ │ │ │ - -- used 0.000227156s (cpu); 0.000219682s (thread); 0s (gc)
│ │ │ │ + -- used 0.000227852s (cpu); 0.00021655s (thread); 0s (gc)
│ │ │ │  i31 : time X = solve(A,B, MaximalRank=>true);
│ │ │ │ - -- used 0.000163036s (cpu); 0.000163226s (thread); 0s (gc)
│ │ │ │ + -- used 0.000112839s (cpu); 0.000112646s (thread); 0s (gc)
│ │ │ │  i32 : norm(A*X-B)
│ │ │ │  
│ │ │ │  o32 = 5.111850690840453e-15
│ │ │ │  
│ │ │ │  o32 : RR (of precision 53)
│ │ │ │  Over higher precision RR or CC, these routines will be much slower than the
│ │ │ │  lower precision LAPACK routines.
│ │ │ │  i33 : N = 100
│ │ │ │  
│ │ │ │  o33 = 100
│ │ │ │  i34 : A = mutableMatrix(CC_100, N, N); fillMatrix A;
│ │ │ │  i36 : B = mutableMatrix(CC_100, N, 2); fillMatrix B;
│ │ │ │  i38 : time X = solve(A,B);
│ │ │ │ - -- used 0.482514s (cpu); 0.3008s (thread); 0s (gc)
│ │ │ │ + -- used 0.135732s (cpu); 0.13574s (thread); 0s (gc)
│ │ │ │  i39 : time X = solve(A,B, MaximalRank=>true);
│ │ │ │ - -- used 0.237275s (cpu); 0.23721s (thread); 0s (gc)
│ │ │ │ + -- used 0.138289s (cpu); 0.138302s (thread); 0s (gc)
│ │ │ │  i40 : norm(A*X-B)
│ │ │ │  
│ │ │ │  o40 = 1.491578274689709814082355885932e-28
│ │ │ │  
│ │ │ │  o40 : RR (of precision 100)
│ │ │ │  Giving the option ClosestFit=>true, in the case when the field is RR or CC,
│ │ │ │  uses a least squares algorithm to find a best fit solution.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_symlink__Directory_lp__String_cm__String_rp.html
│ │ │ @@ -80,93 +80,93 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : src = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-11147-0/0/
    
│ │ │ +o1 = /tmp/M2-12217-0/0/
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-11147-0/1/
    
│ │ │ +o2 = /tmp/M2-12217-0/1/
│ │ │              
    
│ │ │                
    i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-11147-0/0/a/
    
│ │ │ +o3 = /tmp/M2-12217-0/0/a/
│ │ │              
    
│ │ │                
    i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-11147-0/0/b/
    
│ │ │ +o4 = /tmp/M2-12217-0/0/b/
│ │ │              
    
│ │ │                
    i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-11147-0/0/b/c/
    
│ │ │ +o5 = /tmp/M2-12217-0/0/b/c/
│ │ │              
    
│ │ │                
    i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11147-0/0/a/f
│ │ │ +o6 = /tmp/M2-12217-0/0/a/f
│ │ │  
│ │ │  o6 : File
    
│ │ │              
    
│ │ │                
    i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11147-0/0/a/g
│ │ │ +o7 = /tmp/M2-12217-0/0/a/g
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │                
    i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-11147-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-12217-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
    
│ │ │              
    
│ │ │                
    i9 : symlinkDirectory(src,dst,Verbose=>true)
│ │ │ ---symlinking: ../../../0/b/c/g -> /tmp/M2-11147-0/1/b/c/g
│ │ │ ---symlinking: ../../0/a/g -> /tmp/M2-11147-0/1/a/g
│ │ │ ---symlinking: ../../0/a/f -> /tmp/M2-11147-0/1/a/f
    
│ │ │ +--symlinking: ../../0/a/g -> /tmp/M2-12217-0/1/a/g
│ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-12217-0/1/a/f
│ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-12217-0/1/b/c/g
│ │ │              
    
│ │ │                
    i10 : get (dst|"b/c/g")
│ │ │  
│ │ │  o10 = ho there
    
│ │ │              
    
│ │ │                
    i11 : symlinkDirectory(src,dst,Verbose=>true,Undo=>true)
│ │ │ ---unsymlinking: ../../../0/b/c/g -> /tmp/M2-11147-0/1/b/c/g
│ │ │ ---unsymlinking: ../../0/a/g -> /tmp/M2-11147-0/1/a/g
│ │ │ ---unsymlinking: ../../0/a/f -> /tmp/M2-11147-0/1/a/f
    
│ │ │ +--unsymlinking: ../../0/a/g -> /tmp/M2-12217-0/1/a/g
│ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-12217-0/1/a/f
│ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-12217-0/1/b/c/g
│ │ │              
    
│ │ │  Now we remove the files and directories we created.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i12 : rm = d -> if isDirectory d then removeDirectory d else removeFile d
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,53 +30,53 @@
│ │ │ │            o The directory tree rooted at src is duplicated by a directory tree
│ │ │ │              rooted at dst. The files in the source tree are represented by
│ │ │ │              relative symbolic links in the destination tree to the original
│ │ │ │              files in the source tree.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : src = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11147-0/0/
│ │ │ │ +o1 = /tmp/M2-12217-0/0/
│ │ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11147-0/1/
│ │ │ │ +o2 = /tmp/M2-12217-0/1/
│ │ │ │  i3 : makeDirectory (src|"a/")
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11147-0/0/a/
│ │ │ │ +o3 = /tmp/M2-12217-0/0/a/
│ │ │ │  i4 : makeDirectory (src|"b/")
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11147-0/0/b/
│ │ │ │ +o4 = /tmp/M2-12217-0/0/b/
│ │ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-11147-0/0/b/c/
│ │ │ │ +o5 = /tmp/M2-12217-0/0/b/c/
│ │ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11147-0/0/a/f
│ │ │ │ +o6 = /tmp/M2-12217-0/0/a/f
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11147-0/0/a/g
│ │ │ │ +o7 = /tmp/M2-12217-0/0/a/g
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │ │  
│ │ │ │ -o8 = /tmp/M2-11147-0/0/b/c/g
│ │ │ │ +o8 = /tmp/M2-12217-0/0/b/c/g
│ │ │ │  
│ │ │ │  o8 : File
│ │ │ │  i9 : symlinkDirectory(src,dst,Verbose=>true)
│ │ │ │ ---symlinking: ../../../0/b/c/g -> /tmp/M2-11147-0/1/b/c/g
│ │ │ │ ---symlinking: ../../0/a/g -> /tmp/M2-11147-0/1/a/g
│ │ │ │ ---symlinking: ../../0/a/f -> /tmp/M2-11147-0/1/a/f
│ │ │ │ +--symlinking: ../../0/a/g -> /tmp/M2-12217-0/1/a/g
│ │ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-12217-0/1/a/f
│ │ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-12217-0/1/b/c/g
│ │ │ │  i10 : get (dst|"b/c/g")
│ │ │ │  
│ │ │ │  o10 = ho there
│ │ │ │  i11 : symlinkDirectory(src,dst,Verbose=>true,Undo=>true)
│ │ │ │ ---unsymlinking: ../../../0/b/c/g -> /tmp/M2-11147-0/1/b/c/g
│ │ │ │ ---unsymlinking: ../../0/a/g -> /tmp/M2-11147-0/1/a/g
│ │ │ │ ---unsymlinking: ../../0/a/f -> /tmp/M2-11147-0/1/a/f
│ │ │ │ +--unsymlinking: ../../0/a/g -> /tmp/M2-12217-0/1/a/g
│ │ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-12217-0/1/a/f
│ │ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-12217-0/1/b/c/g
│ │ │ │  Now we remove the files and directories we created.
│ │ │ │  i12 : rm = d -> if isDirectory d then removeDirectory d else removeFile d
│ │ │ │  
│ │ │ │  o12 = rm
│ │ │ │  
│ │ │ │  o12 : FunctionClosure
│ │ │ │  i13 : scan(reverse findFiles src, rm)
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_symlink__File.html
│ │ │ @@ -72,15 +72,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,15 +12,15 @@
│ │ │ │            o dst, a _s_t_r_i_n_g
│ │ │ │      * Consequences:
│ │ │ │            o a symbolic link at the location in the directory tree specified by
│ │ │ │              dst is created, pointing to src
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11204-0/0
│ │ │ │ +o1 = /tmp/M2-12334-0/0
│ │ │ │  i2 : symlinkFile("qwert", fn)
│ │ │ │  i3 : fileExists fn
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : readlink fn
│ │ │ │  
│ │ │ │  o4 = qwert
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_temporary__File__Name.html
│ │ │ @@ -64,22 +64,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │  The file name is so unique that even with various suffixes appended, no collision with existing files will occur.  The files will be removed when the program terminates, unless it terminates as the result of an error.        
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11204-0/0
    
│ │ │ +o1 = /tmp/M2-12334-0/0
│ │ │              
    
│ │ │                
    i2 : symlinkFile("qwert", fn)
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : temporaryFileName () | ".tex"
│ │ │  
│ │ │ -o1 = /tmp/M2-12169-0/0.tex
    
│ │ │ +o1 = /tmp/M2-14309-0/0.tex
│ │ │              
    
│ │ │                
    i2 : temporaryFileName () | ".html"
│ │ │  
│ │ │ -o2 = /tmp/M2-12169-0/1.html
    
│ │ │ +o2 = /tmp/M2-14309-0/1.html
│ │ │              
    
│ │ │          
    This function will work under Unix, and also under Windows if you have a directory on the same drive called /tmp.
    
│ │ │          
    If the name of the temporary file will be given to an external program, it may be necessary to concatenate it with rootPath or rootURI to enable the external program to find the file.
    
│ │ │          
    The temporary file name is derived from the value of the environment variable TMPDIR, if it has one.
    
│ │ │          
    If fork is used, then the parent and child Macaulay2 processes will each remove their own temporary files upon termination, with the parent removing any files created before fork was called.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,18 +11,18 @@
│ │ │ │            o a unique temporary file name.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  The file name is so unique that even with various suffixes appended, no
│ │ │ │  collision with existing files will occur. The files will be removed when the
│ │ │ │  program terminates, unless it terminates as the result of an error.
│ │ │ │  i1 : temporaryFileName () | ".tex"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12169-0/0.tex
│ │ │ │ +o1 = /tmp/M2-14309-0/0.tex
│ │ │ │  i2 : temporaryFileName () | ".html"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12169-0/1.html
│ │ │ │ +o2 = /tmp/M2-14309-0/1.html
│ │ │ │  This function will work under Unix, and also under Windows if you have a
│ │ │ │  directory on the same drive called /tmp.
│ │ │ │  If the name of the temporary file will be given to an external program, it may
│ │ │ │  be necessary to concatenate it with _r_o_o_t_P_a_t_h or _r_o_o_t_U_R_I to enable the external
│ │ │ │  program to find the file.
│ │ │ │  The temporary file name is derived from the value of the environment variable
│ │ │ │  TMPDIR, if it has one.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_time.html
│ │ │ @@ -59,15 +59,15 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │  time e evaluates e, prints the amount of cpu time used, and returns the value of e.  The time used by the the current thread and garbage collection during the evaluation of e is also shown.        
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : time 3^30
│ │ │ - -- used 2.18e-05s (cpu); 1.1101e-05s (thread); 0s (gc)
│ │ │ + -- used 1.9043e-05s (cpu); 5.659e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = 205891132094649
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -7,15 +7,15 @@
│ │ │ │      *   Usage:
│ │ │ │              time e
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  time e evaluates e, prints the amount of cpu time used, and returns the value
│ │ │ │  of e. The time used by the the current thread and garbage collection during the
│ │ │ │  evaluation of e is also shown.
│ │ │ │  i1 : time 3^30
│ │ │ │ - -- used 2.18e-05s (cpu); 1.1101e-05s (thread); 0s (gc)
│ │ │ │ + -- used 1.9043e-05s (cpu); 5.659e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = 205891132094649
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_i_m_i_n_g -- time a computation
│ │ │ │      * _c_p_u_T_i_m_e -- seconds of cpu time used since Macaulay2 began
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_i_n_g -- time a computation using time elapsed
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_e -- time a computation including time elapsed
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_timing.html
│ │ │ @@ -54,24 +54,24 @@
│ │ │  timing e evaluates e and returns a list of type Time of the form {t,v}, where t is the number of seconds of cpu timing used, and v is the value of the expression.        
    
│ │ │  The default method for printing such timing results is to display the timing separately in a comment below the computed value.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : timing 3^30
│ │ │  
│ │ │  o1 = 205891132094649
│ │ │ -     -- .000018144 seconds
│ │ │ +     -- .000019906 seconds
│ │ │  
│ │ │  o1 : Time
    
│ │ │              
    
│ │ │                
    i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{.000018144, 205891132094649}
    
│ │ │ +o2 = Time{.000019906, 205891132094649}
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,20 +10,20 @@
│ │ │ │  is the number of seconds of cpu timing used, and v is the value of the
│ │ │ │  expression.
│ │ │ │  The default method for printing such timing results is to display the timing
│ │ │ │  separately in a comment below the computed value.
│ │ │ │  i1 : timing 3^30
│ │ │ │  
│ │ │ │  o1 = 205891132094649
│ │ │ │ -     -- .000018144 seconds
│ │ │ │ +     -- .000019906 seconds
│ │ │ │  
│ │ │ │  o1 : Time
│ │ │ │  i2 : peek oo
│ │ │ │  
│ │ │ │ -o2 = Time{.000018144, 205891132094649}
│ │ │ │ +o2 = Time{.000019906, 205891132094649}
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _T_i_m_e -- the class of all timing results
│ │ │ │      * _t_i_m_e -- time a computation
│ │ │ │      * _c_p_u_T_i_m_e -- seconds of cpu time used since Macaulay2 began
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_i_n_g -- time a computation using time elapsed
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_e -- time a computation including time elapsed
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_version.html
│ │ │ @@ -103,15 +103,15 @@
│ │ │                 "memtailor version" => 1.1
│ │ │                 "mpfi version" => 1.5.4
│ │ │                 "mpfr version" => 4.2.2
│ │ │                 "mpsolve version" => 3.2.2
│ │ │                 "mysql version" => not present
│ │ │                 "normaliz version" => 3.11.0
│ │ │                 "ntl version" => 11.5.1
│ │ │ -               "operating system release" => 6.12.57+deb13-amd64
│ │ │ +               "operating system release" => 6.12.57+deb13-cloud-amd64
│ │ │                 "operating system" => Linux
│ │ │                 "packages" => Style FirstPackage Macaulay2Doc Parsing Classic Browse Benchmark Text SimpleDoc PackageTemplate Saturation PrimaryDecomposition FourierMotzkin Dmodules WeylAlgebras HolonomicSystems BernsteinSato ConnectionMatrices Depth Elimination GenericInitialIdeal IntegralClosure HyperplaneArrangements LexIdeals Markov NoetherNormalization Points ReesAlgebra Regularity SchurRings SymmetricPolynomials SchurFunctors SimplicialComplexes LLLBases TangentCone ChainComplexExtras Varieties Schubert2 PushForward LocalRings PruneComplex BoijSoederberg BGG Bruns InvolutiveBases ConwayPolynomials EdgeIdeals FourTiTwo StatePolytope Polyhedra Truncations Polymake gfanInterface PieriMaps Normaliz Posets XML OpenMath SCSCP RationalPoints MapleInterface ConvexInterface SRdeformations NumericalAlgebraicGeometry BeginningMacaulay2 FormalGroupLaws Graphics WeylGroups HodgeIntegrals Cyclotomic Binomials Kronecker Nauty ToricVectorBundles ModuleDeformations PHCpack SimplicialDecomposability BooleanGB AdjointIdeal Parametrization Serialization NAGtypes NormalToricVarieties DGAlgebras Graphs GraphicalModels BIBasis KustinMiller Units NautyGraphs VersalDeformations CharacteristicClasses RandomIdeals RandomObjects RandomPlaneCurves RandomSpaceCurves RandomGenus14Curves RandomCanonicalCurves RandomCurves TensorComplexes MonomialAlgebras QthPower EliminationMatrices EllipticIntegrals Triplets CompleteIntersectionResolutions EagonResolution MCMApproximations MultiplierIdeals InvariantRing QuillenSuslin EnumerationCurves Book3264Examples WeilDivisors EllipticCurves HighestWeights MinimalPrimes Bertini TorAlgebra Permanents BinomialEdgeIdeals TateOnProducts LatticePolytopes FiniteFittingIdeals HigherCIOperators LieAlgebraRepresentations ConformalBlocks M0nbar AnalyzeSheafOnP1 MultiplierIdealsDim2 RunExternalM2 NumericalSchubertCalculus ToricTopology Cremona Resultants VectorFields SLPexpressions Miura ResidualIntersections Visualize EquivariantGB ExampleSystems RationalMaps FastMinors RandomPoints SwitchingFields SpectralSequences SectionRing OldPolyhedra OldToricVectorBundles K3Carpets ChainComplexOperations NumericalCertification PhylogeneticTrees MonodromySolver ReactionNetworks PackageCitations NumericSolutions GradedLieAlgebras InverseSystems Pullback EngineTests SVDComplexes RandomComplexes CohomCalg Topcom Triangulations ReflexivePolytopesDB AbstractToricVarieties TestIdeals FrobeniusThresholds NonPrincipalTestIdeals Seminormalization AlgebraicSplines TriangularSets Chordal Tropical SymbolicPowers Complexes OldChainComplexes GroebnerWalk RandomMonomialIdeals Matroids NumericalImplicitization NonminimalComplexes CoincidentRootLoci RelativeCanonicalResolution RandomCurvesOverVerySmallFiniteFields StronglyStableIdeals SLnEquivariantMatrices CorrespondenceScrolls NCAlgebra SpaceCurves ExteriorIdeals ToricInvariants SegreClasses SemidefiniteProgramming SumsOfSquares MultiGradedRationalMap AssociativeAlgebras VirtualResolutions Quasidegrees DiffAlg DeterminantalRepresentations FGLM SpechtModule SchurComplexes SimplicialPosets SlackIdeals PositivityToricBundles SparseResultants DecomposableSparseSystems MixedMultiplicity PencilsOfQuadrics ThreadedGB AdjunctionForSurfaces VectorGraphics GKMVarieties MonomialIntegerPrograms NoetherianOperators Hadamard StatGraphs GraphicalModelsMLE EigenSolver MultiplicitySequence ResolutionsOfStanleyReisnerRings NumericalLinearAlgebra ResLengthThree MonomialOrbits MultiprojectiveVarieties SpecialFanoFourfolds RationalPoints2 SuperLinearAlgebra SubalgebraBases AInfinity LinearTruncations ThinSincereQuivers Python BettiCharacters Jets FunctionFieldDesingularization HomotopyLieAlgebra TSpreadIdeals RealRoots ExteriorModules K3Surfaces GroebnerStrata QuaternaryQuartics CotangentSchubert OnlineLookup MergeTeX Probability Isomorphism CodingTheory WhitneyStratifications JSON ForeignFunctions GeometricDecomposability PseudomonomialPrimaryDecomposition PolyominoIdeals MatchingFields CellularResolutions SagbiGbDetection A1BrouwerDegrees QuadraticIdealExamplesByRoos TerraciniLoci MatrixSchubert RInterface OIGroebnerBases PlaneCurveLinearSeries Valuations SchurVeronese VNumber TropicalToric MultigradedBGG AbstractSimplicialComplexes MultigradedImplicitization Msolve Permutations SCMAlgebras NumericalSemigroups ExteriorExtensions Oscillators IncidenceCorrespondenceCohomology ToricHigherDirectImages Brackets IntegerProgramming GameTheory AllMarkovBases Tableaux CpMackeyFunctors JSONRPC MatrixFactorizations PathSignatures
│ │ │                 "pointer size" => 8
│ │ │                 "python version" => 3.13.11
│ │ │                 "readline version" => 8.3
│ │ │                 "scscp version" => not present
│ │ │                 "tbb version" => 2022.1
│ │ │ ├── html2text {}
│ │ │ │ @@ -64,15 +64,15 @@
│ │ │ │                 "memtailor version" => 1.1
│ │ │ │                 "mpfi version" => 1.5.4
│ │ │ │                 "mpfr version" => 4.2.2
│ │ │ │                 "mpsolve version" => 3.2.2
│ │ │ │                 "mysql version" => not present
│ │ │ │                 "normaliz version" => 3.11.0
│ │ │ │                 "ntl version" => 11.5.1
│ │ │ │ -               "operating system release" => 6.12.57+deb13-amd64
│ │ │ │ +               "operating system release" => 6.12.57+deb13-cloud-amd64
│ │ │ │                 "operating system" => Linux
│ │ │ │                 "packages" => Style FirstPackage Macaulay2Doc Parsing Classic
│ │ │ │  Browse Benchmark Text SimpleDoc PackageTemplate Saturation PrimaryDecomposition
│ │ │ │  FourierMotzkin Dmodules WeylAlgebras HolonomicSystems BernsteinSato
│ │ │ │  ConnectionMatrices Depth Elimination GenericInitialIdeal IntegralClosure
│ │ │ │  HyperplaneArrangements LexIdeals Markov NoetherNormalization Points ReesAlgebra
│ │ │ │  Regularity SchurRings SymmetricPolynomials SchurFunctors SimplicialComplexes
│ │ ├── ./usr/share/doc/Macaulay2/Markov/example-output/___Markov.out
│ │ │ @@ -70,15 +70,15 @@
│ │ │       |   1,2,1,2 2,2,1,1    1,2,1,1 2,2,1,2|   1,2,2,2 2,2,2,1    1,2,2,1 2,2,2,2|
│ │ │       +-------------------------------------+-------------------------------------+
│ │ │       |- p       p        + p       p       |- p       p        + p       p       |
│ │ │       |   1,1,2,1 1,2,1,1    1,1,1,1 1,2,2,1|   1,1,2,2 1,2,1,2    1,1,1,2 1,2,2,2|
│ │ │       +-------------------------------------+-------------------------------------+
│ │ │  
│ │ │  i8 : time netList primaryDecomposition J
│ │ │ - -- used 3.38775s (cpu); 1.60175s (thread); 0s (gc)
│ │ │ + -- used 3.77883s (cpu); 1.63154s (thread); 0s (gc)
│ │ │  
│ │ │       +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  o8 = |ideal (p       , p       , p       , p       , p       p        - p       p       , p       p        - p       p       )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
│ │ │       |        1,2,2,2   1,2,2,1   1,2,1,2   1,2,1,1   1,1,2,2 2,1,2,1    1,1,2,1 2,1,2,2   1,1,1,2 2,1,1,1    1,1,1,1 2,1,1,2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
│ │ │       +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │       |ideal (p       , p       , p       , p       , p       p        - p       p       , p       p        - p       p       )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
│ │ │       |        1,2,2,2   1,2,2,1   1,1,2,2   1,1,2,1   1,2,1,2 2,2,1,1    1,2,1,1 2,2,1,2   1,1,1,2 2,1,1,1    1,1,1,1 2,1,1,2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
│ │ ├── ./usr/share/doc/Macaulay2/Markov/html/index.html
│ │ │ @@ -161,15 +161,15 @@
│ │ │          
      
│ │ │            
      This ideal has 5 primary components.  The first is the one that has statistical significance. The significance of the other components is still poorly understood.
      
│ │ │          
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i8 : time netList primaryDecomposition J
│ │ │ - -- used 3.38775s (cpu); 1.60175s (thread); 0s (gc)
│ │ │ + -- used 3.77883s (cpu); 1.63154s (thread); 0s (gc)
│ │ │  
│ │ │       +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  o8 = |ideal (p       , p       , p       , p       , p       p        - p       p       , p       p        - p       p       )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
│ │ │       |        1,2,2,2   1,2,2,1   1,2,1,2   1,2,1,1   1,1,2,2 2,1,2,1    1,1,2,1 2,1,2,2   1,1,1,2 2,1,1,1    1,1,1,1 2,1,1,2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
│ │ │       +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │       |ideal (p       , p       , p       , p       , p       p        - p       p       , p       p        - p       p       )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
│ │ │       |        1,2,2,2   1,2,2,1   1,1,2,2   1,1,2,1   1,2,1,2 2,2,1,1    1,2,1,1 2,2,1,2   1,1,1,2 2,1,1,1    1,1,1,1 2,1,1,2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
│ │ │ ├── html2text {}
│ │ │ │ @@ -102,15 +102,15 @@
│ │ │ │  1,2,2,2|
│ │ │ │       +-------------------------------------+-----------------------------------
│ │ │ │  --+
│ │ │ │  This ideal has 5 primary components. The first is the one that has statistical
│ │ │ │  significance. The significance of the other components is still poorly
│ │ │ │  understood.
│ │ │ │  i8 : time netList primaryDecomposition J
│ │ │ │ - -- used 3.38775s (cpu); 1.60175s (thread); 0s (gc)
│ │ │ │ + -- used 3.77883s (cpu); 1.63154s (thread); 0s (gc)
│ │ │ │  
│ │ │ │       +-------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/___Investigating_sp__A__S__M_spvarieties.out
│ │ │ @@ -212,17 +212,17 @@
│ │ │        | 1 -1 1 |
│ │ │        | 0 1  0 |
│ │ │  
│ │ │                 3       3
│ │ │  o22 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i23 : time schubertRegularity B
│ │ │ - -- used 0.100646s (cpu); 0.0344742s (thread); 0s (gc)
│ │ │ + -- used 0.104907s (cpu); 0.0345186s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = 1
│ │ │  
│ │ │  i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ - -- used 0.0147124s (cpu); 0.0147191s (thread); 0s (gc)
│ │ │ + -- used 0.0177392s (cpu); 0.017746s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = 1
│ │ │  
│ │ │  i25 :
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/___Investigating_spmatrix_sp__Schubert_spvarieties.out
│ │ │ @@ -178,17 +178,17 @@
│ │ │        z   z   z   , z   z   z    - z   z   , z   z   z    - z   z   )
│ │ │         1,2 1,3 2,4   1,2 1,4 2,2    1,2 2,4   1,2 1,3 2,2    1,2 2,3
│ │ │  
│ │ │  o15 : Ideal of QQ[z   ..z   ]
│ │ │                     1,1   5,5
│ │ │  
│ │ │  i16 : time schubertRegularity p
│ │ │ - -- used 0.000285165s (cpu); 0.000279514s (thread); 0s (gc)
│ │ │ + -- used 0.000290339s (cpu); 0.000285063s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 5
│ │ │  
│ │ │  i17 : time regularity comodule I
│ │ │ - -- used 0.0152443s (cpu); 0.0152472s (thread); 0s (gc)
│ │ │ + -- used 0.0164383s (cpu); 0.0164441s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = 5
│ │ │  
│ │ │  i18 :
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/_grothendieck__Polynomial.out
│ │ │ @@ -3,25 +3,25 @@
│ │ │  i1 : w = {2,1,4,3}
│ │ │  
│ │ │  o1 = {2, 1, 4, 3}
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : time grothendieckPolynomial w
│ │ │ - -- used 0.00455946s (cpu); 0.00455689s (thread); 0s (gc)
│ │ │ + -- used 0.00552377s (cpu); 0.00549102s (thread); 0s (gc)
│ │ │  
│ │ │        2        2      2               2
│ │ │  o2 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │  
│ │ │  o2 : QQ[x ..x ]
│ │ │           1   4
│ │ │  
│ │ │  i3 : time grothendieckPolynomial (w,Algorithm=>"PipeDream")
│ │ │ - -- used 0.00228165s (cpu); 0.00228239s (thread); 0s (gc)
│ │ │ + -- used 0.00263547s (cpu); 0.00263637s (thread); 0s (gc)
│ │ │  
│ │ │        2        2      2               2
│ │ │  o3 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │  
│ │ │  o3 : QQ[x ..x ]
│ │ │           1   4
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/html/___Investigating_sp__A__S__M_spvarieties.html
│ │ │ @@ -383,23 +383,23 @@
│ │ │          
      
│ │ │            
      Additionally, this package facilitates investigating homological invariants of ASM ideals such as regularity (schubertRegularity) and codimension (schubertCodim). efficiently by computing the associated invariants for their antidiagonal initial ideals, which are known to be squarefree by [Wei17]. Therefore the extremal Betti numbers (which encode regularity, depth, and projective dimension) of ASM ideals coincide with those of their antidiagonal initial ideals by [CV20].
      
│ │ │          
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i23 : time schubertRegularity B
│ │ │ - -- used 0.100646s (cpu); 0.0344742s (thread); 0s (gc)
│ │ │ + -- used 0.104907s (cpu); 0.0345186s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = 1
      
│ │ │              
      
│ │ │                
      i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ - -- used 0.0147124s (cpu); 0.0147191s (thread); 0s (gc)
│ │ │ + -- used 0.0177392s (cpu); 0.017746s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = 1
      
│ │ │              
      
│ │ │          
      
│ │ │            
      Functions for investigating ASM varieties
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -244,19 +244,19 @@
│ │ │ │  ASM ideals such as regularity (_s_c_h_u_b_e_r_t_R_e_g_u_l_a_r_i_t_y) and codimension
│ │ │ │  (_s_c_h_u_b_e_r_t_C_o_d_i_m). efficiently by computing the associated invariants for their
│ │ │ │  antidiagonal initial ideals, which are known to be squarefree by [Wei17].
│ │ │ │  Therefore the extremal Betti numbers (which encode regularity, depth, and
│ │ │ │  projective dimension) of ASM ideals coincide with those of their antidiagonal
│ │ │ │  initial ideals by [CV20].
│ │ │ │  i23 : time schubertRegularity B
│ │ │ │ - -- used 0.100646s (cpu); 0.0344742s (thread); 0s (gc)
│ │ │ │ + -- used 0.104907s (cpu); 0.0345186s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = 1
│ │ │ │  i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ │ - -- used 0.0147124s (cpu); 0.0147191s (thread); 0s (gc)
│ │ │ │ + -- used 0.0177392s (cpu); 0.017746s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o24 = 1
│ │ │ │  ********** FFuunnccttiioonnss ffoorr iinnvveessttiiggaattiinngg AASSMM vvaarriieettiieess **********
│ │ │ │      * _i_s_P_a_r_t_i_a_l_A_S_M_(_M_a_t_r_i_x_) -- whether a matrix is a partial alternating sign
│ │ │ │        matrix
│ │ │ │      * _p_a_r_t_i_a_l_A_S_M_T_o_A_S_M_(_M_a_t_r_i_x_) -- extend a partial alternating sign matrix to an
│ │ │ │        alternating sign matrix
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/html/___Investigating_spmatrix_sp__Schubert_spvarieties.html
│ │ │ @@ -315,23 +315,23 @@
│ │ │          
      
│ │ │            
      Finally, this package contains functions for investigating homological invariants of matrix Schubert varieties efficiently through combinatorial algorithms produced in [PSW24] via schubertRegularityschubertCodim.
      
│ │ │          
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i16 : time schubertRegularity p
│ │ │ - -- used 0.000285165s (cpu); 0.000279514s (thread); 0s (gc)
│ │ │ + -- used 0.000290339s (cpu); 0.000285063s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 5
      
│ │ │              
      
│ │ │                
      i17 : time regularity comodule I
│ │ │ - -- used 0.0152443s (cpu); 0.0152472s (thread); 0s (gc)
│ │ │ + -- used 0.0164383s (cpu); 0.0164441s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = 5
      
│ │ │              
      
│ │ │          
      
│ │ │            
      Functions for investigating matrix Schubert varieties
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -545,19 +545,19 @@
│ │ │ │  
│ │ │ │  o15 : Ideal of QQ[z   ..z   ]
│ │ │ │                     1,1   5,5
│ │ │ │  Finally, this package contains functions for investigating homological
│ │ │ │  invariants of matrix Schubert varieties efficiently through combinatorial
│ │ │ │  algorithms produced in [PSW24] via _s_c_h_u_b_e_r_t_R_e_g_u_l_a_r_i_t_y_s_c_h_u_b_e_r_t_C_o_d_i_m.
│ │ │ │  i16 : time schubertRegularity p
│ │ │ │ - -- used 0.000285165s (cpu); 0.000279514s (thread); 0s (gc)
│ │ │ │ + -- used 0.000290339s (cpu); 0.000285063s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = 5
│ │ │ │  i17 : time regularity comodule I
│ │ │ │ - -- used 0.0152443s (cpu); 0.0152472s (thread); 0s (gc)
│ │ │ │ + -- used 0.0164383s (cpu); 0.0164441s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 = 5
│ │ │ │  ********** FFuunnccttiioonnss ffoorr iinnvveessttiiggaattiinngg mmaattrriixx SScchhuubbeerrtt vvaarriieettiieess **********
│ │ │ │      * _a_n_t_i_D_i_a_g_I_n_i_t_(_L_i_s_t_) -- compute the (unique) antidiagonal initial ideal of
│ │ │ │        an ASM ideal
│ │ │ │      * _r_a_n_k_T_a_b_l_e_(_L_i_s_t_) -- compute a table of rank conditions that determines the
│ │ │ │        corresponding ASM or matrix Schubert variety
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/html/_grothendieck__Polynomial.html
│ │ │ @@ -80,28 +80,28 @@
│ │ │  
│ │ │  o1 : List
      
│ │ │              
      
│ │ │                
      i2 : time grothendieckPolynomial w
│ │ │ - -- used 0.00455946s (cpu); 0.00455689s (thread); 0s (gc)
│ │ │ + -- used 0.00552377s (cpu); 0.00549102s (thread); 0s (gc)
│ │ │  
│ │ │        2        2      2               2
│ │ │  o2 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │  
│ │ │  o2 : QQ[x ..x ]
│ │ │           1   4
      
│ │ │              
      
│ │ │                
      i3 : time grothendieckPolynomial (w,Algorithm=>"PipeDream")
│ │ │ - -- used 0.00228165s (cpu); 0.00228239s (thread); 0s (gc)
│ │ │ + -- used 0.00263547s (cpu); 0.00263637s (thread); 0s (gc)
│ │ │  
│ │ │        2        2      2               2
│ │ │  o3 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │  
│ │ │  o3 : QQ[x ..x ]
│ │ │           1   4
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,24 +19,24 @@
│ │ │ │  PipeDream.
│ │ │ │  i1 : w = {2,1,4,3}
│ │ │ │  
│ │ │ │  o1 = {2, 1, 4, 3}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : time grothendieckPolynomial w
│ │ │ │ - -- used 0.00455946s (cpu); 0.00455689s (thread); 0s (gc)
│ │ │ │ + -- used 0.00552377s (cpu); 0.00549102s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        2        2      2               2
│ │ │ │  o2 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │ │  
│ │ │ │  o2 : QQ[x ..x ]
│ │ │ │           1   4
│ │ │ │  i3 : time grothendieckPolynomial (w,Algorithm=>"PipeDream")
│ │ │ │ - -- used 0.00228165s (cpu); 0.00228239s (thread); 0s (gc)
│ │ │ │ + -- used 0.00263547s (cpu); 0.00263637s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        2        2      2               2
│ │ │ │  o3 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │ │  
│ │ │ │  o3 : QQ[x ..x ]
│ │ │ │           1   4
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/___Matroid.out
│ │ │ @@ -51,20 +51,20 @@
│ │ │  i9 : keys R10.cache
│ │ │  
│ │ │  o9 = {groundSet, rankFunction, storedRepresentation}
│ │ │  
│ │ │  o9 : List
│ │ │  
│ │ │  i10 : time isWellDefined R10
│ │ │ - -- used 0.050878s (cpu); 0.0526062s (thread); 0s (gc)
│ │ │ + -- used 0.055972s (cpu); 0.0545941s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
│ │ │  
│ │ │  i11 : time fVector R10
│ │ │ - -- used 0.195193s (cpu); 0.0769082s (thread); 0s (gc)
│ │ │ + -- used 0.219357s (cpu); 0.0791931s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = HashTable{0 => 1 }
│ │ │                  1 => 10
│ │ │                  2 => 45
│ │ │                  3 => 75
│ │ │                  4 => 30
│ │ │                  5 => 1
│ │ │ @@ -76,15 +76,15 @@
│ │ │  o12 = {hyperplanes, flatsRelationsTable, rankFunction, ideal, ranks, flats,
│ │ │        -----------------------------------------------------------------------
│ │ │        groundSet, dual, storedRepresentation}
│ │ │  
│ │ │  o12 : List
│ │ │  
│ │ │  i13 : time fVector R10
│ │ │ - -- used 0.000402855s (cpu); 0.000203472s (thread); 0s (gc)
│ │ │ + -- used 0.000356673s (cpu); 0.000195219s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = HashTable{0 => 1 }
│ │ │                  1 => 10
│ │ │                  2 => 45
│ │ │                  3 => 75
│ │ │                  4 => 30
│ │ │                  5 => 1
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_all__Minors.out
│ │ │ @@ -9,15 +9,15 @@
│ │ │  i2 : U25 = uniformMatroid(2,5)
│ │ │  
│ │ │  o2 = a "matroid" of rank 2 on 5 elements
│ │ │  
│ │ │  o2 : Matroid
│ │ │  
│ │ │  i3 : elapsedTime L = allMinors(V, U25);
│ │ │ - -- .0861244s elapsed
│ │ │ + -- .0610617s elapsed
│ │ │  
│ │ │  i4 : #L
│ │ │  
│ │ │  o4 = 64
│ │ │  
│ │ │  i5 : netList L_{0..4}
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_get__Isos.out
│ │ │ @@ -33,14 +33,14 @@
│ │ │  i6 : F7 = specificMatroid "fano"
│ │ │  
│ │ │  o6 = a "matroid" of rank 3 on 7 elements
│ │ │  
│ │ │  o6 : Matroid
│ │ │  
│ │ │  i7 : time autF7 = getIsos(F7, F7);
│ │ │ - -- used 0.142029s (cpu); 0.0648897s (thread); 0s (gc)
│ │ │ + -- used 0.150518s (cpu); 0.0618682s (thread); 0s (gc)
│ │ │  
│ │ │  i8 : #autF7
│ │ │  
│ │ │  o8 = 168
│ │ │  
│ │ │  i9 :
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_has__Minor.out
│ │ │ @@ -9,12 +9,12 @@
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : hasMinor(M4, uniformMatroid(2,4))
│ │ │  
│ │ │  o2 = false
│ │ │  
│ │ │  i3 : time hasMinor(M6, M5)
│ │ │ - -- used 1.78938s (cpu); 1.28303s (thread); 0s (gc)
│ │ │ + -- used 1.79693s (cpu); 1.22252s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_isomorphism_lp__Matroid_cm__Matroid_rp.out
│ │ │ @@ -19,15 +19,15 @@
│ │ │  i4 : minorM6 = minor(M6, set{8}, set{4,5,6,7})
│ │ │  
│ │ │  o4 = a "matroid" of rank 4 on 10 elements
│ │ │  
│ │ │  o4 : Matroid
│ │ │  
│ │ │  i5 : time isomorphism(M5, minorM6)
│ │ │ - -- used 0.0208457s (cpu); 0.0227737s (thread); 0s (gc)
│ │ │ + -- used 0.0120337s (cpu); 0.0136404s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HashTable{0 => 1}
│ │ │                 1 => 0
│ │ │                 2 => 3
│ │ │                 3 => 2
│ │ │                 4 => 6
│ │ │                 5 => 5
│ │ │ @@ -56,15 +56,15 @@
│ │ │  i7 : N = relabel M6
│ │ │  
│ │ │  o7 = a "matroid" of rank 5 on 15 elements
│ │ │  
│ │ │  o7 : Matroid
│ │ │  
│ │ │  i8 : time phi = isomorphism(N,M6)
│ │ │ - -- used 4.74954s (cpu); 3.07529s (thread); 0s (gc)
│ │ │ + -- used 4.58452s (cpu); 2.76376s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = HashTable{0 => 11 }
│ │ │                 1 => 0
│ │ │                 2 => 1
│ │ │                 3 => 6
│ │ │                 4 => 9
│ │ │                 5 => 8
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_quick__Isomorphism__Test.out
│ │ │ @@ -37,15 +37,15 @@
│ │ │  o7 : Matroid
│ │ │  
│ │ │  i8 : R = ZZ[x,y]; tuttePolynomial(M0, R) == tuttePolynomial(M1, R)
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : time quickIsomorphismTest(M0, M1)
│ │ │ - -- used 0.000871805s (cpu); 0.000546956s (thread); 0s (gc)
│ │ │ + -- used 0.00190617s (cpu); 0.000572848s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = false
│ │ │  
│ │ │  i11 : value oo === false
│ │ │  
│ │ │  o11 = true
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_set__Representation.out
│ │ │ @@ -35,15 +35,15 @@
│ │ │  i5 : keys M.cache
│ │ │  
│ │ │  o5 = {groundSet, rankFunction, storedRepresentation}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : elapsedTime fVector M
│ │ │ - -- .0246909s elapsed
│ │ │ + -- .0137558s elapsed
│ │ │  
│ │ │  o6 = HashTable{0 => 1 }
│ │ │                 1 => 6
│ │ │                 2 => 15
│ │ │                 3 => 20
│ │ │                 4 => 1
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/___Matroid.html
│ │ │ @@ -148,23 +148,23 @@
│ │ │  
│ │ │  o9 : List
│ │ │              
      
│ │ │                
      i10 : time isWellDefined R10
│ │ │ - -- used 0.050878s (cpu); 0.0526062s (thread); 0s (gc)
│ │ │ + -- used 0.055972s (cpu); 0.0545941s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
      
│ │ │              
      
│ │ │                
      i11 : time fVector R10
│ │ │ - -- used 0.195193s (cpu); 0.0769082s (thread); 0s (gc)
│ │ │ + -- used 0.219357s (cpu); 0.0791931s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = HashTable{0 => 1 }
│ │ │                  1 => 10
│ │ │                  2 => 45
│ │ │                  3 => 75
│ │ │                  4 => 30
│ │ │                  5 => 1
│ │ │ @@ -182,15 +182,15 @@
│ │ │  
│ │ │  o12 : List
      
│ │ │              
      
│ │ │                
      i13 : time fVector R10
│ │ │ - -- used 0.000402855s (cpu); 0.000203472s (thread); 0s (gc)
│ │ │ + -- used 0.000356673s (cpu); 0.000195219s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = HashTable{0 => 1 }
│ │ │                  1 => 10
│ │ │                  2 => 45
│ │ │                  3 => 75
│ │ │                  4 => 30
│ │ │                  5 => 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -71,19 +71,19 @@
│ │ │ │  o8 : Matroid
│ │ │ │  i9 : keys R10.cache
│ │ │ │  
│ │ │ │  o9 = {groundSet, rankFunction, storedRepresentation}
│ │ │ │  
│ │ │ │  o9 : List
│ │ │ │  i10 : time isWellDefined R10
│ │ │ │ - -- used 0.050878s (cpu); 0.0526062s (thread); 0s (gc)
│ │ │ │ + -- used 0.055972s (cpu); 0.0545941s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = true
│ │ │ │  i11 : time fVector R10
│ │ │ │ - -- used 0.195193s (cpu); 0.0769082s (thread); 0s (gc)
│ │ │ │ + -- used 0.219357s (cpu); 0.0791931s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = HashTable{0 => 1 }
│ │ │ │                  1 => 10
│ │ │ │                  2 => 45
│ │ │ │                  3 => 75
│ │ │ │                  4 => 30
│ │ │ │                  5 => 1
│ │ │ │ @@ -93,15 +93,15 @@
│ │ │ │  
│ │ │ │  o12 = {hyperplanes, flatsRelationsTable, rankFunction, ideal, ranks, flats,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        groundSet, dual, storedRepresentation}
│ │ │ │  
│ │ │ │  o12 : List
│ │ │ │  i13 : time fVector R10
│ │ │ │ - -- used 0.000402855s (cpu); 0.000203472s (thread); 0s (gc)
│ │ │ │ + -- used 0.000356673s (cpu); 0.000195219s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 = HashTable{0 => 1 }
│ │ │ │                  1 => 10
│ │ │ │                  2 => 45
│ │ │ │                  3 => 75
│ │ │ │                  4 => 30
│ │ │ │                  5 => 1
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_all__Minors.html
│ │ │ @@ -92,15 +92,15 @@
│ │ │  
│ │ │  o2 : Matroid
      
│ │ │              
      
│ │ │                
      i3 : elapsedTime L = allMinors(V, U25);
│ │ │ - -- .0861244s elapsed
      
│ │ │ + -- .0610617s elapsed
│ │ │              
      
│ │ │                
      i4 : #L
│ │ │  
│ │ │  o4 = 64
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,15 +27,15 @@
│ │ │ │  o1 : Matroid
│ │ │ │  i2 : U25 = uniformMatroid(2,5)
│ │ │ │  
│ │ │ │  o2 = a "matroid" of rank 2 on 5 elements
│ │ │ │  
│ │ │ │  o2 : Matroid
│ │ │ │  i3 : elapsedTime L = allMinors(V, U25);
│ │ │ │ - -- .0861244s elapsed
│ │ │ │ + -- .0610617s elapsed
│ │ │ │  i4 : #L
│ │ │ │  
│ │ │ │  o4 = 64
│ │ │ │  i5 : netList L_{0..4}
│ │ │ │  
│ │ │ │       +----------+-------+
│ │ │ │  o5 = |set {5, 3}|set {2}|
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_get__Isos.html
│ │ │ @@ -135,15 +135,15 @@
│ │ │  
│ │ │  o6 : Matroid
│ │ │              
      
│ │ │                
      i7 : time autF7 = getIsos(F7, F7);
│ │ │ - -- used 0.142029s (cpu); 0.0648897s (thread); 0s (gc)
      
│ │ │ + -- used 0.150518s (cpu); 0.0618682s (thread); 0s (gc)
│ │ │              
      
│ │ │                
      i8 : #autF7
│ │ │  
│ │ │  o8 = 168
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -51,15 +51,15 @@
│ │ │ │  symmetric group S_7:
│ │ │ │  i6 : F7 = specificMatroid "fano"
│ │ │ │  
│ │ │ │  o6 = a "matroid" of rank 3 on 7 elements
│ │ │ │  
│ │ │ │  o6 : Matroid
│ │ │ │  i7 : time autF7 = getIsos(F7, F7);
│ │ │ │ - -- used 0.142029s (cpu); 0.0648897s (thread); 0s (gc)
│ │ │ │ + -- used 0.150518s (cpu); 0.0618682s (thread); 0s (gc)
│ │ │ │  i8 : #autF7
│ │ │ │  
│ │ │ │  o8 = 168
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_o_m_o_r_p_h_i_s_m_(_M_a_t_r_o_i_d_,_M_a_t_r_o_i_d_) -- computes an isomorphism between
│ │ │ │        isomorphic matroids
│ │ │ │      * _q_u_i_c_k_I_s_o_m_o_r_p_h_i_s_m_T_e_s_t -- quick checks for isomorphism between matroids
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_has__Minor.html
│ │ │ @@ -96,15 +96,15 @@
│ │ │  
│ │ │  o2 = false
│ │ │              
      
│ │ │                
      i3 : time hasMinor(M6, M5)
│ │ │ - -- used 1.78938s (cpu); 1.28303s (thread); 0s (gc)
│ │ │ + -- used 1.79693s (cpu); 1.22252s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
      
│ │ │              
      
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │       elements, a "matroid" of rank 5 on 15 elements)
│ │ │ │  
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : hasMinor(M4, uniformMatroid(2,4))
│ │ │ │  
│ │ │ │  o2 = false
│ │ │ │  i3 : time hasMinor(M6, M5)
│ │ │ │ - -- used 1.78938s (cpu); 1.28303s (thread); 0s (gc)
│ │ │ │ + -- used 1.79693s (cpu); 1.22252s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_i_n_o_r -- minor of matroid
│ │ │ │      * _i_s_B_i_n_a_r_y -- whether a matroid is representable over F_2
│ │ │ │  ********** WWaayyss ttoo uussee hhaassMMiinnoorr:: **********
│ │ │ │      * hasMinor(Matroid,Matroid)
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_isomorphism_lp__Matroid_cm__Matroid_rp.html
│ │ │ @@ -118,15 +118,15 @@
│ │ │  
│ │ │  o4 : Matroid
    
│ │ │              
    
│ │ │                
    i5 : time isomorphism(M5, minorM6)
│ │ │ - -- used 0.0208457s (cpu); 0.0227737s (thread); 0s (gc)
│ │ │ + -- used 0.0120337s (cpu); 0.0136404s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HashTable{0 => 1}
│ │ │                 1 => 0
│ │ │                 2 => 3
│ │ │                 3 => 2
│ │ │                 4 => 6
│ │ │                 5 => 5
│ │ │ @@ -164,15 +164,15 @@
│ │ │  
│ │ │  o7 : Matroid
    
│ │ │              
    
│ │ │                
    i8 : time phi = isomorphism(N,M6)
│ │ │ - -- used 4.74954s (cpu); 3.07529s (thread); 0s (gc)
│ │ │ + -- used 4.58452s (cpu); 2.76376s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = HashTable{0 => 11 }
│ │ │                 1 => 0
│ │ │                 2 => 1
│ │ │                 3 => 6
│ │ │                 4 => 9
│ │ │                 5 => 8
│ │ │ ├── html2text {}
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │  o3 : Sequence
│ │ │ │  i4 : minorM6 = minor(M6, set{8}, set{4,5,6,7})
│ │ │ │  
│ │ │ │  o4 = a "matroid" of rank 4 on 10 elements
│ │ │ │  
│ │ │ │  o4 : Matroid
│ │ │ │  i5 : time isomorphism(M5, minorM6)
│ │ │ │ - -- used 0.0208457s (cpu); 0.0227737s (thread); 0s (gc)
│ │ │ │ + -- used 0.0120337s (cpu); 0.0136404s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = HashTable{0 => 1}
│ │ │ │                 1 => 0
│ │ │ │                 2 => 3
│ │ │ │                 3 => 2
│ │ │ │                 4 => 6
│ │ │ │                 5 => 5
│ │ │ │ @@ -74,15 +74,15 @@
│ │ │ │  o6 : HashTable
│ │ │ │  i7 : N = relabel M6
│ │ │ │  
│ │ │ │  o7 = a "matroid" of rank 5 on 15 elements
│ │ │ │  
│ │ │ │  o7 : Matroid
│ │ │ │  i8 : time phi = isomorphism(N,M6)
│ │ │ │ - -- used 4.74954s (cpu); 3.07529s (thread); 0s (gc)
│ │ │ │ + -- used 4.58452s (cpu); 2.76376s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = HashTable{0 => 11 }
│ │ │ │                 1 => 0
│ │ │ │                 2 => 1
│ │ │ │                 3 => 6
│ │ │ │                 4 => 9
│ │ │ │                 5 => 8
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_quick__Isomorphism__Test.html
│ │ │ @@ -137,15 +137,15 @@
│ │ │  
│ │ │  o9 = true
    
│ │ │              
    
│ │ │                
    i10 : time quickIsomorphismTest(M0, M1)
│ │ │ - -- used 0.000871805s (cpu); 0.000546956s (thread); 0s (gc)
│ │ │ + -- used 0.00190617s (cpu); 0.000572848s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = false
    
│ │ │              
    
│ │ │                
    i11 : value oo === false
│ │ │ ├── html2text {}
│ │ │ │ @@ -51,15 +51,15 @@
│ │ │ │  o7 = a "matroid" of rank 7 on 11 elements
│ │ │ │  
│ │ │ │  o7 : Matroid
│ │ │ │  i8 : R = ZZ[x,y]; tuttePolynomial(M0, R) == tuttePolynomial(M1, R)
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  i10 : time quickIsomorphismTest(M0, M1)
│ │ │ │ - -- used 0.000871805s (cpu); 0.000546956s (thread); 0s (gc)
│ │ │ │ + -- used 0.00190617s (cpu); 0.000572848s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = false
│ │ │ │  i11 : value oo === false
│ │ │ │  
│ │ │ │  o11 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_o_m_o_r_p_h_i_s_m_(_M_a_t_r_o_i_d_,_M_a_t_r_o_i_d_) -- computes an isomorphism between
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_set__Representation.html
│ │ │ @@ -126,15 +126,15 @@
│ │ │  
│ │ │  o5 : List
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime fVector M
│ │ │ - -- .0246909s elapsed
│ │ │ + -- .0137558s elapsed
│ │ │  
│ │ │  o6 = HashTable{0 => 1 }
│ │ │                 1 => 6
│ │ │                 2 => 15
│ │ │                 3 => 20
│ │ │                 4 => 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,15 +48,15 @@
│ │ │ │  o4 : Matrix QQ  <-- QQ
│ │ │ │  i5 : keys M.cache
│ │ │ │  
│ │ │ │  o5 = {groundSet, rankFunction, storedRepresentation}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : elapsedTime fVector M
│ │ │ │ - -- .0246909s elapsed
│ │ │ │ + -- .0137558s elapsed
│ │ │ │  
│ │ │ │  o6 = HashTable{0 => 1 }
│ │ │ │                 1 => 6
│ │ │ │                 2 => 15
│ │ │ │                 3 => 20
│ │ │ │                 4 => 1
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/example-output/___Hybrid.out
│ │ │ @@ -5,16 +5,16 @@
│ │ │  i2 : R = ZZ/101[w..z];
│ │ │  
│ │ │  i3 : I = ideal(w*x^2-42*y*z, x^6+12*w*y+x^3*z, w^2-47*x^4*z-47*x*z^2);
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : elapsedTime minimalPrimes(ideal I_*, Strategy => Hybrid{Linear,Birational,Factorization,DecomposeMonomials}, Verbosity => 2);
│ │ │ -  Strategy: Linear            (time .000976812)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Birational        (time .0138253)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Factorization     (time .000355486)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: DecomposeMonomials(time .000020849)  #primes = 1 #prunedViaCodim = 0
│ │ │ +  Strategy: Linear            (time .00119882)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Birational        (time .0466845)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Factorization     (time .000401013)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: DecomposeMonomials(time .000020574)  #primes = 1 #prunedViaCodim = 0
│ │ │   -- Converting annotated ideals to ideals and selecting minimal primes...
│ │ │ - --  Time taken : .000617178
│ │ │ - -- .0423566s elapsed
│ │ │ + --  Time taken : .00082124
│ │ │ + -- .0310278s elapsed
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/example-output/_radical.out
│ │ │ @@ -30,21 +30,21 @@
│ │ │  
│ │ │               2        2   3     2
│ │ │  o5 = ideal (c , a*c, a , b , a*b )
│ │ │  
│ │ │  o5 : Ideal of R
│ │ │  
│ │ │  i6 : elapsedTime radical(ideal I_*, Strategy => Monomial)
│ │ │ - -- .000461911s elapsed
│ │ │ + -- .000567822s elapsed
│ │ │  
│ │ │  o6 = ideal (a, b, c)
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ - -- .0117995s elapsed
│ │ │ + -- .0143449s elapsed
│ │ │  
│ │ │  o7 = ideal (c, b, a)
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/example-output/_radical__Containment.out
│ │ │ @@ -29,22 +29,22 @@
│ │ │  o5 = 840
│ │ │  
│ │ │  i6 : x_0^(D-1) % I != 0 and x_0^D % I == 0
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : elapsedTime radicalContainment(x_0, I)
│ │ │ - -- .102182s elapsed
│ │ │ + -- .0756223s elapsed
│ │ │  
│ │ │  o7 = true
│ │ │  
│ │ │  i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ - -- .00174825s elapsed
│ │ │ + -- .00184297s elapsed
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ - -- .00126877s elapsed
│ │ │ + -- .00151659s elapsed
│ │ │  
│ │ │  o9 = false
│ │ │  
│ │ │  i10 :
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/html/___Hybrid.html
│ │ │ @@ -72,21 +72,21 @@
│ │ │  
│ │ │  o3 : Ideal of R
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime minimalPrimes(ideal I_*, Strategy => Hybrid{Linear,Birational,Factorization,DecomposeMonomials}, Verbosity => 2);
│ │ │ -  Strategy: Linear            (time .000976812)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Birational        (time .0138253)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Factorization     (time .000355486)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: DecomposeMonomials(time .000020849)  #primes = 1 #prunedViaCodim = 0
│ │ │ +  Strategy: Linear            (time .00119882)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Birational        (time .0466845)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Factorization     (time .000401013)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: DecomposeMonomials(time .000020574)  #primes = 1 #prunedViaCodim = 0
│ │ │   -- Converting annotated ideals to ideals and selecting minimal primes...
│ │ │ - --  Time taken : .000617178
│ │ │ - -- .0423566s elapsed
    
│ │ │ + --  Time taken : .00082124
│ │ │ + -- .0310278s elapsed
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,24 +11,23 @@
│ │ │ │  i1 : debug MinimalPrimes
│ │ │ │  i2 : R = ZZ/101[w..z];
│ │ │ │  i3 : I = ideal(w*x^2-42*y*z, x^6+12*w*y+x^3*z, w^2-47*x^4*z-47*x*z^2);
│ │ │ │  
│ │ │ │  o3 : Ideal of R
│ │ │ │  i4 : elapsedTime minimalPrimes(ideal I_*, Strategy => Hybrid
│ │ │ │  {Linear,Birational,Factorization,DecomposeMonomials}, Verbosity => 2);
│ │ │ │ -  Strategy: Linear            (time .000976812)  #primes = 0 #prunedViaCodim =
│ │ │ │ +  Strategy: Linear            (time .00119882)  #primes = 0 #prunedViaCodim = 0
│ │ │ │ +  Strategy: Birational        (time .0466845)  #primes = 0 #prunedViaCodim = 0
│ │ │ │ +  Strategy: Factorization     (time .000401013)  #primes = 0 #prunedViaCodim =
│ │ │ │  0
│ │ │ │ -  Strategy: Birational        (time .0138253)  #primes = 0 #prunedViaCodim = 0
│ │ │ │ -  Strategy: Factorization     (time .000355486)  #primes = 0 #prunedViaCodim =
│ │ │ │ -0
│ │ │ │ -  Strategy: DecomposeMonomials(time .000020849)  #primes = 1 #prunedViaCodim =
│ │ │ │ +  Strategy: DecomposeMonomials(time .000020574)  #primes = 1 #prunedViaCodim =
│ │ │ │  0
│ │ │ │   -- Converting annotated ideals to ideals and selecting minimal primes...
│ │ │ │ - --  Time taken : .000617178
│ │ │ │ - -- .0423566s elapsed
│ │ │ │ + --  Time taken : .00082124
│ │ │ │ + -- .0310278s elapsed
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _p_r_i_m_a_r_y_D_e_c_o_m_p_o_s_i_t_i_o_n_(_._._._,_S_t_r_a_t_e_g_y_=_>_._._._)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _H_y_b_r_i_d is a _s_e_l_f_ _i_n_i_t_i_a_l_i_z_i_n_g_ _t_y_p_e, with ancestor classes _L_i_s_t <
│ │ │ │  _V_i_s_i_b_l_e_L_i_s_t < _B_a_s_i_c_L_i_s_t < _T_h_i_n_g.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/html/_radical.html
│ │ │ @@ -131,25 +131,25 @@
│ │ │  
│ │ │  o5 : Ideal of R
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime radical(ideal I_*, Strategy => Monomial)
│ │ │ - -- .000461911s elapsed
│ │ │ + -- .000567822s elapsed
│ │ │  
│ │ │  o6 = ideal (a, b, c)
│ │ │  
│ │ │  o6 : Ideal of R
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ - -- .0117995s elapsed
│ │ │ + -- .0143449s elapsed
│ │ │  
│ │ │  o7 = ideal (c, b, a)
│ │ │  
│ │ │  o7 : Ideal of R
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -62,21 +62,21 @@
│ │ │ │  i5 : I = intersect(ideal(a^2,b^2,c), ideal(a,b^3,c^2))
│ │ │ │  
│ │ │ │               2        2   3     2
│ │ │ │  o5 = ideal (c , a*c, a , b , a*b )
│ │ │ │  
│ │ │ │  o5 : Ideal of R
│ │ │ │  i6 : elapsedTime radical(ideal I_*, Strategy => Monomial)
│ │ │ │ - -- .000461911s elapsed
│ │ │ │ + -- .000567822s elapsed
│ │ │ │  
│ │ │ │  o6 = ideal (a, b, c)
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ │ - -- .0117995s elapsed
│ │ │ │ + -- .0143449s elapsed
│ │ │ │  
│ │ │ │  o7 = ideal (c, b, a)
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  For another example, see _P_r_i_m_a_r_y_D_e_c_o_m_p_o_s_i_t_i_o_n.
│ │ │ │  ********** RReeffeerreenncceess **********
│ │ │ │  Eisenbud, Huneke, Vasconcelos, Invent. Math. 110 207-235 (1992).
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/html/_radical__Containment.html
│ │ │ @@ -125,31 +125,31 @@
│ │ │  
│ │ │  o6 = true
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime radicalContainment(x_0, I)
│ │ │ - -- .102182s elapsed
│ │ │ + -- .0756223s elapsed
│ │ │  
│ │ │  o7 = true
    
│ │ │              
    
│ │ │                
    i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ - -- .00174825s elapsed
│ │ │ + -- .00184297s elapsed
│ │ │  
│ │ │  o8 = true
    
│ │ │              
    
│ │ │                
    i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ - -- .00126877s elapsed
│ │ │ + -- .00151659s elapsed
│ │ │  
│ │ │  o9 = false
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -50,23 +50,23 @@
│ │ │ │  i5 : D = product(I_*/degree/sum)
│ │ │ │  
│ │ │ │  o5 = 840
│ │ │ │  i6 : x_0^(D-1) % I != 0 and x_0^D % I == 0
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  i7 : elapsedTime radicalContainment(x_0, I)
│ │ │ │ - -- .102182s elapsed
│ │ │ │ + -- .0756223s elapsed
│ │ │ │  
│ │ │ │  o7 = true
│ │ │ │  i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ │ - -- .00174825s elapsed
│ │ │ │ + -- .00184297s elapsed
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ │ - -- .00126877s elapsed
│ │ │ │ + -- .00151659s elapsed
│ │ │ │  
│ │ │ │  o9 = false
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_a_d_i_c_a_l -- the radical of an ideal
│ │ │ │  ********** WWaayyss ttoo uussee rraaddiiccaallCCoonnttaaiinnmmeenntt:: **********
│ │ │ │      * radicalContainment(Ideal,Ideal)
│ │ │ │      * radicalContainment(RingElement,Ideal)
│ │ ├── ./usr/share/doc/Macaulay2/MixedMultiplicity/example-output/_multi__Rees__Ideal.out
│ │ │ @@ -57,29 +57,29 @@
│ │ │  i9 : J = ideal vars U
│ │ │  
│ │ │  o9 = ideal (a, b, c)
│ │ │  
│ │ │  o9 : Ideal of U
│ │ │  
│ │ │  i10 : time multiReesIdeal J
│ │ │ - -- used 0.102055s (cpu); 0.0642847s (thread); 0s (gc)
│ │ │ + -- used 0.185119s (cpu); 0.082016s (thread); 0s (gc)
│ │ │  
│ │ │                                                                               
│ │ │  o10 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │                  1      2     1      2     1      2     0      2     0      2 
│ │ │        -----------------------------------------------------------------------
│ │ │                      2                 2   2
│ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ │           0      2   1    0 2   0 1    2   0    1 2
│ │ │  
│ │ │  o10 : Ideal of U[X ..X ]
│ │ │                    0   2
│ │ │  
│ │ │  i11 : time multiReesIdeal (J, a)
│ │ │ - -- used 0.0842922s (cpu); 0.0247612s (thread); 0s (gc)
│ │ │ + -- used 0.0930233s (cpu); 0.0265471s (thread); 0s (gc)
│ │ │  
│ │ │                                                                               
│ │ │  o11 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │                  1      2     1      2     1      2     0      2     0      2 
│ │ │        -----------------------------------------------------------------------
│ │ │                      2                 2   2
│ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ ├── ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_multi__Rees__Ideal.html
│ │ │ @@ -173,15 +173,15 @@
│ │ │  
│ │ │  o9 : Ideal of U
    
│ │ │              
    
│ │ │                
    i10 : time multiReesIdeal J
│ │ │ - -- used 0.102055s (cpu); 0.0642847s (thread); 0s (gc)
│ │ │ + -- used 0.185119s (cpu); 0.082016s (thread); 0s (gc)
│ │ │  
│ │ │                                                                               
│ │ │  o10 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │                  1      2     1      2     1      2     0      2     0      2 
│ │ │        -----------------------------------------------------------------------
│ │ │                      2                 2   2
│ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ │ @@ -190,15 +190,15 @@
│ │ │  o10 : Ideal of U[X ..X ]
│ │ │                    0   2
    
│ │ │              
    
│ │ │                
    i11 : time multiReesIdeal (J, a)
│ │ │ - -- used 0.0842922s (cpu); 0.0247612s (thread); 0s (gc)
│ │ │ + -- used 0.0930233s (cpu); 0.0265471s (thread); 0s (gc)
│ │ │  
│ │ │                                                                               
│ │ │  o11 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │                  1      2     1      2     1      2     0      2     0      2 
│ │ │        -----------------------------------------------------------------------
│ │ │                      2                 2   2
│ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ │ ├── html2text {}
│ │ │ │ @@ -79,28 +79,28 @@
│ │ │ │  i8 : U = T/minors(2,m);
│ │ │ │  i9 : J = ideal vars U
│ │ │ │  
│ │ │ │  o9 = ideal (a, b, c)
│ │ │ │  
│ │ │ │  o9 : Ideal of U
│ │ │ │  i10 : time multiReesIdeal J
│ │ │ │ - -- used 0.102055s (cpu); 0.0642847s (thread); 0s (gc)
│ │ │ │ + -- used 0.185119s (cpu); 0.082016s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  
│ │ │ │  o10 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │ │                  1      2     1      2     1      2     0      2     0      2
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │                      2                 2   2
│ │ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ │ │           0      2   1    0 2   0 1    2   0    1 2
│ │ │ │  
│ │ │ │  o10 : Ideal of U[X ..X ]
│ │ │ │                    0   2
│ │ │ │  i11 : time multiReesIdeal (J, a)
│ │ │ │ - -- used 0.0842922s (cpu); 0.0247612s (thread); 0s (gc)
│ │ │ │ + -- used 0.0930233s (cpu); 0.0265471s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  
│ │ │ │  o11 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │ │                  1      2     1      2     1      2     0      2     0      2
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │                      2                 2   2
│ │ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ ├── ./usr/share/doc/Macaulay2/ModuleDeformations/example-output/_deform__M__C__M__Module_lp__Module_rp.out
│ │ │ @@ -40,15 +40,15 @@
│ │ │  
│ │ │  o7 = image | x2 y2 |
│ │ │  
│ │ │                               1
│ │ │  o7 : R-module, submodule of R
│ │ │  
│ │ │  i8 : (S,N) = time deformMCMModule N0 
│ │ │ - -- used 0.481255s (cpu); 0.333746s (thread); 0s (gc)
│ │ │ + -- used 0.553944s (cpu); 0.355909s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = (S, cokernel {6} | x2-xxi_2-xi_1+xi_2^2-yxi_4^2-2xi_3xi_4^2+xi_2xi_4^3
│ │ │                    {8} | xxi_4-y+xi_3                                       
│ │ │       ------------------------------------------------------------------------
│ │ │       xyxi_4+2xxi_3xi_4-xxi_2xi_4^2+y2+yxi_3+xi_3^2-xi_1xi_4^2 |)
│ │ │       -x2-xxi_2-xi_1                                           |
│ │ │  
│ │ │ @@ -70,15 +70,15 @@
│ │ │  o10 = cokernel | x2 y2  |
│ │ │                 | -y -x2 |
│ │ │  
│ │ │                               2
│ │ │  o10 : R-module, quotient of R
│ │ │  
│ │ │  i11 : (S',N') = time deformMCMModule N0'
│ │ │ - -- used 0.648667s (cpu); 0.498693s (thread); 0s (gc)
│ │ │ + -- used 0.760664s (cpu); 0.585757s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = (S', cokernel | x2-xxi_4^3-xxi_2+xi_2xi_4^3-3xi_3xi_4^2+xi_2^2-xi_1
│ │ │                      | xxi_4-y+xi_3                                       
│ │ │        -----------------------------------------------------------------------
│ │ │        x2xi_4^2+xyxi_4+2xxi_3xi_4+y2+yxi_3+xi_3^2 |)
│ │ │        -x2-xxi_2-xi_1                             |
│ │ ├── ./usr/share/doc/Macaulay2/ModuleDeformations/html/_deform__M__C__M__Module_lp__Module_rp.html
│ │ │ @@ -145,15 +145,15 @@
│ │ │                               1
│ │ │  o7 : R-module, submodule of R
    
│ │ │              
    
│ │ │                
    i8 : (S,N) = time deformMCMModule N0 
│ │ │ - -- used 0.481255s (cpu); 0.333746s (thread); 0s (gc)
│ │ │ + -- used 0.553944s (cpu); 0.355909s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = (S, cokernel {6} | x2-xxi_2-xi_1+xi_2^2-yxi_4^2-2xi_3xi_4^2+xi_2xi_4^3
│ │ │                    {8} | xxi_4-y+xi_3                                       
│ │ │       ------------------------------------------------------------------------
│ │ │       xyxi_4+2xxi_3xi_4-xxi_2xi_4^2+y2+yxi_3+xi_3^2-xi_1xi_4^2 |)
│ │ │       -x2-xxi_2-xi_1                                           |
│ │ │  
│ │ │ @@ -186,15 +186,15 @@
│ │ │                               2
│ │ │  o10 : R-module, quotient of R
    
│ │ │              
    
│ │ │                
    i11 : (S',N') = time deformMCMModule N0'
│ │ │ - -- used 0.648667s (cpu); 0.498693s (thread); 0s (gc)
│ │ │ + -- used 0.760664s (cpu); 0.585757s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = (S', cokernel | x2-xxi_4^3-xxi_2+xi_2xi_4^3-3xi_3xi_4^2+xi_2^2-xi_1
│ │ │                      | xxi_4-y+xi_3                                       
│ │ │        -----------------------------------------------------------------------
│ │ │        x2xi_4^2+xyxi_4+2xxi_3xi_4+y2+yxi_3+xi_3^2 |)
│ │ │        -x2-xxi_2-xi_1                             |
│ │ │ ├── html2text {}
│ │ │ │ @@ -70,15 +70,15 @@
│ │ │ │  i7 : N0 = module ideal (x^2,y^2)
│ │ │ │  
│ │ │ │  o7 = image | x2 y2 |
│ │ │ │  
│ │ │ │                               1
│ │ │ │  o7 : R-module, submodule of R
│ │ │ │  i8 : (S,N) = time deformMCMModule N0
│ │ │ │ - -- used 0.481255s (cpu); 0.333746s (thread); 0s (gc)
│ │ │ │ + -- used 0.553944s (cpu); 0.355909s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = (S, cokernel {6} | x2-xxi_2-xi_1+xi_2^2-yxi_4^2-2xi_3xi_4^2+xi_2xi_4^3
│ │ │ │                    {8} | xxi_4-y+xi_3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       xyxi_4+2xxi_3xi_4-xxi_2xi_4^2+y2+yxi_3+xi_3^2-xi_1xi_4^2 |)
│ │ │ │       -x2-xxi_2-xi_1                                           |
│ │ │ │  
│ │ │ │ @@ -103,15 +103,15 @@
│ │ │ │  
│ │ │ │  o10 = cokernel | x2 y2  |
│ │ │ │                 | -y -x2 |
│ │ │ │  
│ │ │ │                               2
│ │ │ │  o10 : R-module, quotient of R
│ │ │ │  i11 : (S',N') = time deformMCMModule N0'
│ │ │ │ - -- used 0.648667s (cpu); 0.498693s (thread); 0s (gc)
│ │ │ │ + -- used 0.760664s (cpu); 0.585757s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = (S', cokernel | x2-xxi_4^3-xxi_2+xi_2xi_4^3-3xi_3xi_4^2+xi_2^2-xi_1
│ │ │ │                      | xxi_4-y+xi_3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        x2xi_4^2+xyxi_4+2xxi_3xi_4+y2+yxi_3+xi_3^2 |)
│ │ │ │        -x2-xxi_2-xi_1                             |
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_dynamic__Flower__Solve.out
│ │ │ @@ -3,27 +3,27 @@
│ │ │  i1 : R = CC[a,b,c,d][x,y];
│ │ │  
│ │ │  i2 : polys = polySystem {a*x+b*y^2,c*x*y+d};
│ │ │  
│ │ │  i3 : (p0, x0) = createSeedPair polys;
│ │ │  
│ │ │  i4 : (L, npaths) = dynamicFlowerSolve(polys.PolyMap,p0,{x0})
│ │ │ - -- .00296506s elapsed
│ │ │ - -- .00282693s elapsed
│ │ │ - -- .000335866s elapsed
│ │ │ - -- .0027534s elapsed
│ │ │ - -- .00295554s elapsed
│ │ │ - -- .000235038s elapsed
│ │ │ - -- .00283445s elapsed
│ │ │ - -- .00295772s elapsed
│ │ │ - -- .000236902s elapsed
│ │ │ - -- .00296333s elapsed
│ │ │ - -- .0029042s elapsed
│ │ │ - -- .000232784s elapsed
│ │ │ ---backup directory created: /tmp/M2-33427-0/1
│ │ │ + -- .00359142s elapsed
│ │ │ + -- .00352355s elapsed
│ │ │ + -- .000396332s elapsed
│ │ │ + -- .00349357s elapsed
│ │ │ + -- .00358184s elapsed
│ │ │ + -- .000337732s elapsed
│ │ │ + -- .0030646s elapsed
│ │ │ + -- .00346339s elapsed
│ │ │ + -- .000302745s elapsed
│ │ │ + -- .0033665s elapsed
│ │ │ + -- .0034413s elapsed
│ │ │ + -- .000296716s elapsed
│ │ │ +--backup directory created: /tmp/M2-49481-0/1
│ │ │    H01: 1
│ │ │    H10: 1
│ │ │  number of paths tracked: 2
│ │ │  found 1 points in the fiber so far
│ │ │    H01: 1
│ │ │    H10: 1
│ │ │  number of paths tracked: 4
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_monodromy__Group.out
│ │ │ @@ -15,128 +15,128 @@
│ │ │  
│ │ │  i7 : dLoss = diff(varMatrix, gateMatrix{{loss}});
│ │ │  
│ │ │  i8 : G = gateSystem(paramMatrix,varMatrix,transpose dLoss);
│ │ │  
│ │ │  i9 : monodromyGroup(G,"msOptions" => {NumberOfEdges=>10})
│ │ │  
│ │ │ -o9 = {{2, 0, 11, 3, 10, 4, 14, 6, 5, 1, 13, 9, 7, 8, 16, 15, 12, 17, 18, 19,
│ │ │ +o9 = {{16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18,
│ │ │ +     4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 4, 6}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16,
│ │ │ +     17, 3}, {3, 16, 14, 4, 5, 2, 7, 8, 1, 11, 0, 12, 10, 15, 13, 6, 17, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 9, 4, 6}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18,
│ │ │ +     18, 19, 20}, {12, 16, 14, 19, 4, 7, 6, 11, 1, 9, 0, 5, 10, 2, 13, 20, 8,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 19, 15, 17, 3}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
│ │ │ +     18, 3, 15, 17}, {16, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 0, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 16, 17, 18, 19, 20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8,
│ │ │ +     3, 13, 15, 6, 9, 4}, {0, 1, 3, 19, 12, 5, 2, 7, 15, 8, 10, 11, 17, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 15, 12, 17, 18, 19, 20}, {0, 1, 2, 3, 4, 11, 6, 5, 8, 9, 10, 7, 12,
│ │ │ +     14, 20, 16, 18, 9, 4, 6}, {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     13, 14, 15, 16, 17, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10,
│ │ │ +     2, 13, 3, 8, 15, 18, 19, 20}, {12, 16, 14, 17, 4, 7, 6, 11, 1, 9, 0, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {3, 1, 12, 4, 0, 11, 13, 5, 2, 16,
│ │ │ +     10, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 7, 8, 15, 14, 6, 17, 9, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15,
│ │ │ +     7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {1, 7, 12, 19, 16, 4, 0, 6,
│ │ │ +     11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {4, 16, 14, 3, 5, 2, 7, 8, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     2, 13, 11, 9, 8, 10, 5, 20, 14, 18, 3, 15, 17}, {1, 16, 12, 19, 11, 15,
│ │ │ +     11, 0, 12, 10, 6, 13, 15, 9, 17, 18, 19, 20}, {12, 16, 7, 17, 10, 9, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14, 18, 9, 4, 6}, {2, 0, 1, 3, 6, 5,
│ │ │ +     4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 11, 2, 3, 10, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 7, 10, 4, 13, 11, 14, 8, 16, 15, 12, 17, 18, 19, 20}, {3, 16, 12, 4,
│ │ │ +     14, 6, 8, 1, 5, 9, 12, 13, 7, 15, 16, 17, 18, 19, 20}, {16, 14, 17, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 11, 14, 5, 2, 1, 0, 7, 8, 15, 13, 6, 17, 9, 18, 19, 20}, {12, 1, 7,
│ │ │ +     8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9, 4}, {2, 0, 11, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     17, 16, 9, 0, 4, 11, 13, 10, 6, 5, 2, 14, 3, 8, 15, 20, 18, 19}, {0, 1,
│ │ │ +     10, 4, 14, 6, 5, 1, 13, 9, 7, 8, 16, 15, 12, 17, 18, 19, 20}, {0, 1, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6},
│ │ │ +     19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {17, 16, 14, 9, 8, 7, 12, 11, 1, 2, 0, 5, 10, 3, 13, 4, 15, 6, 20, 18,
│ │ │ +     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19}, {1, 16, 12, 19, 11, 15, 5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14, 18,
│ │ │ +     1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15, 17, 3},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 4, 6}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
│ │ │ +     {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 19, 20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12,
│ │ │ +     20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     17, 18, 19, 20}, {0, 9, 2, 3, 10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15,
│ │ │ +     19, 20}, {0, 1, 2, 3, 4, 11, 6, 5, 8, 9, 10, 7, 12, 13, 14, 15, 16, 17,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 17, 18, 19, 20}, {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0,
│ │ │ +     18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 19, 13, 20, 6, 9, 4}, {2, 1, 0, 3, 4, 5, 6, 7, 13, 9, 10, 11, 16, 8,
│ │ │ +     16, 18, 9, 4, 6}, {3, 1, 12, 4, 0, 11, 13, 5, 2, 16, 10, 7, 8, 15, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     14, 15, 12, 17, 18, 19, 20}, {12, 16, 14, 19, 4, 3, 6, 15, 1, 9, 0, 17,
│ │ │ +     6, 17, 9, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 2, 13, 20, 8, 18, 7, 11, 5}, {0, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10,
│ │ │ +     14, 20, 16, 18, 9, 4, 6}, {1, 7, 12, 19, 16, 4, 0, 6, 2, 13, 11, 9, 8,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 5, 13, 14, 3, 16, 15, 6, 9, 4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10,
│ │ │ +     10, 5, 20, 14, 18, 3, 15, 17}, {1, 16, 12, 19, 11, 15, 5, 17, 2, 7, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {12, 16, 7, 17, 10, 9, 14, 4,
│ │ │ +     3, 8, 10, 13, 20, 14, 18, 9, 4, 6}, {2, 0, 1, 3, 6, 5, 9, 7, 10, 4, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 1, 2, 3, 9, 11, 4, 5,
│ │ │ +     11, 14, 8, 16, 15, 12, 17, 18, 19, 20}, {3, 16, 12, 4, 10, 11, 14, 5, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     8, 6, 10, 7, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {16, 7, 12, 19, 9, 1,
│ │ │ +     1, 0, 7, 8, 15, 13, 6, 17, 9, 18, 19, 20}, {12, 1, 7, 17, 16, 9, 0, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     4, 10, 2, 6, 11, 14, 8, 0, 5, 20, 13, 18, 3, 15, 17}, {0, 1, 2, 3, 4, 5,
│ │ │ +     11, 13, 10, 6, 5, 2, 14, 3, 8, 15, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12, 16, 3, 19,
│ │ │ +     5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {17, 16, 14, 9, 8, 7,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 11, 14, 5, 15, 1, 0, 7, 17, 2, 13, 20, 8, 18, 9, 4, 6}, {0, 9, 2, 3,
│ │ │ +     12, 11, 1, 2, 0, 5, 10, 3, 13, 4, 15, 6, 20, 18, 19}, {1, 16, 12, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20}, {0, 1, 2,
│ │ │ +     11, 15, 5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14, 18, 9, 4, 6}, {0, 1, 2, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12,
│ │ │ +     4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {2, 0, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     1, 3, 19, 0, 11, 13, 5, 15, 16, 10, 7, 17, 2, 14, 20, 8, 18, 9, 4, 6},
│ │ │ +     3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 18, 19, 20}, {0, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4,
│ │ │ +     2, 3, 10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     6}, {4, 1, 2, 3, 0, 5, 13, 7, 8, 16, 10, 11, 12, 6, 14, 15, 9, 17, 18,
│ │ │ +     {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16,
│ │ │ +     4}, {2, 1, 0, 3, 4, 5, 6, 7, 13, 9, 10, 11, 16, 8, 14, 15, 12, 17, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 9, 4, 6}, {0, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3,
│ │ │ +     19, 20}, {12, 16, 14, 19, 4, 3, 6, 15, 1, 9, 0, 17, 10, 2, 13, 20, 8,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 15, 6, 9, 4}, {0, 14, 12, 19, 11, 15, 5, 17, 2, 7, 1, 3, 8, 13, 10,
│ │ │ +     18, 7, 11, 5}, {0, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 16, 18, 9, 4, 6}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16,
│ │ │ +     16, 15, 6, 9, 4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 12, 17, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17,
│ │ │ +     18, 16, 19, 15, 17, 3}, {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     13, 14, 20, 16, 18, 9, 4, 6}, {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5,
│ │ │ +     13, 3, 8, 15, 20, 18, 19}, {0, 1, 2, 3, 9, 11, 4, 5, 8, 6, 10, 7, 12,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 0, 10, 19, 13, 20, 6, 9, 4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11,
│ │ │ +     13, 14, 15, 16, 17, 18, 19, 20}, {16, 7, 12, 19, 9, 1, 4, 10, 2, 6, 11,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 13, 14, 18, 16, 19, 15, 17, 3}, {3, 16, 14, 4, 5, 2, 7, 8, 1, 11, 0,
│ │ │ +     14, 8, 0, 5, 20, 13, 18, 3, 15, 17}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 10, 15, 13, 6, 17, 9, 18, 19, 20}, {12, 16, 14, 19, 4, 7, 6, 11, 1,
│ │ │ +     11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12, 16, 3, 19, 10, 11, 14, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 0, 5, 10, 2, 13, 20, 8, 18, 3, 15, 17}, {16, 1, 7, 17, 12, 18, 2, 19,
│ │ │ +     15, 1, 0, 7, 17, 2, 13, 20, 8, 18, 9, 4, 6}, {0, 9, 2, 3, 10, 5, 14, 7,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 8, 10, 20, 5, 0, 14, 3, 13, 15, 6, 9, 4}, {0, 1, 3, 19, 12, 5, 2, 7,
│ │ │ +     8, 1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20}, {0, 1, 2, 3, 4, 5, 6,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 8, 10, 11, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {12, 16, 7, 17, 10, 9,
│ │ │ +     7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12, 1, 3, 19, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     14, 4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 18, 19, 20}, {12, 16, 14, 17, 4,
│ │ │ +     11, 13, 5, 15, 16, 10, 7, 17, 2, 14, 20, 8, 18, 9, 4, 6}, {0, 1, 3, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     7, 6, 11, 1, 9, 0, 5, 10, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 1, 3, 19,
│ │ │ +     12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {4, 1, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1, 2,
│ │ │ +     3, 0, 5, 13, 7, 8, 16, 10, 11, 12, 6, 14, 15, 9, 17, 18, 19, 20}, {0, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {4,
│ │ │ +     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 14, 3, 5, 2, 7, 8, 1, 11, 0, 12, 10, 6, 13, 15, 9, 17, 18, 19, 20},
│ │ │ +     1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3, 16, 15, 6, 9, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18,
│ │ │ +     {0, 14, 12, 19, 11, 15, 5, 17, 2, 7, 1, 3, 8, 13, 10, 20, 16, 18, 9, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19}, {0, 11, 2, 3, 10, 4, 14, 6, 8, 1, 5, 9, 12, 13, 7, 15, 16, 17, 18,
│ │ │ +     6}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 20,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19, 20}, {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13,
│ │ │ +     18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 6, 9, 4}}
│ │ │ +     18, 9, 4, 6}}
│ │ │  
│ │ │  o9 : List
│ │ │  
│ │ │  i10 :
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/html/_dynamic__Flower__Solve.html
│ │ │ @@ -96,27 +96,27 @@
│ │ │              
    		
│ │ │                
    i3 : (p0, x0) = createSeedPair polys;
    
│ │ │              
    
│ │ │                
    i4 : (L, npaths) = dynamicFlowerSolve(polys.PolyMap,p0,{x0})
│ │ │ - -- .00296506s elapsed
│ │ │ - -- .00282693s elapsed
│ │ │ - -- .000335866s elapsed
│ │ │ - -- .0027534s elapsed
│ │ │ - -- .00295554s elapsed
│ │ │ - -- .000235038s elapsed
│ │ │ - -- .00283445s elapsed
│ │ │ - -- .00295772s elapsed
│ │ │ - -- .000236902s elapsed
│ │ │ - -- .00296333s elapsed
│ │ │ - -- .0029042s elapsed
│ │ │ - -- .000232784s elapsed
│ │ │ ---backup directory created: /tmp/M2-33427-0/1
│ │ │ + -- .00359142s elapsed
│ │ │ + -- .00352355s elapsed
│ │ │ + -- .000396332s elapsed
│ │ │ + -- .00349357s elapsed
│ │ │ + -- .00358184s elapsed
│ │ │ + -- .000337732s elapsed
│ │ │ + -- .0030646s elapsed
│ │ │ + -- .00346339s elapsed
│ │ │ + -- .000302745s elapsed
│ │ │ + -- .0033665s elapsed
│ │ │ + -- .0034413s elapsed
│ │ │ + -- .000296716s elapsed
│ │ │ +--backup directory created: /tmp/M2-49481-0/1
│ │ │    H01: 1
│ │ │    H10: 1
│ │ │  number of paths tracked: 2
│ │ │  found 1 points in the fiber so far
│ │ │    H01: 1
│ │ │    H10: 1
│ │ │  number of paths tracked: 4
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,27 +22,27 @@
│ │ │ │            o npaths, an _i_n_t_e_g_e_r,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Output is verbose. For other dynamic strategies, see _M_o_n_o_d_r_o_m_y_S_o_l_v_e_r_O_p_t_i_o_n_s.
│ │ │ │  i1 : R = CC[a,b,c,d][x,y];
│ │ │ │  i2 : polys = polySystem {a*x+b*y^2,c*x*y+d};
│ │ │ │  i3 : (p0, x0) = createSeedPair polys;
│ │ │ │  i4 : (L, npaths) = dynamicFlowerSolve(polys.PolyMap,p0,{x0})
│ │ │ │ - -- .00296506s elapsed
│ │ │ │ - -- .00282693s elapsed
│ │ │ │ - -- .000335866s elapsed
│ │ │ │ - -- .0027534s elapsed
│ │ │ │ - -- .00295554s elapsed
│ │ │ │ - -- .000235038s elapsed
│ │ │ │ - -- .00283445s elapsed
│ │ │ │ - -- .00295772s elapsed
│ │ │ │ - -- .000236902s elapsed
│ │ │ │ - -- .00296333s elapsed
│ │ │ │ - -- .0029042s elapsed
│ │ │ │ - -- .000232784s elapsed
│ │ │ │ ---backup directory created: /tmp/M2-33427-0/1
│ │ │ │ + -- .00359142s elapsed
│ │ │ │ + -- .00352355s elapsed
│ │ │ │ + -- .000396332s elapsed
│ │ │ │ + -- .00349357s elapsed
│ │ │ │ + -- .00358184s elapsed
│ │ │ │ + -- .000337732s elapsed
│ │ │ │ + -- .0030646s elapsed
│ │ │ │ + -- .00346339s elapsed
│ │ │ │ + -- .000302745s elapsed
│ │ │ │ + -- .0033665s elapsed
│ │ │ │ + -- .0034413s elapsed
│ │ │ │ + -- .000296716s elapsed
│ │ │ │ +--backup directory created: /tmp/M2-49481-0/1
│ │ │ │    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/MonomialIntegerPrograms/dump/rawdocumentation.dump
│ │ │ @@ -1,8 +1,8 @@
│ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sun Dec 14 14:09:53 2025
│ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sun Dec 14 14:09:54 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │  #:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=12
│ │ │  R3JhZGVkQmV0dGlz
│ │ ├── ./usr/share/doc/Macaulay2/Msolve/example-output/___Msolve.out
│ │ │ @@ -9,15 +9,15 @@
│ │ │  i2 : I = ideal(x, y, z)
│ │ │  
│ │ │  o2 = ideal (x, y, z)
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : msolveGB(I, Verbosity => 2, Threads => 6) 
│ │ │ - -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-53632-0/0-in.ms -o /tmp/M2-53632-0/0-out.ms
│ │ │ + -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-84650-0/0-in.ms -o /tmp/M2-84650-0/0-out.ms
│ │ │  
│ │ │  --------------- INPUT DATA ---------------
│ │ │  #variables                       3
│ │ │  #equations                       3
│ │ │  #invalid equations               0
│ │ │  field characteristic             0
│ │ │  homogeneous input?               1
│ │ │ @@ -28,15 +28,15 @@
│ │ │  initial hash table size     131072 (2^17)
│ │ │  max pair selection             ALL
│ │ │  reduce gb                        1
│ │ │  #threads                         6
│ │ │  info level                       2
│ │ │  generate pbm files               0
│ │ │  ------------------------------------------
│ │ │ -Initial prime = 1196244169
│ │ │ +Initial prime = 1266683059
│ │ │  
│ │ │  Legend for f4 information
│ │ │  --------------------------------------------------------
│ │ │  deg       current degree of pairs selected in this round
│ │ │  sel       number of pairs selected in this round
│ │ │  pairs     total number of pairs in pair list
│ │ │  mat       matrix dimensions (# rows x # columns)
│ │ │ @@ -46,25 +46,25 @@
│ │ │  time(rd)  time of the current f4 round in seconds given
│ │ │            for real and cpu time
│ │ │  --------------------------------------------------------
│ │ │  
│ │ │  deg     sel   pairs        mat          density            new data         time(rd) in sec (real|cpu)
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │ -reduce final basis        3 x 3          33.33%        3 new       0 zero         0.02 | 0.07         
│ │ │ +reduce final basis        3 x 3          33.33%        3 new       0 zero         0.00 | 0.00         
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │ -overall(elapsed)        0.06 sec
│ │ │ -overall(cpu)            0.17 sec
│ │ │ +overall(elapsed)        0.00 sec
│ │ │ +overall(cpu)            0.00 sec
│ │ │  select                  0.00 sec   0.0%
│ │ │ -symbolic prep.          0.00 sec   0.0%
│ │ │ -update                  0.03 sec  57.3%
│ │ │ -convert                 0.02 sec  42.4%
│ │ │ -linear algebra          0.00 sec   0.0%
│ │ │ +symbolic prep.          0.00 sec   0.3%
│ │ │ +update                  0.00 sec  73.3%
│ │ │ +convert                 0.00 sec   7.6%
│ │ │ +linear algebra          0.00 sec   1.0%
│ │ │  reduce gb               0.00 sec   0.0%
│ │ │  -----------------------------------------
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  size of basis                     3
│ │ │  #terms in basis                   3
│ │ │  #pairs reduced                    0
│ │ │ @@ -78,18 +78,18 @@
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  [3]
│ │ │  #polynomials to lift              3
│ │ │  -----------------------------------------
│ │ │ -New prime = 1107170621
│ │ │ +New prime = 1184773069
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │ -multi-mod overall(elapsed)      0.02 sec
│ │ │ +multi-mod overall(elapsed)      0.00 sec
│ │ │  learning phase                  0.00 Gops/sec
│ │ │  application phase               0.00 Gops/sec
│ │ │  -----------------------------------------
│ │ │  
│ │ │  multi-modular steps
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  {1}{2}<100.00%> 
│ │ │ @@ -105,15 +105,15 @@
│ │ │  ---------------- TIMINGS ----------------
│ │ │  CRT     (elapsed)               0.00 sec
│ │ │  ratrecon(elapsed)               0.00 sec
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ------------------------------------------------------------------------------------
│ │ │ -msolve overall time           0.19 sec (elapsed) /  0.49 sec (cpu)
│ │ │ +msolve overall time           0.01 sec (elapsed) /  0.04 sec (cpu)
│ │ │  ------------------------------------------------------------------------------------
│ │ │  
│ │ │  o3 = | z y x |
│ │ │  
│ │ │               1      3
│ │ │  o3 : Matrix R  <-- R
│ │ ├── ./usr/share/doc/Macaulay2/Msolve/html/index.html
│ │ │ @@ -78,15 +78,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │              
    
│ │ │                
    i3 : msolveGB(I, Verbosity => 2, Threads => 6) 
│ │ │ - -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-53632-0/0-in.ms -o /tmp/M2-53632-0/0-out.ms
│ │ │ + -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-84650-0/0-in.ms -o /tmp/M2-84650-0/0-out.ms
│ │ │  
│ │ │  --------------- INPUT DATA ---------------
│ │ │  #variables                       3
│ │ │  #equations                       3
│ │ │  #invalid equations               0
│ │ │  field characteristic             0
│ │ │  homogeneous input?               1
│ │ │ @@ -97,15 +97,15 @@
│ │ │  initial hash table size     131072 (2^17)
│ │ │  max pair selection             ALL
│ │ │  reduce gb                        1
│ │ │  #threads                         6
│ │ │  info level                       2
│ │ │  generate pbm files               0
│ │ │  ------------------------------------------
│ │ │ -Initial prime = 1196244169
│ │ │ +Initial prime = 1266683059
│ │ │  
│ │ │  Legend for f4 information
│ │ │  --------------------------------------------------------
│ │ │  deg       current degree of pairs selected in this round
│ │ │  sel       number of pairs selected in this round
│ │ │  pairs     total number of pairs in pair list
│ │ │  mat       matrix dimensions (# rows x # columns)
│ │ │ @@ -115,25 +115,25 @@
│ │ │  time(rd)  time of the current f4 round in seconds given
│ │ │            for real and cpu time
│ │ │  --------------------------------------------------------
│ │ │  
│ │ │  deg     sel   pairs        mat          density            new data         time(rd) in sec (real|cpu)
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │ -reduce final basis        3 x 3          33.33%        3 new       0 zero         0.02 | 0.07         
│ │ │ +reduce final basis        3 x 3          33.33%        3 new       0 zero         0.00 | 0.00         
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │ -overall(elapsed)        0.06 sec
│ │ │ -overall(cpu)            0.17 sec
│ │ │ +overall(elapsed)        0.00 sec
│ │ │ +overall(cpu)            0.00 sec
│ │ │  select                  0.00 sec   0.0%
│ │ │ -symbolic prep.          0.00 sec   0.0%
│ │ │ -update                  0.03 sec  57.3%
│ │ │ -convert                 0.02 sec  42.4%
│ │ │ -linear algebra          0.00 sec   0.0%
│ │ │ +symbolic prep.          0.00 sec   0.3%
│ │ │ +update                  0.00 sec  73.3%
│ │ │ +convert                 0.00 sec   7.6%
│ │ │ +linear algebra          0.00 sec   1.0%
│ │ │  reduce gb               0.00 sec   0.0%
│ │ │  -----------------------------------------
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  size of basis                     3
│ │ │  #terms in basis                   3
│ │ │  #pairs reduced                    0
│ │ │ @@ -147,18 +147,18 @@
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  [3]
│ │ │  #polynomials to lift              3
│ │ │  -----------------------------------------
│ │ │ -New prime = 1107170621
│ │ │ +New prime = 1184773069
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │ -multi-mod overall(elapsed)      0.02 sec
│ │ │ +multi-mod overall(elapsed)      0.00 sec
│ │ │  learning phase                  0.00 Gops/sec
│ │ │  application phase               0.00 Gops/sec
│ │ │  -----------------------------------------
│ │ │  
│ │ │  multi-modular steps
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  {1}{2}<100.00%> 
│ │ │ @@ -174,15 +174,15 @@
│ │ │  ---------------- TIMINGS ----------------
│ │ │  CRT     (elapsed)               0.00 sec
│ │ │  ratrecon(elapsed)               0.00 sec
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ------------------------------------------------------------------------------------
│ │ │ -msolve overall time           0.19 sec (elapsed) /  0.49 sec (cpu)
│ │ │ +msolve overall time           0.01 sec (elapsed) /  0.04 sec (cpu)
│ │ │  ------------------------------------------------------------------------------------
│ │ │  
│ │ │  o3 = | z y x |
│ │ │  
│ │ │               1      3
│ │ │  o3 : Matrix R  <-- R
    
│ │ │              
    
│ │ │                
    i5 : peek componentsOfKernel(2, F)
│ │ │  warning: computation begun over finite field. resulting polynomials may not lie in the ideal
│ │ │  computing total degree: 1
│ │ │  number of monomials = 6
│ │ │  number of distinct multidegrees = 6
│ │ │ - -- .00186857s elapsed
│ │ │ + -- .00246071s elapsed
│ │ │  computing total degree: 2
│ │ │  number of monomials = 21
│ │ │  number of distinct multidegrees = 18
│ │ │ - -- .00833047s elapsed
│ │ │ + -- .0106961s elapsed
│ │ │  
│ │ │  o5 = MutableHashTable{{0, 1, 0, 0, 1} => {}                   }
│ │ │                        {0, 1, 0, 1, 0} => {}
│ │ │                        {0, 1, 1, 0, 0} => {}
│ │ │                        {0, 2, 0, 0, 2} => {}
│ │ │                        {0, 2, 0, 1, 1} => {}
│ │ │                        {0, 2, 0, 2, 0} => {}
│ │ │ ├── html2text {}
│ │ │ │ @@ -51,19 +51,19 @@
│ │ │ │  o4 : RingMap S <-- R
│ │ │ │  i5 : peek componentsOfKernel(2, F)
│ │ │ │  warning: computation begun over finite field. resulting polynomials may not lie
│ │ │ │  in the ideal
│ │ │ │  computing total degree: 1
│ │ │ │  number of monomials = 6
│ │ │ │  number of distinct multidegrees = 6
│ │ │ │ - -- .00186857s elapsed
│ │ │ │ + -- .00246071s elapsed
│ │ │ │  computing total degree: 2
│ │ │ │  number of monomials = 21
│ │ │ │  number of distinct multidegrees = 18
│ │ │ │ - -- .00833047s elapsed
│ │ │ │ + -- .0106961s elapsed
│ │ │ │  
│ │ │ │  o5 = MutableHashTable{{0, 1, 0, 0, 1} => {}                   }
│ │ │ │                        {0, 1, 0, 1, 0} => {}
│ │ │ │                        {0, 1, 1, 0, 0} => {}
│ │ │ │                        {0, 2, 0, 0, 2} => {}
│ │ │ │                        {0, 2, 0, 1, 1} => {}
│ │ │ │                        {0, 2, 0, 2, 0} => {}
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_j__Mult.out
│ │ │ @@ -9,25 +9,25 @@
│ │ │  i2 : I = ideal"xy,yz,zx"
│ │ │  
│ │ │  o2 = ideal (x*y, y*z, x*z)
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : elapsedTime jMult I
│ │ │ - -- .0234547s elapsed
│ │ │ + -- .027638s elapsed
│ │ │  
│ │ │  o3 = 2
│ │ │  
│ │ │  i4 : elapsedTime monjMult I
│ │ │ - -- .107713s elapsed
│ │ │ + -- .0842469s elapsed
│ │ │  
│ │ │  o4 = 2
│ │ │  
│ │ │  i5 : elapsedTime multiplicitySequence I
│ │ │ - -- .181349s elapsed
│ │ │ + -- .140034s elapsed
│ │ │  
│ │ │  o5 = HashTable{2 => 3}
│ │ │                 3 => 2
│ │ │  
│ │ │  o5 : HashTable
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_mon__Analytic__Spread.out
│ │ │ @@ -10,12 +10,12 @@
│ │ │  
│ │ │               2        3
│ │ │  o2 = ideal (x , x*y, y )
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : elapsedTime monAnalyticSpread I
│ │ │ - -- .171404s elapsed
│ │ │ + -- .0976458s elapsed
│ │ │  
│ │ │  o3 = 2
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_monj__Mult.out
│ │ │ @@ -13,17 +13,17 @@
│ │ │       ------------------------------------------------------------------------
│ │ │        10 11   8 12   9 11   10 10   11 9   12 8
│ │ │       x  y  , x y  , x y  , x  y  , x  y , x  y )
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : elapsedTime monjMult I
│ │ │ - -- .119902s elapsed
│ │ │ + -- .106255s elapsed
│ │ │  
│ │ │  o3 = 192
│ │ │  
│ │ │  i4 : elapsedTime jMult I
│ │ │ - -- 1.54599s elapsed
│ │ │ + -- 1.37349s elapsed
│ │ │  
│ │ │  o4 = 192
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_j__Mult.html
│ │ │ @@ -88,31 +88,31 @@
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime jMult I
│ │ │ - -- .0234547s elapsed
│ │ │ + -- .027638s elapsed
│ │ │  
│ │ │  o3 = 2
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime monjMult I
│ │ │ - -- .107713s elapsed
│ │ │ + -- .0842469s elapsed
│ │ │  
│ │ │  o4 = 2
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime multiplicitySequence I
│ │ │ - -- .181349s elapsed
│ │ │ + -- .140034s elapsed
│ │ │  
│ │ │  o5 = HashTable{2 => 3}
│ │ │                 3 => 2
│ │ │  
│ │ │  o5 : HashTable
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime monAnalyticSpread I
│ │ │ - -- .171404s elapsed
│ │ │ + -- .0976458s elapsed
│ │ │  
│ │ │  o3 = 2
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,15 +22,15 @@
│ │ │ │  i2 : I = ideal"x2,xy,y3"
│ │ │ │  
│ │ │ │               2        3
│ │ │ │  o2 = ideal (x , x*y, y )
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : elapsedTime monAnalyticSpread I
│ │ │ │ - -- .171404s elapsed
│ │ │ │ + -- .0976458s elapsed
│ │ │ │  
│ │ │ │  o3 = 2
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _N_P -- the Newton polyhedron of a monomial ideal
│ │ │ │  ********** WWaayyss ttoo uussee mmoonnAAnnaallyyttiiccSSpprreeaadd:: **********
│ │ │ │      * monAnalyticSpread(Ideal)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_monj__Mult.html
│ │ │ @@ -92,23 +92,23 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │              
    
│ │ │                
    i3 : elapsedTime monjMult I
│ │ │ - -- .119902s elapsed
│ │ │ + -- .106255s elapsed
│ │ │  
│ │ │  o3 = 192
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime jMult I
│ │ │ - -- 1.54599s elapsed
│ │ │ + -- 1.37349s elapsed
│ │ │  
│ │ │  o4 = 192
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,19 +24,19 @@
│ │ │ │  o2 = ideal (x y  , x y  , x y  , x y  , x y  , x y  , x y  , x y  , x y  ,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        10 11   8 12   9 11   10 10   11 9   12 8
│ │ │ │       x  y  , x y  , x y  , x  y  , x  y , x  y )
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : elapsedTime monjMult I
│ │ │ │ - -- .119902s elapsed
│ │ │ │ + -- .106255s elapsed
│ │ │ │  
│ │ │ │  o3 = 192
│ │ │ │  i4 : elapsedTime jMult I
│ │ │ │ - -- 1.54599s elapsed
│ │ │ │ + -- 1.37349s elapsed
│ │ │ │  
│ │ │ │  o4 = 192
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_u_l_t_i_p_l_i_c_i_t_y_S_e_q_u_e_n_c_e -- the multiplicity sequence of an ideal
│ │ │ │      * _j_M_u_l_t -- the j-multiplicity of an ideal
│ │ │ │      * _m_o_n_R_e_d_u_c_t_i_o_n -- the minimal monomial reduction of a monomial ideal
│ │ │ │      * _N_P -- the Newton polyhedron of a monomial ideal
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Embedded__Projective__Variety_sp_eq_eq_eq_gt_sp__Embedded__Projective__Variety.out
│ │ │ @@ -13,15 +13,15 @@
│ │ │  o4 : ProjectiveVariety, curve in PP^8
│ │ │  
│ │ │  i5 : ? X
│ │ │  
│ │ │  o5 = curve in PP^8 cut out by 17 hypersurfaces of degrees 1^2 2^15 
│ │ │  
│ │ │  i6 : time f = X ===> Y;
│ │ │ - -- used 3.0434s (cpu); 1.80018s (thread); 0s (gc)
│ │ │ + -- used 4.13499s (cpu); 2.08126s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : MultirationalMap (automorphism of PP^8)
│ │ │  
│ │ │  i7 : f X
│ │ │  
│ │ │  o7 = Y
│ │ │  
│ │ │ @@ -38,15 +38,15 @@
│ │ │  o9 : ProjectiveVariety, 6-dimensional subvariety of PP^8
│ │ │  
│ │ │  i10 : W = random({{2},{1}},Y);
│ │ │  
│ │ │  o10 : ProjectiveVariety, 6-dimensional subvariety of PP^8
│ │ │  
│ │ │  i11 : time g = V ===> W;
│ │ │ - -- used 3.14065s (cpu); 1.91214s (thread); 0s (gc)
│ │ │ + -- used 3.69451s (cpu); 2.10155s (thread); 0s (gc)
│ │ │  
│ │ │  o11 : MultirationalMap (automorphism of PP^8)
│ │ │  
│ │ │  i12 : g||W
│ │ │  
│ │ │  o12 = multi-rational map consisting of one single rational map
│ │ │        source variety: 6-dimensional subvariety of PP^8 cut out by 2 hypersurfaces of degrees 1^1 2^1 
│ │ │ @@ -129,15 +129,15 @@
│ │ │  o15 : ProjectiveVariety, 6-dimensional subvariety of PP^9
│ │ │  
│ │ │  i16 : ? Z
│ │ │  
│ │ │  o16 = 6-dimensional subvariety of PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │  
│ │ │  i17 : time h = Z ===> GG_K(1,4)
│ │ │ - -- used 8.07742s (cpu); 4.95873s (thread); 0s (gc)
│ │ │ + -- used 6.52083s (cpu); 4.57396s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = h
│ │ │  
│ │ │  o17 : MultirationalMap (isomorphism from PP^9 to PP^9)
│ │ │  
│ │ │  i18 : h || GG_K(1,4)
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp^_st_st_sp__Multiprojective__Variety.out
│ │ │ @@ -7,15 +7,15 @@
│ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to PP^2 x PP^4)
│ │ │  
│ │ │  i3 : Y = projectiveVariety ideal(random({1,1},ring target Phi), random({1,1},ring target Phi));
│ │ │  
│ │ │  o3 : ProjectiveVariety, 4-dimensional subvariety of PP^2 x PP^4
│ │ │  
│ │ │  i4 : time X = Phi^* Y;
│ │ │ - -- used 5.11368s (cpu); 3.88996s (thread); 0s (gc)
│ │ │ + -- used 4.40229s (cpu); 3.72029s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : ProjectiveVariety, curve in PP^3 x PP^2 x PP^4 (subvariety of codimension 2 in threefold in PP^3 x PP^2 x PP^4 cut out by 12 hypersurfaces of multi-degrees (0,0,2)^1 (0,1,1)^2 (1,0,1)^7 (1,1,0)^2 )
│ │ │  
│ │ │  i5 : dim X, degree X, degrees X
│ │ │  
│ │ │  o5 = (1, 31, {({0, 0, 2}, 1), ({0, 0, 3}, 4), ({0, 1, 1}, 4), ({0, 4, 1}, 1),
│ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp__Multiprojective__Variety.out
│ │ │ @@ -11,26 +11,26 @@
│ │ │  o3 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^7 to PP^7 x PP^7)
│ │ │  
│ │ │  i4 : Z = source Phi;
│ │ │  
│ │ │  o4 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^7
│ │ │  
│ │ │  i5 : time Phi Z;
│ │ │ - -- used 0.0951343s (cpu); 0.0963862s (thread); 0s (gc)
│ │ │ + -- used 0.130263s (cpu); 0.111376s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^7
│ │ │  
│ │ │  i6 : dim oo, degree oo, degrees oo
│ │ │  
│ │ │  o6 = (4, 80, {({0, 2}, 5), ({1, 1}, 33), ({2, 0}, 5)})
│ │ │  
│ │ │  o6 : Sequence
│ │ │  
│ │ │  i7 : time Phi (point Z + point Z + point Z)
│ │ │ - -- used 2.05743s (cpu); 1.42394s (thread); 0s (gc)
│ │ │ + -- used 2.51637s (cpu); 1.40924s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = 0-dimensional subvariety of PP^7 x PP^7 cut out by 22 hypersurfaces of multi-degrees (0,1)^5 (0,2)^3 (1,0)^5 (1,1)^6 (2,0)^3 
│ │ │  
│ │ │  o7 : ProjectiveVariety, 0-dimensional subvariety of PP^7 x PP^7
│ │ │  
│ │ │  i8 : dim oo, degree oo, degrees oo
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multirational__Map_cm__Option_rp.out
│ │ │ @@ -11,22 +11,22 @@
│ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │       source variety: threefold in PP^4 x PP^4 cut out by 13 hypersurfaces of
│ │ │       target variety: hypersurface in PP^4 defined by a form of degree 2
│ │ │       ------------------------------------------------------------------------
│ │ │       multi-degrees (0,2)^1 (1,1)^3 (2,1)^8 (4,0)^1
│ │ │  
│ │ │  i4 : time degree(Phi,Strategy=>"random point")
│ │ │ - -- used 3.60186s (cpu); 2.37753s (thread); 0s (gc)
│ │ │ + -- used 4.33556s (cpu); 2.47925s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 2
│ │ │  
│ │ │  i5 : time degree(Phi,Strategy=>"0-th projective degree")
│ │ │ - -- used 0.300764s (cpu); 0.253978s (thread); 0s (gc)
│ │ │ + -- used 0.35768s (cpu); 0.283995s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
│ │ │  
│ │ │  i6 : time degree Phi
│ │ │ - -- used 0.32363s (cpu); 0.259653s (thread); 0s (gc)
│ │ │ + -- used 0.332341s (cpu); 0.262557s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = 2
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multirational__Map_rp.out
│ │ │ @@ -3,12 +3,12 @@
│ │ │  i1 : ZZ/300007[x_0..x_3], f = rationalMap {x_2^2-x_1*x_3, x_1*x_2-x_0*x_3, x_1^2-x_0*x_2}, g = rationalMap {x_1^2-x_0*x_2, x_0*x_3, x_1*x_3, x_2*x_3, x_3^2};
│ │ │  
│ │ │  i2 : Phi = last graph rationalMap {f,g};
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to PP^2 x PP^4)
│ │ │  
│ │ │  i3 : time degree Phi
│ │ │ - -- used 0.39932s (cpu); 0.353204s (thread); 0s (gc)
│ │ │ + -- used 0.574524s (cpu); 0.371837s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = 1
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_describe_lp__Multirational__Map_rp.out
│ │ │ @@ -1,52 +1,52 @@
│ │ │  -- -*- M2-comint -*- hash: 11533721324852072161
│ │ │  
│ │ │  i1 : Phi = multirationalMap graph rationalMap PP_(ZZ/65521)^(1,4);
│ │ │  
│ │ │  o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^4 x PP^5)
│ │ │  
│ │ │  i2 : time ? Phi
│ │ │ - -- used 0.000917731s (cpu); 0.000161313s (thread); 0s (gc)
│ │ │ + -- used 0.000922781s (cpu); 0.000167482s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9
│ │ │       target variety: PP^4 x PP^5
│ │ │       ------------------------------------------------------------------------
│ │ │       hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
│ │ │  
│ │ │  i3 : image Phi;
│ │ │  
│ │ │  o3 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^5
│ │ │  
│ │ │  i4 : time ? Phi
│ │ │ - -- used 0.00272072s (cpu); 0.000248957s (thread); 0s (gc)
│ │ │ + -- used 0.0010021s (cpu); 0.000243723s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │       target variety: PP^4 x PP^5
│ │ │       dominance: false
│ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │  
│ │ │  i5 : time describe Phi
│ │ │ - -- used 1.4658s (cpu); 1.13064s (thread); 0s (gc)
│ │ │ + -- used 1.25956s (cpu); 1.04959s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │       target variety: PP^4 x PP^5
│ │ │       base locus: empty subscheme of PP^4 x PP^5
│ │ │       dominance: false
│ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │       multidegree: {51, 51, 51, 51, 51}
│ │ │       degree: 1
│ │ │       degree sequence (map 1/2): [(1,0), (0,2)]
│ │ │       degree sequence (map 2/2): [(0,1), (2,0)]
│ │ │       coefficient ring: ZZ/65521
│ │ │  
│ │ │  i6 : time ? Phi
│ │ │ - -- used 0.000169538s (cpu); 0.000607139s (thread); 0s (gc)
│ │ │ + -- used 0.000130585s (cpu); 0.000457767s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │       target variety: PP^4 x PP^5
│ │ │       base locus: empty subscheme of PP^4 x PP^5
│ │ │       dominance: false
│ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_graph_lp__Multirational__Map_rp.out
│ │ │ @@ -3,45 +3,45 @@
│ │ │  i1 : Phi = rationalMap(PP_(ZZ/333331)^(1,4),Dominant=>true)
│ │ │  
│ │ │  o1 = Phi
│ │ │  
│ │ │  o1 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │  
│ │ │  i2 : time (Phi1,Phi2) = graph Phi
│ │ │ - -- used 0.0895117s (cpu); 0.0398444s (thread); 0s (gc)
│ │ │ + -- used 0.11519s (cpu); 0.0525209s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = (Phi1, Phi2)
│ │ │  
│ │ │  o2 : Sequence
│ │ │  
│ │ │  i3 : Phi1;
│ │ │  
│ │ │  o3 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^4)
│ │ │  
│ │ │  i4 : Phi2;
│ │ │  
│ │ │  o4 : MultirationalMap (dominant rational map from 4-dimensional subvariety of PP^4 x PP^5 to hypersurface in PP^5)
│ │ │  
│ │ │  i5 : time (Phi21,Phi22) = graph Phi2
│ │ │ - -- used 0.032293s (cpu); 0.0340052s (thread); 0s (gc)
│ │ │ + -- used 0.0750556s (cpu); 0.0584397s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = (Phi21, Phi22)
│ │ │  
│ │ │  o5 : Sequence
│ │ │  
│ │ │  i6 : Phi21;
│ │ │  
│ │ │  o6 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to 4-dimensional subvariety of PP^4 x PP^5)
│ │ │  
│ │ │  i7 : Phi22;
│ │ │  
│ │ │  o7 : MultirationalMap (dominant rational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to hypersurface in PP^5)
│ │ │  
│ │ │  i8 : time (Phi211,Phi212) = graph Phi21
│ │ │ - -- used 0.194002s (cpu); 0.149271s (thread); 0s (gc)
│ │ │ + -- used 0.347284s (cpu); 0.207232s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = (Phi211, Phi212)
│ │ │  
│ │ │  o8 : Sequence
│ │ │  
│ │ │  i9 : Phi211;
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_image_lp__Multirational__Map_rp.out
│ │ │ @@ -11,25 +11,25 @@
│ │ │  o3 : RationalMap (quadratic rational map from PP^4 to PP^4)
│ │ │  
│ │ │  i4 : Phi = rationalMap {f,g};
│ │ │  
│ │ │  o4 : MultirationalMap (rational map from PP^4 to PP^7 x PP^4)
│ │ │  
│ │ │  i5 : time Z = image Phi;
│ │ │ - -- used 0.200211s (cpu); 0.133927s (thread); 0s (gc)
│ │ │ + -- used 0.185535s (cpu); 0.12657s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
│ │ │  
│ │ │  i6 : dim Z, degree Z, degrees Z
│ │ │  
│ │ │  o6 = (4, 151, {({1, 1}, 4), ({1, 2}, 3), ({2, 0}, 5), ({2, 1}, 13)})
│ │ │  
│ │ │  o6 : Sequence
│ │ │  
│ │ │  i7 : time Z' = projectiveVariety (map segre target Phi) image(segre Phi,"F4");
│ │ │ - -- used 5.791s (cpu); 2.94859s (thread); 0s (gc)
│ │ │ + -- used 10.319s (cpu); 2.72855s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
│ │ │  
│ │ │  i8 : assert(Z == Z')
│ │ │  
│ │ │  i9 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_inverse2.out
│ │ │ @@ -4,25 +4,25 @@
│ │ │  
│ │ │  i2 : -- map defined by the cubics through the secant variety to the rational normal curve of degree 6
│ │ │       Phi = multirationalMap rationalMap(ring PP_K^6,ring GG_K(2,4),gens ideal PP_K([6],2));
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from PP^6 to GG(2,4))
│ │ │  
│ │ │  i3 : time Psi = inverse2 Phi;
│ │ │ - -- used 0.36801s (cpu); 0.291283s (thread); 0s (gc)
│ │ │ + -- used 0.408651s (cpu); 0.322879s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : MultirationalMap (birational map from GG(2,4) to PP^6)
│ │ │  
│ │ │  i4 : assert(Phi * Psi == 1)
│ │ │  
│ │ │  i5 : Phi' = Phi || Phi;
│ │ │  
│ │ │  o5 : MultirationalMap (rational map from PP^6 x PP^6 to GG(2,4) x GG(2,4))
│ │ │  
│ │ │  i6 : time Psi' = inverse2 Phi';
│ │ │ - -- used 1.53154s (cpu); 1.09582s (thread); 0s (gc)
│ │ │ + -- used 1.20589s (cpu); 1.03972s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : MultirationalMap (birational map from GG(2,4) x GG(2,4) to PP^6 x PP^6)
│ │ │  
│ │ │  i7 : assert(Phi' * Psi' == 1)
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_inverse_lp__Multirational__Map_rp.out
│ │ │ @@ -7,33 +7,33 @@
│ │ │  
│ │ │  i2 : -- we see Phi as a dominant map
│ │ │       Phi = rationalMap(Phi,image Phi);
│ │ │  
│ │ │  o2 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │  
│ │ │  i3 : time inverse Phi;
│ │ │ - -- used 0.151859s (cpu); 0.0747115s (thread); 0s (gc)
│ │ │ + -- used 0.114552s (cpu); 0.0638305s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : MultirationalMap (birational map from hypersurface in PP^5 to PP^4)
│ │ │  
│ │ │  i4 : Psi = last graph Phi;
│ │ │  
│ │ │  o4 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 to hypersurface in PP^5)
│ │ │  
│ │ │  i5 : time inverse Psi;
│ │ │ - -- used 0.1777s (cpu); 0.0987744s (thread); 0s (gc)
│ │ │ + -- used 0.295237s (cpu); 0.112553s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : MultirationalMap (birational map from hypersurface in PP^5 to 4-dimensional subvariety of PP^4 x PP^5)
│ │ │  
│ │ │  i6 : Eta = first graph Psi;
│ │ │  
│ │ │  o6 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to 4-dimensional subvariety of PP^4 x PP^5)
│ │ │  
│ │ │  i7 : time inverse Eta;
│ │ │ - -- used 0.442368s (cpu); 0.288571s (thread); 0s (gc)
│ │ │ + -- used 0.568476s (cpu); 0.328152s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 to 4-dimensional subvariety of PP^4 x PP^5 x PP^5)
│ │ │  
│ │ │  i8 : assert(Phi * Phi^-1 == 1 and Phi^-1 * Phi == 1)
│ │ │  
│ │ │  i9 : assert(Psi * Psi^-1 == 1 and Psi^-1 * Psi == 1)
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Isomorphism_lp__Multirational__Map_rp.out
│ │ │ @@ -6,32 +6,32 @@
│ │ │  o2 : RationalMap (quadratic rational map from PP^3 to PP^2)
│ │ │  
│ │ │  i3 : Phi = rationalMap {f,f};
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from PP^3 to PP^2 x PP^2)
│ │ │  
│ │ │  i4 : time isIsomorphism Phi
│ │ │ - -- used 0.00299491s (cpu); 9.007e-06s (thread); 0s (gc)
│ │ │ + -- used 0.00336491s (cpu); 8.46e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = false
│ │ │  
│ │ │  i5 : Psi = first graph Phi;
│ │ │  
│ │ │  o5 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 to PP^3)
│ │ │  
│ │ │  i6 : time isIsomorphism Psi
│ │ │ - -- used 0.331015s (cpu); 0.180919s (thread); 0s (gc)
│ │ │ + -- used 0.423703s (cpu); 0.190694s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = false
│ │ │  
│ │ │  i7 : Eta = first graph Psi;
│ │ │  
│ │ │  o7 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 x PP^3 to threefold in PP^3 x PP^2 x PP^2)
│ │ │  
│ │ │  i8 : time isIsomorphism Eta
│ │ │ - -- used 1.56012s (cpu); 0.826624s (thread); 0s (gc)
│ │ │ + -- used 1.69466s (cpu); 0.865677s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : assert(o8 and (not o6) and (not o4))
│ │ │  
│ │ │  i10 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Morphism_lp__Multirational__Map_rp.out
│ │ │ @@ -3,24 +3,24 @@
│ │ │  i1 : ZZ/300007[a..e], f = first graph rationalMap ideal(c^2-b*d,b*c-a*d,b^2-a*c,e), g = rationalMap submatrix(matrix f,{0..2});
│ │ │  
│ │ │  i2 : Phi = rationalMap {f,g};
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^7 to PP^4 x PP^2)
│ │ │  
│ │ │  i3 : time isMorphism Phi
│ │ │ - -- used 0.369946s (cpu); 0.279856s (thread); 0s (gc)
│ │ │ + -- used 0.506281s (cpu); 0.258799s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = false
│ │ │  
│ │ │  i4 : time Psi = first graph Phi;
│ │ │ - -- used 0.170151s (cpu); 0.094037s (thread); 0s (gc)
│ │ │ + -- used 0.128697s (cpu); 0.0729765s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^7 x PP^4 x PP^2 to 4-dimensional subvariety of PP^4 x PP^7)
│ │ │  
│ │ │  i5 : time isMorphism Psi
│ │ │ - -- used 4.39879s (cpu); 3.33728s (thread); 0s (gc)
│ │ │ + -- used 3.59347s (cpu); 2.98975s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
│ │ │  
│ │ │  i6 : assert((not o3) and o5)
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_linearly__Normal__Embedding.out
│ │ │ @@ -3,24 +3,24 @@
│ │ │  i1 : K = ZZ/333331;
│ │ │  
│ │ │  i2 : X = PP_K^(1,7); -- rational normal curve of degree 7
│ │ │  
│ │ │  o2 : ProjectiveVariety, curve in PP^7
│ │ │  
│ │ │  i3 : time f = linearlyNormalEmbedding X;
│ │ │ - -- used 0.010893s (cpu); 0.00980925s (thread); 0s (gc)
│ │ │ + -- used 0.108782s (cpu); 0.0340201s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : MultirationalMap (automorphism of X)
│ │ │  
│ │ │  i4 : Y = (rationalMap {for i to 3 list random(1,ring ambient X)}) X; -- an isomorphic projection of X in PP^3
│ │ │  
│ │ │  o4 : ProjectiveVariety, curve in PP^3
│ │ │  
│ │ │  i5 : time g = linearlyNormalEmbedding Y;
│ │ │ - -- used 0.564105s (cpu); 0.442475s (thread); 0s (gc)
│ │ │ + -- used 0.515086s (cpu); 0.434805s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : MultirationalMap (birational map from Y to curve in PP^7)
│ │ │  
│ │ │  i6 : assert(isIsomorphism g)
│ │ │  
│ │ │  i7 : describe g
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multidegree_lp__Multirational__Map_rp.out
│ │ │ @@ -3,15 +3,15 @@
│ │ │  i1 : ZZ/300007[x_0..x_3], f = rationalMap {x_2^2-x_1*x_3, x_1*x_2-x_0*x_3, x_1^2-x_0*x_2}, g = rationalMap {x_1^2-x_0*x_2, x_0*x_3, x_1*x_3, x_2*x_3, x_3^2};
│ │ │  
│ │ │  i2 : Phi = last graph rationalMap {f,g};
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to PP^2 x PP^4)
│ │ │  
│ │ │  i3 : time multidegree Phi
│ │ │ - -- used 0.521458s (cpu); 0.386885s (thread); 0s (gc)
│ │ │ + -- used 0.50863s (cpu); 0.364535s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = {66, 46, 31, 20}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : (degree source Phi,degree image Phi)
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multidegree_lp__Z__Z_cm__Multirational__Map_rp.out
│ │ │ @@ -1,21 +1,21 @@
│ │ │  -- -*- M2-comint -*- hash: 16199733219210081214
│ │ │  
│ │ │  i1 : Phi = last graph rationalMap PP_(ZZ/300007)^(1,4);
│ │ │  
│ │ │  o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^5)
│ │ │  
│ │ │  i2 : for i in {4,3,2,1,0} list time multidegree(i,Phi)
│ │ │ - -- used 0.00397921s (cpu); 0.00133095s (thread); 0s (gc)
│ │ │ - -- used 0.272866s (cpu); 0.142784s (thread); 0s (gc)
│ │ │ - -- used 0.237166s (cpu); 0.173525s (thread); 0s (gc)
│ │ │ - -- used 0.202997s (cpu); 0.139949s (thread); 0s (gc)
│ │ │ - -- used 0.187465s (cpu); 0.110073s (thread); 0s (gc)
│ │ │ + -- used 0.00399531s (cpu); 0.0012992s (thread); 0s (gc)
│ │ │ + -- used 0.221072s (cpu); 0.147087s (thread); 0s (gc)
│ │ │ + -- used 0.243298s (cpu); 0.168701s (thread); 0s (gc)
│ │ │ + -- used 0.221095s (cpu); 0.152101s (thread); 0s (gc)
│ │ │ + -- used 0.191018s (cpu); 0.12145s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = {51, 28, 14, 6, 2}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : time assert(oo == multidegree Phi)
│ │ │ - -- used 0.127059s (cpu); 0.0818662s (thread); 0s (gc)
│ │ │ + -- used 0.2182s (cpu); 0.0975932s (thread); 0s (gc)
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_point_lp__Multiprojective__Variety_rp.out
│ │ │ @@ -3,26 +3,26 @@
│ │ │  i1 : K = ZZ/1000003;
│ │ │  
│ │ │  i2 : X = PP_K^({1,1,2},{3,2,3});
│ │ │  
│ │ │  o2 : ProjectiveVariety, 4-dimensional subvariety of PP^3 x PP^2 x PP^9
│ │ │  
│ │ │  i3 : time p := point X
│ │ │ - -- used 0.0161552s (cpu); 0.0167468s (thread); 0s (gc)
│ │ │ + -- used 0.0319696s (cpu); 0.0192418s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = point of coordinates ([421369, 39917, -212481, 1],[-128795, -176966, 1],[3870, -390108, -496127, -308581, 46649, 164926, -446111, 48038, 415309, 1])
│ │ │  
│ │ │  o3 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9
│ │ │  
│ │ │  i4 : Y = random({2,1,2},X);
│ │ │  
│ │ │  o4 : ProjectiveVariety, hypersurface in PP^3 x PP^2 x PP^9
│ │ │  
│ │ │  i5 : time q = point Y
│ │ │ - -- used 1.74407s (cpu); 0.993938s (thread); 0s (gc)
│ │ │ + -- used 1.52734s (cpu); 0.994378s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = q
│ │ │  
│ │ │  o5 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9
│ │ │  
│ │ │  i6 : assert(isSubset(p,X) and isSubset(q,Y))
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_segre_lp__Multirational__Map_rp.out
│ │ │ @@ -15,15 +15,15 @@
│ │ │  o4 : RationalMap (quadratic rational map from PP^4 to PP^4)
│ │ │  
│ │ │  i5 : Phi = rationalMap {f,g,h};
│ │ │  
│ │ │  o5 : MultirationalMap (rational map from PP^4 to hypersurface in PP^5 x PP^4 x PP^4)
│ │ │  
│ │ │  i6 : time segre Phi;
│ │ │ - -- used 0.750404s (cpu); 0.533826s (thread); 0s (gc)
│ │ │ + -- used 1.11502s (cpu); 0.590124s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : RationalMap (rational map from PP^4 to PP^149)
│ │ │  
│ │ │  i7 : describe segre Phi
│ │ │  
│ │ │  o7 = rational map defined by forms of degree 6
│ │ │       source variety: PP^4
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_show_lp__Multirational__Map_rp.out
│ │ │ @@ -3,15 +3,15 @@
│ │ │  i1 : Phi = inverse first graph last graph rationalMap PP_(ZZ/33331)^(1,3)
│ │ │  
│ │ │  o1 = Phi
│ │ │  
│ │ │  o1 : MultirationalMap (birational map from threefold in PP^3 x PP^2 to threefold in PP^3 x PP^2 x PP^2)
│ │ │  
│ │ │  i2 : time describe Phi
│ │ │ - -- used 0.273463s (cpu); 0.167098s (thread); 0s (gc)
│ │ │ + -- used 0.225094s (cpu); 0.149449s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = multi-rational map consisting of 3 rational maps
│ │ │       source variety: threefold in PP^3 x PP^2 cut out by 2 hypersurfaces of multi-degree (1,1)
│ │ │       target variety: threefold in PP^3 x PP^2 x PP^2 cut out by 7 hypersurfaces of multi-degrees (0,1,1)^3 (1,0,1)^2 (1,1,0)^2 
│ │ │       base locus: empty subscheme of PP^3 x PP^2
│ │ │       dominance: true
│ │ │       multidegree: {10, 14, 19, 25}
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Embedded__Projective__Variety_sp_eq_eq_eq_gt_sp__Embedded__Projective__Variety.html
│ │ │ @@ -103,15 +103,15 @@
│ │ │  
│ │ │  o5 = curve in PP^8 cut out by 17 hypersurfaces of degrees 1^2 2^15 
    
│ │ │              
    
│ │ │                
    i6 : time f = X ===> Y;
│ │ │ - -- used 3.0434s (cpu); 1.80018s (thread); 0s (gc)
│ │ │ + -- used 4.13499s (cpu); 2.08126s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : MultirationalMap (automorphism of PP^8)
    
│ │ │              
    
│ │ │                
    i7 : f X
│ │ │ @@ -143,15 +143,15 @@
│ │ │  
│ │ │  o10 : ProjectiveVariety, 6-dimensional subvariety of PP^8
    
│ │ │              
    
│ │ │                
    i11 : time g = V ===> W;
│ │ │ - -- used 3.14065s (cpu); 1.91214s (thread); 0s (gc)
│ │ │ + -- used 3.69451s (cpu); 2.10155s (thread); 0s (gc)
│ │ │  
│ │ │  o11 : MultirationalMap (automorphism of PP^8)
    
│ │ │              
    
│ │ │                
    i12 : g||W
│ │ │ @@ -252,15 +252,15 @@
│ │ │  
│ │ │  o16 = 6-dimensional subvariety of PP^9 cut out by 5 hypersurfaces of degree 2
    
│ │ │              
    
│ │ │                
    i17 : time h = Z ===> GG_K(1,4)
│ │ │ - -- used 8.07742s (cpu); 4.95873s (thread); 0s (gc)
│ │ │ + -- used 6.52083s (cpu); 4.57396s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = h
│ │ │  
│ │ │  o17 : MultirationalMap (isomorphism from PP^9 to PP^9)
    
│ │ │              
    
│ │ │                
    i4 : time X = Phi^* Y;
│ │ │ - -- used 5.11368s (cpu); 3.88996s (thread); 0s (gc)
│ │ │ + -- used 4.40229s (cpu); 3.72029s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : ProjectiveVariety, curve in PP^3 x PP^2 x PP^4 (subvariety of codimension 2 in threefold in PP^3 x PP^2 x PP^4 cut out by 12 hypersurfaces of multi-degrees (0,0,2)^1 (0,1,1)^2 (1,0,1)^7 (1,1,0)^2 )
    
│ │ │              
    
│ │ │                
    i5 : dim X, degree X, degrees X
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to
│ │ │ │  PP^2 x PP^4)
│ │ │ │  i3 : Y = projectiveVariety ideal(random({1,1},ring target Phi), random(
│ │ │ │  {1,1},ring target Phi));
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, 4-dimensional subvariety of PP^2 x PP^4
│ │ │ │  i4 : time X = Phi^* Y;
│ │ │ │ - -- used 5.11368s (cpu); 3.88996s (thread); 0s (gc)
│ │ │ │ + -- used 4.40229s (cpu); 3.72029s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : ProjectiveVariety, curve in PP^3 x PP^2 x PP^4 (subvariety of codimension
│ │ │ │  2 in threefold in PP^3 x PP^2 x PP^4 cut out by 12 hypersurfaces of multi-
│ │ │ │  degrees (0,0,2)^1 (0,1,1)^2 (1,0,1)^7 (1,1,0)^2 )
│ │ │ │  i5 : dim X, degree X, degrees X
│ │ │ │  
│ │ │ │  o5 = (1, 31, {({0, 0, 2}, 1), ({0, 0, 3}, 4), ({0, 1, 1}, 4), ({0, 4, 1}, 1),
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp__Multiprojective__Variety.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │  
│ │ │  o4 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^7
    
│ │ │              
    
│ │ │                
    i5 : time Phi Z;
│ │ │ - -- used 0.0951343s (cpu); 0.0963862s (thread); 0s (gc)
│ │ │ + -- used 0.130263s (cpu); 0.111376s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^7
    
│ │ │              
    
│ │ │                
    i6 : dim oo, degree oo, degrees oo
│ │ │ @@ -112,15 +112,15 @@
│ │ │  
│ │ │  o6 : Sequence
    
│ │ │              
    
│ │ │                
    i7 : time Phi (point Z + point Z + point Z)
│ │ │ - -- used 2.05743s (cpu); 1.42394s (thread); 0s (gc)
│ │ │ + -- used 2.51637s (cpu); 1.40924s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = 0-dimensional subvariety of PP^7 x PP^7 cut out by 22 hypersurfaces of multi-degrees (0,1)^5 (0,2)^3 (1,0)^5 (1,1)^6 (2,0)^3 
│ │ │  
│ │ │  o7 : ProjectiveVariety, 0-dimensional subvariety of PP^7 x PP^7
    
│ │ │              
    
│ │ │                
    i4 : time degree(Phi,Strategy=>"random point")
│ │ │ - -- used 3.60186s (cpu); 2.37753s (thread); 0s (gc)
│ │ │ + -- used 4.33556s (cpu); 2.47925s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 2
    
│ │ │              
    
│ │ │                
    i5 : time degree(Phi,Strategy=>"0-th projective degree")
│ │ │ - -- used 0.300764s (cpu); 0.253978s (thread); 0s (gc)
│ │ │ + -- used 0.35768s (cpu); 0.283995s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
    
│ │ │              
    
│ │ │                
    i6 : time degree Phi
│ │ │ - -- used 0.32363s (cpu); 0.259653s (thread); 0s (gc)
│ │ │ + -- used 0.332341s (cpu); 0.262557s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = 2
    
│ │ │              
    
│ │ │          
    Note, as in the example above, that calculation times may vary depending on the strategy used.
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,23 +27,23 @@
│ │ │ │  
│ │ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: threefold in PP^4 x PP^4 cut out by 13 hypersurfaces of
│ │ │ │       target variety: hypersurface in PP^4 defined by a form of degree 2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       multi-degrees (0,2)^1 (1,1)^3 (2,1)^8 (4,0)^1
│ │ │ │  i4 : time degree(Phi,Strategy=>"random point")
│ │ │ │ - -- used 3.60186s (cpu); 2.37753s (thread); 0s (gc)
│ │ │ │ + -- used 4.33556s (cpu); 2.47925s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 2
│ │ │ │  i5 : time degree(Phi,Strategy=>"0-th projective degree")
│ │ │ │ - -- used 0.300764s (cpu); 0.253978s (thread); 0s (gc)
│ │ │ │ + -- used 0.35768s (cpu); 0.283995s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 2
│ │ │ │  i6 : time degree Phi
│ │ │ │ - -- used 0.32363s (cpu); 0.259653s (thread); 0s (gc)
│ │ │ │ + -- used 0.332341s (cpu); 0.262557s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = 2
│ │ │ │  Note, as in the example above, that calculation times may vary depending on the
│ │ │ │  strategy used.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_e_g_r_e_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- degree of a multi-rational map
│ │ │ │      * _d_e_g_r_e_e_M_a_p_(_R_a_t_i_o_n_a_l_M_a_p_) -- degree of a rational map
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_degree_lp__Multirational__Map_rp.html
│ │ │ @@ -81,15 +81,15 @@
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to PP^2 x PP^4)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time degree Phi
│ │ │ - -- used 0.39932s (cpu); 0.353204s (thread); 0s (gc)
│ │ │ + -- used 0.574524s (cpu); 0.371837s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = 1
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,15 +18,15 @@
│ │ │ │  x_1^2-x_0*x_2}, g = rationalMap {x_1^2-x_0*x_2, x_0*x_3, x_1*x_3, x_2*x_3,
│ │ │ │  x_3^2};
│ │ │ │  i2 : Phi = last graph rationalMap {f,g};
│ │ │ │  
│ │ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to
│ │ │ │  PP^2 x PP^4)
│ │ │ │  i3 : time degree Phi
│ │ │ │ - -- used 0.39932s (cpu); 0.353204s (thread); 0s (gc)
│ │ │ │ + -- used 0.574524s (cpu); 0.371837s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = 1
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_e_g_r_e_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_,_O_p_t_i_o_n_) -- degree of a multi-rational map using a
│ │ │ │        probabilistic approach
│ │ │ │      * _d_e_g_r_e_e_(_R_a_t_i_o_n_a_l_M_a_p_) -- degree of a rational map
│ │ │ │      * _m_u_l_t_i_d_e_g_r_e_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- projective degrees of a multi-rational
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_describe_lp__Multirational__Map_rp.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │  
│ │ │  o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^4 x PP^5)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time ? Phi
│ │ │ - -- used 0.000917731s (cpu); 0.000161313s (thread); 0s (gc)
│ │ │ + -- used 0.000922781s (cpu); 0.000167482s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9
│ │ │       target variety: PP^4 x PP^5
│ │ │       ------------------------------------------------------------------------
│ │ │       hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
    
│ │ │              
│ │ │ @@ -96,27 +96,27 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^5
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time ? Phi
│ │ │ - -- used 0.00272072s (cpu); 0.000248957s (thread); 0s (gc)
│ │ │ + -- used 0.0010021s (cpu); 0.000243723s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │       target variety: PP^4 x PP^5
│ │ │       dominance: false
│ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time describe Phi
│ │ │ - -- used 1.4658s (cpu); 1.13064s (thread); 0s (gc)
│ │ │ + -- used 1.25956s (cpu); 1.04959s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │       target variety: PP^4 x PP^5
│ │ │       base locus: empty subscheme of PP^4 x PP^5
│ │ │       dominance: false
│ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │ @@ -126,15 +126,15 @@
│ │ │       degree sequence (map 2/2): [(0,1), (2,0)]
│ │ │       coefficient ring: ZZ/65521
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time ? Phi
│ │ │ - -- used 0.000169538s (cpu); 0.000607139s (thread); 0s (gc)
│ │ │ + -- used 0.000130585s (cpu); 0.000457767s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │       target variety: PP^4 x PP^5
│ │ │       base locus: empty subscheme of PP^4 x PP^5
│ │ │       dominance: false
│ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,36 +16,36 @@
│ │ │ │  ? Phi is a lite version of describe Phi. The latter has a different behavior
│ │ │ │  than _d_e_s_c_r_i_b_e_(_R_a_t_i_o_n_a_l_M_a_p_), since it performs computations.
│ │ │ │  i1 : Phi = multirationalMap graph rationalMap PP_(ZZ/65521)^(1,4);
│ │ │ │  
│ │ │ │  o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^5 to PP^4 x PP^5)
│ │ │ │  i2 : time ? Phi
│ │ │ │ - -- used 0.000917731s (cpu); 0.000161313s (thread); 0s (gc)
│ │ │ │ + -- used 0.000922781s (cpu); 0.000167482s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = multi-rational map consisting of 2 rational maps
│ │ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9
│ │ │ │       target variety: PP^4 x PP^5
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
│ │ │ │  i3 : image Phi;
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^5
│ │ │ │  i4 : time ? Phi
│ │ │ │ - -- used 0.00272072s (cpu); 0.000248957s (thread); 0s (gc)
│ │ │ │ + -- used 0.0010021s (cpu); 0.000243723s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = multi-rational map consisting of 2 rational maps
│ │ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9
│ │ │ │  hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
│ │ │ │       target variety: PP^4 x PP^5
│ │ │ │       dominance: false
│ │ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces
│ │ │ │  of multi-degrees (0,2)^1 (1,1)^8
│ │ │ │  i5 : time describe Phi
│ │ │ │ - -- used 1.4658s (cpu); 1.13064s (thread); 0s (gc)
│ │ │ │ + -- used 1.25956s (cpu); 1.04959s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = multi-rational map consisting of 2 rational maps
│ │ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9
│ │ │ │  hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
│ │ │ │       target variety: PP^4 x PP^5
│ │ │ │       base locus: empty subscheme of PP^4 x PP^5
│ │ │ │       dominance: false
│ │ │ │ @@ -53,15 +53,15 @@
│ │ │ │  of multi-degrees (0,2)^1 (1,1)^8
│ │ │ │       multidegree: {51, 51, 51, 51, 51}
│ │ │ │       degree: 1
│ │ │ │       degree sequence (map 1/2): [(1,0), (0,2)]
│ │ │ │       degree sequence (map 2/2): [(0,1), (2,0)]
│ │ │ │       coefficient ring: ZZ/65521
│ │ │ │  i6 : time ? Phi
│ │ │ │ - -- used 0.000169538s (cpu); 0.000607139s (thread); 0s (gc)
│ │ │ │ + -- used 0.000130585s (cpu); 0.000457767s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = multi-rational map consisting of 2 rational maps
│ │ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9
│ │ │ │  hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
│ │ │ │       target variety: PP^4 x PP^5
│ │ │ │       base locus: empty subscheme of PP^4 x PP^5
│ │ │ │       dominance: false
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_graph_lp__Multirational__Map_rp.html
│ │ │ @@ -83,15 +83,15 @@
│ │ │  
│ │ │  o1 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time (Phi1,Phi2) = graph Phi
│ │ │ - -- used 0.0895117s (cpu); 0.0398444s (thread); 0s (gc)
│ │ │ + -- used 0.11519s (cpu); 0.0525209s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = (Phi1, Phi2)
│ │ │  
│ │ │  o2 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -107,15 +107,15 @@
│ │ │  
│ │ │  o4 : MultirationalMap (dominant rational map from 4-dimensional subvariety of PP^4 x PP^5 to hypersurface in PP^5)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time (Phi21,Phi22) = graph Phi2
│ │ │ - -- used 0.032293s (cpu); 0.0340052s (thread); 0s (gc)
│ │ │ + -- used 0.0750556s (cpu); 0.0584397s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = (Phi21, Phi22)
│ │ │  
│ │ │  o5 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -131,15 +131,15 @@
│ │ │  
│ │ │  o7 : MultirationalMap (dominant rational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to hypersurface in PP^5)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : time (Phi211,Phi212) = graph Phi21
│ │ │ - -- used 0.194002s (cpu); 0.149271s (thread); 0s (gc)
│ │ │ + -- used 0.347284s (cpu); 0.207232s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = (Phi211, Phi212)
│ │ │  
│ │ │  o8 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,43 +19,43 @@
│ │ │ │  Phi)^-1 * (last graph Phi) == Phi are always satisfied.
│ │ │ │  i1 : Phi = rationalMap(PP_(ZZ/333331)^(1,4),Dominant=>true)
│ │ │ │  
│ │ │ │  o1 = Phi
│ │ │ │  
│ │ │ │  o1 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │ │  i2 : time (Phi1,Phi2) = graph Phi
│ │ │ │ - -- used 0.0895117s (cpu); 0.0398444s (thread); 0s (gc)
│ │ │ │ + -- used 0.11519s (cpu); 0.0525209s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = (Phi1, Phi2)
│ │ │ │  
│ │ │ │  o2 : Sequence
│ │ │ │  i3 : Phi1;
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^5 to PP^4)
│ │ │ │  i4 : Phi2;
│ │ │ │  
│ │ │ │  o4 : MultirationalMap (dominant rational map from 4-dimensional subvariety of
│ │ │ │  PP^4 x PP^5 to hypersurface in PP^5)
│ │ │ │  i5 : time (Phi21,Phi22) = graph Phi2
│ │ │ │ - -- used 0.032293s (cpu); 0.0340052s (thread); 0s (gc)
│ │ │ │ + -- used 0.0750556s (cpu); 0.0584397s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = (Phi21, Phi22)
│ │ │ │  
│ │ │ │  o5 : Sequence
│ │ │ │  i6 : Phi21;
│ │ │ │  
│ │ │ │  o6 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^5 x PP^5 to 4-dimensional subvariety of PP^4 x PP^5)
│ │ │ │  i7 : Phi22;
│ │ │ │  
│ │ │ │  o7 : MultirationalMap (dominant rational map from 4-dimensional subvariety of
│ │ │ │  PP^4 x PP^5 x PP^5 to hypersurface in PP^5)
│ │ │ │  i8 : time (Phi211,Phi212) = graph Phi21
│ │ │ │ - -- used 0.194002s (cpu); 0.149271s (thread); 0s (gc)
│ │ │ │ + -- used 0.347284s (cpu); 0.207232s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = (Phi211, Phi212)
│ │ │ │  
│ │ │ │  o8 : Sequence
│ │ │ │  i9 : Phi211;
│ │ │ │  
│ │ │ │  o9 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_image_lp__Multirational__Map_rp.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │  
│ │ │  o4 : MultirationalMap (rational map from PP^4 to PP^7 x PP^4)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time Z = image Phi;
│ │ │ - -- used 0.200211s (cpu); 0.133927s (thread); 0s (gc)
│ │ │ + -- used 0.185535s (cpu); 0.12657s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : dim Z, degree Z, degrees Z
│ │ │ @@ -115,15 +115,15 @@
│ │ │            
│ │ │          
│ │ │          
    Alternatively, the calculation can be performed using the Segre embedding as follows:
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,15 +19,15 @@
│ │ │ │  x_1^2-x_0*x_2}, g = rationalMap {x_1^2-x_0*x_2, x_0*x_3, x_1*x_3, x_2*x_3,
│ │ │ │  x_3^2};
│ │ │ │  i2 : Phi = last graph rationalMap {f,g};
│ │ │ │  
│ │ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to
│ │ │ │  PP^2 x PP^4)
│ │ │ │  i3 : time multidegree Phi
│ │ │ │ - -- used 0.521458s (cpu); 0.386885s (thread); 0s (gc)
│ │ │ │ + -- used 0.50863s (cpu); 0.364535s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = {66, 46, 31, 20}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : (degree source Phi,degree image Phi)
│ │ │ │  
│ │ │ │  o4 = (66, 20)
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_multidegree_lp__Z__Z_cm__Multirational__Map_rp.html
│ │ │ @@ -77,29 +77,29 @@
│ │ │  
│ │ │  o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^5)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : time Z' = projectiveVariety (map segre target Phi) image(segre Phi,"F4");
│ │ │ - -- used 5.791s (cpu); 2.94859s (thread); 0s (gc)
│ │ │ + -- used 10.319s (cpu); 2.72855s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
    
│ │ │              
    
│ │ │                
    i8 : assert(Z == Z')
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,26 +23,26 @@
│ │ │ │  3*x_2^2+2*x_1*x_3+x_0*x_4, 2*x_1*x_2-2*x_0*x_3, -x_1^2+x_0*x_2};
│ │ │ │  
│ │ │ │  o3 : RationalMap (quadratic rational map from PP^4 to PP^4)
│ │ │ │  i4 : Phi = rationalMap {f,g};
│ │ │ │  
│ │ │ │  o4 : MultirationalMap (rational map from PP^4 to PP^7 x PP^4)
│ │ │ │  i5 : time Z = image Phi;
│ │ │ │ - -- used 0.200211s (cpu); 0.133927s (thread); 0s (gc)
│ │ │ │ + -- used 0.185535s (cpu); 0.12657s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
│ │ │ │  i6 : dim Z, degree Z, degrees Z
│ │ │ │  
│ │ │ │  o6 = (4, 151, {({1, 1}, 4), ({1, 2}, 3), ({2, 0}, 5), ({2, 1}, 13)})
│ │ │ │  
│ │ │ │  o6 : Sequence
│ │ │ │  Alternatively, the calculation can be performed using the Segre embedding as
│ │ │ │  follows:
│ │ │ │  i7 : time Z' = projectiveVariety (map segre target Phi) image(segre Phi,"F4");
│ │ │ │ - -- used 5.791s (cpu); 2.94859s (thread); 0s (gc)
│ │ │ │ + -- used 10.319s (cpu); 2.72855s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
│ │ │ │  i8 : assert(Z == Z')
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_ _M_u_l_t_i_p_r_o_j_e_c_t_i_v_e_V_a_r_i_e_t_y -- direct image via a multi-
│ │ │ │        rational map
│ │ │ │      * _i_m_a_g_e_(_R_a_t_i_o_n_a_l_M_a_p_) -- closure of the image of a rational map
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_inverse2.html
│ │ │ @@ -83,15 +83,15 @@
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from PP^6 to GG(2,4))
│ │ │              
    
│ │ │                
    i3 : time Psi = inverse2 Phi;
│ │ │ - -- used 0.36801s (cpu); 0.291283s (thread); 0s (gc)
│ │ │ + -- used 0.408651s (cpu); 0.322879s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : MultirationalMap (birational map from GG(2,4) to PP^6)
    
│ │ │              
    
│ │ │                
    i4 : assert(Phi * Psi == 1)
    
│ │ │ @@ -103,15 +103,15 @@
│ │ │  
│ │ │  o5 : MultirationalMap (rational map from PP^6 x PP^6 to GG(2,4) x GG(2,4))
│ │ │              
    
│ │ │                
    i6 : time Psi' = inverse2 Phi';
│ │ │ - -- used 1.53154s (cpu); 1.09582s (thread); 0s (gc)
│ │ │ + -- used 1.20589s (cpu); 1.03972s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : MultirationalMap (birational map from GG(2,4) x GG(2,4) to PP^6 x PP^6)
    
│ │ │              
    
│ │ │                
    i7 : assert(Phi' * Psi' == 1)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,23 +24,23 @@
│ │ │ │  i2 : -- map defined by the cubics through the secant variety to the rational
│ │ │ │  normal curve of degree 6
│ │ │ │       Phi = multirationalMap rationalMap(ring PP_K^6,ring GG_K(2,4),gens ideal
│ │ │ │  PP_K([6],2));
│ │ │ │  
│ │ │ │  o2 : MultirationalMap (rational map from PP^6 to GG(2,4))
│ │ │ │  i3 : time Psi = inverse2 Phi;
│ │ │ │ - -- used 0.36801s (cpu); 0.291283s (thread); 0s (gc)
│ │ │ │ + -- used 0.408651s (cpu); 0.322879s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (birational map from GG(2,4) to PP^6)
│ │ │ │  i4 : assert(Phi * Psi == 1)
│ │ │ │  i5 : Phi' = Phi || Phi;
│ │ │ │  
│ │ │ │  o5 : MultirationalMap (rational map from PP^6 x PP^6 to GG(2,4) x GG(2,4))
│ │ │ │  i6 : time Psi' = inverse2 Phi';
│ │ │ │ - -- used 1.53154s (cpu); 1.09582s (thread); 0s (gc)
│ │ │ │ + -- used 1.20589s (cpu); 1.03972s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 : MultirationalMap (birational map from GG(2,4) x GG(2,4) to PP^6 x PP^6)
│ │ │ │  i7 : assert(Phi' * Psi' == 1)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_n_v_e_r_s_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- inverse of a birational map
│ │ │ │      * _M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_ _<_=_=_>_ _M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p -- equality of multi-rational maps
│ │ │ │        with checks on internal data
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_inverse_lp__Multirational__Map_rp.html
│ │ │ @@ -88,45 +88,45 @@
│ │ │  
│ │ │  o2 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │              
    
│ │ │                
    i3 : time inverse Phi;
│ │ │ - -- used 0.151859s (cpu); 0.0747115s (thread); 0s (gc)
│ │ │ + -- used 0.114552s (cpu); 0.0638305s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : MultirationalMap (birational map from hypersurface in PP^5 to PP^4)
    
│ │ │              
    
│ │ │                
    i4 : Psi = last graph Phi;
│ │ │  
│ │ │  o4 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 to hypersurface in PP^5)
    
│ │ │              
    
│ │ │                
    i5 : time inverse Psi;
│ │ │ - -- used 0.1777s (cpu); 0.0987744s (thread); 0s (gc)
│ │ │ + -- used 0.295237s (cpu); 0.112553s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : MultirationalMap (birational map from hypersurface in PP^5 to 4-dimensional subvariety of PP^4 x PP^5)
    
│ │ │              
    
│ │ │                
    i6 : Eta = first graph Psi;
│ │ │  
│ │ │  o6 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to 4-dimensional subvariety of PP^4 x PP^5)
    
│ │ │              
    
│ │ │                
    i7 : time inverse Eta;
│ │ │ - -- used 0.442368s (cpu); 0.288571s (thread); 0s (gc)
│ │ │ + -- used 0.568476s (cpu); 0.328152s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 to 4-dimensional subvariety of PP^4 x PP^5 x PP^5)
    
│ │ │              
    
│ │ │                
    i8 : assert(Phi * Phi^-1 == 1 and Phi^-1 * Phi == 1)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,32 +24,32 @@
│ │ │ │  
│ │ │ │  o1 : MultirationalMap (rational map from PP^4 to PP^5)
│ │ │ │  i2 : -- we see Phi as a dominant map
│ │ │ │       Phi = rationalMap(Phi,image Phi);
│ │ │ │  
│ │ │ │  o2 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │ │  i3 : time inverse Phi;
│ │ │ │ - -- used 0.151859s (cpu); 0.0747115s (thread); 0s (gc)
│ │ │ │ + -- used 0.114552s (cpu); 0.0638305s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (birational map from hypersurface in PP^5 to PP^4)
│ │ │ │  i4 : Psi = last graph Phi;
│ │ │ │  
│ │ │ │  o4 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^5 to hypersurface in PP^5)
│ │ │ │  i5 : time inverse Psi;
│ │ │ │ - -- used 0.1777s (cpu); 0.0987744s (thread); 0s (gc)
│ │ │ │ + -- used 0.295237s (cpu); 0.112553s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : MultirationalMap (birational map from hypersurface in PP^5 to 4-
│ │ │ │  dimensional subvariety of PP^4 x PP^5)
│ │ │ │  i6 : Eta = first graph Psi;
│ │ │ │  
│ │ │ │  o6 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^5 x PP^5 to 4-dimensional subvariety of PP^4 x PP^5)
│ │ │ │  i7 : time inverse Eta;
│ │ │ │ - -- used 0.442368s (cpu); 0.288571s (thread); 0s (gc)
│ │ │ │ + -- used 0.568476s (cpu); 0.328152s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^5 to 4-dimensional subvariety of PP^4 x PP^5 x PP^5)
│ │ │ │  i8 : assert(Phi * Phi^-1 == 1 and Phi^-1 * Phi == 1)
│ │ │ │  i9 : assert(Psi * Psi^-1 == 1 and Psi^-1 * Psi == 1)
│ │ │ │  i10 : assert(Eta * Eta^-1 == 1 and Eta^-1 * Eta == 1)
│ │ │ │  ********** RReeffeerreenncceess **********
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_is__Isomorphism_lp__Multirational__Map_rp.html
│ │ │ @@ -83,45 +83,45 @@
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from PP^3 to PP^2 x PP^2)
│ │ │              
    
│ │ │                
    i4 : time isIsomorphism Phi
│ │ │ - -- used 0.00299491s (cpu); 9.007e-06s (thread); 0s (gc)
│ │ │ + -- used 0.00336491s (cpu); 8.46e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = false
    
│ │ │              
    
│ │ │                
    i5 : Psi = first graph Phi;
│ │ │  
│ │ │  o5 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 to PP^3)
    
│ │ │              
    
│ │ │                
    i6 : time isIsomorphism Psi
│ │ │ - -- used 0.331015s (cpu); 0.180919s (thread); 0s (gc)
│ │ │ + -- used 0.423703s (cpu); 0.190694s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = false
    
│ │ │              
    
│ │ │                
    i7 : Eta = first graph Psi;
│ │ │  
│ │ │  o7 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 x PP^3 to threefold in PP^3 x PP^2 x PP^2)
    
│ │ │              
    
│ │ │                
    i8 : time isIsomorphism Eta
│ │ │ - -- used 1.56012s (cpu); 0.826624s (thread); 0s (gc)
│ │ │ + -- used 1.69466s (cpu); 0.865677s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
    
│ │ │              
    
│ │ │                
    i9 : assert(o8 and (not o6) and (not o4))
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,31 +17,31 @@
│ │ │ │       ZZ/33331[a..d]; f = rationalMap {c^2-b*d,b*c-a*d,b^2-a*c};
│ │ │ │  
│ │ │ │  o2 : RationalMap (quadratic rational map from PP^3 to PP^2)
│ │ │ │  i3 : Phi = rationalMap {f,f};
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (rational map from PP^3 to PP^2 x PP^2)
│ │ │ │  i4 : time isIsomorphism Phi
│ │ │ │ - -- used 0.00299491s (cpu); 9.007e-06s (thread); 0s (gc)
│ │ │ │ + -- used 0.00336491s (cpu); 8.46e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = false
│ │ │ │  i5 : Psi = first graph Phi;
│ │ │ │  
│ │ │ │  o5 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 to
│ │ │ │  PP^3)
│ │ │ │  i6 : time isIsomorphism Psi
│ │ │ │ - -- used 0.331015s (cpu); 0.180919s (thread); 0s (gc)
│ │ │ │ + -- used 0.423703s (cpu); 0.190694s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = false
│ │ │ │  i7 : Eta = first graph Psi;
│ │ │ │  
│ │ │ │  o7 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 x
│ │ │ │  PP^3 to threefold in PP^3 x PP^2 x PP^2)
│ │ │ │  i8 : time isIsomorphism Eta
│ │ │ │ - -- used 1.56012s (cpu); 0.826624s (thread); 0s (gc)
│ │ │ │ + -- used 1.69466s (cpu); 0.865677s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  i9 : assert(o8 and (not o6) and (not o4))
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_n_v_e_r_s_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- inverse of a birational map
│ │ │ │      * _i_s_M_o_r_p_h_i_s_m_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- whether a multi-rational map is a
│ │ │ │        morphism
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_is__Morphism_lp__Multirational__Map_rp.html
│ │ │ @@ -80,31 +80,31 @@
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^7 to PP^4 x PP^2)
│ │ │              
    
│ │ │                
    i3 : time isMorphism Phi
│ │ │ - -- used 0.369946s (cpu); 0.279856s (thread); 0s (gc)
│ │ │ + -- used 0.506281s (cpu); 0.258799s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = false
    
│ │ │              
    
│ │ │                
    i4 : time Psi = first graph Phi;
│ │ │ - -- used 0.170151s (cpu); 0.094037s (thread); 0s (gc)
│ │ │ + -- used 0.128697s (cpu); 0.0729765s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^7 x PP^4 x PP^2 to 4-dimensional subvariety of PP^4 x PP^7)
    
│ │ │              
    
│ │ │                
    i5 : time isMorphism Psi
│ │ │ - -- used 4.39879s (cpu); 3.33728s (thread); 0s (gc)
│ │ │ + -- used 3.59347s (cpu); 2.98975s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
    
│ │ │              
    
│ │ │                
    i6 : assert((not o3) and o5)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,24 +17,24 @@
│ │ │ │  i1 : ZZ/300007[a..e], f = first graph rationalMap ideal(c^2-b*d,b*c-a*d,b^2-
│ │ │ │  a*c,e), g = rationalMap submatrix(matrix f,{0..2});
│ │ │ │  i2 : Phi = rationalMap {f,g};
│ │ │ │  
│ │ │ │  o2 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^7 to PP^4 x PP^2)
│ │ │ │  i3 : time isMorphism Phi
│ │ │ │ - -- used 0.369946s (cpu); 0.279856s (thread); 0s (gc)
│ │ │ │ + -- used 0.506281s (cpu); 0.258799s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : time Psi = first graph Phi;
│ │ │ │ - -- used 0.170151s (cpu); 0.094037s (thread); 0s (gc)
│ │ │ │ + -- used 0.128697s (cpu); 0.0729765s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^7 x PP^4 x PP^2 to 4-dimensional subvariety of PP^4 x PP^7)
│ │ │ │  i5 : time isMorphism Psi
│ │ │ │ - -- used 4.39879s (cpu); 3.33728s (thread); 0s (gc)
│ │ │ │ + -- used 3.59347s (cpu); 2.98975s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  i6 : assert((not o3) and o5)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_I_s_o_m_o_r_p_h_i_s_m_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- whether a birational map is an
│ │ │ │        isomorphism
│ │ │ │      * _i_s_M_o_r_p_h_i_s_m_(_R_a_t_i_o_n_a_l_M_a_p_) -- whether a rational map is a morphism
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_linearly__Normal__Embedding.html
│ │ │ @@ -79,30 +79,30 @@
│ │ │  
│ │ │  o2 : ProjectiveVariety, curve in PP^7
│ │ │              
    
│ │ │                
    i3 : time f = linearlyNormalEmbedding X;
│ │ │ - -- used 0.010893s (cpu); 0.00980925s (thread); 0s (gc)
│ │ │ + -- used 0.108782s (cpu); 0.0340201s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : MultirationalMap (automorphism of X)
    
│ │ │              
    
│ │ │                
    i4 : Y = (rationalMap {for i to 3 list random(1,ring ambient X)}) X; -- an isomorphic projection of X in PP^3
│ │ │  
│ │ │  o4 : ProjectiveVariety, curve in PP^3
    
│ │ │              
    
│ │ │                
    i5 : time g = linearlyNormalEmbedding Y;
│ │ │ - -- used 0.564105s (cpu); 0.442475s (thread); 0s (gc)
│ │ │ + -- used 0.515086s (cpu); 0.434805s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : MultirationalMap (birational map from Y to curve in PP^7)
    
│ │ │              
    
│ │ │                
    i6 : assert(isIsomorphism g)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,23 +13,23 @@
│ │ │ │              is a linear projection
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : K = ZZ/333331;
│ │ │ │  i2 : X = PP_K^(1,7); -- rational normal curve of degree 7
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, curve in PP^7
│ │ │ │  i3 : time f = linearlyNormalEmbedding X;
│ │ │ │ - -- used 0.010893s (cpu); 0.00980925s (thread); 0s (gc)
│ │ │ │ + -- used 0.108782s (cpu); 0.0340201s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (automorphism of X)
│ │ │ │  i4 : Y = (rationalMap {for i to 3 list random(1,ring ambient X)}) X; -- an
│ │ │ │  isomorphic projection of X in PP^3
│ │ │ │  
│ │ │ │  o4 : ProjectiveVariety, curve in PP^3
│ │ │ │  i5 : time g = linearlyNormalEmbedding Y;
│ │ │ │ - -- used 0.564105s (cpu); 0.442475s (thread); 0s (gc)
│ │ │ │ + -- used 0.515086s (cpu); 0.434805s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : MultirationalMap (birational map from Y to curve in PP^7)
│ │ │ │  i6 : assert(isIsomorphism g)
│ │ │ │  i7 : describe g
│ │ │ │  
│ │ │ │  o7 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: curve in PP^3 cut out by 6 hypersurfaces of degree 4
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_multidegree_lp__Multirational__Map_rp.html
│ │ │ @@ -81,15 +81,15 @@
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to PP^2 x PP^4)
│ │ │              
    
│ │ │                
    i3 : time multidegree Phi
│ │ │ - -- used 0.521458s (cpu); 0.386885s (thread); 0s (gc)
│ │ │ + -- used 0.50863s (cpu); 0.364535s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = {66, 46, 31, 20}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
    
│ │ │                
    i2 : for i in {4,3,2,1,0} list time multidegree(i,Phi)
│ │ │ - -- used 0.00397921s (cpu); 0.00133095s (thread); 0s (gc)
│ │ │ - -- used 0.272866s (cpu); 0.142784s (thread); 0s (gc)
│ │ │ - -- used 0.237166s (cpu); 0.173525s (thread); 0s (gc)
│ │ │ - -- used 0.202997s (cpu); 0.139949s (thread); 0s (gc)
│ │ │ - -- used 0.187465s (cpu); 0.110073s (thread); 0s (gc)
│ │ │ + -- used 0.00399531s (cpu); 0.0012992s (thread); 0s (gc)
│ │ │ + -- used 0.221072s (cpu); 0.147087s (thread); 0s (gc)
│ │ │ + -- used 0.243298s (cpu); 0.168701s (thread); 0s (gc)
│ │ │ + -- used 0.221095s (cpu); 0.152101s (thread); 0s (gc)
│ │ │ + -- used 0.191018s (cpu); 0.12145s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = {51, 28, 14, 6, 2}
│ │ │  
│ │ │  o2 : List
    
│ │ │              
    
│ │ │                
    i3 : time assert(oo == multidegree Phi)
│ │ │ - -- used 0.127059s (cpu); 0.0818662s (thread); 0s (gc)
    
│ │ │ + -- used 0.2182s (cpu); 0.0975932s (thread); 0s (gc)
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    References
    
│ │ │  ArXiv preprint: Computations with rational maps between multi-projective varieties.      
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,25 +17,25 @@
│ │ │ │  This is calculated by means of the inverse image of an appropriate random
│ │ │ │  subvariety of the target.
│ │ │ │  i1 : Phi = last graph rationalMap PP_(ZZ/300007)^(1,4);
│ │ │ │  
│ │ │ │  o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^5 to PP^5)
│ │ │ │  i2 : for i in {4,3,2,1,0} list time multidegree(i,Phi)
│ │ │ │ - -- used 0.00397921s (cpu); 0.00133095s (thread); 0s (gc)
│ │ │ │ - -- used 0.272866s (cpu); 0.142784s (thread); 0s (gc)
│ │ │ │ - -- used 0.237166s (cpu); 0.173525s (thread); 0s (gc)
│ │ │ │ - -- used 0.202997s (cpu); 0.139949s (thread); 0s (gc)
│ │ │ │ - -- used 0.187465s (cpu); 0.110073s (thread); 0s (gc)
│ │ │ │ + -- used 0.00399531s (cpu); 0.0012992s (thread); 0s (gc)
│ │ │ │ + -- used 0.221072s (cpu); 0.147087s (thread); 0s (gc)
│ │ │ │ + -- used 0.243298s (cpu); 0.168701s (thread); 0s (gc)
│ │ │ │ + -- used 0.221095s (cpu); 0.152101s (thread); 0s (gc)
│ │ │ │ + -- used 0.191018s (cpu); 0.12145s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = {51, 28, 14, 6, 2}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  i3 : time assert(oo == multidegree Phi)
│ │ │ │ - -- used 0.127059s (cpu); 0.0818662s (thread); 0s (gc)
│ │ │ │ + -- used 0.2182s (cpu); 0.0975932s (thread); 0s (gc)
│ │ │ │  ********** RReeffeerreenncceess **********
│ │ │ │  ArXiv preprint: _C_o_m_p_u_t_a_t_i_o_n_s_ _w_i_t_h_ _r_a_t_i_o_n_a_l_ _m_a_p_s_ _b_e_t_w_e_e_n_ _m_u_l_t_i_-_p_r_o_j_e_c_t_i_v_e
│ │ │ │  _v_a_r_i_e_t_i_e_s.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_u_l_t_i_d_e_g_r_e_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- projective degrees of a multi-rational
│ │ │ │        map
│ │ │ │      * _p_r_o_j_e_c_t_i_v_e_D_e_g_r_e_e_s_(_R_a_t_i_o_n_a_l_M_a_p_) -- projective degrees of a rational map
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_point_lp__Multiprojective__Variety_rp.html
│ │ │ @@ -80,15 +80,15 @@
│ │ │  
│ │ │  o2 : ProjectiveVariety, 4-dimensional subvariety of PP^3 x PP^2 x PP^9    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time p := point X
│ │ │ - -- used 0.0161552s (cpu); 0.0167468s (thread); 0s (gc)
│ │ │ + -- used 0.0319696s (cpu); 0.0192418s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = point of coordinates ([421369, 39917, -212481, 1],[-128795, -176966, 1],[3870, -390108, -496127, -308581, 46649, 164926, -446111, 48038, 415309, 1])
│ │ │  
│ │ │  o3 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o4 : ProjectiveVariety, hypersurface in PP^3 x PP^2 x PP^9
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time q = point Y
│ │ │ - -- used 1.74407s (cpu); 0.993938s (thread); 0s (gc)
│ │ │ + -- used 1.52734s (cpu); 0.994378s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = q
│ │ │  
│ │ │  o5 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,25 +14,25 @@
│ │ │ │            o a _m_u_l_t_i_-_p_r_o_j_e_c_t_i_v_e_ _v_a_r_i_e_t_y, a random rational point on $X$
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : K = ZZ/1000003;
│ │ │ │  i2 : X = PP_K^({1,1,2},{3,2,3});
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, 4-dimensional subvariety of PP^3 x PP^2 x PP^9
│ │ │ │  i3 : time p := point X
│ │ │ │ - -- used 0.0161552s (cpu); 0.0167468s (thread); 0s (gc)
│ │ │ │ + -- used 0.0319696s (cpu); 0.0192418s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = point of coordinates ([421369, 39917, -212481, 1],[-128795, -176966, 1],
│ │ │ │  [3870, -390108, -496127, -308581, 46649, 164926, -446111, 48038, 415309, 1])
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9
│ │ │ │  i4 : Y = random({2,1,2},X);
│ │ │ │  
│ │ │ │  o4 : ProjectiveVariety, hypersurface in PP^3 x PP^2 x PP^9
│ │ │ │  i5 : time q = point Y
│ │ │ │ - -- used 1.74407s (cpu); 0.993938s (thread); 0s (gc)
│ │ │ │ + -- used 1.52734s (cpu); 0.994378s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = q
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9
│ │ │ │  i6 : assert(isSubset(p,X) and isSubset(q,Y))
│ │ │ │  The list of homogeneous coordinates can be obtained with the operator |-.
│ │ │ │  i7 : |- p
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_segre_lp__Multirational__Map_rp.html
│ │ │ @@ -101,15 +101,15 @@
│ │ │  
│ │ │  o5 : MultirationalMap (rational map from PP^4 to hypersurface in PP^5 x PP^4 x PP^4)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time segre Phi;
│ │ │ - -- used 0.750404s (cpu); 0.533826s (thread); 0s (gc)
│ │ │ + -- used 1.11502s (cpu); 0.590124s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : RationalMap (rational map from PP^4 to PP^149)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : describe segre Phi
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  
│ │ │ │  o4 : RationalMap (quadratic rational map from PP^4 to PP^4)
│ │ │ │  i5 : Phi = rationalMap {f,g,h};
│ │ │ │  
│ │ │ │  o5 : MultirationalMap (rational map from PP^4 to hypersurface in PP^5 x PP^4 x
│ │ │ │  PP^4)
│ │ │ │  i6 : time segre Phi;
│ │ │ │ - -- used 0.750404s (cpu); 0.533826s (thread); 0s (gc)
│ │ │ │ + -- used 1.11502s (cpu); 0.590124s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 : RationalMap (rational map from PP^4 to PP^149)
│ │ │ │  i7 : describe segre Phi
│ │ │ │  
│ │ │ │  o7 = rational map defined by forms of degree 6
│ │ │ │       source variety: PP^4
│ │ │ │       target variety: PP^149
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_show_lp__Multirational__Map_rp.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │  
│ │ │  o1 : MultirationalMap (birational map from threefold in PP^3 x PP^2 to threefold in PP^3 x PP^2 x PP^2)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time describe Phi
│ │ │ - -- used 0.273463s (cpu); 0.167098s (thread); 0s (gc)
│ │ │ + -- used 0.225094s (cpu); 0.149449s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = multi-rational map consisting of 3 rational maps
│ │ │       source variety: threefold in PP^3 x PP^2 cut out by 2 hypersurfaces of multi-degree (1,1)
│ │ │       target variety: threefold in PP^3 x PP^2 x PP^2 cut out by 7 hypersurfaces of multi-degrees (0,1,1)^3 (1,0,1)^2 (1,1,0)^2 
│ │ │       base locus: empty subscheme of PP^3 x PP^2
│ │ │       dominance: true
│ │ │       multidegree: {10, 14, 19, 25}
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │  i1 : Phi = inverse first graph last graph rationalMap PP_(ZZ/33331)^(1,3)
│ │ │ │  
│ │ │ │  o1 = Phi
│ │ │ │  
│ │ │ │  o1 : MultirationalMap (birational map from threefold in PP^3 x PP^2 to
│ │ │ │  threefold in PP^3 x PP^2 x PP^2)
│ │ │ │  i2 : time describe Phi
│ │ │ │ - -- used 0.273463s (cpu); 0.167098s (thread); 0s (gc)
│ │ │ │ + -- used 0.225094s (cpu); 0.149449s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = multi-rational map consisting of 3 rational maps
│ │ │ │       source variety: threefold in PP^3 x PP^2 cut out by 2 hypersurfaces of
│ │ │ │  multi-degree (1,1)
│ │ │ │       target variety: threefold in PP^3 x PP^2 x PP^2 cut out by 7 hypersurfaces
│ │ │ │  of multi-degrees (0,1,1)^3 (1,0,1)^2 (1,1,0)^2
│ │ │ │       base locus: empty subscheme of PP^3 x PP^2
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out
│ │ │ @@ -26,22 +26,22 @@
│ │ │  
│ │ │  i7 : connected = buildGraphFilter {"Connectivity" => 0, "NegateConnectivity" => true};
│ │ │  
│ │ │  i8 : prob = n -> log(n)/n;
│ │ │  
│ │ │  i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), connected))
│ │ │  
│ │ │ -o9 = (61, 77, 85, 95, 93, 96, 96, 95, 98, 96, 99, 97, 96, 96, 97, 98, 98, 96,
│ │ │ +o9 = (67, 75, 93, 93, 96, 92, 96, 91, 96, 97, 93, 95, 96, 98, 97, 96, 97, 97,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     98, 100, 97, 99, 100, 99, 100, 98, 97, 96, 99)
│ │ │ +     98, 96, 99, 99, 98, 98, 97, 99, 99, 99, 100)
│ │ │  
│ │ │  o9 : Sequence
│ │ │  
│ │ │  i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (11, 6, 4, 5, 4, 2, 0, 2, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1,
│ │ │ +o10 = (19, 16, 7, 4, 3, 3, 2, 1, 2, 0, 1, 1, 2, 2, 1, 0, 1, 3, 1, 0, 0, 0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      1, 1, 0, 0, 0, 1)
│ │ │ +      1, 0, 0, 0, 0, 0)
│ │ │  
│ │ │  o10 : Sequence
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/example-output/_generate__Random__Graphs.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  
│ │ │  o1 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : generateRandomGraphs(5, 5)
│ │ │  
│ │ │ -o2 = {DSc, DR_, D|o, Ddw, D^k}
│ │ │ +o2 = {Dhk, DdG, DPK, Dk[, DES}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │  
│ │ │  o3 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/example-output/_generate__Random__Regular__Graphs.out
│ │ │ @@ -1,21 +1,21 @@
│ │ │  -- -*- M2-comint -*- hash: 1729831171060067675
│ │ │  
│ │ │  i1 : R = QQ[a..e];
│ │ │  
│ │ │  i2 : generateRandomRegularGraphs(R, 3, 2)
│ │ │  
│ │ │ -o2 = {Graph{"edges" => {{a, b}, {a, c}, {b, d}, {c, e}, {d, e}}},
│ │ │ +o2 = {Graph{"edges" => {{b, c}, {a, d}, {b, d}, {a, e}, {c, e}}},
│ │ │              "ring" => R                                          
│ │ │              "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │       Graph{"edges" => {{a, b}, {b, c}, {a, d}, {c, e}, {d, 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}, {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 0.000476844s (cpu); 0.000401593s (thread); 0s (gc)
│ │ │ + -- used 0.000672568s (cpu); 0.000645755s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : time (graphComplement \ G);
│ │ │ - -- used 0.147573s (cpu); 0.079233s (thread); 0s (gc)
│ │ │ + -- used 0.162324s (cpu); 0.0773518s (thread); 0s (gc)
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html
│ │ │ @@ -117,28 +117,28 @@
│ │ │                
    i8 : prob = n -> log(n)/n;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), connected))
│ │ │  
│ │ │ -o9 = (61, 77, 85, 95, 93, 96, 96, 95, 98, 96, 99, 97, 96, 96, 97, 98, 98, 96,
│ │ │ +o9 = (67, 75, 93, 93, 96, 92, 96, 91, 96, 97, 93, 95, 96, 98, 97, 96, 97, 97,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     98, 100, 97, 99, 100, 99, 100, 98, 97, 96, 99)
│ │ │ +     98, 96, 99, 99, 98, 98, 97, 99, 99, 99, 100)
│ │ │  
│ │ │  o9 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (11, 6, 4, 5, 4, 2, 0, 2, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1,
│ │ │ +o10 = (19, 16, 7, 4, 3, 3, 2, 1, 2, 0, 1, 1, 2, 2, 1, 0, 1, 3, 1, 0, 0, 0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      1, 1, 0, 0, 0, 1)
│ │ │ +      1, 0, 0, 0, 0, 0)
│ │ │  
│ │ │  o10 : Sequence
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -38,25 +38,25 @@
│ │ │ │  connected, at least as $n$ tends to infinity.
│ │ │ │  i7 : connected = buildGraphFilter {"Connectivity" => 0, "NegateConnectivity" =>
│ │ │ │  true};
│ │ │ │  i8 : prob = n -> log(n)/n;
│ │ │ │  i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)),
│ │ │ │  connected))
│ │ │ │  
│ │ │ │ -o9 = (61, 77, 85, 95, 93, 96, 96, 95, 98, 96, 99, 97, 96, 96, 97, 98, 98, 96,
│ │ │ │ +o9 = (67, 75, 93, 93, 96, 92, 96, 91, 96, 97, 93, 95, 96, 98, 97, 96, 97, 97,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     98, 100, 97, 99, 100, 99, 100, 98, 97, 96, 99)
│ │ │ │ +     98, 96, 99, 99, 98, 98, 97, 99, 99, 99, 100)
│ │ │ │  
│ │ │ │  o9 : Sequence
│ │ │ │  i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2),
│ │ │ │  connected))
│ │ │ │  
│ │ │ │ -o10 = (11, 6, 4, 5, 4, 2, 0, 2, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1,
│ │ │ │ +o10 = (19, 16, 7, 4, 3, 3, 2, 1, 2, 0, 1, 1, 2, 2, 1, 0, 1, 3, 1, 0, 0, 0, 0,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      1, 1, 0, 0, 0, 1)
│ │ │ │ +      1, 0, 0, 0, 0, 0)
│ │ │ │  
│ │ │ │  o10 : Sequence
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _b_u_i_l_d_G_r_a_p_h_F_i_l_t_e_r -- creates the appropriate filter string for use with
│ │ │ │        filterGraphs and countGraphs
│ │ │ │      * _f_i_l_t_e_r_G_r_a_p_h_s -- filters (i.e., selects) graphs in a list for given
│ │ │ │        properties
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/html/_generate__Random__Graphs.html
│ │ │ @@ -100,15 +100,15 @@
│ │ │  o1 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : generateRandomGraphs(5, 5)
│ │ │  
│ │ │ -o2 = {DSc, DR_, D|o, Ddw, D^k}
│ │ │ +o2 = {Dhk, DdG, DPK, Dk[, DES}
│ │ │  
│ │ │  o2 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ ├── html2text {}
│ │ │ │ @@ -37,15 +37,15 @@
│ │ │ │  i1 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ │  
│ │ │ │  o1 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : generateRandomGraphs(5, 5)
│ │ │ │  
│ │ │ │ -o2 = {DSc, DR_, D|o, Ddw, D^k}
│ │ │ │ +o2 = {Dhk, DdG, DPK, Dk[, DES}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ │  
│ │ │ │  o3 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │ │  
│ │ │ │  o3 : List
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/html/_generate__Random__Regular__Graphs.html
│ │ │ @@ -87,23 +87,23 @@
│ │ │                
    i1 : R = QQ[a..e];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : generateRandomRegularGraphs(R, 3, 2)
│ │ │  
│ │ │ -o2 = {Graph{"edges" => {{a, b}, {a, c}, {b, d}, {c, e}, {d, e}}},
│ │ │ +o2 = {Graph{"edges" => {{b, c}, {a, d}, {b, d}, {a, e}, {c, e}}},
│ │ │              "ring" => R                                          
│ │ │              "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │       Graph{"edges" => {{a, b}, {b, c}, {a, d}, {c, e}, {d, 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}, {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, b}, {a, c}, {b, d}, {c, e}, {d, e}}},
│ │ │ │ +o2 = {Graph{"edges" => {{b, c}, {a, d}, {b, d}, {a, e}, {c, e}}},
│ │ │ │              "ring" => R
│ │ │ │              "vertices" => {a, b, c, d, e}
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       Graph{"edges" => {{a, b}, {b, c}, {a, d}, {c, e}, {d, 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}, {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 0.000476844s (cpu); 0.000401593s (thread); 0s (gc)
    
│ │ │ + -- used 0.000672568s (cpu); 0.000645755s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time (graphComplement \ G);
│ │ │ - -- used 0.147573s (cpu); 0.079233s (thread); 0s (gc)
    
│ │ │ + -- used 0.162324s (cpu); 0.0773518s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,17 +41,17 @@
│ │ │ │  
│ │ │ │  o3 = DUW
│ │ │ │  Batch calls can be performed considerably faster when using the List input
│ │ │ │  format. However, care should be taken as the returned list is entirely in
│ │ │ │  Graph6 or Sparse6 format.
│ │ │ │  i4 : G = generateBipartiteGraphs 7;
│ │ │ │  i5 : time graphComplement G;
│ │ │ │ - -- used 0.000476844s (cpu); 0.000401593s (thread); 0s (gc)
│ │ │ │ + -- used 0.000672568s (cpu); 0.000645755s (thread); 0s (gc)
│ │ │ │  i6 : time (graphComplement \ G);
│ │ │ │ - -- used 0.147573s (cpu); 0.079233s (thread); 0s (gc)
│ │ │ │ + -- used 0.162324s (cpu); 0.0773518s (thread); 0s (gc)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_o_m_p_l_e_m_e_n_t_G_r_a_p_h -- returns the complement of a graph or hypergraph
│ │ │ │  ********** WWaayyss ttoo uussee ggrraapphhCCoommpplleemmeenntt:: **********
│ │ │ │      * graphComplement(Graph)
│ │ │ │      * graphComplement(List)
│ │ │ │      * graphComplement(String)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out
│ │ │ @@ -26,22 +26,22 @@
│ │ │  
│ │ │  i7 : connected = buildGraphFilter {"Connectivity" => 0, "NegateConnectivity" => true};
│ │ │  
│ │ │  i8 : prob = n -> log(n)/n;
│ │ │  
│ │ │  i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), connected))
│ │ │  
│ │ │ -o9 = (72, 79, 88, 91, 95, 95, 93, 96, 97, 99, 97, 95, 96, 97, 95, 98, 99, 99,
│ │ │ +o9 = (70, 80, 87, 93, 91, 95, 96, 91, 98, 92, 96, 94, 97, 97, 98, 95, 96, 97,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     94, 99, 96, 98, 98, 97, 97, 98, 98, 94, 98)
│ │ │ +     93, 99, 99, 98, 98, 99, 98, 98, 100, 97, 98)
│ │ │  
│ │ │  o9 : Sequence
│ │ │  
│ │ │  i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (18, 11, 7, 2, 7, 3, 5, 3, 1, 1, 1, 2, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,
│ │ │ +o10 = (24, 8, 4, 1, 5, 7, 3, 2, 1, 1, 1, 1, 1, 0, 2, 2, 0, 1, 0, 1, 0, 1, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      0, 1, 1, 1, 0, 0)
│ │ │ +      1, 1, 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 = {DU?, Dt_, DFw, DTG, DxC}
│ │ │ +o2 = {DDs, DQG, DL[, DVg, Dck}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │  
│ │ │  o3 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Random__Regular__Graphs.out
│ │ │ @@ -1,9 +1,9 @@
│ │ │  -- -*- M2-comint -*- hash: 1331287392268
│ │ │  
│ │ │  i1 : generateRandomRegularGraphs(5, 3, 2)
│ │ │  
│ │ │ -o1 = {DMg, DLo, DLo}
│ │ │ +o1 = {D[S, DLo, Dbg}
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_graph__Complement.out
│ │ │ @@ -13,13 +13,13 @@
│ │ │             4 => {2, 1}
│ │ │  
│ │ │  o2 : Graph
│ │ │  
│ │ │  i3 : G = generateBipartiteGraphs 7;
│ │ │  
│ │ │  i4 : time graphComplement G;
│ │ │ - -- used 0.00101801s (cpu); 0.000811832s (thread); 0s (gc)
│ │ │ + -- used 0.000667562s (cpu); 0.000536291s (thread); 0s (gc)
│ │ │  
│ │ │  i5 : time (graphComplement \ G);
│ │ │ - -- used 0.146022s (cpu); 0.0727874s (thread); 0s (gc)
│ │ │ + -- used 0.150279s (cpu); 0.0756606s (thread); 0s (gc)
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html
│ │ │ @@ -117,28 +117,28 @@
│ │ │                
      i8 : prob = n -> log(n)/n;
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), connected))
│ │ │  
│ │ │ -o9 = (72, 79, 88, 91, 95, 95, 93, 96, 97, 99, 97, 95, 96, 97, 95, 98, 99, 99,
│ │ │ +o9 = (70, 80, 87, 93, 91, 95, 96, 91, 98, 92, 96, 94, 97, 97, 98, 95, 96, 97,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     94, 99, 96, 98, 98, 97, 97, 98, 98, 94, 98)
│ │ │ +     93, 99, 99, 98, 98, 99, 98, 98, 100, 97, 98)
│ │ │  
│ │ │  o9 : Sequence
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (18, 11, 7, 2, 7, 3, 5, 3, 1, 1, 1, 2, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,
│ │ │ +o10 = (24, 8, 4, 1, 5, 7, 3, 2, 1, 1, 1, 1, 1, 0, 2, 2, 0, 1, 0, 1, 0, 1, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      0, 1, 1, 1, 0, 0)
│ │ │ +      1, 1, 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 = (72, 79, 88, 91, 95, 95, 93, 96, 97, 99, 97, 95, 96, 97, 95, 98, 99, 99,
│ │ │ │ +o9 = (70, 80, 87, 93, 91, 95, 96, 91, 98, 92, 96, 94, 97, 97, 98, 95, 96, 97,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     94, 99, 96, 98, 98, 97, 97, 98, 98, 94, 98)
│ │ │ │ +     93, 99, 99, 98, 98, 99, 98, 98, 100, 97, 98)
│ │ │ │  
│ │ │ │  o9 : Sequence
│ │ │ │  i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2),
│ │ │ │  connected))
│ │ │ │  
│ │ │ │ -o10 = (18, 11, 7, 2, 7, 3, 5, 3, 1, 1, 1, 2, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,
│ │ │ │ +o10 = (24, 8, 4, 1, 5, 7, 3, 2, 1, 1, 1, 1, 1, 0, 2, 2, 0, 1, 0, 1, 0, 1, 0,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      0, 1, 1, 1, 0, 0)
│ │ │ │ +      1, 1, 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 = {DU?, Dt_, DFw, DTG, DxC}
│ │ │ +o2 = {DDs, DQG, DL[, DVg, Dck}
│ │ │  
│ │ │  o2 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,15 +30,15 @@
│ │ │ │  i1 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ │  
│ │ │ │  o1 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : generateRandomGraphs(5, 5)
│ │ │ │  
│ │ │ │ -o2 = {DU?, Dt_, DFw, DTG, DxC}
│ │ │ │ +o2 = {DDs, DQG, DL[, DVg, Dck}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ │  
│ │ │ │  o3 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │ │  
│ │ │ │  o3 : List
│ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Random__Regular__Graphs.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │            
    This method generates a specified number of random graphs on a given number of vertices with a given regularity. Note that some graphs may be isomorphic.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : generateRandomRegularGraphs(5, 3, 2)
│ │ │  
│ │ │ -o1 = {DMg, DLo, DLo}
│ │ │ +o1 = {D[S, DLo, Dbg}
│ │ │  
│ │ │  o1 : List
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,15 +18,15 @@
│ │ │ │      * Outputs:
│ │ │ │            o G, a _l_i_s_t, the randomly generated regular graphs
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This method generates a specified number of random graphs on a given number of
│ │ │ │  vertices with a given regularity. Note that some graphs may be isomorphic.
│ │ │ │  i1 : generateRandomRegularGraphs(5, 3, 2)
│ │ │ │  
│ │ │ │ -o1 = {DMg, DLo, DLo}
│ │ │ │ +o1 = {D[S, DLo, Dbg}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  The number of vertices $n$ must be positive as nauty cannot handle graphs with
│ │ │ │  zero vertices.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _g_e_n_e_r_a_t_e_R_a_n_d_o_m_G_r_a_p_h_s -- generates random graphs on a given number of
│ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/html/_graph__Complement.html
│ │ │ @@ -110,21 +110,21 @@
│ │ │              
│ │ │                
    i3 : G = generateBipartiteGraphs 7;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time graphComplement G;
│ │ │ - -- used 0.00101801s (cpu); 0.000811832s (thread); 0s (gc)
    
│ │ │ + -- used 0.000667562s (cpu); 0.000536291s (thread); 0s (gc)    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time (graphComplement \ G);
│ │ │ - -- used 0.146022s (cpu); 0.0727874s (thread); 0s (gc)
    
│ │ │ + -- used 0.150279s (cpu); 0.0756606s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    
│ │ │            
    Ways to use graphComplement:
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -38,17 +38,17 @@
│ │ │ │  
│ │ │ │  o2 : Graph
│ │ │ │  Batch calls can be performed considerably faster when using the List input
│ │ │ │  format. However, care should be taken as the returned list is entirely in
│ │ │ │  Graph6 or Sparse6 format.
│ │ │ │  i3 : G = generateBipartiteGraphs 7;
│ │ │ │  i4 : time graphComplement G;
│ │ │ │ - -- used 0.00101801s (cpu); 0.000811832s (thread); 0s (gc)
│ │ │ │ + -- used 0.000667562s (cpu); 0.000536291s (thread); 0s (gc)
│ │ │ │  i5 : time (graphComplement \ G);
│ │ │ │ - -- used 0.146022s (cpu); 0.0727874s (thread); 0s (gc)
│ │ │ │ + -- used 0.150279s (cpu); 0.0756606s (thread); 0s (gc)
│ │ │ │  ********** WWaayyss ttoo uussee ggrraapphhCCoommpplleemmeenntt:: **********
│ │ │ │      * graphComplement(Graph)
│ │ │ │      * graphComplement(List)
│ │ │ │      * graphComplement(String)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _g_r_a_p_h_C_o_m_p_l_e_m_e_n_t is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s.
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.out
│ │ │ @@ -47,15 +47,15 @@
│ │ │  o4 : Ideal of R
│ │ │  
│ │ │  i5 : isPrimary Q
│ │ │  
│ │ │  o5 = true
│ │ │  
│ │ │  i6 : elapsedTime noetherianOperators(Q, Strategy => "PunctualQuot")
│ │ │ - -- .113138s elapsed
│ │ │ + -- .0874811s elapsed
│ │ │  
│ │ │  o6 = {| 1 |, | dx_1 |, | dx_2 |, | dx_1^2 |, | dx_1dx_2 |, | dx_2^2 |, |
│ │ │       ------------------------------------------------------------------------
│ │ │       2x_1x_3dx_1^3+3x_2x_3dx_1^2dx_2-3x_3x_4dx_1dx_2^2-2x_1x_4dx_2^3 |}
│ │ │  
│ │ │  o6 : List
│ │ ├── ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.html
│ │ │ @@ -120,15 +120,15 @@
│ │ │  
│ │ │  o5 = true
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : elapsedTime noetherianOperators(Q, Strategy => "PunctualQuot")
│ │ │ - -- .113138s elapsed
│ │ │ + -- .0874811s elapsed
│ │ │  
│ │ │  o6 = {| 1 |, | dx_1 |, | dx_2 |, | dx_1^2 |, | dx_1dx_2 |, | dx_2^2 |, |
│ │ │       ------------------------------------------------------------------------
│ │ │       2x_1x_3dx_1^3+3x_2x_3dx_1^2dx_2-3x_3x_4dx_1dx_2^2-2x_1x_4dx_2^3 |}
│ │ │  
│ │ │  o6 : List
    
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -51,15 +51,15 @@
│ │ │ │         1 2 3    2 3
│ │ │ │  
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : isPrimary Q
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  i6 : elapsedTime noetherianOperators(Q, Strategy => "PunctualQuot")
│ │ │ │ - -- .113138s elapsed
│ │ │ │ + -- .0874811s elapsed
│ │ │ │  
│ │ │ │  o6 = {| 1 |, | dx_1 |, | dx_2 |, | dx_1^2 |, | dx_1dx_2 |, | dx_2^2 |, |
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       2x_1x_3dx_1^3+3x_2x_3dx_1^2dx_2-3x_3x_4dx_1dx_2^2-2x_1x_4dx_2^3 |}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Chow_spring.out
│ │ │ @@ -78,15 +78,15 @@
│ │ │  i13 : for i to dim X list hilbertFunction (i, A1)
│ │ │  
│ │ │  o13 = {1, 2, 3, 3, 2, 1}
│ │ │  
│ │ │  o13 : List
│ │ │  
│ │ │  i14 : Y = time smoothFanoToricVariety(5,100);
│ │ │ - -- used 0.195947s (cpu); 0.196132s (thread); 0s (gc)
│ │ │ + -- used 0.241848s (cpu); 0.242193s (thread); 0s (gc)
│ │ │  
│ │ │  i15 : A2 = intersectionRing Y;
│ │ │  
│ │ │  i16 : assert (# rays Y === numgens A2)
│ │ │  
│ │ │  i17 : ideal A2
│ │ │  
│ │ │ @@ -110,19 +110,19 @@
│ │ │          2                 2   2                       2   2                             2             2                      2     3      2
│ │ │        (t  + t t , t t  + t , t  + t t , t t , t t  + t , t  - t t  - 3t t   + t t   + 2t  , - t t  + t  + 2t t  , t t , - t t   + t  , t t  )
│ │ │          3    3 5   3 5    5   5    5 6   3 6   5 6    6   8    8 9     8 10    9 10     10     8 9    9     9 10   8 9     8 10    10   8 10
│ │ │  
│ │ │  o18 : QuotientRing
│ │ │  
│ │ │  i19 : for i to dim Y list time hilbertFunction (i, A2)
│ │ │ - -- used 0.00192857s (cpu); 0.00120085s (thread); 0s (gc)
│ │ │ - -- used 2.3484e-05s (cpu); 9.2934e-05s (thread); 0s (gc)
│ │ │ - -- used 9.298e-06s (cpu); 7.0111e-05s (thread); 0s (gc)
│ │ │ - -- used 8.586e-06s (cpu); 6.6936e-05s (thread); 0s (gc)
│ │ │ - -- used 8.175e-06s (cpu); 7.2777e-05s (thread); 0s (gc)
│ │ │ - -- used 8.416e-06s (cpu); 6.7105e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00395355s (cpu); 0.00182835s (thread); 0s (gc)
│ │ │ + -- used 2.9369e-05s (cpu); 0.000103096s (thread); 0s (gc)
│ │ │ + -- used 1.0757e-05s (cpu); 8.2799e-05s (thread); 0s (gc)
│ │ │ + -- used 9.926e-06s (cpu); 7.5793e-05s (thread); 0s (gc)
│ │ │ + -- used 9.64e-06s (cpu); 7.7367e-05s (thread); 0s (gc)
│ │ │ + -- used 9.662e-06s (cpu); 7.1582e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o19 = {1, 6, 13, 13, 6, 1}
│ │ │  
│ │ │  o19 : List
│ │ │  
│ │ │  i20 :
│ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_is__Well__Defined_lp__Normal__Toric__Variety_rp.out
│ │ │ @@ -1,29 +1,29 @@
│ │ │  -- -*- M2-comint -*- hash: 16408385764843695632
│ │ │  
│ │ │  i1 : assert all (5, d -> isWellDefined toricProjectiveSpace (d+1))
│ │ │  
│ │ │  i2 : setRandomSeed (currentTime ());
│ │ │ - -- setting random seed to 1765726817
│ │ │ + -- setting random seed to 1767265970
│ │ │  
│ │ │  i3 : a = sort apply (3, i -> random (7))
│ │ │  
│ │ │ -o3 = {2, 4, 5}
│ │ │ +o3 = {0, 3, 4}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : assert isWellDefined kleinschmidt (4,a)
│ │ │  
│ │ │  i5 : q = sort apply (5, j -> random (1,9));
│ │ │  
│ │ │  i6 : while not all (subsets (q,#q-1), s -> gcd s === 1) do q = sort apply (5, j -> random (1,9));
│ │ │  
│ │ │  i7 : q
│ │ │  
│ │ │ -o7 = {5, 7, 7, 9, 9}
│ │ │ +o7 = {4, 6, 7, 9, 9}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : assert isWellDefined weightedProjectiveSpace q
│ │ │  
│ │ │  i9 : X = new MutableHashTable;
│ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_monomials_lp__Toric__Divisor_rp.out
│ │ │ @@ -6,61 +6,61 @@
│ │ │  
│ │ │  o2 = 5*PP2
│ │ │            0
│ │ │  
│ │ │  o2 : ToricDivisor on PP2
│ │ │  
│ │ │  i3 : M1 = elapsedTime monomials D1
│ │ │ - -- .0534782s elapsed
│ │ │ + -- .0375704s elapsed
│ │ │  
│ │ │         5     4     4   2 3       3   2 3   3 2     2 2   2   2   3 2   4   
│ │ │  o3 = {x , x x , x x , x x , x x x , x x , x x , x x x , x x x , x x , x x ,
│ │ │         2   1 2   0 2   1 2   0 1 2   0 2   1 2   0 1 2   0 1 2   0 2   1 2 
│ │ │       ------------------------------------------------------------------------
│ │ │          3     2 2     3       4     5     4   2 3   3 2   4     5
│ │ │       x x x , x x x , x x x , x x , x , x x , x x , x x , x x , x }
│ │ │        0 1 2   0 1 2   0 1 2   0 2   1   0 1   0 1   0 1   0 1   0
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : elapsedTime assert (set M1 === set first entries basis(degree D1, ring variety D1))
│ │ │ - -- .0017525s elapsed
│ │ │ + -- .00125292s elapsed
│ │ │  
│ │ │  i5 : FF2 = hirzebruchSurface 2;
│ │ │  
│ │ │  i6 : D2 = 2*FF2_0 + 3 * FF2_1
│ │ │  
│ │ │  o6 = 2*FF2  + 3*FF2
│ │ │            0        1
│ │ │  
│ │ │  o6 : ToricDivisor on FF2
│ │ │  
│ │ │  i7 : M2 = elapsedTime monomials D2
│ │ │ - -- .0763581s elapsed
│ │ │ + -- .0323888s elapsed
│ │ │  
│ │ │         2     3 2     3     2 3
│ │ │  o7 = {x x , x x , x x x , x x }
│ │ │         1 3   1 2   0 1 2   0 1
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : elapsedTime assert (set M2 === set first entries basis (degree D2, ring variety D2))
│ │ │ - -- .00167424s elapsed
│ │ │ + -- .00121086s elapsed
│ │ │  
│ │ │  i9 : X = kleinschmidt (5, {1,2,3});
│ │ │  
│ │ │  i10 : D3 = 3*X_0 + 5*X_1
│ │ │  
│ │ │  o10 = 3*X  + 5*X
│ │ │           0      1
│ │ │  
│ │ │  o10 : ToricDivisor on X
│ │ │  
│ │ │  i11 : m3 = elapsedTime # monomials D3
│ │ │ - -- 40.8741s elapsed
│ │ │ + -- 28.5986s elapsed
│ │ │  
│ │ │  o11 = 7909
│ │ │  
│ │ │  i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety D3))
│ │ │ - -- .0293459s elapsed
│ │ │ + -- .0309196s elapsed
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Fan_rp.out
│ │ │ @@ -24,19 +24,19 @@
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : X = normalToricVariety F;
│ │ │  
│ │ │  i5 : assert (transpose matrix rays X == rays F and max X == sort maxCones F)
│ │ │  
│ │ │  i6 : X1 = time normalToricVariety ({{-1,-1},{1,0},{0,1}}, {{0,1},{1,2},{0,2}})
│ │ │ - -- used 2.2001e-05s (cpu); 1.9186e-05s (thread); 0s (gc)
│ │ │ + -- used 3.9133e-05s (cpu); 2.9122e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = X1
│ │ │  
│ │ │  o6 : NormalToricVariety
│ │ │  
│ │ │  i7 : X2 = time normalToricVariety fan {posHull matrix {{-1,1},{-1,0}}, posHull matrix {{1,0},{0,1}}, posHull matrix{{-1,0},{-1,1}}};
│ │ │ - -- used 0.0417375s (cpu); 0.0417407s (thread); 0s (gc)
│ │ │ + -- used 0.0531533s (cpu); 0.0531637s (thread); 0s (gc)
│ │ │  
│ │ │  i8 : assert (sort rays X1 == sort rays X2 and max X1 == max X2)
│ │ │  
│ │ │  i9 :
│ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Polyhedron_rp.out
│ │ │ @@ -88,15 +88,15 @@
│ │ │  o18 = | 0 1 0 |
│ │ │        | 0 0 1 |
│ │ │  
│ │ │                 2       3
│ │ │  o18 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i19 : X1 = time normalToricVariety convexHull (vertMatrix);
│ │ │ - -- used 0.0209352s (cpu); 0.020935s (thread); 0s (gc)
│ │ │ + -- used 0.0270248s (cpu); 0.0270239s (thread); 0s (gc)
│ │ │  
│ │ │  i20 : X2 = time normalToricVariety vertMatrix;
│ │ │ - -- used 0.00216116s (cpu); 0.00216187s (thread); 0s (gc)
│ │ │ + -- used 0.00264935s (cpu); 0.00265533s (thread); 0s (gc)
│ │ │  
│ │ │  i21 : assert (set rays X2 === set rays X1 and max X1 === max X2)
│ │ │  
│ │ │  i22 :
│ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Chow_spring.html
│ │ │ @@ -207,15 +207,15 @@
│ │ │          
    
│ │ │            
    We end with a slightly larger example.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -254,20 +254,20 @@
│ │ │  
│ │ │  o18 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i14 : Y = time smoothFanoToricVariety(5,100);
│ │ │ - -- used 0.195947s (cpu); 0.196132s (thread); 0s (gc)
    
│ │ │ + -- used 0.241848s (cpu); 0.242193s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i15 : A2 = intersectionRing Y;
    
│ │ │              
    
│ │ │                
    i19 : for i to dim Y list time hilbertFunction (i, A2)
│ │ │ - -- used 0.00192857s (cpu); 0.00120085s (thread); 0s (gc)
│ │ │ - -- used 2.3484e-05s (cpu); 9.2934e-05s (thread); 0s (gc)
│ │ │ - -- used 9.298e-06s (cpu); 7.0111e-05s (thread); 0s (gc)
│ │ │ - -- used 8.586e-06s (cpu); 6.6936e-05s (thread); 0s (gc)
│ │ │ - -- used 8.175e-06s (cpu); 7.2777e-05s (thread); 0s (gc)
│ │ │ - -- used 8.416e-06s (cpu); 6.7105e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00395355s (cpu); 0.00182835s (thread); 0s (gc)
│ │ │ + -- used 2.9369e-05s (cpu); 0.000103096s (thread); 0s (gc)
│ │ │ + -- used 1.0757e-05s (cpu); 8.2799e-05s (thread); 0s (gc)
│ │ │ + -- used 9.926e-06s (cpu); 7.5793e-05s (thread); 0s (gc)
│ │ │ + -- used 9.64e-06s (cpu); 7.7367e-05s (thread); 0s (gc)
│ │ │ + -- used 9.662e-06s (cpu); 7.1582e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o19 = {1, 6, 13, 13, 6, 1}
│ │ │  
│ │ │  o19 : List
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -96,15 +96,15 @@
│ │ │ │  i13 : for i to dim X list hilbertFunction (i, A1)
│ │ │ │  
│ │ │ │  o13 = {1, 2, 3, 3, 2, 1}
│ │ │ │  
│ │ │ │  o13 : List
│ │ │ │  We end with a slightly larger example.
│ │ │ │  i14 : Y = time smoothFanoToricVariety(5,100);
│ │ │ │ - -- used 0.195947s (cpu); 0.196132s (thread); 0s (gc)
│ │ │ │ + -- used 0.241848s (cpu); 0.242193s (thread); 0s (gc)
│ │ │ │  i15 : A2 = intersectionRing Y;
│ │ │ │  i16 : assert (# rays Y === numgens A2)
│ │ │ │  i17 : ideal A2
│ │ │ │  
│ │ │ │  o17 = ideal (t t , t t , t t , t t , t t , t t , t t , t t , t t t  ,
│ │ │ │                2 3   2 5   4 5   3 6   4 6   1 7   7 9   8 9   0 1 10
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ @@ -129,20 +129,20 @@
│ │ │ │        (t  + t t , t t  + t , t  + t t , t t , t t  + t , t  - t t  - 3t t   + t
│ │ │ │  t   + 2t  , - t t  + t  + 2t t  , t t , - t t   + t  , t t  )
│ │ │ │          3    3 5   3 5    5   5    5 6   3 6   5 6    6   8    8 9     8 10
│ │ │ │  9 10     10     8 9    9     9 10   8 9     8 10    10   8 10
│ │ │ │  
│ │ │ │  o18 : QuotientRing
│ │ │ │  i19 : for i to dim Y list time hilbertFunction (i, A2)
│ │ │ │ - -- used 0.00192857s (cpu); 0.00120085s (thread); 0s (gc)
│ │ │ │ - -- used 2.3484e-05s (cpu); 9.2934e-05s (thread); 0s (gc)
│ │ │ │ - -- used 9.298e-06s (cpu); 7.0111e-05s (thread); 0s (gc)
│ │ │ │ - -- used 8.586e-06s (cpu); 6.6936e-05s (thread); 0s (gc)
│ │ │ │ - -- used 8.175e-06s (cpu); 7.2777e-05s (thread); 0s (gc)
│ │ │ │ - -- used 8.416e-06s (cpu); 6.7105e-05s (thread); 0s (gc)
│ │ │ │ + -- used 0.00395355s (cpu); 0.00182835s (thread); 0s (gc)
│ │ │ │ + -- used 2.9369e-05s (cpu); 0.000103096s (thread); 0s (gc)
│ │ │ │ + -- used 1.0757e-05s (cpu); 8.2799e-05s (thread); 0s (gc)
│ │ │ │ + -- used 9.926e-06s (cpu); 7.5793e-05s (thread); 0s (gc)
│ │ │ │ + -- used 9.64e-06s (cpu); 7.7367e-05s (thread); 0s (gc)
│ │ │ │ + -- used 9.662e-06s (cpu); 7.1582e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o19 = {1, 6, 13, 13, 6, 1}
│ │ │ │  
│ │ │ │  o19 : List
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _w_o_r_k_i_n_g_ _w_i_t_h_ _s_h_e_a_v_e_s -- information about coherent sheaves and total
│ │ │ │        coordinate rings (a.k.a. Cox rings)
│ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_is__Well__Defined_lp__Normal__Toric__Variety_rp.html
│ │ │ @@ -93,22 +93,22 @@
│ │ │          
    
│ │ │            
    The second examples show that a randomly selected Kleinschmidt toric variety and a weighted projective space are also well-defined.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i2 : setRandomSeed (currentTime ());
│ │ │ - -- setting random seed to 1765726817
    
│ │ │ + -- setting random seed to 1767265970
│ │ │              
    
│ │ │                
    i3 : a = sort apply (3, i -> random (7))
│ │ │  
│ │ │ -o3 = {2, 4, 5}
│ │ │ +o3 = {0, 3, 4}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
    
│ │ │                
    i4 : assert isWellDefined kleinschmidt (4,a)
    
│ │ │ @@ -126,15 +126,15 @@
│ │ │                
    i6 : while not all (subsets (q,#q-1), s -> gcd s === 1) do q = sort apply (5, j -> random (1,9));
    
│ │ │              
    
│ │ │                
    i7 : q
│ │ │  
│ │ │ -o7 = {5, 7, 7, 9, 9}
│ │ │ +o7 = {4, 6, 7, 9, 9}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │                
    i8 : assert isWellDefined weightedProjectiveSpace q
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,27 +28,27 @@
│ │ │ │      * the intersection of the cones associated to two elements of coneList is a
│ │ │ │        face of each cone.
│ │ │ │  The first examples illustrate that small projective spaces are well-defined.
│ │ │ │  i1 : assert all (5, d -> isWellDefined toricProjectiveSpace (d+1))
│ │ │ │  The second examples show that a randomly selected Kleinschmidt toric variety
│ │ │ │  and a weighted projective space are also well-defined.
│ │ │ │  i2 : setRandomSeed (currentTime ());
│ │ │ │ - -- setting random seed to 1765726817
│ │ │ │ + -- setting random seed to 1767265970
│ │ │ │  i3 : a = sort apply (3, i -> random (7))
│ │ │ │  
│ │ │ │ -o3 = {2, 4, 5}
│ │ │ │ +o3 = {0, 3, 4}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : assert isWellDefined kleinschmidt (4,a)
│ │ │ │  i5 : q = sort apply (5, j -> random (1,9));
│ │ │ │  i6 : while not all (subsets (q,#q-1), s -> gcd s === 1) do q = sort apply (5, j
│ │ │ │  -> random (1,9));
│ │ │ │  i7 : q
│ │ │ │  
│ │ │ │ -o7 = {5, 7, 7, 9, 9}
│ │ │ │ +o7 = {4, 6, 7, 9, 9}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  i8 : assert isWellDefined weightedProjectiveSpace q
│ │ │ │  The next ten examples illustrate various ways that two lists can fail to define
│ │ │ │  a normal toric variety. By making the current debugging level greater than one,
│ │ │ │  one gets some addition information about the nature of the failure.
│ │ │ │  i9 : X = new MutableHashTable;
│ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_monomials_lp__Toric__Divisor_rp.html
│ │ │ @@ -96,15 +96,15 @@
│ │ │  
│ │ │  o2 : ToricDivisor on PP2
│ │ │              
    
│ │ │                
    i3 : M1 = elapsedTime monomials D1
│ │ │ - -- .0534782s elapsed
│ │ │ + -- .0375704s elapsed
│ │ │  
│ │ │         5     4     4   2 3       3   2 3   3 2     2 2   2   2   3 2   4   
│ │ │  o3 = {x , x x , x x , x x , x x x , x x , x x , x x x , x x x , x x , x x ,
│ │ │         2   1 2   0 2   1 2   0 1 2   0 2   1 2   0 1 2   0 1 2   0 2   1 2 
│ │ │       ------------------------------------------------------------------------
│ │ │          3     2 2     3       4     5     4   2 3   3 2   4     5
│ │ │       x x x , x x x , x x x , x x , x , x x , x x , x x , x x , x }
│ │ │ @@ -112,15 +112,15 @@
│ │ │  
│ │ │  o3 : List
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime assert (set M1 === set first entries basis(degree D1, ring variety D1))
│ │ │ - -- .0017525s elapsed
    
│ │ │ + -- .00125292s elapsed
│ │ │              
    
│ │ │          
    
│ │ │            
    Toric varieties of Picard-rank 2 are slightly more interesting.
    
│ │ │          
    
│ │ │          
│ │ │ @@ -138,27 +138,27 @@
│ │ │  
│ │ │  o6 : ToricDivisor on FF2
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -171,23 +171,23 @@
│ │ │  
│ │ │  o10 : ToricDivisor on X
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : M2 = elapsedTime monomials D2
│ │ │ - -- .0763581s elapsed
│ │ │ + -- .0323888s elapsed
│ │ │  
│ │ │         2     3 2     3     2 3
│ │ │  o7 = {x x , x x , x x x , x x }
│ │ │         1 3   1 2   0 1 2   0 1
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │                
    i8 : elapsedTime assert (set M2 === set first entries basis (degree D2, ring variety D2))
│ │ │ - -- .00167424s elapsed
    
│ │ │ + -- .00121086s elapsed
│ │ │              
    
│ │ │                
    i9 : X = kleinschmidt (5, {1,2,3});
    
│ │ │              
    
│ │ │                
    i11 : m3 = elapsedTime # monomials D3
│ │ │ - -- 40.8741s elapsed
│ │ │ + -- 28.5986s elapsed
│ │ │  
│ │ │  o11 = 7909
    
│ │ │              
    
│ │ │                
    i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety D3))
│ │ │ - -- .0293459s elapsed
    
│ │ │ + -- .0309196s elapsed
│ │ │              
    
│ │ │          
    
│ │ │            
    By exploiting latticePoints, this method function avoids using the basis function.
    
│ │ │          
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,61 +27,61 @@
│ │ │ │  i2 : D1 = 5*PP2_0
│ │ │ │  
│ │ │ │  o2 = 5*PP2
│ │ │ │            0
│ │ │ │  
│ │ │ │  o2 : ToricDivisor on PP2
│ │ │ │  i3 : M1 = elapsedTime monomials D1
│ │ │ │ - -- .0534782s elapsed
│ │ │ │ + -- .0375704s elapsed
│ │ │ │  
│ │ │ │         5     4     4   2 3       3   2 3   3 2     2 2   2   2   3 2   4
│ │ │ │  o3 = {x , x x , x x , x x , x x x , x x , x x , x x x , x x x , x x , x x ,
│ │ │ │         2   1 2   0 2   1 2   0 1 2   0 2   1 2   0 1 2   0 1 2   0 2   1 2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          3     2 2     3       4     5     4   2 3   3 2   4     5
│ │ │ │       x x x , x x x , x x x , x x , x , x x , x x , x x , x x , x }
│ │ │ │        0 1 2   0 1 2   0 1 2   0 2   1   0 1   0 1   0 1   0 1   0
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : elapsedTime assert (set M1 === set first entries basis(degree D1, ring
│ │ │ │  variety D1))
│ │ │ │ - -- .0017525s elapsed
│ │ │ │ + -- .00125292s elapsed
│ │ │ │  Toric varieties of Picard-rank 2 are slightly more interesting.
│ │ │ │  i5 : FF2 = hirzebruchSurface 2;
│ │ │ │  i6 : D2 = 2*FF2_0 + 3 * FF2_1
│ │ │ │  
│ │ │ │  o6 = 2*FF2  + 3*FF2
│ │ │ │            0        1
│ │ │ │  
│ │ │ │  o6 : ToricDivisor on FF2
│ │ │ │  i7 : M2 = elapsedTime monomials D2
│ │ │ │ - -- .0763581s elapsed
│ │ │ │ + -- .0323888s elapsed
│ │ │ │  
│ │ │ │         2     3 2     3     2 3
│ │ │ │  o7 = {x x , x x , x x x , x x }
│ │ │ │         1 3   1 2   0 1 2   0 1
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  i8 : elapsedTime assert (set M2 === set first entries basis (degree D2, ring
│ │ │ │  variety D2))
│ │ │ │ - -- .00167424s elapsed
│ │ │ │ + -- .00121086s elapsed
│ │ │ │  i9 : X = kleinschmidt (5, {1,2,3});
│ │ │ │  i10 : D3 = 3*X_0 + 5*X_1
│ │ │ │  
│ │ │ │  o10 = 3*X  + 5*X
│ │ │ │           0      1
│ │ │ │  
│ │ │ │  o10 : ToricDivisor on X
│ │ │ │  i11 : m3 = elapsedTime # monomials D3
│ │ │ │ - -- 40.8741s elapsed
│ │ │ │ + -- 28.5986s elapsed
│ │ │ │  
│ │ │ │  o11 = 7909
│ │ │ │  i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety
│ │ │ │  D3))
│ │ │ │ - -- .0293459s elapsed
│ │ │ │ + -- .0309196s elapsed
│ │ │ │  By exploiting _l_a_t_t_i_c_e_P_o_i_n_t_s, this method function avoids using the _b_a_s_i_s
│ │ │ │  function.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _w_o_r_k_i_n_g_ _w_i_t_h_ _d_i_v_i_s_o_r_s -- information about toric divisors and their
│ │ │ │        related groups
│ │ │ │      * _r_i_n_g_(_N_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_) -- make the total coordinate ring (a.k.a. Cox
│ │ │ │        ring)
│ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Fan_rp.html
│ │ │ @@ -125,25 +125,25 @@
│ │ │          
    
│ │ │            
    The recommended method for creating a NormalToricVariety from a fan is normalToricVariety(List,List).  In fact, this package avoids using objects from the Polyhedra package whenever possible.   Here is a trivial example, namely projective 2-space, illustrating the substantial increase in time resulting from the use of a Polyhedra fan.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,22 +48,22 @@
│ │ │ │  i5 : assert (transpose matrix rays X == rays F and max X == sort maxCones F)
│ │ │ │  The recommended method for creating a _N_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y from a fan is
│ │ │ │  _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_(_L_i_s_t_,_L_i_s_t_). In fact, this package avoids using objects from
│ │ │ │  the _P_o_l_y_h_e_d_r_a package whenever possible. Here is a trivial example, namely
│ │ │ │  projective 2-space, illustrating the substantial increase in time resulting
│ │ │ │  from the use of a _P_o_l_y_h_e_d_r_a fan.
│ │ │ │  i6 : X1 = time normalToricVariety ({{-1,-1},{1,0},{0,1}}, {{0,1},{1,2},{0,2}})
│ │ │ │ - -- used 2.2001e-05s (cpu); 1.9186e-05s (thread); 0s (gc)
│ │ │ │ + -- used 3.9133e-05s (cpu); 2.9122e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = X1
│ │ │ │  
│ │ │ │  o6 : NormalToricVariety
│ │ │ │  i7 : X2 = time normalToricVariety fan {posHull matrix {{-1,1},{-1,0}}, posHull
│ │ │ │  matrix {{1,0},{0,1}}, posHull matrix{{-1,0},{-1,1}}};
│ │ │ │ - -- used 0.0417375s (cpu); 0.0417407s (thread); 0s (gc)
│ │ │ │ + -- used 0.0531533s (cpu); 0.0531637s (thread); 0s (gc)
│ │ │ │  i8 : assert (sort rays X1 == sort rays X2 and max X1 == max X2)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_a_k_i_n_g_ _n_o_r_m_a_l_ _t_o_r_i_c_ _v_a_r_i_e_t_i_e_s -- information about the basic constructors
│ │ │ │      * _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y -- make a normal toric variety
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_(_F_a_n_) -- make a normal toric variety from a 'Polyhedra'
│ │ │ │        fan
│ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Polyhedron_rp.html
│ │ │ @@ -233,21 +233,21 @@
│ │ │                 2       3
│ │ │  o18 : Matrix ZZ  <-- ZZ
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -102,17 +102,17 @@
│ │ │ │  
│ │ │ │  o18 = | 0 1 0 |
│ │ │ │        | 0 0 1 |
│ │ │ │  
│ │ │ │                 2       3
│ │ │ │  o18 : Matrix ZZ  <-- ZZ
│ │ │ │  i19 : X1 = time normalToricVariety convexHull (vertMatrix);
│ │ │ │ - -- used 0.0209352s (cpu); 0.020935s (thread); 0s (gc)
│ │ │ │ + -- used 0.0270248s (cpu); 0.0270239s (thread); 0s (gc)
│ │ │ │  i20 : X2 = time normalToricVariety vertMatrix;
│ │ │ │ - -- used 0.00216116s (cpu); 0.00216187s (thread); 0s (gc)
│ │ │ │ + -- used 0.00264935s (cpu); 0.00265533s (thread); 0s (gc)
│ │ │ │  i21 : assert (set rays X2 === set rays X1 and max X1 === max X2)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_a_k_i_n_g_ _n_o_r_m_a_l_ _t_o_r_i_c_ _v_a_r_i_e_t_i_e_s -- information about the basic constructors
│ │ │ │      * _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_(_M_a_t_r_i_x_) -- make a normal toric variety from a polytope
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_(_P_o_l_y_h_e_d_r_o_n_) -- make a normal toric variety from a
│ │ │ │        'Polyhedra' polyhedron
│ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/___Convert__To__Cone.out
│ │ │ @@ -21,21 +21,21 @@
│ │ │  
│ │ │  i4 : (numericalHilbertFunction(F, I, 3, Verbose => false)).hilbertFunctionValue == 0
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 : T = numericalHilbertFunction(F, I, 3, ConvertToCone => true)
│ │ │  Sampling image points ...
│ │ │ -     -- used .0884675 seconds
│ │ │ +     -- used .0873637 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .00923927 seconds
│ │ │ +     -- used .0256302 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00487238 seconds
│ │ │ +     -- used .00304949 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000642635 seconds
│ │ │ +     -- used .000463706 seconds
│ │ │  
│ │ │  o5 = a "numerical interpolation table", indicating
│ │ │       the space of degree 3 forms in the ideal of the image has dimension 3
│ │ │  
│ │ │  o5 : NumericalInterpolationTable
│ │ │  
│ │ │  i6 : extractImageEquations(T, AttemptZZ => true)
│ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_extract__Image__Equations.out
│ │ │ @@ -11,21 +11,21 @@
│ │ │  o2 = | s3 s2t st2 t3 |
│ │ │  
│ │ │               1      4
│ │ │  o2 : Matrix R  <-- R
│ │ │  
│ │ │  i3 : extractImageEquations(F, ideal 0_R, 2, AttemptZZ => true)
│ │ │  Sampling image points ...
│ │ │ -     -- used .00368858 seconds
│ │ │ +     -- used .00413492 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .00273464 seconds
│ │ │ +     -- used .00298022 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00106818 seconds
│ │ │ +     -- used .00114695 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000315712 seconds
│ │ │ +     -- used .000333959 seconds
│ │ │  
│ │ │  o3 = | y_1^2-y_0y_2 y_1y_2-y_0y_3 y_2^2-y_1y_3 |
│ │ │  
│ │ │                            1                   3
│ │ │  o3 : Matrix (CC  [y ..y ])  <-- (CC  [y ..y ])
│ │ │                 53  0   3           53  0   3
│ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Hilbert__Function.out
│ │ │ @@ -11,40 +11,40 @@
│ │ │  o2 = | s3 s2t st2 t3 |
│ │ │  
│ │ │               1      4
│ │ │  o2 : Matrix R  <-- R
│ │ │  
│ │ │  i3 : numericalHilbertFunction(F, ideal 0_R, 4)
│ │ │  Sampling image points ...
│ │ │ -     -- used .0123198 seconds
│ │ │ +     -- used .0139219 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .0108776 seconds
│ │ │ +     -- used .0132317 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00719824 seconds
│ │ │ +     -- used .00708841 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000762781 seconds
│ │ │ +     -- used .000853255 seconds
│ │ │  
│ │ │  o3 = a "numerical interpolation table", indicating
│ │ │       the space of degree 4 forms in the ideal of the image has dimension 22
│ │ │  
│ │ │  o3 : NumericalInterpolationTable
│ │ │  
│ │ │  i4 : R = CC[x_(1,1)..x_(2,4)];
│ │ │  
│ │ │  i5 : F = (minors(2, genericMatrix(R, 2, 4)))_*;
│ │ │  
│ │ │  i6 : S = numericalImageSample(F, ideal 0_R, 60);
│ │ │  
│ │ │  i7 : numericalHilbertFunction(F, ideal 0_R, S, 2, UseSLP => true)
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .0026712 seconds
│ │ │ +     -- used .00311279 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00778446 seconds
│ │ │ +     -- used .0079176 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000791835 seconds
│ │ │ +     -- used .00102141 seconds
│ │ │  
│ │ │  o7 = a "numerical interpolation table", indicating
│ │ │       the space of degree 2 forms in the ideal of the image has dimension 1
│ │ │  
│ │ │  o7 : NumericalInterpolationTable
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Image__Dim.out
│ │ │ @@ -20,12 +20,12 @@
│ │ │  
│ │ │  i8 : F = sum(1..14, i -> basis(4, R, Variables=>toList(a_(i,1)..a_(i,5))));
│ │ │  
│ │ │               1      70
│ │ │  o8 : Matrix R  <-- R
│ │ │  
│ │ │  i9 : time numericalImageDim(F, ideal 0_R)
│ │ │ - -- used 0.0641647s (cpu); 0.064162s (thread); 0s (gc)
│ │ │ + -- used 0.0702269s (cpu); 0.0702192s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = 69
│ │ │  
│ │ │  i10 :
│ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_real__Point.out
│ │ │ @@ -31,15 +31,15 @@
│ │ │  o5 : Ideal of R
│ │ │  
│ │ │  i6 : I = I1 + I2;
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : elapsedTime p = realPoint(I, Iterations => 100)
│ │ │ - -- .638062s elapsed
│ │ │ + -- .489357s elapsed
│ │ │  
│ │ │  o7 = p
│ │ │  
│ │ │  o7 : Point
│ │ │  
│ │ │  i8 : matrix pack(5, p#Coordinates)
│ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/___Convert__To__Cone.html
│ │ │ @@ -100,21 +100,21 @@
│ │ │  o4 = true
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,21 +33,21 @@
│ │ │ │  o3 : Ideal of R
│ │ │ │  i4 : (numericalHilbertFunction(F, I, 3, Verbose => false)).hilbertFunctionValue
│ │ │ │  == 0
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  i5 : T = numericalHilbertFunction(F, I, 3, ConvertToCone => true)
│ │ │ │  Sampling image points ...
│ │ │ │ -     -- used .0884675 seconds
│ │ │ │ +     -- used .0873637 seconds
│ │ │ │  Creating interpolation matrix ...
│ │ │ │ -     -- used .00923927 seconds
│ │ │ │ +     -- used .0256302 seconds
│ │ │ │  Performing normalization preconditioning ...
│ │ │ │ -     -- used .00487238 seconds
│ │ │ │ +     -- used .00304949 seconds
│ │ │ │  Computing numerical kernel ...
│ │ │ │ -     -- used .000642635 seconds
│ │ │ │ +     -- used .000463706 seconds
│ │ │ │  
│ │ │ │  o5 = a "numerical interpolation table", indicating
│ │ │ │       the space of degree 3 forms in the ideal of the image has dimension 3
│ │ │ │  
│ │ │ │  o5 : NumericalInterpolationTable
│ │ │ │  i6 : extractImageEquations(T, AttemptZZ => true)
│ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_extract__Image__Equations.html
│ │ │ @@ -102,21 +102,21 @@
│ │ │  o2 : Matrix R  <-- R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -38,21 +38,21 @@
│ │ │ │  
│ │ │ │  o2 = | s3 s2t st2 t3 |
│ │ │ │  
│ │ │ │               1      4
│ │ │ │  o2 : Matrix R  <-- R
│ │ │ │  i3 : extractImageEquations(F, ideal 0_R, 2, AttemptZZ => true)
│ │ │ │  Sampling image points ...
│ │ │ │ -     -- used .00368858 seconds
│ │ │ │ +     -- used .00413492 seconds
│ │ │ │  Creating interpolation matrix ...
│ │ │ │ -     -- used .00273464 seconds
│ │ │ │ +     -- used .00298022 seconds
│ │ │ │  Performing normalization preconditioning ...
│ │ │ │ -     -- used .00106818 seconds
│ │ │ │ +     -- used .00114695 seconds
│ │ │ │  Computing numerical kernel ...
│ │ │ │ -     -- used .000315712 seconds
│ │ │ │ +     -- used .000333959 seconds
│ │ │ │  
│ │ │ │  o3 = | y_1^2-y_0y_2 y_1y_2-y_0y_3 y_2^2-y_1y_3 |
│ │ │ │  
│ │ │ │                            1                   3
│ │ │ │  o3 : Matrix (CC  [y ..y ])  <-- (CC  [y ..y ])
│ │ │ │                 53  0   3           53  0   3
│ │ │ │  Here is how to do the same computation symbolically.
│ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Hilbert__Function.html
│ │ │ @@ -107,21 +107,21 @@
│ │ │  o2 : Matrix R  <-- R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -147,19 +147,19 @@
│ │ │                
    i6 : S = numericalImageSample(F, ideal 0_R, 60);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -57,39 +57,39 @@
│ │ │ │  
│ │ │ │  o2 = | s3 s2t st2 t3 |
│ │ │ │  
│ │ │ │               1      4
│ │ │ │  o2 : Matrix R  <-- R
│ │ │ │  i3 : numericalHilbertFunction(F, ideal 0_R, 4)
│ │ │ │  Sampling image points ...
│ │ │ │ -     -- used .0123198 seconds
│ │ │ │ +     -- used .0139219 seconds
│ │ │ │  Creating interpolation matrix ...
│ │ │ │ -     -- used .0108776 seconds
│ │ │ │ +     -- used .0132317 seconds
│ │ │ │  Performing normalization preconditioning ...
│ │ │ │ -     -- used .00719824 seconds
│ │ │ │ +     -- used .00708841 seconds
│ │ │ │  Computing numerical kernel ...
│ │ │ │ -     -- used .000762781 seconds
│ │ │ │ +     -- used .000853255 seconds
│ │ │ │  
│ │ │ │  o3 = a "numerical interpolation table", indicating
│ │ │ │       the space of degree 4 forms in the ideal of the image has dimension 22
│ │ │ │  
│ │ │ │  o3 : NumericalInterpolationTable
│ │ │ │  The following example computes the dimension of Plücker quadrics in the
│ │ │ │  defining ideal of the Grassmannian $Gr(2,4)$ of $P^1$'s in $P^3$, in the
│ │ │ │  ambient space $P^5$.
│ │ │ │  i4 : R = CC[x_(1,1)..x_(2,4)];
│ │ │ │  i5 : F = (minors(2, genericMatrix(R, 2, 4)))_*;
│ │ │ │  i6 : S = numericalImageSample(F, ideal 0_R, 60);
│ │ │ │  i7 : numericalHilbertFunction(F, ideal 0_R, S, 2, UseSLP => true)
│ │ │ │  Creating interpolation matrix ...
│ │ │ │ -     -- used .0026712 seconds
│ │ │ │ +     -- used .00311279 seconds
│ │ │ │  Performing normalization preconditioning ...
│ │ │ │ -     -- used .00778446 seconds
│ │ │ │ +     -- used .0079176 seconds
│ │ │ │  Computing numerical kernel ...
│ │ │ │ -     -- used .000791835 seconds
│ │ │ │ +     -- used .00102141 seconds
│ │ │ │  
│ │ │ │  o7 = a "numerical interpolation table", indicating
│ │ │ │       the space of degree 2 forms in the ideal of the image has dimension 1
│ │ │ │  
│ │ │ │  o7 : NumericalInterpolationTable
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _N_u_m_e_r_i_c_a_l_I_n_t_e_r_p_o_l_a_t_i_o_n_T_a_b_l_e -- the class of all
│ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Image__Dim.html
│ │ │ @@ -140,15 +140,15 @@
│ │ │               1      70
│ │ │  o8 : Matrix R  <-- R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i6 : X1 = time normalToricVariety ({{-1,-1},{1,0},{0,1}}, {{0,1},{1,2},{0,2}})
│ │ │ - -- used 2.2001e-05s (cpu); 1.9186e-05s (thread); 0s (gc)
│ │ │ + -- used 3.9133e-05s (cpu); 2.9122e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = X1
│ │ │  
│ │ │  o6 : NormalToricVariety
    
│ │ │              
    
│ │ │                
    i7 : X2 = time normalToricVariety fan {posHull matrix {{-1,1},{-1,0}}, posHull matrix {{1,0},{0,1}}, posHull matrix{{-1,0},{-1,1}}};
│ │ │ - -- used 0.0417375s (cpu); 0.0417407s (thread); 0s (gc)
    
│ │ │ + -- used 0.0531533s (cpu); 0.0531637s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i8 : assert (sort rays X1 == sort rays X2 and max X1 == max X2)
    
│ │ │              
    
│ │ │                
    i19 : X1 = time normalToricVariety convexHull (vertMatrix);
│ │ │ - -- used 0.0209352s (cpu); 0.020935s (thread); 0s (gc)
    
│ │ │ + -- used 0.0270248s (cpu); 0.0270239s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i20 : X2 = time normalToricVariety vertMatrix;
│ │ │ - -- used 0.00216116s (cpu); 0.00216187s (thread); 0s (gc)
    
│ │ │ + -- used 0.00264935s (cpu); 0.00265533s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i21 : assert (set rays X2 === set rays X1 and max X1 === max X2)
    
│ │ │              
    
│ │ │                
    i5 : T = numericalHilbertFunction(F, I, 3, ConvertToCone => true)
│ │ │  Sampling image points ...
│ │ │ -     -- used .0884675 seconds
│ │ │ +     -- used .0873637 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .00923927 seconds
│ │ │ +     -- used .0256302 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00487238 seconds
│ │ │ +     -- used .00304949 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000642635 seconds
│ │ │ +     -- used .000463706 seconds
│ │ │  
│ │ │  o5 = a "numerical interpolation table", indicating
│ │ │       the space of degree 3 forms in the ideal of the image has dimension 3
│ │ │  
│ │ │  o5 : NumericalInterpolationTable
    
│ │ │              
    
│ │ │                
    i3 : extractImageEquations(F, ideal 0_R, 2, AttemptZZ => true)
│ │ │  Sampling image points ...
│ │ │ -     -- used .00368858 seconds
│ │ │ +     -- used .00413492 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .00273464 seconds
│ │ │ +     -- used .00298022 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00106818 seconds
│ │ │ +     -- used .00114695 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000315712 seconds
│ │ │ +     -- used .000333959 seconds
│ │ │  
│ │ │  o3 = | y_1^2-y_0y_2 y_1y_2-y_0y_3 y_2^2-y_1y_3 |
│ │ │  
│ │ │                            1                   3
│ │ │  o3 : Matrix (CC  [y ..y ])  <-- (CC  [y ..y ])
│ │ │                 53  0   3           53  0   3
    
│ │ │              
    
│ │ │                
    i3 : numericalHilbertFunction(F, ideal 0_R, 4)
│ │ │  Sampling image points ...
│ │ │ -     -- used .0123198 seconds
│ │ │ +     -- used .0139219 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .0108776 seconds
│ │ │ +     -- used .0132317 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00719824 seconds
│ │ │ +     -- used .00708841 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000762781 seconds
│ │ │ +     -- used .000853255 seconds
│ │ │  
│ │ │  o3 = a "numerical interpolation table", indicating
│ │ │       the space of degree 4 forms in the ideal of the image has dimension 22
│ │ │  
│ │ │  o3 : NumericalInterpolationTable
    
│ │ │              
    
│ │ │                
    i7 : numericalHilbertFunction(F, ideal 0_R, S, 2, UseSLP => true)
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .0026712 seconds
│ │ │ +     -- used .00311279 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00778446 seconds
│ │ │ +     -- used .0079176 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000791835 seconds
│ │ │ +     -- used .00102141 seconds
│ │ │  
│ │ │  o7 = a "numerical interpolation table", indicating
│ │ │       the space of degree 2 forms in the ideal of the image has dimension 1
│ │ │  
│ │ │  o7 : NumericalInterpolationTable
    
│ │ │              
    
│ │ │                
    i9 : time numericalImageDim(F, ideal 0_R)
│ │ │ - -- used 0.0641647s (cpu); 0.064162s (thread); 0s (gc)
│ │ │ + -- used 0.0702269s (cpu); 0.0702192s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = 69
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -43,15 +43,15 @@
│ │ │ │  201-222. We numerically verify this below.
│ │ │ │  i7 : R = CC[a_(1,1)..a_(14,5)];
│ │ │ │  i8 : F = sum(1..14, i -> basis(4, R, Variables=>toList(a_(i,1)..a_(i,5))));
│ │ │ │  
│ │ │ │               1      70
│ │ │ │  o8 : Matrix R  <-- R
│ │ │ │  i9 : time numericalImageDim(F, ideal 0_R)
│ │ │ │ - -- used 0.0641647s (cpu); 0.064162s (thread); 0s (gc)
│ │ │ │ + -- used 0.0702269s (cpu); 0.0702192s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 = 69
│ │ │ │  ********** WWaayyss ttoo uussee nnuummeerriiccaallIImmaaggeeDDiimm:: **********
│ │ │ │      * numericalImageDim(List,Ideal)
│ │ │ │      * numericalImageDim(List,Ideal,Point)
│ │ │ │      * numericalImageDim(Matrix,Ideal)
│ │ │ │      * numericalImageDim(Matrix,Ideal,Point)
│ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_real__Point.html
│ │ │ @@ -132,15 +132,15 @@
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : elapsedTime p = realPoint(I, Iterations => 100)
│ │ │ - -- .638062s elapsed
│ │ │ + -- .489357s elapsed
│ │ │  
│ │ │  o7 = p
│ │ │  
│ │ │  o7 : Point
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -49,15 +49,15 @@
│ │ │ │  i5 : I2 = ideal apply(entries transpose A, row -> sum(row, v -> v^2) - 1);
│ │ │ │  
│ │ │ │  o5 : Ideal of R
│ │ │ │  i6 : I = I1 + I2;
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : elapsedTime p = realPoint(I, Iterations => 100)
│ │ │ │ - -- .638062s elapsed
│ │ │ │ + -- .489357s elapsed
│ │ │ │  
│ │ │ │  o7 = p
│ │ │ │  
│ │ │ │  o7 : Point
│ │ │ │  i8 : matrix pack(5, p#Coordinates)
│ │ │ │  
│ │ │ │  o8 = | .722359  .289465  -.295808  .591752  -.454678 |
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSchubertCalculus/example-output/_set__Verbose__Level.out
│ │ │ @@ -52,92 +52,92 @@
│ │ │  
│ │ │  i4 : assert all(S,s->checkIncidenceSolution(s,SchPblm))
│ │ │  
│ │ │  i5 : setVerboseLevel 1; 
│ │ │  
│ │ │  i6 : S = solveSchubertProblem(SchPblm,2,4)
│ │ │  -- playCheckers
│ │ │ --- cpu time = .00800386
│ │ │ +-- cpu time = .0119375
│ │ │  -- making a recursive call to resolveNode
│ │ │  -- playCheckers
│ │ │ --- cpu time = .00400121
│ │ │ +-- cpu time = .00801091
│ │ │  -- making a recursive call to resolveNode
│ │ │  -- playCheckers
│ │ │ --- cpu time = .00400194
│ │ │ +-- cpu time = 0
│ │ │  resolveNode reached node of no remaining conditions
│ │ │ --- time to make equations: .00479222
│ │ │ +-- time to make equations: .00797632
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0200437 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}]
│ │ │ --- time of performing one checker move: .01877
│ │ │ --- time of performing one checker move: .0033589
│ │ │ --- time of performing one checker move: .00399169
│ │ │ --- time to make equations: .00398021
│ │ │ + -- trackHomotopy time = .00834612 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}]
│ │ │ +-- time of performing one checker move: .0240051
│ │ │ +-- time of performing one checker move: .00396347
│ │ │ +-- time of performing one checker move: 0
│ │ │ +-- time to make equations: .00814426
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0063706 sec. for [{1, 2, 3, 0}, {1, infinity, infinity, 2}]
│ │ │ --- time of performing one checker move: .0199622
│ │ │ --- time to make equations: .00399873
│ │ │ + -- trackHomotopy time = .00809053 sec. for [{1, 2, 3, 0}, {1, infinity, infinity, 2}]
│ │ │ +-- time of performing one checker move: .0245578
│ │ │ +-- time to make equations: .00801786
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0665117 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 2}]
│ │ │ --- time of performing one checker move: .105096
│ │ │ --- time to make equations: .00399983
│ │ │ + -- trackHomotopy time = .022167 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 2}]
│ │ │ +-- time of performing one checker move: .100524
│ │ │ +-- time to make equations: .00803429
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0060564 sec. for [{2, 3, 1, 0}, {2, infinity, infinity, 1}]
│ │ │ --- time of performing one checker move: .0159996
│ │ │ --- time to make equations: .0996534
│ │ │ + -- trackHomotopy time = .00670951 sec. for [{2, 3, 1, 0}, {2, infinity, infinity, 1}]
│ │ │ +-- time of performing one checker move: .0160682
│ │ │ +-- time to make equations: .0948392
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .00690418 sec. for [{0, 1, 2, 3}, {infinity, 1, 2, infinity}]
│ │ │ --- time of performing one checker move: .107588
│ │ │ --- time to make equations: .011977
│ │ │ + -- trackHomotopy time = .00868241 sec. for [{0, 1, 2, 3}, {infinity, 1, 2, infinity}]
│ │ │ +-- time of performing one checker move: .106909
│ │ │ +-- time to make equations: .0158668
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0490889 sec. for [{0, 1, 3, 2}, {infinity, 1, infinity, 2}]
│ │ │ --- time of performing one checker move: .107635
│ │ │ --- time of performing one checker move: 0
│ │ │ + -- trackHomotopy time = .022452 sec. for [{0, 1, 3, 2}, {infinity, 1, infinity, 2}]
│ │ │ +-- time of performing one checker move: .107199
│ │ │  -- time of performing one checker move: 0
│ │ │ --- time to make equations: .00799983
│ │ │ -Setup time: 0
│ │ │ +-- time of performing one checker move: .00408582
│ │ │ +-- time to make equations: .0159268
│ │ │ +Setup time: 1
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0551627 sec. for [{1, 3, 2, 0}, {infinity, 3, infinity, 1}]
│ │ │ --- time of performing one checker move: .108513
│ │ │ + -- trackHomotopy time = .0233384 sec. for [{1, 3, 2, 0}, {infinity, 3, infinity, 1}]
│ │ │ +-- time of performing one checker move: .111805
│ │ │  -- making a recursive call to resolveNode
│ │ │  -- playCheckers
│ │ │ --- cpu time = .00664616
│ │ │ +-- cpu time = .00802985
│ │ │  -- making a recursive call to resolveNode
│ │ │  -- playCheckers
│ │ │  -- cpu time = 0
│ │ │  resolveNode reached node of no remaining conditions
│ │ │ --- time to make equations: .00400042
│ │ │ +-- time to make equations: .00800243
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .00628713 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}]
│ │ │ --- time of performing one checker move: .0997514
│ │ │ --- time of performing one checker move: .00399816
│ │ │ --- time to make equations: .00400027
│ │ │ + -- trackHomotopy time = .00937261 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}]
│ │ │ +-- time of performing one checker move: .111982
│ │ │ +-- time of performing one checker move: .00400104
│ │ │ +-- time to make equations: .00802771
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .00611061 sec. for [{0, 2, 3, 1}, {0, infinity, infinity, 2}]
│ │ │ --- time of performing one checker move: .0159585
│ │ │ --- time of performing one checker move: .0923873
│ │ │ + -- trackHomotopy time = .00767974 sec. for [{0, 2, 3, 1}, {0, infinity, infinity, 2}]
│ │ │ +-- time of performing one checker move: .0199981
│ │ │ +-- time of performing one checker move: .104028
│ │ │  -- time of performing one checker move: 0
│ │ │ --- time of performing one checker move: .003972
│ │ │ +-- time of performing one checker move: .00402681
│ │ │ +-- time of performing one checker move: .00401052
│ │ │  -- time of performing one checker move: 0
│ │ │ --- time of performing one checker move: .00399838
│ │ │ +-- time of performing one checker move: .00397717
│ │ │  -- time of performing one checker move: 0
│ │ │ --- time of performing one checker move: .00400062
│ │ │ --- time to make equations: .0120026
│ │ │ +-- time to make equations: .0159139
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .07797 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 3}]
│ │ │ --- time of performing one checker move: .113179
│ │ │ --- time of performing one checker move: 0
│ │ │ + -- trackHomotopy time = .0361492 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 3}]
│ │ │ +-- time of performing one checker move: .121209
│ │ │ +-- time of performing one checker move: .00395312
│ │ │  
│ │ │  o6 = {| -1.65573-.600637ii .0201935+.0437095ii   |, | -.154703+.175591ii 
│ │ │        | -1.23037-1.66989ii -.0308057-.00120618ii |  | -.801221-.0354303ii
│ │ │        | 1.35971-.743988ii  -.0713133-.049047ii   |  | .325581-2.08048ii  
│ │ │        | -.397038-1.8974ii  .0102261-.024397ii    |  | -.475895-.209388ii 
│ │ │       ------------------------------------------------------------------------
│ │ │       .0376857+.0683239ii   |}
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSchubertCalculus/html/_set__Verbose__Level.html
│ │ │ @@ -147,92 +147,92 @@
│ │ │                
    i5 : setVerboseLevel 1; 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : S = solveSchubertProblem(SchPblm,2,4)
│ │ │  -- playCheckers
│ │ │ --- cpu time = .00800386
│ │ │ +-- cpu time = .0119375
│ │ │  -- making a recursive call to resolveNode
│ │ │  -- playCheckers
│ │ │ --- cpu time = .00400121
│ │ │ +-- cpu time = .00801091
│ │ │  -- making a recursive call to resolveNode
│ │ │  -- playCheckers
│ │ │ --- cpu time = .00400194
│ │ │ +-- cpu time = 0
│ │ │  resolveNode reached node of no remaining conditions
│ │ │ --- time to make equations: .00479222
│ │ │ +-- time to make equations: .00797632
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0200437 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}]
│ │ │ --- time of performing one checker move: .01877
│ │ │ --- time of performing one checker move: .0033589
│ │ │ --- time of performing one checker move: .00399169
│ │ │ --- time to make equations: .00398021
│ │ │ + -- trackHomotopy time = .00834612 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}]
│ │ │ +-- time of performing one checker move: .0240051
│ │ │ +-- time of performing one checker move: .00396347
│ │ │ +-- time of performing one checker move: 0
│ │ │ +-- time to make equations: .00814426
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0063706 sec. for [{1, 2, 3, 0}, {1, infinity, infinity, 2}]
│ │ │ --- time of performing one checker move: .0199622
│ │ │ --- time to make equations: .00399873
│ │ │ + -- trackHomotopy time = .00809053 sec. for [{1, 2, 3, 0}, {1, infinity, infinity, 2}]
│ │ │ +-- time of performing one checker move: .0245578
│ │ │ +-- time to make equations: .00801786
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0665117 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 2}]
│ │ │ --- time of performing one checker move: .105096
│ │ │ --- time to make equations: .00399983
│ │ │ + -- trackHomotopy time = .022167 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 2}]
│ │ │ +-- time of performing one checker move: .100524
│ │ │ +-- time to make equations: .00803429
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0060564 sec. for [{2, 3, 1, 0}, {2, infinity, infinity, 1}]
│ │ │ --- time of performing one checker move: .0159996
│ │ │ --- time to make equations: .0996534
│ │ │ + -- trackHomotopy time = .00670951 sec. for [{2, 3, 1, 0}, {2, infinity, infinity, 1}]
│ │ │ +-- time of performing one checker move: .0160682
│ │ │ +-- time to make equations: .0948392
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .00690418 sec. for [{0, 1, 2, 3}, {infinity, 1, 2, infinity}]
│ │ │ --- time of performing one checker move: .107588
│ │ │ --- time to make equations: .011977
│ │ │ + -- trackHomotopy time = .00868241 sec. for [{0, 1, 2, 3}, {infinity, 1, 2, infinity}]
│ │ │ +-- time of performing one checker move: .106909
│ │ │ +-- time to make equations: .0158668
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0490889 sec. for [{0, 1, 3, 2}, {infinity, 1, infinity, 2}]
│ │ │ --- time of performing one checker move: .107635
│ │ │ --- time of performing one checker move: 0
│ │ │ + -- trackHomotopy time = .022452 sec. for [{0, 1, 3, 2}, {infinity, 1, infinity, 2}]
│ │ │ +-- time of performing one checker move: .107199
│ │ │  -- time of performing one checker move: 0
│ │ │ --- time to make equations: .00799983
│ │ │ -Setup time: 0
│ │ │ +-- time of performing one checker move: .00408582
│ │ │ +-- time to make equations: .0159268
│ │ │ +Setup time: 1
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0551627 sec. for [{1, 3, 2, 0}, {infinity, 3, infinity, 1}]
│ │ │ --- time of performing one checker move: .108513
│ │ │ + -- trackHomotopy time = .0233384 sec. for [{1, 3, 2, 0}, {infinity, 3, infinity, 1}]
│ │ │ +-- time of performing one checker move: .111805
│ │ │  -- making a recursive call to resolveNode
│ │ │  -- playCheckers
│ │ │ --- cpu time = .00664616
│ │ │ +-- cpu time = .00802985
│ │ │  -- making a recursive call to resolveNode
│ │ │  -- playCheckers
│ │ │  -- cpu time = 0
│ │ │  resolveNode reached node of no remaining conditions
│ │ │ --- time to make equations: .00400042
│ │ │ +-- time to make equations: .00800243
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .00628713 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}]
│ │ │ --- time of performing one checker move: .0997514
│ │ │ --- time of performing one checker move: .00399816
│ │ │ --- time to make equations: .00400027
│ │ │ + -- trackHomotopy time = .00937261 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}]
│ │ │ +-- time of performing one checker move: .111982
│ │ │ +-- time of performing one checker move: .00400104
│ │ │ +-- time to make equations: .00802771
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .00611061 sec. for [{0, 2, 3, 1}, {0, infinity, infinity, 2}]
│ │ │ --- time of performing one checker move: .0159585
│ │ │ --- time of performing one checker move: .0923873
│ │ │ + -- trackHomotopy time = .00767974 sec. for [{0, 2, 3, 1}, {0, infinity, infinity, 2}]
│ │ │ +-- time of performing one checker move: .0199981
│ │ │ +-- time of performing one checker move: .104028
│ │ │  -- time of performing one checker move: 0
│ │ │ --- time of performing one checker move: .003972
│ │ │ +-- time of performing one checker move: .00402681
│ │ │ +-- time of performing one checker move: .00401052
│ │ │  -- time of performing one checker move: 0
│ │ │ --- time of performing one checker move: .00399838
│ │ │ +-- time of performing one checker move: .00397717
│ │ │  -- time of performing one checker move: 0
│ │ │ --- time of performing one checker move: .00400062
│ │ │ --- time to make equations: .0120026
│ │ │ +-- time to make equations: .0159139
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .07797 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 3}]
│ │ │ --- time of performing one checker move: .113179
│ │ │ --- time of performing one checker move: 0
│ │ │ + -- trackHomotopy time = .0361492 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 3}]
│ │ │ +-- time of performing one checker move: .121209
│ │ │ +-- time of performing one checker move: .00395312
│ │ │  
│ │ │  o6 = {| -1.65573-.600637ii .0201935+.0437095ii   |, | -.154703+.175591ii 
│ │ │        | -1.23037-1.66989ii -.0308057-.00120618ii |  | -.801221-.0354303ii
│ │ │        | 1.35971-.743988ii  -.0713133-.049047ii   |  | .325581-2.08048ii  
│ │ │        | -.397038-1.8974ii  .0102261-.024397ii    |  | -.475895-.209388ii 
│ │ │       ------------------------------------------------------------------------
│ │ │       .0376857+.0683239ii   |}
│ │ │ ├── html2text {}
│ │ │ │ @@ -65,102 +65,102 @@
│ │ │ │       -.0336427+.0141017ii  |
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : assert all(S,s->checkIncidenceSolution(s,SchPblm))
│ │ │ │  i5 : setVerboseLevel 1;
│ │ │ │  i6 : S = solveSchubertProblem(SchPblm,2,4)
│ │ │ │  -- playCheckers
│ │ │ │ --- cpu time = .00800386
│ │ │ │ +-- cpu time = .0119375
│ │ │ │  -- making a recursive call to resolveNode
│ │ │ │  -- playCheckers
│ │ │ │ --- cpu time = .00400121
│ │ │ │ +-- cpu time = .00801091
│ │ │ │  -- making a recursive call to resolveNode
│ │ │ │  -- playCheckers
│ │ │ │ --- cpu time = .00400194
│ │ │ │ +-- cpu time = 0
│ │ │ │  resolveNode reached node of no remaining conditions
│ │ │ │ --- time to make equations: .00479222
│ │ │ │ +-- time to make equations: .00797632
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .0200437 sec. for [{0, 1, 2, 3}, {0, infinity, 2,
│ │ │ │ + -- trackHomotopy time = .00834612 sec. for [{0, 1, 2, 3}, {0, infinity, 2,
│ │ │ │  infinity}]
│ │ │ │ --- time of performing one checker move: .01877
│ │ │ │ --- time of performing one checker move: .0033589
│ │ │ │ --- time of performing one checker move: .00399169
│ │ │ │ --- time to make equations: .00398021
│ │ │ │ +-- time of performing one checker move: .0240051
│ │ │ │ +-- time of performing one checker move: .00396347
│ │ │ │ +-- time of performing one checker move: 0
│ │ │ │ +-- time to make equations: .00814426
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .0063706 sec. for [{1, 2, 3, 0}, {1, infinity,
│ │ │ │ + -- trackHomotopy time = .00809053 sec. for [{1, 2, 3, 0}, {1, infinity,
│ │ │ │  infinity, 2}]
│ │ │ │ --- time of performing one checker move: .0199622
│ │ │ │ --- time to make equations: .00399873
│ │ │ │ +-- time of performing one checker move: .0245578
│ │ │ │ +-- time to make equations: .00801786
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .0665117 sec. for [{1, 3, 2, 0}, {1, infinity,
│ │ │ │ + -- trackHomotopy time = .022167 sec. for [{1, 3, 2, 0}, {1, infinity,
│ │ │ │  infinity, 2}]
│ │ │ │ --- time of performing one checker move: .105096
│ │ │ │ --- time to make equations: .00399983
│ │ │ │ +-- time of performing one checker move: .100524
│ │ │ │ +-- time to make equations: .00803429
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .0060564 sec. for [{2, 3, 1, 0}, {2, infinity,
│ │ │ │ + -- trackHomotopy time = .00670951 sec. for [{2, 3, 1, 0}, {2, infinity,
│ │ │ │  infinity, 1}]
│ │ │ │ --- time of performing one checker move: .0159996
│ │ │ │ --- time to make equations: .0996534
│ │ │ │ +-- time of performing one checker move: .0160682
│ │ │ │ +-- time to make equations: .0948392
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .00690418 sec. for [{0, 1, 2, 3}, {infinity, 1, 2,
│ │ │ │ + -- trackHomotopy time = .00868241 sec. for [{0, 1, 2, 3}, {infinity, 1, 2,
│ │ │ │  infinity}]
│ │ │ │ --- time of performing one checker move: .107588
│ │ │ │ --- time to make equations: .011977
│ │ │ │ +-- time of performing one checker move: .106909
│ │ │ │ +-- time to make equations: .0158668
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .0490889 sec. for [{0, 1, 3, 2}, {infinity, 1,
│ │ │ │ + -- trackHomotopy time = .022452 sec. for [{0, 1, 3, 2}, {infinity, 1,
│ │ │ │  infinity, 2}]
│ │ │ │ --- time of performing one checker move: .107635
│ │ │ │ --- time of performing one checker move: 0
│ │ │ │ +-- time of performing one checker move: .107199
│ │ │ │  -- time of performing one checker move: 0
│ │ │ │ --- time to make equations: .00799983
│ │ │ │ -Setup time: 0
│ │ │ │ +-- time of performing one checker move: .00408582
│ │ │ │ +-- time to make equations: .0159268
│ │ │ │ +Setup time: 1
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .0551627 sec. for [{1, 3, 2, 0}, {infinity, 3,
│ │ │ │ + -- trackHomotopy time = .0233384 sec. for [{1, 3, 2, 0}, {infinity, 3,
│ │ │ │  infinity, 1}]
│ │ │ │ --- time of performing one checker move: .108513
│ │ │ │ +-- time of performing one checker move: .111805
│ │ │ │  -- making a recursive call to resolveNode
│ │ │ │  -- playCheckers
│ │ │ │ --- cpu time = .00664616
│ │ │ │ +-- cpu time = .00802985
│ │ │ │  -- making a recursive call to resolveNode
│ │ │ │  -- playCheckers
│ │ │ │  -- cpu time = 0
│ │ │ │  resolveNode reached node of no remaining conditions
│ │ │ │ --- time to make equations: .00400042
│ │ │ │ +-- time to make equations: .00800243
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .00628713 sec. for [{0, 1, 2, 3}, {0, infinity, 2,
│ │ │ │ + -- trackHomotopy time = .00937261 sec. for [{0, 1, 2, 3}, {0, infinity, 2,
│ │ │ │  infinity}]
│ │ │ │ --- time of performing one checker move: .0997514
│ │ │ │ --- time of performing one checker move: .00399816
│ │ │ │ --- time to make equations: .00400027
│ │ │ │ +-- time of performing one checker move: .111982
│ │ │ │ +-- time of performing one checker move: .00400104
│ │ │ │ +-- time to make equations: .00802771
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .00611061 sec. for [{0, 2, 3, 1}, {0, infinity,
│ │ │ │ + -- trackHomotopy time = .00767974 sec. for [{0, 2, 3, 1}, {0, infinity,
│ │ │ │  infinity, 2}]
│ │ │ │ --- time of performing one checker move: .0159585
│ │ │ │ --- time of performing one checker move: .0923873
│ │ │ │ +-- time of performing one checker move: .0199981
│ │ │ │ +-- time of performing one checker move: .104028
│ │ │ │  -- time of performing one checker move: 0
│ │ │ │ --- time of performing one checker move: .003972
│ │ │ │ +-- time of performing one checker move: .00402681
│ │ │ │ +-- time of performing one checker move: .00401052
│ │ │ │  -- time of performing one checker move: 0
│ │ │ │ --- time of performing one checker move: .00399838
│ │ │ │ +-- time of performing one checker move: .00397717
│ │ │ │  -- time of performing one checker move: 0
│ │ │ │ --- time of performing one checker move: .00400062
│ │ │ │ --- time to make equations: .0120026
│ │ │ │ +-- time to make equations: .0159139
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .07797 sec. for [{1, 3, 2, 0}, {1, infinity, infinity,
│ │ │ │ -3}]
│ │ │ │ --- time of performing one checker move: .113179
│ │ │ │ --- time of performing one checker move: 0
│ │ │ │ + -- trackHomotopy time = .0361492 sec. for [{1, 3, 2, 0}, {1, infinity,
│ │ │ │ +infinity, 3}]
│ │ │ │ +-- time of performing one checker move: .121209
│ │ │ │ +-- time of performing one checker move: .00395312
│ │ │ │  
│ │ │ │  o6 = {| -1.65573-.600637ii .0201935+.0437095ii   |, | -.154703+.175591ii
│ │ │ │        | -1.23037-1.66989ii -.0308057-.00120618ii |  | -.801221-.0354303ii
│ │ │ │        | 1.35971-.743988ii  -.0713133-.049047ii   |  | .325581-2.08048ii
│ │ │ │        | -.397038-1.8974ii  .0102261-.024397ii    |  | -.475895-.209388ii
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       .0376857+.0683239ii   |}
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/___Lab__Book__Protocol.out
│ │ │ @@ -14,35 +14,35 @@
│ │ │  
│ │ │  i4 : LL7a=select(LL7,L->not knownExample L);#LL7a
│ │ │  
│ │ │  o5 = 2
│ │ │  
│ │ │  i6 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0,Verbose=>true))
│ │ │  unfolding
│ │ │ - -- .146189s elapsed
│ │ │ + -- .114155s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .126375s elapsed
│ │ │ + -- .0969787s elapsed
│ │ │  next gb
│ │ │ - -- .000828707s elapsed
│ │ │ + -- .000812128s elapsed
│ │ │  true
│ │ │  unfolding
│ │ │ - -- .190454s elapsed
│ │ │ + -- .0951526s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .151259s elapsed
│ │ │ + -- .090677s elapsed
│ │ │  next gb
│ │ │ - -- .000705507s elapsed
│ │ │ + -- .000598621s elapsed
│ │ │  true
│ │ │ - -- 1.58898s elapsed
│ │ │ + -- 1.2188s elapsed
│ │ │  
│ │ │  o6 = {}
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0))
│ │ │ - -- 1.5313s elapsed
│ │ │ + -- 1.1263s elapsed
│ │ │  
│ │ │  o7 = {}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : LL7b=={}
│ │ │  
│ │ │ @@ -75,23 +75,23 @@
│ │ │  
│ │ │  o10 : Sequence
│ │ │  
│ │ │  i11 : elapsedTime nonWeierstrassSemigroups(m,g,Verbose=>true)
│ │ │  (13, 1)
│ │ │  {5, 8, 11, 12}
│ │ │  unfolding
│ │ │ - -- .207946s elapsed
│ │ │ + -- .192485s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .134913s elapsed
│ │ │ + -- .111815s elapsed
│ │ │  next gb
│ │ │ - -- .000914878s elapsed
│ │ │ + -- .000896207s elapsed
│ │ │  true
│ │ │ - -- .683163s elapsed
│ │ │ + -- .526892s elapsed
│ │ │  (5, 8,  all semigroups are smoothable)
│ │ │ - -- .716582s elapsed
│ │ │ + -- .560064s elapsed
│ │ │  
│ │ │  o11 = {}
│ │ │  
│ │ │  o11 : List
│ │ │  
│ │ │  i12 : L={6,8,9,11}
│ │ │  
│ │ │ @@ -100,22 +100,22 @@
│ │ │  o12 : List
│ │ │  
│ │ │  i13 : genus L
│ │ │  
│ │ │  o13 = 8
│ │ │  
│ │ │  i14 : isWeierstrassSemigroup(L,0.2,Verbose=>true)
│ │ │ - -- .0484596s elapsed
│ │ │ + -- .075721s elapsed
│ │ │  6
│ │ │  false
│ │ │  5
│ │ │  false
│ │ │  4
│ │ │  decompose
│ │ │ - -- .370099s elapsed
│ │ │ + -- .296697s elapsed
│ │ │  number of components: 2
│ │ │  support c, codim c: {(1, 1), (16, 3)}
│ │ │  {0, -1}
│ │ │  
│ │ │  o14 = true
│ │ │  
│ │ │  i15 :
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/_heuristic__Smoothness.out
│ │ │ @@ -10,15 +10,15 @@
│ │ │   -- setting random seed to 1644814534404491274313411285186041988099567563905780374824086062516559438
│ │ │  
│ │ │  i4 : elapsedTime tally apply(10,i-> (
│ │ │               c=minors(2,random(S^2,S^{3:-2}));
│ │ │               c=sub(c,x_0=>1);
│ │ │               R=kk[support c];c=sub(c,R);
│ │ │               heuristicSmoothness c))
│ │ │ - -- 3.62892s elapsed
│ │ │ + -- 2.98967s elapsed
│ │ │  
│ │ │  o4 = Tally{false => 6}
│ │ │             true => 4
│ │ │  
│ │ │  o4 : Tally
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/_is__Smoothable__Semigroup.out
│ │ │ @@ -7,17 +7,17 @@
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : genus L
│ │ │  
│ │ │  o2 = 8
│ │ │  
│ │ │  i3 : elapsedTime isSmoothableSemigroup(L,0.30,0)
│ │ │ - -- 1.05053s elapsed
│ │ │ + -- .789243s elapsed
│ │ │  
│ │ │  o3 = false
│ │ │  
│ │ │  i4 : elapsedTime isSmoothableSemigroup(L,0.14,0)
│ │ │ - -- 4.34178s elapsed
│ │ │ + -- 3.33826s elapsed
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/_is__Weierstrass__Semigroup.out
│ │ │ @@ -7,12 +7,12 @@
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : genus L
│ │ │  
│ │ │  o2 = 8
│ │ │  
│ │ │  i3 : elapsedTime isWeierstrassSemigroup(L,0.15)
│ │ │ - -- 4.3542s elapsed
│ │ │ + -- 3.02136s elapsed
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/_non__Weierstrass__Semigroups.out
│ │ │ @@ -1,12 +1,12 @@
│ │ │  -- -*- M2-comint -*- hash: 6860996532851631556
│ │ │  
│ │ │  i1 : elapsedTime nonWeierstrassSemigroups(6,7)
│ │ │  (6, 7,  all semigroups are smoothable)
│ │ │ - -- 1.35152s elapsed
│ │ │ + -- 1.17924s elapsed
│ │ │  
│ │ │  o1 = {}
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : LLdifficult={{6, 8, 9, 11}}
│ │ │  
│ │ │ @@ -14,61 +14,61 @@
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : elapsedTime nonWeierstrassSemigroups(6,8,LLdifficult,Verbose=>true)
│ │ │  (17, 5)
│ │ │  {6, 7, 8, 17}
│ │ │  unfolding
│ │ │ - -- .514484s elapsed
│ │ │ + -- .348027s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .187751s elapsed
│ │ │ + -- .143963s elapsed
│ │ │  next gb
│ │ │ - -- .00166673s elapsed
│ │ │ + -- .00194033s elapsed
│ │ │  true
│ │ │ - -- 1.09437s elapsed
│ │ │ + -- .809672s elapsed
│ │ │  {6, 7, 9, 17}
│ │ │  unfolding
│ │ │ - -- .483431s elapsed
│ │ │ + -- .334811s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .204984s elapsed
│ │ │ + -- .209567s elapsed
│ │ │  next gb
│ │ │ - -- .0026011s elapsed
│ │ │ + -- .00364996s elapsed
│ │ │  decompose
│ │ │ - -- .13498s elapsed
│ │ │ + -- .130883s elapsed
│ │ │  number of components: 2
│ │ │  support c, codim c: {(2, 2), (5, 2)}
│ │ │  {0, -1}
│ │ │ - -- 3.11814s elapsed
│ │ │ + -- 2.38925s elapsed
│ │ │  {6, 8, 9, 10}
│ │ │  unfolding
│ │ │ - -- .139467s elapsed
│ │ │ + -- .120298s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .154163s elapsed
│ │ │ + -- .0883283s elapsed
│ │ │  next gb
│ │ │ - -- .000485587s elapsed
│ │ │ + -- .000442622s elapsed
│ │ │  true
│ │ │ - -- .687947s elapsed
│ │ │ + -- .583484s elapsed
│ │ │  {6, 8, 10, 11, 13}
│ │ │  unfolding
│ │ │ - -- .550391s elapsed
│ │ │ + -- .448255s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .233636s elapsed
│ │ │ + -- .177409s elapsed
│ │ │  next gb
│ │ │ - -- .00395566s elapsed
│ │ │ + -- .00458298s elapsed
│ │ │  decompose
│ │ │ - -- .975039s elapsed
│ │ │ + -- .811769s elapsed
│ │ │  number of components: 1
│ │ │  support c, codim c: {(5, 1)}
│ │ │  {-1}
│ │ │ - -- 2.73979s elapsed
│ │ │ - -- 7.64035s elapsed
│ │ │ + -- 2.28763s elapsed
│ │ │ + -- 6.07018s elapsed
│ │ │  0
│ │ │  
│ │ │  {}
│ │ │ - -- .000003787s elapsed
│ │ │ - -- 7.67511s elapsed
│ │ │ + -- .000003683s elapsed
│ │ │ + -- 6.10673s elapsed
│ │ │  
│ │ │  o3 = {{6, 8, 9, 11}}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/html/___Lab__Book__Protocol.html
│ │ │ @@ -96,38 +96,38 @@
│ │ │  o5 = 2
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0,Verbose=>true))
│ │ │  unfolding
│ │ │ - -- .146189s elapsed
│ │ │ + -- .114155s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .126375s elapsed
│ │ │ + -- .0969787s elapsed
│ │ │  next gb
│ │ │ - -- .000828707s elapsed
│ │ │ + -- .000812128s elapsed
│ │ │  true
│ │ │  unfolding
│ │ │ - -- .190454s elapsed
│ │ │ + -- .0951526s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .151259s elapsed
│ │ │ + -- .090677s elapsed
│ │ │  next gb
│ │ │ - -- .000705507s elapsed
│ │ │ + -- .000598621s elapsed
│ │ │  true
│ │ │ - -- 1.58898s elapsed
│ │ │ + -- 1.2188s elapsed
│ │ │  
│ │ │  o6 = {}
│ │ │  
│ │ │  o6 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0))
│ │ │ - -- 1.5313s elapsed
│ │ │ + -- 1.1263s elapsed
│ │ │  
│ │ │  o7 = {}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -184,23 +184,23 @@
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : elapsedTime nonWeierstrassSemigroups(m,g,Verbose=>true)
│ │ │  (13, 1)
│ │ │  {5, 8, 11, 12}
│ │ │  unfolding
│ │ │ - -- .207946s elapsed
│ │ │ + -- .192485s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .134913s elapsed
│ │ │ + -- .111815s elapsed
│ │ │  next gb
│ │ │ - -- .000914878s elapsed
│ │ │ + -- .000896207s elapsed
│ │ │  true
│ │ │ - -- .683163s elapsed
│ │ │ + -- .526892s elapsed
│ │ │  (5, 8,  all semigroups are smoothable)
│ │ │ - -- .716582s elapsed
│ │ │ + -- .560064s elapsed
│ │ │  
│ │ │  o11 = {}
│ │ │  
│ │ │  o11 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ @@ -223,22 +223,22 @@
│ │ │  
│ │ │  o13 = 8
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : isWeierstrassSemigroup(L,0.2,Verbose=>true)
│ │ │ - -- .0484596s elapsed
│ │ │ + -- .075721s elapsed
│ │ │  6
│ │ │  false
│ │ │  5
│ │ │  false
│ │ │  4
│ │ │  decompose
│ │ │ - -- .370099s elapsed
│ │ │ + -- .296697s elapsed
│ │ │  number of components: 2
│ │ │  support c, codim c: {(1, 1), (16, 3)}
│ │ │  {0, -1}
│ │ │  
│ │ │  o14 = true
    
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,34 +26,34 @@
│ │ │ │  o3 = 39
│ │ │ │  i4 : LL7a=select(LL7,L->not knownExample L);#LL7a
│ │ │ │  
│ │ │ │  o5 = 2
│ │ │ │  i6 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup
│ │ │ │  (L,0.25,0,Verbose=>true))
│ │ │ │  unfolding
│ │ │ │ - -- .146189s elapsed
│ │ │ │ + -- .114155s elapsed
│ │ │ │  flatteningRelations
│ │ │ │ - -- .126375s elapsed
│ │ │ │ + -- .0969787s elapsed
│ │ │ │  next gb
│ │ │ │ - -- .000828707s elapsed
│ │ │ │ + -- .000812128s elapsed
│ │ │ │  true
│ │ │ │  unfolding
│ │ │ │ - -- .190454s elapsed
│ │ │ │ + -- .0951526s elapsed
│ │ │ │  flatteningRelations
│ │ │ │ - -- .151259s elapsed
│ │ │ │ + -- .090677s elapsed
│ │ │ │  next gb
│ │ │ │ - -- .000705507s elapsed
│ │ │ │ + -- .000598621s elapsed
│ │ │ │  true
│ │ │ │ - -- 1.58898s elapsed
│ │ │ │ + -- 1.2188s elapsed
│ │ │ │  
│ │ │ │  o6 = {}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0))
│ │ │ │ - -- 1.5313s elapsed
│ │ │ │ + -- 1.1263s elapsed
│ │ │ │  
│ │ │ │  o7 = {}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  i8 : LL7b=={}
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │ @@ -92,23 +92,23 @@
│ │ │ │  o10 = (5, 8)
│ │ │ │  
│ │ │ │  o10 : Sequence
│ │ │ │  i11 : elapsedTime nonWeierstrassSemigroups(m,g,Verbose=>true)
│ │ │ │  (13, 1)
│ │ │ │  {5, 8, 11, 12}
│ │ │ │  unfolding
│ │ │ │ - -- .207946s elapsed
│ │ │ │ + -- .192485s elapsed
│ │ │ │  flatteningRelations
│ │ │ │ - -- .134913s elapsed
│ │ │ │ + -- .111815s elapsed
│ │ │ │  next gb
│ │ │ │ - -- .000914878s elapsed
│ │ │ │ + -- .000896207s elapsed
│ │ │ │  true
│ │ │ │ - -- .683163s elapsed
│ │ │ │ + -- .526892s elapsed
│ │ │ │  (5, 8,  all semigroups are smoothable)
│ │ │ │ - -- .716582s elapsed
│ │ │ │ + -- .560064s elapsed
│ │ │ │  
│ │ │ │  o11 = {}
│ │ │ │  
│ │ │ │  o11 : List
│ │ │ │  In the verbose mode we get timings of various computation steps and further
│ │ │ │  information. The first line, (13,1), indicates that there 13 semigroups of
│ │ │ │  multiplicity 5 and genus 8 of which only 1 is not flagged as smoothable by the
│ │ │ │ @@ -120,22 +120,22 @@
│ │ │ │  o12 = {6, 8, 9, 11}
│ │ │ │  
│ │ │ │  o12 : List
│ │ │ │  i13 : genus L
│ │ │ │  
│ │ │ │  o13 = 8
│ │ │ │  i14 : isWeierstrassSemigroup(L,0.2,Verbose=>true)
│ │ │ │ - -- .0484596s elapsed
│ │ │ │ + -- .075721s elapsed
│ │ │ │  6
│ │ │ │  false
│ │ │ │  5
│ │ │ │  false
│ │ │ │  4
│ │ │ │  decompose
│ │ │ │ - -- .370099s elapsed
│ │ │ │ + -- .296697s elapsed
│ │ │ │  number of components: 2
│ │ │ │  support c, codim c: {(1, 1), (16, 3)}
│ │ │ │  {0, -1}
│ │ │ │  
│ │ │ │  o14 = true
│ │ │ │  The first integer, 6, tells that in this attempt deformation parameters of
│ │ │ │  degree >= 6 were used and no smooth fiber was found. Finally with all
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/html/_heuristic__Smoothness.html
│ │ │ @@ -94,15 +94,15 @@
│ │ │            
│ │ │              
│ │ │                
    i4 : elapsedTime tally apply(10,i-> (
│ │ │               c=minors(2,random(S^2,S^{3:-2}));
│ │ │               c=sub(c,x_0=>1);
│ │ │               R=kk[support c];c=sub(c,R);
│ │ │               heuristicSmoothness c))
│ │ │ - -- 3.62892s elapsed
│ │ │ + -- 2.98967s elapsed
│ │ │  
│ │ │  o4 = Tally{false => 6}
│ │ │             true => 4
│ │ │  
│ │ │  o4 : Tally
    
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,15 +27,15 @@
│ │ │ │   -- setting random seed to
│ │ │ │  1644814534404491274313411285186041988099567563905780374824086062516559438
│ │ │ │  i4 : elapsedTime tally apply(10,i-> (
│ │ │ │               c=minors(2,random(S^2,S^{3:-2}));
│ │ │ │               c=sub(c,x_0=>1);
│ │ │ │               R=kk[support c];c=sub(c,R);
│ │ │ │               heuristicSmoothness c))
│ │ │ │ - -- 3.62892s elapsed
│ │ │ │ + -- 2.98967s elapsed
│ │ │ │  
│ │ │ │  o4 = Tally{false => 6}
│ │ │ │             true => 4
│ │ │ │  
│ │ │ │  o4 : Tally
│ │ │ │  ********** WWaayyss ttoo uussee hheeuurriissttiiccSSmmooootthhnneessss:: **********
│ │ │ │      * heuristicSmoothness(Ideal)
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/html/_is__Smoothable__Semigroup.html
│ │ │ @@ -95,23 +95,23 @@
│ │ │  
│ │ │  o2 = 8
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime isSmoothableSemigroup(L,0.30,0)
│ │ │ - -- 1.05053s elapsed
│ │ │ + -- .789243s elapsed
│ │ │  
│ │ │  o3 = false
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : elapsedTime isSmoothableSemigroup(L,0.14,0)
│ │ │ - -- 4.34178s elapsed
│ │ │ + -- 3.33826s elapsed
│ │ │  
│ │ │  o4 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,19 +29,19 @@
│ │ │ │  o1 = {6, 8, 9, 11}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : genus L
│ │ │ │  
│ │ │ │  o2 = 8
│ │ │ │  i3 : elapsedTime isSmoothableSemigroup(L,0.30,0)
│ │ │ │ - -- 1.05053s elapsed
│ │ │ │ + -- .789243s elapsed
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : elapsedTime isSmoothableSemigroup(L,0.14,0)
│ │ │ │ - -- 4.34178s elapsed
│ │ │ │ + -- 3.33826s elapsed
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_a_k_e_U_n_f_o_l_d_i_n_g -- Makes the universal homogeneous unfolding of an ideal
│ │ │ │        with positive degree parameters
│ │ │ │      * _f_l_a_t_t_e_n_i_n_g_R_e_l_a_t_i_o_n_s -- Compute the flattening relations of an unfolding
│ │ │ │      * _g_e_t_F_l_a_t_F_a_m_i_l_y -- Compute the flat family depending on a subset of
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/html/_is__Weierstrass__Semigroup.html
│ │ │ @@ -94,15 +94,15 @@
│ │ │  
│ │ │  o2 = 8
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime isWeierstrassSemigroup(L,0.15)
│ │ │ - -- 4.3542s elapsed
│ │ │ + -- 3.02136s elapsed
│ │ │  
│ │ │  o3 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  o1 = {6, 8, 9, 11}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : genus L
│ │ │ │  
│ │ │ │  o2 = 8
│ │ │ │  i3 : elapsedTime isWeierstrassSemigroup(L,0.15)
│ │ │ │ - -- 4.3542s elapsed
│ │ │ │ + -- 3.02136s elapsed
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_a_k_e_U_n_f_o_l_d_i_n_g -- Makes the universal homogeneous unfolding of an ideal
│ │ │ │        with positive degree parameters
│ │ │ │      * _f_l_a_t_t_e_n_i_n_g_R_e_l_a_t_i_o_n_s -- Compute the flattening relations of an unfolding
│ │ │ │      * _g_e_t_F_l_a_t_F_a_m_i_l_y -- Compute the flat family depending on a subset of
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/html/_non__Weierstrass__Semigroups.html
│ │ │ @@ -79,15 +79,15 @@
│ │ │            
    We test which semigroups of multiplicity m and genus g are smoothable. If no smoothing was found then L is a candidate for a non Weierstrass semigroup. In this search certain semigroups L in LLdifficult, where the computation is particular heavy are excluded.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -101,62 +101,62 @@
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : elapsedTime nonWeierstrassSemigroups(6,7)
│ │ │  (6, 7,  all semigroups are smoothable)
│ │ │ - -- 1.35152s elapsed
│ │ │ + -- 1.17924s elapsed
│ │ │  
│ │ │  o1 = {}
│ │ │  
│ │ │  o1 : List
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime nonWeierstrassSemigroups(6,8,LLdifficult,Verbose=>true)
│ │ │  (17, 5)
│ │ │  {6, 7, 8, 17}
│ │ │  unfolding
│ │ │ - -- .514484s elapsed
│ │ │ + -- .348027s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .187751s elapsed
│ │ │ + -- .143963s elapsed
│ │ │  next gb
│ │ │ - -- .00166673s elapsed
│ │ │ + -- .00194033s elapsed
│ │ │  true
│ │ │ - -- 1.09437s elapsed
│ │ │ + -- .809672s elapsed
│ │ │  {6, 7, 9, 17}
│ │ │  unfolding
│ │ │ - -- .483431s elapsed
│ │ │ + -- .334811s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .204984s elapsed
│ │ │ + -- .209567s elapsed
│ │ │  next gb
│ │ │ - -- .0026011s elapsed
│ │ │ + -- .00364996s elapsed
│ │ │  decompose
│ │ │ - -- .13498s elapsed
│ │ │ + -- .130883s elapsed
│ │ │  number of components: 2
│ │ │  support c, codim c: {(2, 2), (5, 2)}
│ │ │  {0, -1}
│ │ │ - -- 3.11814s elapsed
│ │ │ + -- 2.38925s elapsed
│ │ │  {6, 8, 9, 10}
│ │ │  unfolding
│ │ │ - -- .139467s elapsed
│ │ │ + -- .120298s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .154163s elapsed
│ │ │ + -- .0883283s elapsed
│ │ │  next gb
│ │ │ - -- .000485587s elapsed
│ │ │ + -- .000442622s elapsed
│ │ │  true
│ │ │ - -- .687947s elapsed
│ │ │ + -- .583484s elapsed
│ │ │  {6, 8, 10, 11, 13}
│ │ │  unfolding
│ │ │ - -- .550391s elapsed
│ │ │ + -- .448255s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .233636s elapsed
│ │ │ + -- .177409s elapsed
│ │ │  next gb
│ │ │ - -- .00395566s elapsed
│ │ │ + -- .00458298s elapsed
│ │ │  decompose
│ │ │ - -- .975039s elapsed
│ │ │ + -- .811769s elapsed
│ │ │  number of components: 1
│ │ │  support c, codim c: {(5, 1)}
│ │ │  {-1}
│ │ │ - -- 2.73979s elapsed
│ │ │ - -- 7.64035s elapsed
│ │ │ + -- 2.28763s elapsed
│ │ │ + -- 6.07018s elapsed
│ │ │  0
│ │ │  
│ │ │  {}
│ │ │ - -- .000003787s elapsed
│ │ │ - -- 7.67511s elapsed
│ │ │ + -- .000003683s elapsed
│ │ │ + -- 6.10673s elapsed
│ │ │  
│ │ │  o3 = {{6, 8, 9, 11}}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,76 +22,76 @@
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  We test which semigroups of multiplicity m and genus g are smoothable. If no
│ │ │ │  smoothing was found then L is a candidate for a non Weierstrass semigroup. In
│ │ │ │  this search certain semigroups L in LLdifficult, where the computation is
│ │ │ │  particular heavy are excluded.
│ │ │ │  i1 : elapsedTime nonWeierstrassSemigroups(6,7)
│ │ │ │  (6, 7,  all semigroups are smoothable)
│ │ │ │ - -- 1.35152s elapsed
│ │ │ │ + -- 1.17924s elapsed
│ │ │ │  
│ │ │ │  o1 = {}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : LLdifficult={{6, 8, 9, 11}}
│ │ │ │  
│ │ │ │  o2 = {{6, 8, 9, 11}}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  i3 : elapsedTime nonWeierstrassSemigroups(6,8,LLdifficult,Verbose=>true)
│ │ │ │  (17, 5)
│ │ │ │  {6, 7, 8, 17}
│ │ │ │  unfolding
│ │ │ │ - -- .514484s elapsed
│ │ │ │ + -- .348027s elapsed
│ │ │ │  flatteningRelations
│ │ │ │ - -- .187751s elapsed
│ │ │ │ + -- .143963s elapsed
│ │ │ │  next gb
│ │ │ │ - -- .00166673s elapsed
│ │ │ │ + -- .00194033s elapsed
│ │ │ │  true
│ │ │ │ - -- 1.09437s elapsed
│ │ │ │ + -- .809672s elapsed
│ │ │ │  {6, 7, 9, 17}
│ │ │ │  unfolding
│ │ │ │ - -- .483431s elapsed
│ │ │ │ + -- .334811s elapsed
│ │ │ │  flatteningRelations
│ │ │ │ - -- .204984s elapsed
│ │ │ │ + -- .209567s elapsed
│ │ │ │  next gb
│ │ │ │ - -- .0026011s elapsed
│ │ │ │ + -- .00364996s elapsed
│ │ │ │  decompose
│ │ │ │ - -- .13498s elapsed
│ │ │ │ + -- .130883s elapsed
│ │ │ │  number of components: 2
│ │ │ │  support c, codim c: {(2, 2), (5, 2)}
│ │ │ │  {0, -1}
│ │ │ │ - -- 3.11814s elapsed
│ │ │ │ + -- 2.38925s elapsed
│ │ │ │  {6, 8, 9, 10}
│ │ │ │  unfolding
│ │ │ │ - -- .139467s elapsed
│ │ │ │ + -- .120298s elapsed
│ │ │ │  flatteningRelations
│ │ │ │ - -- .154163s elapsed
│ │ │ │ + -- .0883283s elapsed
│ │ │ │  next gb
│ │ │ │ - -- .000485587s elapsed
│ │ │ │ + -- .000442622s elapsed
│ │ │ │  true
│ │ │ │ - -- .687947s elapsed
│ │ │ │ + -- .583484s elapsed
│ │ │ │  {6, 8, 10, 11, 13}
│ │ │ │  unfolding
│ │ │ │ - -- .550391s elapsed
│ │ │ │ + -- .448255s elapsed
│ │ │ │  flatteningRelations
│ │ │ │ - -- .233636s elapsed
│ │ │ │ + -- .177409s elapsed
│ │ │ │  next gb
│ │ │ │ - -- .00395566s elapsed
│ │ │ │ + -- .00458298s elapsed
│ │ │ │  decompose
│ │ │ │ - -- .975039s elapsed
│ │ │ │ + -- .811769s elapsed
│ │ │ │  number of components: 1
│ │ │ │  support c, codim c: {(5, 1)}
│ │ │ │  {-1}
│ │ │ │ - -- 2.73979s elapsed
│ │ │ │ - -- 7.64035s elapsed
│ │ │ │ + -- 2.28763s elapsed
│ │ │ │ + -- 6.07018s elapsed
│ │ │ │  0
│ │ │ │  
│ │ │ │  {}
│ │ │ │ - -- .000003787s elapsed
│ │ │ │ - -- 7.67511s elapsed
│ │ │ │ + -- .000003683s elapsed
│ │ │ │ + -- 6.10673s elapsed
│ │ │ │  
│ │ │ │  o3 = {{6, 8, 9, 11}}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  In the verbose mode we get timings of various computation steps and further
│ │ │ │  information. The first line, (17,5), indicates that there 17 semigroups of
│ │ │ │  multiplicity 6 and genus 8 of which only 5 is not flagged as smoothable by the
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Free__O__I__Module__Map.out
│ │ │ @@ -8,15 +8,15 @@
│ │ │  
│ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(3,{2},1)+x_(2,2)*x_(2,1)*e_(3,{1,3},2);
│ │ │  
│ │ │  i5 : C = oiRes({b}, 2)
│ │ │  
│ │ │  o5 = 0: (e0, {3}, {-2})
│ │ │       1: (e1, {5, 5}, {-3, -4})
│ │ │ -     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-2, -5, -4, -3, -5, -4, -5, -3, -4})
│ │ │ +     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-5, -5, -5, -2, -4, -4, -3, -3, -4})
│ │ │  
│ │ │  o5 : OIResolution
│ │ │  
│ │ │  i6 : phi = C.dd_1
│ │ │  
│ │ │  o6 = Source: (e1, {5, 5}, {-3, -4}) Target: (e0, {3}, {-2})
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Free__O__I__Module__Map_sp__Vector__In__Width.out
│ │ │ @@ -8,15 +8,15 @@
│ │ │  
│ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(3,{2},1)+x_(2,2)*x_(2,1)*e_(3,{1,3},2);
│ │ │  
│ │ │  i5 : C = oiRes({b}, 2)
│ │ │  
│ │ │  o5 = 0: (e0, {3}, {-2})
│ │ │       1: (e1, {5, 5}, {-4, -3})
│ │ │ -     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-3, -4, -5, -4, -3, -2, -4, -5, -5})
│ │ │ +     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-3, -4, -4, -5, -4, -5, -5, -3, -2})
│ │ │  
│ │ │  o5 : OIResolution
│ │ │  
│ │ │  i6 : phi = C.dd_1
│ │ │  
│ │ │  o6 = Source: (e1, {5, 5}, {-4, -3}) Target: (e0, {3}, {-2})
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___O__I__Resolution.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │  
│ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │  
│ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │  
│ │ │  i5 : time C = oiRes({b}, 1)
│ │ │ - -- used 0.0819263s (cpu); 0.0819238s (thread); 0s (gc)
│ │ │ + -- used 0.0941109s (cpu); 0.0941099s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4, 4}, {-4, -4})
│ │ │  
│ │ │  o5 : OIResolution
│ │ │  
│ │ │  i6 : C.dd_0
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___O__I__Resolution_sp_us_sp__Z__Z.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │  
│ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │  
│ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │  
│ │ │  i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.215335s (cpu); 0.118692s (thread); 0s (gc)
│ │ │ + -- used 0.18889s (cpu); 0.111741s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : C_0
│ │ │  
│ │ │  o6 = Basis symbol: e0
│ │ │       Basis element widths: {2}
│ │ │       Degree shifts: {-2}
│ │ │       Polynomial OI-algebra: (2, x, QQ, RowUpColUp)
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Top__Nonminimal.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │  
│ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │  
│ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │  
│ │ │  i5 : time oiRes({b}, 2, TopNonminimal => true)
│ │ │ - -- used 0.476094s (cpu); 0.290539s (thread); 0s (gc)
│ │ │ + -- used 0.496292s (cpu); 0.309998s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4}, {-4})
│ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │  
│ │ │  o5 : OIResolution
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_describe__Full.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │  
│ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │  
│ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │  
│ │ │  i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.0881844s (cpu); 0.0881837s (thread); 0s (gc)
│ │ │ + -- used 0.228812s (cpu); 0.122847s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : describeFull C
│ │ │  
│ │ │  o6 = 0: Module: Basis symbol: e0
│ │ │                  Basis element widths: {2}
│ │ │                  Degree shifts: {-2}
│ │ │                  Polynomial OI-algebra: (2, x, QQ, RowUpColUp)
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_describe_lp__Free__O__I__Module__Map_rp.out
│ │ │ @@ -10,21 +10,18 @@
│ │ │  
│ │ │  i5 : C = oiRes({b}, 2);
│ │ │  
│ │ │  i6 : phi = C.dd_1;
│ │ │  
│ │ │  i7 : describe phi
│ │ │  
│ │ │ -o7 = Source: (e1, {5, 5}, {-3, -4}) Target: (e0, {3}, {-2})
│ │ │ -     Basis element images: {-x   e0              + x   e0              +
│ │ │ -                              2,2  5,{1, 3, 5},1    2,2  5,{1, 3, 4},1  
│ │ │ +o7 = Source: (e1, {5, 5}, {-4, -3}) Target: (e0, {3}, {-2})
│ │ │ +     Basis element images: {x   x   e0              - x   x   e0             
│ │ │ +                             2,3 1,1  5,{2, 4, 5},1    2,4 1,1  5,{2, 3, 5},1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     x   e0              - x   e0             , x   x   e0              -
│ │ │ -      2,3  5,{1, 2, 5},1    2,3  5,{1, 2, 4},1   2,3 1,1  5,{2, 4, 5},1  
│ │ │ +     - x   x   e0              + x   x   e0             , -x   e0        
│ │ │ +        2,3 1,2  5,{1, 4, 5},1    2,4 1,2  5,{1, 3, 5},1    2,2  5,{1, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     x   x   e0              - x   x   e0              + x   x   e0        
│ │ │ -      2,4 1,1  5,{2, 3, 5},1    2,3 1,2  5,{1, 4, 5},1    2,4 1,2  5,{1, 3,
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -         }
│ │ │ -     5},1
│ │ │ +          + x   e0              + x   e0              - x   e0             }
│ │ │ +     5},1    2,2  5,{1, 3, 4},1    2,3  5,{1, 2, 5},1    2,3  5,{1, 2, 4},1
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_describe_lp__O__I__Resolution_rp.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │  
│ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │  
│ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │  
│ │ │  i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.0801109s (cpu); 0.0801106s (thread); 0s (gc)
│ │ │ + -- used 0.0975082s (cpu); 0.0975089s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : describe C
│ │ │  
│ │ │  o6 = 0: Module: Basis symbol: e0
│ │ │                  Basis element widths: {2}
│ │ │                  Degree shifts: {-2}
│ │ │                  Polynomial OI-algebra: (2, x, QQ, RowUpColUp)
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_get__Schreyer__Map.out
│ │ │ @@ -17,21 +17,21 @@
│ │ │        2,3 2,1 1,2 3,{1},2
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : G' = oiSyz(G, d)
│ │ │  
│ │ │  o6 = {x   d           - x   d           + 1d             , x   d             
│ │ │ -       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   2,4 4,{1, 2, 3},2
│ │ │ +       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   1,2 4,{1, 3, 4},2
│ │ │       ------------------------------------------------------------------------
│ │ │ -     - x   d             , x   d              - x   d              -
│ │ │ -        2,3 4,{1, 2, 4},2   1,2 4,{1, 3, 4},2    1,1 4,{2, 3, 4},2  
│ │ │ +     - x   d              - x   d             , x   d              -
│ │ │ +        1,1 4,{2, 3, 4},2    1,3 4,{1, 2, 4},2   2,4 4,{1, 2, 3},2  
│ │ │       ------------------------------------------------------------------------
│ │ │       x   d             }
│ │ │ -      1,3 4,{1, 2, 4},2
│ │ │ +      2,3 4,{1, 2, 4},2
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : H = getFreeOIModule G'#0
│ │ │  
│ │ │  o7 = Basis symbol: d
│ │ │       Basis element widths: {2, 3}
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_image_lp__Free__O__I__Module__Map_rp.out
│ │ │ @@ -10,19 +10,19 @@
│ │ │  
│ │ │  i5 : C = oiRes({b}, 2);
│ │ │  
│ │ │  i6 : phi = C.dd_1;
│ │ │  
│ │ │  i7 : image phi
│ │ │  
│ │ │ -o7 = {x   x   e0              - x   x   e0              - x   x   e0        
│ │ │ -       2,3 1,1  5,{2, 4, 5},1    2,4 1,1  5,{2, 3, 5},1    2,3 1,2  5,{1, 4,
│ │ │ +o7 = {-x   e0              + x   e0              + x   e0              -
│ │ │ +        2,2  5,{1, 3, 5},1    2,2  5,{1, 3, 4},1    2,3  5,{1, 2, 5},1  
│ │ │       ------------------------------------------------------------------------
│ │ │ -          + x   x   e0             , -x   e0              + x   e0        
│ │ │ -     5},1    2,4 1,2  5,{1, 3, 5},1    2,2  5,{1, 3, 5},1    2,2  5,{1, 3,
│ │ │ +     x   e0             , x   x   e0              - x   x   e0              -
│ │ │ +      2,3  5,{1, 2, 4},1   2,3 1,1  5,{2, 4, 5},1    2,4 1,1  5,{2, 3, 5},1  
│ │ │       ------------------------------------------------------------------------
│ │ │ -          + x   e0              - x   e0             }
│ │ │ -     4},1    2,3  5,{1, 2, 5},1    2,3  5,{1, 2, 4},1
│ │ │ +     x   x   e0              + x   x   e0             }
│ │ │ +      2,3 1,2  5,{1, 4, 5},1    2,4 1,2  5,{1, 3, 5},1
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_is__Complex.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │  
│ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │  
│ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │  
│ │ │  i5 : time C = oiRes({b}, 2, TopNonminimal => true)
│ │ │ - -- used 0.309561s (cpu); 0.253624s (thread); 0s (gc)
│ │ │ + -- used 0.35275s (cpu); 0.268012s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4}, {-4})
│ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │  
│ │ │  o5 : OIResolution
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_is__O__I__G__B.out
│ │ │ @@ -15,15 +15,15 @@
│ │ │  i8 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
│ │ │  
│ │ │  i10 : isOIGB {b1, b2}
│ │ │  
│ │ │  o10 = false
│ │ │  
│ │ │  i11 : time B = oiGB {b1, b2}
│ │ │ - -- used 0.0233297s (cpu); 0.0233295s (thread); 0s (gc)
│ │ │ + -- used 0.0339962s (cpu); 0.0340002s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │          1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3 
│ │ │        -----------------------------------------------------------------------
│ │ │        x   x   x   e           - x   x   x   e          }
│ │ │         2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_minimize__O__I__G__B.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │  i5 : installGeneratorsInWidth(F, 3);
│ │ │  
│ │ │  i6 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);
│ │ │  
│ │ │  i8 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
│ │ │  
│ │ │  i10 : time B = oiGB {b1, b2}
│ │ │ - -- used 0.0253589s (cpu); 0.0253584s (thread); 0s (gc)
│ │ │ + -- used 0.0293353s (cpu); 0.0293339s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │          1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3 
│ │ │        -----------------------------------------------------------------------
│ │ │        x   x   x   e           - x   x   x   e          }
│ │ │         2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ │  
│ │ │ @@ -41,18 +41,17 @@
│ │ │             - x   x   e          }
│ │ │        3},3    2,1 1,2 3,{1, 3},3
│ │ │  
│ │ │  o13 : List
│ │ │  
│ │ │  i14 : minimizeOIGB C -- an element gets removed
│ │ │  
│ │ │ -                                                                        
│ │ │ -o14 = {x   x   e        + x   x   e          , x   x   x   e           -
│ │ │ -        1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3   2,3 2,2 1,1 3,{2, 3},3  
│ │ │ +                                                                   2
│ │ │ +o14 = {x   e        + x   e       , x   x   x   e           - x   x   e     
│ │ │ +        1,1 1,{1},1    2,1 1,{1},2   2,3 2,2 1,1 3,{2, 3},3    2,1 1,2 3,{1,
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2
│ │ │ -      x   x   e          , x   e        + x   e       }
│ │ │ -       2,1 1,2 3,{1, 3},3   1,1 1,{1},1    2,1 1,{1},2
│ │ │ +          , x   x   e        + x   x   e          }
│ │ │ +      3},3   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3
│ │ │  
│ │ │  o14 : List
│ │ │  
│ │ │  i15 :
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_net_lp__Free__O__I__Module__Map_rp.out
│ │ │ @@ -10,10 +10,10 @@
│ │ │  
│ │ │  i5 : C = oiRes({b}, 2);
│ │ │  
│ │ │  i6 : phi = C.dd_1;
│ │ │  
│ │ │  i7 : net phi
│ │ │  
│ │ │ -o7 = Source: (e1, {5, 5}, {-3, -4}) Target: (e0, {3}, {-2})
│ │ │ +o7 = Source: (e1, {5, 5}, {-4, -3}) Target: (e0, {3}, {-2})
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_net_lp__O__I__Resolution_rp.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │  
│ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │  
│ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │  
│ │ │  i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.218164s (cpu); 0.115767s (thread); 0s (gc)
│ │ │ + -- used 0.229739s (cpu); 0.121814s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : net C
│ │ │  
│ │ │  o6 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4, 4}, {-4, -4})
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_oi__G__B.out
│ │ │ @@ -9,15 +9,15 @@
│ │ │  i4 : installGeneratorsInWidth(F, 2);
│ │ │  
│ │ │  i5 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);
│ │ │  
│ │ │  i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
│ │ │  
│ │ │  i9 : time oiGB {b1, b2}
│ │ │ - -- used 0.0270916s (cpu); 0.0270906s (thread); 0s (gc)
│ │ │ + -- used 0.0345071s (cpu); 0.034507s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │         1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3 
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x   x   e           - x   x   x   e          }
│ │ │        2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_oi__Res.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │  
│ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │  
│ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │  
│ │ │  i5 : time oiRes({b}, 2, TopNonminimal => true)
│ │ │ - -- used 0.425811s (cpu); 0.2725s (thread); 0s (gc)
│ │ │ + -- used 0.482852s (cpu); 0.292221s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4}, {-4})
│ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │  
│ │ │  o5 : OIResolution
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_oi__Syz.out
│ │ │ @@ -17,18 +17,18 @@
│ │ │        2,3 2,1 1,2 3,{1},2
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : oiSyz(G, d)
│ │ │  
│ │ │  o6 = {x   d           - x   d           + 1d             , x   d             
│ │ │ -       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   2,4 4,{1, 2, 3},2
│ │ │ +       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   1,2 4,{1, 3, 4},2
│ │ │       ------------------------------------------------------------------------
│ │ │ -     - x   d             , x   d              - x   d              -
│ │ │ -        2,3 4,{1, 2, 4},2   1,2 4,{1, 3, 4},2    1,1 4,{2, 3, 4},2  
│ │ │ +     - x   d              - x   d             , x   d              -
│ │ │ +        1,1 4,{2, 3, 4},2    1,3 4,{1, 2, 4},2   2,4 4,{1, 2, 3},2  
│ │ │       ------------------------------------------------------------------------
│ │ │       x   d             }
│ │ │ -      1,3 4,{1, 2, 4},2
│ │ │ +      2,3 4,{1, 2, 4},2
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_reduce__O__I__G__B.out
│ │ │ @@ -9,15 +9,15 @@
│ │ │  i4 : installGeneratorsInWidth(F, 2);
│ │ │  
│ │ │  i5 : use F_1; b1 = x_(2,1)*e_(1,{1},2)+x_(1,1)*e_(1,{1},2);
│ │ │  
│ │ │  i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(1,2)*e_(2,{2},2);
│ │ │  
│ │ │  i9 : time B = oiGB({b1, b2}, Strategy => FastNonminimal)
│ │ │ - -- used 0.127003s (cpu); 0.127006s (thread); 0s (gc)
│ │ │ + -- used 0.188856s (cpu); 0.13352s (thread); 0s (gc)
│ │ │  
│ │ │                                                                         
│ │ │  o9 = {x   e        + x   e       , x   x   e        + x   x   e       ,
│ │ │         2,1 1,{1},2    1,1 1,{1},2   1,2 1,1 2,{2},1    2,2 1,2 2,{2},2 
│ │ │       ------------------------------------------------------------------------
│ │ │        2                  2
│ │ │       x   x   e        - x   x   e       }
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Free__O__I__Module__Map.html
│ │ │ @@ -79,15 +79,15 @@
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : C = oiRes({b}, 2)
│ │ │  
│ │ │  o5 = 0: (e0, {3}, {-2})
│ │ │       1: (e1, {5, 5}, {-3, -4})
│ │ │ -     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-2, -5, -4, -3, -5, -4, -5, -3, -4})
│ │ │ +     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-5, -5, -5, -2, -4, -4, -3, -3, -4})
│ │ │  
│ │ │  o5 : OIResolution
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : phi = C.dd_1
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,15 +17,15 @@
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,2}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 3);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(3,{2},1)+x_(2,2)*x_(2,1)*e_(3,{1,3},2);
│ │ │ │  i5 : C = oiRes({b}, 2)
│ │ │ │  
│ │ │ │  o5 = 0: (e0, {3}, {-2})
│ │ │ │       1: (e1, {5, 5}, {-3, -4})
│ │ │ │ -     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-2, -5, -4, -3, -5, -4, -5, -3, -4})
│ │ │ │ +     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-5, -5, -5, -2, -4, -4, -3, -3, -4})
│ │ │ │  
│ │ │ │  o5 : OIResolution
│ │ │ │  i6 : phi = C.dd_1
│ │ │ │  
│ │ │ │  o6 = Source: (e1, {5, 5}, {-3, -4}) Target: (e0, {3}, {-2})
│ │ │ │  
│ │ │ │  o6 : FreeOIModuleMap
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Free__O__I__Module__Map_sp__Vector__In__Width.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : C = oiRes({b}, 2)
│ │ │  
│ │ │  o5 = 0: (e0, {3}, {-2})
│ │ │       1: (e1, {5, 5}, {-4, -3})
│ │ │ -     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-3, -4, -5, -4, -3, -2, -4, -5, -5})
│ │ │ +     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-3, -4, -4, -5, -4, -5, -5, -3, -2})
│ │ │  
│ │ │  o5 : OIResolution
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : phi = C.dd_1
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,15 +19,15 @@
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,2}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 3);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(3,{2},1)+x_(2,2)*x_(2,1)*e_(3,{1,3},2);
│ │ │ │  i5 : C = oiRes({b}, 2)
│ │ │ │  
│ │ │ │  o5 = 0: (e0, {3}, {-2})
│ │ │ │       1: (e1, {5, 5}, {-4, -3})
│ │ │ │ -     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-3, -4, -5, -4, -3, -2, -4, -5, -5})
│ │ │ │ +     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-3, -4, -4, -5, -4, -5, -5, -3, -2})
│ │ │ │  
│ │ │ │  o5 : OIResolution
│ │ │ │  i6 : phi = C.dd_1
│ │ │ │  
│ │ │ │  o6 = Source: (e1, {5, 5}, {-4, -3}) Target: (e0, {3}, {-2})
│ │ │ │  
│ │ │ │  o6 : FreeOIModuleMap
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___O__I__Resolution.html
│ │ │ @@ -74,15 +74,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time C = oiRes({b}, 1)
│ │ │ - -- used 0.0819263s (cpu); 0.0819238s (thread); 0s (gc)
│ │ │ + -- used 0.0941109s (cpu); 0.0941099s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4, 4}, {-4, -4})
│ │ │  
│ │ │  o5 : OIResolution
    
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,15 +11,15 @@
│ │ │ │  complex, use _i_s_C_o_m_p_l_e_x. To get the $n$th differential in an OI-resolution C,
│ │ │ │  use C.dd_n.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time C = oiRes({b}, 1)
│ │ │ │ - -- used 0.0819263s (cpu); 0.0819238s (thread); 0s (gc)
│ │ │ │ + -- used 0.0941109s (cpu); 0.0941099s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │ │       1: (e1, {4, 4}, {-4, -4})
│ │ │ │  
│ │ │ │  o5 : OIResolution
│ │ │ │  i6 : C.dd_0
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___O__I__Resolution_sp_us_sp__Z__Z.html
│ │ │ @@ -92,15 +92,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.215335s (cpu); 0.118692s (thread); 0s (gc)
    
│ │ │ + -- used 0.18889s (cpu); 0.111741s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : C_0
│ │ │  
│ │ │  o6 = Basis symbol: e0
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,15 +16,15 @@
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Returns the free OI-module of $C$ in homological degree $n$.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time C = oiRes({b}, 1);
│ │ │ │ - -- used 0.215335s (cpu); 0.118692s (thread); 0s (gc)
│ │ │ │ + -- used 0.18889s (cpu); 0.111741s (thread); 0s (gc)
│ │ │ │  i6 : C_0
│ │ │ │  
│ │ │ │  o6 = Basis symbol: e0
│ │ │ │       Basis element widths: {2}
│ │ │ │       Degree shifts: {-2}
│ │ │ │       Polynomial OI-algebra: (2, x, QQ, RowUpColUp)
│ │ │ │       Monomial order: Lex
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Top__Nonminimal.html
│ │ │ @@ -74,15 +74,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time oiRes({b}, 2, TopNonminimal => true)
│ │ │ - -- used 0.476094s (cpu); 0.290539s (thread); 0s (gc)
│ │ │ + -- used 0.496292s (cpu); 0.309998s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4}, {-4})
│ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │  
│ │ │  o5 : OIResolution
    
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,15 +11,15 @@
│ │ │ │  homological degree $n-1$ to be minimized. Therefore, use TopNonminimal => true
│ │ │ │  for no minimization of the basis in degree $n-1$.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time oiRes({b}, 2, TopNonminimal => true)
│ │ │ │ - -- used 0.476094s (cpu); 0.290539s (thread); 0s (gc)
│ │ │ │ + -- used 0.496292s (cpu); 0.309998s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │ │       1: (e1, {4}, {-4})
│ │ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │ │  
│ │ │ │  o5 : OIResolution
│ │ │ │  ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd TTooppNNoonnmmiinniimmaall:: **********
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_describe__Full.html
│ │ │ @@ -90,15 +90,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.0881844s (cpu); 0.0881837s (thread); 0s (gc)
    
│ │ │ + -- used 0.228812s (cpu); 0.122847s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : describeFull C
│ │ │  
│ │ │  o6 = 0: Module: Basis symbol: e0
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,15 +14,15 @@
│ │ │ │  Displays the free OI-modules and describes the differentials of an OI-
│ │ │ │  resolution.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time C = oiRes({b}, 1);
│ │ │ │ - -- used 0.0881844s (cpu); 0.0881837s (thread); 0s (gc)
│ │ │ │ + -- used 0.228812s (cpu); 0.122847s (thread); 0s (gc)
│ │ │ │  i6 : describeFull C
│ │ │ │  
│ │ │ │  o6 = 0: Module: Basis symbol: e0
│ │ │ │                  Basis element widths: {2}
│ │ │ │                  Degree shifts: {-2}
│ │ │ │                  Polynomial OI-algebra: (2, x, QQ, RowUpColUp)
│ │ │ │                  Monomial order: Lex
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_describe_lp__Free__O__I__Module__Map_rp.html
│ │ │ @@ -102,26 +102,23 @@
│ │ │                
    i6 : phi = C.dd_1;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : describe phi
│ │ │  
│ │ │ -o7 = Source: (e1, {5, 5}, {-3, -4}) Target: (e0, {3}, {-2})
│ │ │ -     Basis element images: {-x   e0              + x   e0              +
│ │ │ -                              2,2  5,{1, 3, 5},1    2,2  5,{1, 3, 4},1  
│ │ │ +o7 = Source: (e1, {5, 5}, {-4, -3}) Target: (e0, {3}, {-2})
│ │ │ +     Basis element images: {x   x   e0              - x   x   e0             
│ │ │ +                             2,3 1,1  5,{2, 4, 5},1    2,4 1,1  5,{2, 3, 5},1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     x   e0              - x   e0             , x   x   e0              -
│ │ │ -      2,3  5,{1, 2, 5},1    2,3  5,{1, 2, 4},1   2,3 1,1  5,{2, 4, 5},1  
│ │ │ +     - x   x   e0              + x   x   e0             , -x   e0        
│ │ │ +        2,3 1,2  5,{1, 4, 5},1    2,4 1,2  5,{1, 3, 5},1    2,2  5,{1, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     x   x   e0              - x   x   e0              + x   x   e0        
│ │ │ -      2,4 1,1  5,{2, 3, 5},1    2,3 1,2  5,{1, 4, 5},1    2,4 1,2  5,{1, 3,
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -         }
│ │ │ -     5},1
    
│ │ │ +          + x   e0              + x   e0              - x   e0             }
│ │ │ +     5},1    2,2  5,{1, 3, 4},1    2,3  5,{1, 2, 5},1    2,3  5,{1, 2, 4},1
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    
│ │ │            
    Ways to use this method:
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,24 +18,21 @@
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,2}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 3);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(3,{2},1)+x_(2,2)*x_(2,1)*e_(3,{1,3},2);
│ │ │ │  i5 : C = oiRes({b}, 2);
│ │ │ │  i6 : phi = C.dd_1;
│ │ │ │  i7 : describe phi
│ │ │ │  
│ │ │ │ -o7 = Source: (e1, {5, 5}, {-3, -4}) Target: (e0, {3}, {-2})
│ │ │ │ -     Basis element images: {-x   e0              + x   e0              +
│ │ │ │ -                              2,2  5,{1, 3, 5},1    2,2  5,{1, 3, 4},1
│ │ │ │ +o7 = Source: (e1, {5, 5}, {-4, -3}) Target: (e0, {3}, {-2})
│ │ │ │ +     Basis element images: {x   x   e0              - x   x   e0
│ │ │ │ +                             2,3 1,1  5,{2, 4, 5},1    2,4 1,1  5,{2, 3, 5},1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     x   e0              - x   e0             , x   x   e0              -
│ │ │ │ -      2,3  5,{1, 2, 5},1    2,3  5,{1, 2, 4},1   2,3 1,1  5,{2, 4, 5},1
│ │ │ │ +     - x   x   e0              + x   x   e0             , -x   e0
│ │ │ │ +        2,3 1,2  5,{1, 4, 5},1    2,4 1,2  5,{1, 3, 5},1    2,2  5,{1, 3,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     x   x   e0              - x   x   e0              + x   x   e0
│ │ │ │ -      2,4 1,1  5,{2, 3, 5},1    2,3 1,2  5,{1, 4, 5},1    2,4 1,2  5,{1, 3,
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -         }
│ │ │ │ -     5},1
│ │ │ │ +          + x   e0              + x   e0              - x   e0             }
│ │ │ │ +     5},1    2,2  5,{1, 3, 4},1    2,3  5,{1, 2, 5},1    2,3  5,{1, 2, 4},1
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _d_e_s_c_r_i_b_e_(_F_r_e_e_O_I_M_o_d_u_l_e_M_a_p_) -- display a free OI-module map
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/OIGroebnerBases.m2:1979:0.
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_describe_lp__O__I__Resolution_rp.html
│ │ │ @@ -91,15 +91,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.0801109s (cpu); 0.0801106s (thread); 0s (gc)
    
│ │ │ + -- used 0.0975082s (cpu); 0.0975089s (thread); 0s (gc)    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : describe C
│ │ │  
│ │ │  o6 = 0: Module: Basis symbol: e0
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,15 +14,15 @@
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Displays the free OI-modules and differentials of an OI-resolution.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time C = oiRes({b}, 1);
│ │ │ │ - -- used 0.0801109s (cpu); 0.0801106s (thread); 0s (gc)
│ │ │ │ + -- used 0.0975082s (cpu); 0.0975089s (thread); 0s (gc)
│ │ │ │  i6 : describe C
│ │ │ │  
│ │ │ │  o6 = 0: Module: Basis symbol: e0
│ │ │ │                  Basis element widths: {2}
│ │ │ │                  Degree shifts: {-2}
│ │ │ │                  Polynomial OI-algebra: (2, x, QQ, RowUpColUp)
│ │ │ │                  Monomial order: Lex
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_get__Schreyer__Map.html
│ │ │ @@ -105,21 +105,21 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : G' = oiSyz(G, d)
│ │ │  
│ │ │  o6 = {x   d           - x   d           + 1d             , x   d             
│ │ │ -       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   2,4 4,{1, 2, 3},2
│ │ │ +       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   1,2 4,{1, 3, 4},2
│ │ │       ------------------------------------------------------------------------
│ │ │ -     - x   d             , x   d              - x   d              -
│ │ │ -        2,3 4,{1, 2, 4},2   1,2 4,{1, 3, 4},2    1,1 4,{2, 3, 4},2  
│ │ │ +     - x   d              - x   d             , x   d              -
│ │ │ +        1,1 4,{2, 3, 4},2    1,3 4,{1, 2, 4},2   2,4 4,{1, 2, 3},2  
│ │ │       ------------------------------------------------------------------------
│ │ │       x   d             }
│ │ │ -      1,3 4,{1, 2, 4},2
│ │ │ +      2,3 4,{1, 2, 4},2
│ │ │  
│ │ │  o6 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : H = getFreeOIModule G'#0
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,21 +29,21 @@
│ │ │ │       x   x   x   e       }
│ │ │ │        2,3 2,1 1,2 3,{1},2
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : G' = oiSyz(G, d)
│ │ │ │  
│ │ │ │  o6 = {x   d           - x   d           + 1d             , x   d
│ │ │ │ -       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   2,4 4,{1, 2, 3},2
│ │ │ │ +       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   1,2 4,{1, 3, 4},2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     - x   d             , x   d              - x   d              -
│ │ │ │ -        2,3 4,{1, 2, 4},2   1,2 4,{1, 3, 4},2    1,1 4,{2, 3, 4},2
│ │ │ │ +     - x   d              - x   d             , x   d              -
│ │ │ │ +        1,1 4,{2, 3, 4},2    1,3 4,{1, 2, 4},2   2,4 4,{1, 2, 3},2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       x   d             }
│ │ │ │ -      1,3 4,{1, 2, 4},2
│ │ │ │ +      2,3 4,{1, 2, 4},2
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : H = getFreeOIModule G'#0
│ │ │ │  
│ │ │ │  o7 = Basis symbol: d
│ │ │ │       Basis element widths: {2, 3}
│ │ │ │       Degree shifts: {-2, -3}
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_image_lp__Free__O__I__Module__Map_rp.html
│ │ │ @@ -102,22 +102,22 @@
│ │ │                
    i6 : phi = C.dd_1;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : image phi
│ │ │  
│ │ │ -o7 = {x   x   e0              - x   x   e0              - x   x   e0        
│ │ │ -       2,3 1,1  5,{2, 4, 5},1    2,4 1,1  5,{2, 3, 5},1    2,3 1,2  5,{1, 4,
│ │ │ +o7 = {-x   e0              + x   e0              + x   e0              -
│ │ │ +        2,2  5,{1, 3, 5},1    2,2  5,{1, 3, 4},1    2,3  5,{1, 2, 5},1  
│ │ │       ------------------------------------------------------------------------
│ │ │ -          + x   x   e0             , -x   e0              + x   e0        
│ │ │ -     5},1    2,4 1,2  5,{1, 3, 5},1    2,2  5,{1, 3, 5},1    2,2  5,{1, 3,
│ │ │ +     x   e0             , x   x   e0              - x   x   e0              -
│ │ │ +      2,3  5,{1, 2, 4},1   2,3 1,1  5,{2, 4, 5},1    2,4 1,1  5,{2, 3, 5},1  
│ │ │       ------------------------------------------------------------------------
│ │ │ -          + x   e0              - x   e0             }
│ │ │ -     4},1    2,3  5,{1, 2, 5},1    2,3  5,{1, 2, 4},1
│ │ │ +     x   x   e0              + x   x   e0             }
│ │ │ +      2,3 1,2  5,{1, 4, 5},1    2,4 1,2  5,{1, 3, 5},1
│ │ │  
│ │ │  o7 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,22 +18,22 @@
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,2}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 3);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(3,{2},1)+x_(2,2)*x_(2,1)*e_(3,{1,3},2);
│ │ │ │  i5 : C = oiRes({b}, 2);
│ │ │ │  i6 : phi = C.dd_1;
│ │ │ │  i7 : image phi
│ │ │ │  
│ │ │ │ -o7 = {x   x   e0              - x   x   e0              - x   x   e0
│ │ │ │ -       2,3 1,1  5,{2, 4, 5},1    2,4 1,1  5,{2, 3, 5},1    2,3 1,2  5,{1, 4,
│ │ │ │ +o7 = {-x   e0              + x   e0              + x   e0              -
│ │ │ │ +        2,2  5,{1, 3, 5},1    2,2  5,{1, 3, 4},1    2,3  5,{1, 2, 5},1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -          + x   x   e0             , -x   e0              + x   e0
│ │ │ │ -     5},1    2,4 1,2  5,{1, 3, 5},1    2,2  5,{1, 3, 5},1    2,2  5,{1, 3,
│ │ │ │ +     x   e0             , x   x   e0              - x   x   e0              -
│ │ │ │ +      2,3  5,{1, 2, 4},1   2,3 1,1  5,{2, 4, 5},1    2,4 1,1  5,{2, 3, 5},1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -          + x   e0              - x   e0             }
│ │ │ │ -     4},1    2,3  5,{1, 2, 5},1    2,3  5,{1, 2, 4},1
│ │ │ │ +     x   x   e0              + x   x   e0             }
│ │ │ │ +      2,3 1,2  5,{1, 4, 5},1    2,4 1,2  5,{1, 3, 5},1
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _i_m_a_g_e_(_F_r_e_e_O_I_M_o_d_u_l_e_M_a_p_) -- get the basis element images of a free OI-
│ │ │ │        module map
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_is__Complex.html
│ │ │ @@ -94,15 +94,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time C = oiRes({b}, 2, TopNonminimal => true)
│ │ │ - -- used 0.309561s (cpu); 0.253624s (thread); 0s (gc)
│ │ │ + -- used 0.35275s (cpu); 0.268012s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4}, {-4})
│ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │  
│ │ │  o5 : OIResolution
    
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,15 +17,15 @@
│ │ │ │  option must be either true or false, depending on whether one wants debug
│ │ │ │  information printed.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time C = oiRes({b}, 2, TopNonminimal => true)
│ │ │ │ - -- used 0.309561s (cpu); 0.253624s (thread); 0s (gc)
│ │ │ │ + -- used 0.35275s (cpu); 0.268012s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │ │       1: (e1, {4}, {-4})
│ │ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │ │  
│ │ │ │  o5 : OIResolution
│ │ │ │  i6 : isComplex C
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_is__O__I__G__B.html
│ │ │ @@ -116,15 +116,15 @@
│ │ │  
│ │ │  o10 = false    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : time B = oiGB {b1, b2}
│ │ │ - -- used 0.0233297s (cpu); 0.0233295s (thread); 0s (gc)
│ │ │ + -- used 0.0339962s (cpu); 0.0340002s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │          1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3 
│ │ │        -----------------------------------------------------------------------
│ │ │        x   x   x   e           - x   x   x   e          }
│ │ │         2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,15 +24,15 @@
│ │ │ │  i5 : installGeneratorsInWidth(F, 3);
│ │ │ │  i6 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);
│ │ │ │  i8 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
│ │ │ │  i10 : isOIGB {b1, b2}
│ │ │ │  
│ │ │ │  o10 = false
│ │ │ │  i11 : time B = oiGB {b1, b2}
│ │ │ │ - -- used 0.0233297s (cpu); 0.0233295s (thread); 0s (gc)
│ │ │ │ + -- used 0.0339962s (cpu); 0.0340002s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │ │          1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        x   x   x   e           - x   x   x   e          }
│ │ │ │         2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_minimize__O__I__G__B.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │              
│ │ │                
    i8 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : time B = oiGB {b1, b2}
│ │ │ - -- used 0.0253589s (cpu); 0.0253584s (thread); 0s (gc)
│ │ │ + -- used 0.0293353s (cpu); 0.0293339s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │          1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3 
│ │ │        -----------------------------------------------------------------------
│ │ │        x   x   x   e           - x   x   x   e          }
│ │ │         2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ │  
│ │ │ @@ -148,21 +148,20 @@
│ │ │  o13 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : minimizeOIGB C -- an element gets removed
│ │ │  
│ │ │ -                                                                        
│ │ │ -o14 = {x   x   e        + x   x   e          , x   x   x   e           -
│ │ │ -        1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3   2,3 2,2 1,1 3,{2, 3},3  
│ │ │ +                                                                   2
│ │ │ +o14 = {x   e        + x   e       , x   x   x   e           - x   x   e     
│ │ │ +        1,1 1,{1},1    2,1 1,{1},2   2,3 2,2 1,1 3,{2, 3},3    2,1 1,2 3,{1,
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2
│ │ │ -      x   x   e          , x   e        + x   e       }
│ │ │ -       2,1 1,2 3,{1, 3},3   1,1 1,{1},1    2,1 1,{1},2
│ │ │ +          , x   x   e        + x   x   e          }
│ │ │ +      3},3   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3
│ │ │  
│ │ │  o14 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,15 +21,15 @@
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1,2}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 1);
│ │ │ │  i4 : installGeneratorsInWidth(F, 2);
│ │ │ │  i5 : installGeneratorsInWidth(F, 3);
│ │ │ │  i6 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);
│ │ │ │  i8 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
│ │ │ │  i10 : time B = oiGB {b1, b2}
│ │ │ │ - -- used 0.0253589s (cpu); 0.0253584s (thread); 0s (gc)
│ │ │ │ + -- used 0.0293353s (cpu); 0.0293339s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │ │          1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        x   x   x   e           - x   x   x   e          }
│ │ │ │         2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ │ │  
│ │ │ │ @@ -49,21 +49,20 @@
│ │ │ │                    2
│ │ │ │             - x   x   e          }
│ │ │ │        3},3    2,1 1,2 3,{1, 3},3
│ │ │ │  
│ │ │ │  o13 : List
│ │ │ │  i14 : minimizeOIGB C -- an element gets removed
│ │ │ │  
│ │ │ │ -
│ │ │ │ -o14 = {x   x   e        + x   x   e          , x   x   x   e           -
│ │ │ │ -        1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3   2,3 2,2 1,1 3,{2, 3},3
│ │ │ │ +                                                                   2
│ │ │ │ +o14 = {x   e        + x   e       , x   x   x   e           - x   x   e
│ │ │ │ +        1,1 1,{1},1    2,1 1,{1},2   2,3 2,2 1,1 3,{2, 3},3    2,1 1,2 3,{1,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -           2
│ │ │ │ -      x   x   e          , x   e        + x   e       }
│ │ │ │ -       2,1 1,2 3,{1, 3},3   1,1 1,{1},1    2,1 1,{1},2
│ │ │ │ +          , x   x   e        + x   x   e          }
│ │ │ │ +      3},3   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3
│ │ │ │  
│ │ │ │  o14 : List
│ │ │ │  ********** WWaayyss ttoo uussee mmiinniimmiizzeeOOIIGGBB:: **********
│ │ │ │      * minimizeOIGB(List)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _m_i_n_i_m_i_z_e_O_I_G_B is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s.
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__Free__O__I__Module__Map_rp.html
│ │ │ @@ -102,15 +102,15 @@
│ │ │                
    i6 : phi = C.dd_1;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : net phi
│ │ │  
│ │ │ -o7 = Source: (e1, {5, 5}, {-3, -4}) Target: (e0, {3}, {-2})
    
│ │ │ +o7 = Source: (e1, {5, 5}, {-4, -3}) Target: (e0, {3}, {-2})    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    
│ │ │            
    Ways to use this method:
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,13 +18,13 @@
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,2}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 3);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(3,{2},1)+x_(2,2)*x_(2,1)*e_(3,{1,3},2);
│ │ │ │  i5 : C = oiRes({b}, 2);
│ │ │ │  i6 : phi = C.dd_1;
│ │ │ │  i7 : net phi
│ │ │ │  
│ │ │ │ -o7 = Source: (e1, {5, 5}, {-3, -4}) Target: (e0, {3}, {-2})
│ │ │ │ +o7 = Source: (e1, {5, 5}, {-4, -3}) Target: (e0, {3}, {-2})
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _n_e_t_(_F_r_e_e_O_I_M_o_d_u_l_e_M_a_p_) -- display a free OI-module map source and target
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/OIGroebnerBases.m2:1931:0.
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__O__I__Resolution_rp.html
│ │ │ @@ -91,15 +91,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.218164s (cpu); 0.115767s (thread); 0s (gc)
    
│ │ │ + -- used 0.229739s (cpu); 0.121814s (thread); 0s (gc)    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : net C
│ │ │  
│ │ │  o6 = 0: (e0, {2}, {-2})
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │  Displays the basis element widths and degree shifts of the free OI-modules in
│ │ │ │  an OI-resolution.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time C = oiRes({b}, 1);
│ │ │ │ - -- used 0.218164s (cpu); 0.115767s (thread); 0s (gc)
│ │ │ │ + -- used 0.229739s (cpu); 0.121814s (thread); 0s (gc)
│ │ │ │  i6 : net C
│ │ │ │  
│ │ │ │  o6 = 0: (e0, {2}, {-2})
│ │ │ │       1: (e1, {4, 4}, {-4, -4})
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _n_e_t_(_O_I_R_e_s_o_l_u_t_i_o_n_) -- display an OI-resolution
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__G__B.html
│ │ │ @@ -111,15 +111,15 @@
│ │ │              
│ │ │                
    i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : time oiGB {b1, b2}
│ │ │ - -- used 0.0270916s (cpu); 0.0270906s (thread); 0s (gc)
│ │ │ + -- used 0.0345071s (cpu); 0.034507s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │         1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3 
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x   x   e           - x   x   x   e          }
│ │ │        2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,15 +28,15 @@
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1,2}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 1);
│ │ │ │  i4 : installGeneratorsInWidth(F, 2);
│ │ │ │  i5 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);
│ │ │ │  i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
│ │ │ │  i9 : time oiGB {b1, b2}
│ │ │ │ - -- used 0.0270916s (cpu); 0.0270906s (thread); 0s (gc)
│ │ │ │ + -- used 0.0345071s (cpu); 0.034507s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │ │         1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       x   x   x   e           - x   x   x   e          }
│ │ │ │        2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__Res.html
│ │ │ @@ -106,15 +106,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time oiRes({b}, 2, TopNonminimal => true)
│ │ │ - -- used 0.425811s (cpu); 0.2725s (thread); 0s (gc)
│ │ │ + -- used 0.482852s (cpu); 0.292221s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4}, {-4})
│ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │  
│ │ │  o5 : OIResolution
    
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,15 +33,15 @@
│ │ │ │  Therefore, use TopNonminimal => true for no minimization of the basis in degree
│ │ │ │  $n-1$.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time oiRes({b}, 2, TopNonminimal => true)
│ │ │ │ - -- used 0.425811s (cpu); 0.2725s (thread); 0s (gc)
│ │ │ │ + -- used 0.482852s (cpu); 0.292221s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │ │       1: (e1, {4}, {-4})
│ │ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │ │  
│ │ │ │  o5 : OIResolution
│ │ │ │  ********** WWaayyss ttoo uussee ooiiRReess:: **********
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__Syz.html
│ │ │ @@ -119,21 +119,21 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : oiSyz(G, d)
│ │ │  
│ │ │  o6 = {x   d           - x   d           + 1d             , x   d             
│ │ │ -       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   2,4 4,{1, 2, 3},2
│ │ │ +       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   1,2 4,{1, 3, 4},2
│ │ │       ------------------------------------------------------------------------
│ │ │ -     - x   d             , x   d              - x   d              -
│ │ │ -        2,3 4,{1, 2, 4},2   1,2 4,{1, 3, 4},2    1,1 4,{2, 3, 4},2  
│ │ │ +     - x   d              - x   d             , x   d              -
│ │ │ +        1,1 4,{2, 3, 4},2    1,3 4,{1, 2, 4},2   2,4 4,{1, 2, 3},2  
│ │ │       ------------------------------------------------------------------------
│ │ │       x   d             }
│ │ │ -      1,3 4,{1, 2, 4},2
│ │ │ +      2,3 4,{1, 2, 4},2
│ │ │  
│ │ │  o6 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    References:
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,21 +48,21 @@
│ │ │ │       x   x   x   e       }
│ │ │ │        2,3 2,1 1,2 3,{1},2
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : oiSyz(G, d)
│ │ │ │  
│ │ │ │  o6 = {x   d           - x   d           + 1d             , x   d
│ │ │ │ -       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   2,4 4,{1, 2, 3},2
│ │ │ │ +       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   1,2 4,{1, 3, 4},2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     - x   d             , x   d              - x   d              -
│ │ │ │ -        2,3 4,{1, 2, 4},2   1,2 4,{1, 3, 4},2    1,1 4,{2, 3, 4},2
│ │ │ │ +     - x   d              - x   d             , x   d              -
│ │ │ │ +        1,1 4,{2, 3, 4},2    1,3 4,{1, 2, 4},2   2,4 4,{1, 2, 3},2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       x   d             }
│ │ │ │ -      1,3 4,{1, 2, 4},2
│ │ │ │ +      2,3 4,{1, 2, 4},2
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  RReeffeerreenncceess::
│ │ │ │  [1] M. Morrow and U. Nagel, Computing Gröbner Bases and Free Resolutions of
│ │ │ │  OI-Modules, Preprint, arXiv:2303.06725, 2023.
│ │ │ │  ********** WWaayyss ttoo uussee ooiiSSyyzz:: **********
│ │ │ │      * oiSyz(List,Symbol)
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_reduce__O__I__G__B.html
│ │ │ @@ -104,15 +104,15 @@
│ │ │              
│ │ │                
    i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(1,2)*e_(2,{2},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : time B = oiGB({b1, b2}, Strategy => FastNonminimal)
│ │ │ - -- used 0.127003s (cpu); 0.127006s (thread); 0s (gc)
│ │ │ + -- used 0.188856s (cpu); 0.13352s (thread); 0s (gc)
│ │ │  
│ │ │                                                                         
│ │ │  o9 = {x   e        + x   e       , x   x   e        + x   x   e       ,
│ │ │         2,1 1,{1},2    1,1 1,{1},2   1,2 1,1 2,{2},1    2,2 1,2 2,{2},2 
│ │ │       ------------------------------------------------------------------------
│ │ │        2                  2
│ │ │       x   x   e        - x   x   e       }
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1,2}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 1);
│ │ │ │  i4 : installGeneratorsInWidth(F, 2);
│ │ │ │  i5 : use F_1; b1 = x_(2,1)*e_(1,{1},2)+x_(1,1)*e_(1,{1},2);
│ │ │ │  i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(1,2)*e_(2,{2},2);
│ │ │ │  i9 : time B = oiGB({b1, b2}, Strategy => FastNonminimal)
│ │ │ │ - -- used 0.127003s (cpu); 0.127006s (thread); 0s (gc)
│ │ │ │ + -- used 0.188856s (cpu); 0.13352s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  
│ │ │ │  o9 = {x   e        + x   e       , x   x   e        + x   x   e       ,
│ │ │ │         2,1 1,{1},2    1,1 1,{1},2   1,2 1,1 2,{2},1    2,2 1,2 2,{2},2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        2                  2
│ │ │ │       x   x   e        - x   x   e       }
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___Fast__Nonminimal.out
│ │ │ @@ -9,25 +9,25 @@
│ │ │  i2 : S = ring I
│ │ │  
│ │ │  o2 = S
│ │ │  
│ │ │  o2 : PolynomialRing
│ │ │  
│ │ │  i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ - -- 2.28836s elapsed
│ │ │ + -- 2.51763s elapsed
│ │ │  
│ │ │        1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o3 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S   <-- 0
│ │ │                                                                                                           
│ │ │       0      1       2        3        4         5         6         7         8        9        10      11
│ │ │  
│ │ │  o3 : ChainComplex
│ │ │  
│ │ │  i4 : elapsedTime C1 = res ideal(I_*)
│ │ │ - -- 1.58344s elapsed
│ │ │ + -- 1.35367s elapsed
│ │ │  
│ │ │        1      35      140      385      819      1080      819      385      140      35      1
│ │ │  o4 = S  <-- S   <-- S    <-- S    <-- S    <-- S     <-- S    <-- S    <-- S    <-- S   <-- S  <-- 0
│ │ │                                                                                                      
│ │ │       0      1       2        3        4        5         6        7        8        9       10     11
│ │ │  
│ │ │  o4 : ChainComplex
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_betti_lp..._cm__Minimize_eq_gt..._rp.out
│ │ │ @@ -9,15 +9,15 @@
│ │ │  i2 : S = ring I
│ │ │  
│ │ │  o2 = S
│ │ │  
│ │ │  o2 : PolynomialRing
│ │ │  
│ │ │  i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ - -- 1.96896s elapsed
│ │ │ + -- 2.57819s 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 0.912289s (cpu); 0.749709s (thread); 0s (gc)
│ │ │ - -- used 0.493728s (cpu); 0.422643s (thread); 0s (gc)
│ │ │ + -- used 1.14031s (cpu); 0.992318s (thread); 0s (gc)
│ │ │ + -- used 0.653268s (cpu); 0.572673s (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)
│ │ │ - -- 2.28836s elapsed
│ │ │ + -- 2.51763s elapsed
│ │ │  
│ │ │        1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o3 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S   <-- 0
│ │ │                                                                                                           
│ │ │       0      1       2        3        4         5         6         7         8        9        10      11
│ │ │  
│ │ │  o3 : ChainComplex
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : elapsedTime C1 = res ideal(I_*)
│ │ │ - -- 1.58344s elapsed
│ │ │ + -- 1.35367s elapsed
│ │ │  
│ │ │        1      35      140      385      819      1080      819      385      140      35      1
│ │ │  o4 = S  <-- S   <-- S    <-- S    <-- S    <-- S     <-- S    <-- S    <-- S    <-- S   <-- S  <-- 0
│ │ │                                                                                                      
│ │ │       0      1       2        3        4        5         6        7        8        9       10     11
│ │ │  
│ │ │  o4 : ChainComplex
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,28 +29,28 @@
│ │ │ │  0,5   1,5   2,5   3,5   4,5   0,6   1,6   2,6   3,6   4,6   5,6
│ │ │ │  i2 : S = ring I
│ │ │ │  
│ │ │ │  o2 = S
│ │ │ │  
│ │ │ │  o2 : PolynomialRing
│ │ │ │  i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ │ - -- 2.28836s elapsed
│ │ │ │ + -- 2.51763s elapsed
│ │ │ │  
│ │ │ │        1      35      241      841      1781      2464      2294      1432
│ │ │ │  576      135      14
│ │ │ │  o3 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S
│ │ │ │  <-- S    <-- S   <-- 0
│ │ │ │  
│ │ │ │  
│ │ │ │       0      1       2        3        4         5         6         7         8
│ │ │ │  9        10      11
│ │ │ │  
│ │ │ │  o3 : ChainComplex
│ │ │ │  i4 : elapsedTime C1 = res ideal(I_*)
│ │ │ │ - -- 1.58344s elapsed
│ │ │ │ + -- 1.35367s elapsed
│ │ │ │  
│ │ │ │        1      35      140      385      819      1080      819      385      140
│ │ │ │  35      1
│ │ │ │  o4 = S  <-- S   <-- S    <-- S    <-- S    <-- S     <-- S    <-- S    <-- S
│ │ │ │  <-- S   <-- S  <-- 0
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/html/_betti_lp..._cm__Minimize_eq_gt..._rp.html
│ │ │ @@ -88,15 +88,15 @@
│ │ │  
│ │ │  o2 : PolynomialRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ - -- 1.96896s elapsed
│ │ │ + -- 2.57819s elapsed
│ │ │  
│ │ │        1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o3 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S   <-- 0
│ │ │                                                                                                           
│ │ │       0      1       2        3        4         5         6         7         8        9        10      11
│ │ │  
│ │ │  o3 : ChainComplex
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │  0,5   1,5   2,5   3,5   4,5   0,6   1,6   2,6   3,6   4,6   5,6
│ │ │ │  i2 : S = ring I
│ │ │ │  
│ │ │ │  o2 = S
│ │ │ │  
│ │ │ │  o2 : PolynomialRing
│ │ │ │  i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ │ - -- 1.96896s elapsed
│ │ │ │ + -- 2.57819s 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 0.912289s (cpu); 0.749709s (thread); 0s (gc)
│ │ │ - -- used 0.493728s (cpu); 0.422643s (thread); 0s (gc)
│ │ │ + -- used 1.14031s (cpu); 0.992318s (thread); 0s (gc)
│ │ │ + -- used 0.653268s (cpu); 0.572673s (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 0.912289s (cpu); 0.749709s (thread); 0s (gc)
│ │ │ │ - -- used 0.493728s (cpu); 0.422643s (thread); 0s (gc)
│ │ │ │ + -- used 1.14031s (cpu); 0.992318s (thread); 0s (gc)
│ │ │ │ + -- used 0.653268s (cpu); 0.572673s (thread); 0s (gc)
│ │ │ │  -- computation started:
│ │ │ │  -- computation interrupted
│ │ │ │  -- continuing the computation
│ │ │ │  -- computation complete
│ │ │ │   4      11      89      122      40
│ │ │ │  R  <-- R   <-- R   <-- R    <-- R   <-- 0
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.out
│ │ │ @@ -182,25 +182,25 @@
│ │ │  o15 = 4
│ │ │  
│ │ │  i16 : for G in Gs list (
│ │ │            IG = oscQuadrics(G, R);
│ │ │            elapsedTime comps := decompose IG;
│ │ │            {comps/codim, comps/degree}
│ │ │            );
│ │ │ - -- .280854s elapsed
│ │ │ - -- .263174s elapsed
│ │ │ - -- .501439s elapsed
│ │ │ - -- .231638s elapsed
│ │ │ - -- .342923s elapsed
│ │ │ - -- .34852s elapsed
│ │ │ - -- .619954s elapsed
│ │ │ - -- .456734s elapsed
│ │ │ - -- .648967s elapsed
│ │ │ - -- .367209s elapsed
│ │ │ - -- .250328s elapsed
│ │ │ + -- .23795s elapsed
│ │ │ + -- .277177s elapsed
│ │ │ + -- .477459s elapsed
│ │ │ + -- .252176s elapsed
│ │ │ + -- .259965s elapsed
│ │ │ + -- .309722s elapsed
│ │ │ + -- .534615s elapsed
│ │ │ + -- .45729s elapsed
│ │ │ + -- .476608s elapsed
│ │ │ + -- .311675s elapsed
│ │ │ + -- .215434s elapsed
│ │ │  
│ │ │  i17 : netList oo
│ │ │  
│ │ │        +---------------+---------------+
│ │ │  o17 = |{3, 4, 4}      |{2, 3, 5}      |
│ │ │        +---------------+---------------+
│ │ │        |{3, 4, 4}      |{2, 3, 5}      |
│ │ │ @@ -242,75 +242,75 @@
│ │ │  o22 = 15
│ │ │  
│ │ │  i23 : allcomps = for G in Gs list (
│ │ │            IG = oscQuadrics(G, R);
│ │ │            elapsedTime comps := decompose IG;
│ │ │            {comps/codim, comps/degree}
│ │ │            );
│ │ │ - -- .536601s elapsed
│ │ │ - -- .816725s elapsed
│ │ │ - -- 1.00632s elapsed
│ │ │ - -- 1.37164s elapsed
│ │ │ - -- .718355s elapsed
│ │ │ - -- 1.00236s elapsed
│ │ │ - -- 1.18305s elapsed
│ │ │ - -- 1.31195s elapsed
│ │ │ - -- .727806s elapsed
│ │ │ - -- .774958s elapsed
│ │ │ - -- .333808s elapsed
│ │ │ - -- .5035s elapsed
│ │ │ - -- .490585s elapsed
│ │ │ - -- .637754s elapsed
│ │ │ - -- .880246s elapsed
│ │ │ - -- 1.2462s elapsed
│ │ │ - -- .940742s elapsed
│ │ │ - -- 1.01084s elapsed
│ │ │ - -- 1.44951s elapsed
│ │ │ - -- 1.31812s elapsed
│ │ │ - -- 1.00162s elapsed
│ │ │ - -- 1.3528s elapsed
│ │ │ - -- 1.90187s elapsed
│ │ │ - -- 1.44103s elapsed
│ │ │ - -- .432654s elapsed
│ │ │ - -- .573037s elapsed
│ │ │ - -- 1.40316s elapsed
│ │ │ - -- .640916s elapsed
│ │ │ - -- .945512s elapsed
│ │ │ - -- .952893s elapsed
│ │ │ - -- 1.08722s elapsed
│ │ │ - -- .825525s elapsed
│ │ │ - -- .672893s elapsed
│ │ │ - -- 1.13429s elapsed
│ │ │ - -- .81054s elapsed
│ │ │ - -- 1.14908s elapsed
│ │ │ - -- 1.49209s elapsed
│ │ │ - -- 1.09431s elapsed
│ │ │ - -- 1.18828s elapsed
│ │ │ - -- .875916s elapsed
│ │ │ - -- .6899s elapsed
│ │ │ - -- 1.08621s elapsed
│ │ │ - -- 1.59117s elapsed
│ │ │ - -- 1.88666s elapsed
│ │ │ - -- 1.39064s elapsed
│ │ │ - -- 1.01948s elapsed
│ │ │ - -- 1.47672s elapsed
│ │ │ - -- 1.19724s elapsed
│ │ │ - -- .910329s elapsed
│ │ │ - -- 1.02771s elapsed
│ │ │ - -- 1.06676s elapsed
│ │ │ - -- .986144s elapsed
│ │ │ - -- .739995s elapsed
│ │ │ - -- 1.05189s elapsed
│ │ │ - -- .696168s elapsed
│ │ │ - -- 1.4284s elapsed
│ │ │ - -- 1.25123s elapsed
│ │ │ - -- 1.4209s elapsed
│ │ │ - -- .786654s elapsed
│ │ │ - -- .485612s elapsed
│ │ │ - -- .40177s elapsed
│ │ │ + -- .400714s elapsed
│ │ │ + -- .461488s elapsed
│ │ │ + -- .820326s elapsed
│ │ │ + -- 1.10107s elapsed
│ │ │ + -- .660756s elapsed
│ │ │ + -- .798543s elapsed
│ │ │ + -- .922777s elapsed
│ │ │ + -- .991239s elapsed
│ │ │ + -- .688378s elapsed
│ │ │ + -- .68108s elapsed
│ │ │ + -- .350612s elapsed
│ │ │ + -- .39335s elapsed
│ │ │ + -- .415792s elapsed
│ │ │ + -- .608201s elapsed
│ │ │ + -- .829812s elapsed
│ │ │ + -- 1.10574s elapsed
│ │ │ + -- .961423s elapsed
│ │ │ + -- .854882s elapsed
│ │ │ + -- 1.12791s elapsed
│ │ │ + -- .993617s elapsed
│ │ │ + -- .753886s elapsed
│ │ │ + -- .905183s elapsed
│ │ │ + -- 1.21524s elapsed
│ │ │ + -- 1.22511s elapsed
│ │ │ + -- .508206s elapsed
│ │ │ + -- .647445s elapsed
│ │ │ + -- 1.20647s elapsed
│ │ │ + -- .625125s elapsed
│ │ │ + -- .528831s elapsed
│ │ │ + -- .757064s elapsed
│ │ │ + -- .984286s elapsed
│ │ │ + -- .766153s elapsed
│ │ │ + -- .523541s elapsed
│ │ │ + -- .956656s elapsed
│ │ │ + -- .744038s elapsed
│ │ │ + -- 1.03215s elapsed
│ │ │ + -- .920567s elapsed
│ │ │ + -- 1.10685s elapsed
│ │ │ + -- 1.1441s elapsed
│ │ │ + -- .713198s elapsed
│ │ │ + -- .629638s elapsed
│ │ │ + -- 1.05435s elapsed
│ │ │ + -- 1.33413s elapsed
│ │ │ + -- 1.67678s elapsed
│ │ │ + -- 1.04044s elapsed
│ │ │ + -- 1.02841s elapsed
│ │ │ + -- 1.44305s elapsed
│ │ │ + -- 1.17036s elapsed
│ │ │ + -- .921718s elapsed
│ │ │ + -- .977289s elapsed
│ │ │ + -- 1.17595s elapsed
│ │ │ + -- .796097s elapsed
│ │ │ + -- .840068s elapsed
│ │ │ + -- .982611s elapsed
│ │ │ + -- .5935s elapsed
│ │ │ + -- 1.12292s elapsed
│ │ │ + -- 1.20133s elapsed
│ │ │ + -- 1.26307s elapsed
│ │ │ + -- .711245s elapsed
│ │ │ + -- .465595s elapsed
│ │ │ + -- .366061s elapsed
│ │ │  
│ │ │  i24 : netList ({{"codimensions", "degrees"}} | allcomps)
│ │ │  
│ │ │        +------------------------+------------------------+
│ │ │  o24 = |codimensions            |degrees                 |
│ │ │        +------------------------+------------------------+
│ │ │        |{3, 5, 5}               |{2, 4, 6}               |
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.2_co_spa_sp__K5_spand_sppentagon_spglued_spalong_span_spedge.out
│ │ │ @@ -39,15 +39,15 @@
│ │ │       .98, .98, .101, -.98, -.298, .393, .201, .201, .201, -.995, -.201,
│ │ │       ------------------------------------------------------------------------
│ │ │       .954}}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : elapsedTime stablesolsPent = showExoticSolutions Pent
│ │ │ - -- .662s elapsed
│ │ │ + -- .955s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │                                                  1 => {0, 2}
│ │ │                                                  2 => {1, 3}
│ │ │                                                  3 => {2, 4}
│ │ │                                                  4 => {0, 3}
│ │ │  +----+-----+-----+----+-----+-----+-----+-----+
│ │ │  |.309|-.809|-.809|.309|.951 |.588 |-.588|-.951|
│ │ │ @@ -60,15 +60,15 @@
│ │ │  +---+---+---+---+
│ │ │  |72 |144|216|288|
│ │ │  +---+---+---+---+
│ │ │  |288|216|144|72 |
│ │ │  +---+---+---+---+
│ │ │  |0  |0  |0  |0  |
│ │ │  +---+---+---+---+
│ │ │ - -- .721s elapsed
│ │ │ + -- 1s elapsed
│ │ │  
│ │ │  o6 = {{.309, -.809, -.809, .309, .951, .588, -.588, -.951}, {.309, -.809,
│ │ │       ------------------------------------------------------------------------
│ │ │       -.809, .309, -.951, -.588, .588, .951}, {1, 1, 1, 1, 0, 0, 0, 0}}
│ │ │  
│ │ │  o6 : List
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/___S__C__T_spgraphs_spwith_spexotic_spsolutions.out
│ │ │ @@ -44,19 +44,19 @@
│ │ │  
│ │ │  i5 : printingPrecision = 3
│ │ │  
│ │ │  o5 = 3
│ │ │  
│ │ │  i6 : for G in Gs list showExoticSolutions G;
│ │ │  warning: some solutions are not regular: {37, 38, 40, 41, 42, 43, 44, 45, 46, 48, 50, 52, 53, 54, 55, 59, 60, 64, 65, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 81, 83, 85, 86, 87, 88, 89}
│ │ │ - -- .763s elapsed
│ │ │ + -- .798s elapsed
│ │ │  warning: some solutions are not regular: {43, 44, 47, 49, 50, 51, 52, 53, 55, 57, 59, 61, 62, 63, 64, 65, 66, 68, 72, 74, 76, 77, 78, 79, 80, 84, 85, 87, 88, 89, 90, 91, 97}
│ │ │ - -- .659s elapsed
│ │ │ - -- .88s elapsed
│ │ │ - -- 1.02s elapsed
│ │ │ + -- .69s elapsed
│ │ │ + -- .907s elapsed
│ │ │ + -- 1.13s 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|
│ │ │ @@ -70,19 +70,19 @@
│ │ │  |144|288|72 |216|
│ │ │  +---+---+---+---+
│ │ │  |0  |0  |0  |0  |
│ │ │  +---+---+---+---+
│ │ │  |216|72 |288|144|
│ │ │  +---+---+---+---+
│ │ │   -- 1.3s elapsed
│ │ │ - -- 1.31s elapsed
│ │ │ + -- 1.37s elapsed
│ │ │  warning: some solutions are not regular: {27, 31, 32, 34, 37, 38, 44, 46, 47, 53, 54, 56, 57, 59, 60}
│ │ │ - -- 1.68s elapsed
│ │ │ + -- 1.75s elapsed
│ │ │  warning: some solutions are not regular: {16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 34}
│ │ │ - -- 1.32s elapsed
│ │ │ - -- 1.4s elapsed
│ │ │ + -- 1.47s elapsed
│ │ │ + -- 1.5s elapsed
│ │ │  warning: some solutions are not regular: {26, 29, 30, 32, 33}
│ │ │ - -- 1.79s elapsed
│ │ │ + -- 1.6s elapsed
│ │ │  warning: some solutions are not regular: {38, 40, 42, 53, 54, 55, 62, 63, 67, 72, 77, 78}
│ │ │ - -- 1.59s elapsed
│ │ │ + -- 1.42s elapsed
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/_get__Linearly__Stable__Solutions.out
│ │ │ @@ -1,15 +1,15 @@
│ │ │  -- -*- M2-comint -*- hash: 1729328129346969841
│ │ │  
│ │ │  i1 : G = graph({0,1,2,3}, {{0,1},{1,2},{2,3},{0,3}});
│ │ │  
│ │ │  i2 : getLinearlyStableSolutions(G)
│ │ │  -- warning: experimental computation over inexact field begun
│ │ │  --          results not reliable (one warning given per session)
│ │ │ - -- .250068s elapsed
│ │ │ + -- .256068s elapsed
│ │ │  warning: some solutions are not regular: {4, 5, 7, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21}
│ │ │  
│ │ │  o2 = {{1, 1, 1, 0, 0, 0}}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/_show__Exotic__Solutions.out
│ │ │ @@ -9,15 +9,15 @@
│ │ │             4 => {0, 3}
│ │ │  
│ │ │  o1 : Graph
│ │ │  
│ │ │  i2 : showExoticSolutions G
│ │ │  -- warning: experimental computation over inexact field begun
│ │ │  --          results not reliable (one warning given per session)
│ │ │ - -- .943746s elapsed
│ │ │ + -- 1.05722s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │                                                  1 => {0, 2}
│ │ │                                                  2 => {1, 3}
│ │ │                                                  3 => {2, 4}
│ │ │                                                  4 => {0, 3}
│ │ │  +-------+--------+--------+-------+--------+--------+--------+--------+
│ │ │  |.309017|-.809017|-.809017|.309017|.951057 |.587785 |-.587785|-.951057|
│ │ │ @@ -50,14 +50,14 @@
│ │ │             2 => {1, 3, 4}
│ │ │             3 => {2, 4}
│ │ │             4 => {0, 2, 3}
│ │ │  
│ │ │  o3 : Graph
│ │ │  
│ │ │  i4 : showExoticSolutions G
│ │ │ - -- 1.24035s elapsed
│ │ │ + -- 1.24998s elapsed
│ │ │  
│ │ │  o4 = {{1, 1, 1, 1, 0, 0, 0, 0}}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.html
│ │ │ @@ -295,25 +295,25 @@
│ │ │            
│ │ │              
│ │ │                
    i16 : for G in Gs list (
│ │ │            IG = oscQuadrics(G, R);
│ │ │            elapsedTime comps := decompose IG;
│ │ │            {comps/codim, comps/degree}
│ │ │            );
│ │ │ - -- .280854s elapsed
│ │ │ - -- .263174s elapsed
│ │ │ - -- .501439s elapsed
│ │ │ - -- .231638s elapsed
│ │ │ - -- .342923s elapsed
│ │ │ - -- .34852s elapsed
│ │ │ - -- .619954s elapsed
│ │ │ - -- .456734s elapsed
│ │ │ - -- .648967s elapsed
│ │ │ - -- .367209s elapsed
│ │ │ - -- .250328s elapsed
    
│ │ │ + -- .23795s elapsed
│ │ │ + -- .277177s elapsed
│ │ │ + -- .477459s elapsed
│ │ │ + -- .252176s elapsed
│ │ │ + -- .259965s elapsed
│ │ │ + -- .309722s elapsed
│ │ │ + -- .534615s elapsed
│ │ │ + -- .45729s elapsed
│ │ │ + -- .476608s elapsed
│ │ │ + -- .311675s elapsed
│ │ │ + -- .215434s elapsed
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i17 : netList oo
│ │ │  
│ │ │        +---------------+---------------+
│ │ │ @@ -380,75 +380,75 @@
│ │ │            
│ │ │              
│ │ │                
    i23 : allcomps = for G in Gs list (
│ │ │            IG = oscQuadrics(G, R);
│ │ │            elapsedTime comps := decompose IG;
│ │ │            {comps/codim, comps/degree}
│ │ │            );
│ │ │ - -- .536601s elapsed
│ │ │ - -- .816725s elapsed
│ │ │ - -- 1.00632s elapsed
│ │ │ - -- 1.37164s elapsed
│ │ │ - -- .718355s elapsed
│ │ │ - -- 1.00236s elapsed
│ │ │ - -- 1.18305s elapsed
│ │ │ - -- 1.31195s elapsed
│ │ │ - -- .727806s elapsed
│ │ │ - -- .774958s elapsed
│ │ │ - -- .333808s elapsed
│ │ │ - -- .5035s elapsed
│ │ │ - -- .490585s elapsed
│ │ │ - -- .637754s elapsed
│ │ │ - -- .880246s elapsed
│ │ │ - -- 1.2462s elapsed
│ │ │ - -- .940742s elapsed
│ │ │ - -- 1.01084s elapsed
│ │ │ - -- 1.44951s elapsed
│ │ │ - -- 1.31812s elapsed
│ │ │ - -- 1.00162s elapsed
│ │ │ - -- 1.3528s elapsed
│ │ │ - -- 1.90187s elapsed
│ │ │ - -- 1.44103s elapsed
│ │ │ - -- .432654s elapsed
│ │ │ - -- .573037s elapsed
│ │ │ - -- 1.40316s elapsed
│ │ │ - -- .640916s elapsed
│ │ │ - -- .945512s elapsed
│ │ │ - -- .952893s elapsed
│ │ │ - -- 1.08722s elapsed
│ │ │ - -- .825525s elapsed
│ │ │ - -- .672893s elapsed
│ │ │ - -- 1.13429s elapsed
│ │ │ - -- .81054s elapsed
│ │ │ - -- 1.14908s elapsed
│ │ │ - -- 1.49209s elapsed
│ │ │ - -- 1.09431s elapsed
│ │ │ - -- 1.18828s elapsed
│ │ │ - -- .875916s elapsed
│ │ │ - -- .6899s elapsed
│ │ │ - -- 1.08621s elapsed
│ │ │ - -- 1.59117s elapsed
│ │ │ - -- 1.88666s elapsed
│ │ │ - -- 1.39064s elapsed
│ │ │ - -- 1.01948s elapsed
│ │ │ - -- 1.47672s elapsed
│ │ │ - -- 1.19724s elapsed
│ │ │ - -- .910329s elapsed
│ │ │ - -- 1.02771s elapsed
│ │ │ - -- 1.06676s elapsed
│ │ │ - -- .986144s elapsed
│ │ │ - -- .739995s elapsed
│ │ │ - -- 1.05189s elapsed
│ │ │ - -- .696168s elapsed
│ │ │ - -- 1.4284s elapsed
│ │ │ - -- 1.25123s elapsed
│ │ │ - -- 1.4209s elapsed
│ │ │ - -- .786654s elapsed
│ │ │ - -- .485612s elapsed
│ │ │ - -- .40177s elapsed
    
│ │ │ + -- .400714s elapsed
│ │ │ + -- .461488s elapsed
│ │ │ + -- .820326s elapsed
│ │ │ + -- 1.10107s elapsed
│ │ │ + -- .660756s elapsed
│ │ │ + -- .798543s elapsed
│ │ │ + -- .922777s elapsed
│ │ │ + -- .991239s elapsed
│ │ │ + -- .688378s elapsed
│ │ │ + -- .68108s elapsed
│ │ │ + -- .350612s elapsed
│ │ │ + -- .39335s elapsed
│ │ │ + -- .415792s elapsed
│ │ │ + -- .608201s elapsed
│ │ │ + -- .829812s elapsed
│ │ │ + -- 1.10574s elapsed
│ │ │ + -- .961423s elapsed
│ │ │ + -- .854882s elapsed
│ │ │ + -- 1.12791s elapsed
│ │ │ + -- .993617s elapsed
│ │ │ + -- .753886s elapsed
│ │ │ + -- .905183s elapsed
│ │ │ + -- 1.21524s elapsed
│ │ │ + -- 1.22511s elapsed
│ │ │ + -- .508206s elapsed
│ │ │ + -- .647445s elapsed
│ │ │ + -- 1.20647s elapsed
│ │ │ + -- .625125s elapsed
│ │ │ + -- .528831s elapsed
│ │ │ + -- .757064s elapsed
│ │ │ + -- .984286s elapsed
│ │ │ + -- .766153s elapsed
│ │ │ + -- .523541s elapsed
│ │ │ + -- .956656s elapsed
│ │ │ + -- .744038s elapsed
│ │ │ + -- 1.03215s elapsed
│ │ │ + -- .920567s elapsed
│ │ │ + -- 1.10685s elapsed
│ │ │ + -- 1.1441s elapsed
│ │ │ + -- .713198s elapsed
│ │ │ + -- .629638s elapsed
│ │ │ + -- 1.05435s elapsed
│ │ │ + -- 1.33413s elapsed
│ │ │ + -- 1.67678s elapsed
│ │ │ + -- 1.04044s elapsed
│ │ │ + -- 1.02841s elapsed
│ │ │ + -- 1.44305s elapsed
│ │ │ + -- 1.17036s elapsed
│ │ │ + -- .921718s elapsed
│ │ │ + -- .977289s elapsed
│ │ │ + -- 1.17595s elapsed
│ │ │ + -- .796097s elapsed
│ │ │ + -- .840068s elapsed
│ │ │ + -- .982611s elapsed
│ │ │ + -- .5935s elapsed
│ │ │ + -- 1.12292s elapsed
│ │ │ + -- 1.20133s elapsed
│ │ │ + -- 1.26307s elapsed
│ │ │ + -- .711245s elapsed
│ │ │ + -- .465595s elapsed
│ │ │ + -- .366061s elapsed
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i24 : netList ({{"codimensions", "degrees"}} | allcomps)
│ │ │  
│ │ │        +------------------------+------------------------+
│ │ │ ├── html2text {}
│ │ │ │ @@ -180,25 +180,25 @@
│ │ │ │  
│ │ │ │  o15 = 4
│ │ │ │  i16 : for G in Gs list (
│ │ │ │            IG = oscQuadrics(G, R);
│ │ │ │            elapsedTime comps := decompose IG;
│ │ │ │            {comps/codim, comps/degree}
│ │ │ │            );
│ │ │ │ - -- .280854s elapsed
│ │ │ │ - -- .263174s elapsed
│ │ │ │ - -- .501439s elapsed
│ │ │ │ - -- .231638s elapsed
│ │ │ │ - -- .342923s elapsed
│ │ │ │ - -- .34852s elapsed
│ │ │ │ - -- .619954s elapsed
│ │ │ │ - -- .456734s elapsed
│ │ │ │ - -- .648967s elapsed
│ │ │ │ - -- .367209s elapsed
│ │ │ │ - -- .250328s elapsed
│ │ │ │ + -- .23795s elapsed
│ │ │ │ + -- .277177s elapsed
│ │ │ │ + -- .477459s elapsed
│ │ │ │ + -- .252176s elapsed
│ │ │ │ + -- .259965s elapsed
│ │ │ │ + -- .309722s elapsed
│ │ │ │ + -- .534615s elapsed
│ │ │ │ + -- .45729s elapsed
│ │ │ │ + -- .476608s elapsed
│ │ │ │ + -- .311675s elapsed
│ │ │ │ + -- .215434s elapsed
│ │ │ │  i17 : netList oo
│ │ │ │  
│ │ │ │        +---------------+---------------+
│ │ │ │  o17 = |{3, 4, 4}      |{2, 3, 5}      |
│ │ │ │        +---------------+---------------+
│ │ │ │        |{3, 4, 4}      |{2, 3, 5}      |
│ │ │ │        +---------------+---------------+
│ │ │ │ @@ -233,75 +233,75 @@
│ │ │ │  
│ │ │ │  o22 = 15
│ │ │ │  i23 : allcomps = for G in Gs list (
│ │ │ │            IG = oscQuadrics(G, R);
│ │ │ │            elapsedTime comps := decompose IG;
│ │ │ │            {comps/codim, comps/degree}
│ │ │ │            );
│ │ │ │ - -- .536601s elapsed
│ │ │ │ - -- .816725s elapsed
│ │ │ │ - -- 1.00632s elapsed
│ │ │ │ - -- 1.37164s elapsed
│ │ │ │ - -- .718355s elapsed
│ │ │ │ - -- 1.00236s elapsed
│ │ │ │ - -- 1.18305s elapsed
│ │ │ │ - -- 1.31195s elapsed
│ │ │ │ - -- .727806s elapsed
│ │ │ │ - -- .774958s elapsed
│ │ │ │ - -- .333808s elapsed
│ │ │ │ - -- .5035s elapsed
│ │ │ │ - -- .490585s elapsed
│ │ │ │ - -- .637754s elapsed
│ │ │ │ - -- .880246s elapsed
│ │ │ │ - -- 1.2462s elapsed
│ │ │ │ - -- .940742s elapsed
│ │ │ │ - -- 1.01084s elapsed
│ │ │ │ - -- 1.44951s elapsed
│ │ │ │ - -- 1.31812s elapsed
│ │ │ │ - -- 1.00162s elapsed
│ │ │ │ - -- 1.3528s elapsed
│ │ │ │ - -- 1.90187s elapsed
│ │ │ │ - -- 1.44103s elapsed
│ │ │ │ - -- .432654s elapsed
│ │ │ │ - -- .573037s elapsed
│ │ │ │ - -- 1.40316s elapsed
│ │ │ │ - -- .640916s elapsed
│ │ │ │ - -- .945512s elapsed
│ │ │ │ - -- .952893s elapsed
│ │ │ │ - -- 1.08722s elapsed
│ │ │ │ - -- .825525s elapsed
│ │ │ │ - -- .672893s elapsed
│ │ │ │ - -- 1.13429s elapsed
│ │ │ │ - -- .81054s elapsed
│ │ │ │ - -- 1.14908s elapsed
│ │ │ │ - -- 1.49209s elapsed
│ │ │ │ - -- 1.09431s elapsed
│ │ │ │ - -- 1.18828s elapsed
│ │ │ │ - -- .875916s elapsed
│ │ │ │ - -- .6899s elapsed
│ │ │ │ - -- 1.08621s elapsed
│ │ │ │ - -- 1.59117s elapsed
│ │ │ │ - -- 1.88666s elapsed
│ │ │ │ - -- 1.39064s elapsed
│ │ │ │ - -- 1.01948s elapsed
│ │ │ │ - -- 1.47672s elapsed
│ │ │ │ - -- 1.19724s elapsed
│ │ │ │ - -- .910329s elapsed
│ │ │ │ - -- 1.02771s elapsed
│ │ │ │ - -- 1.06676s elapsed
│ │ │ │ - -- .986144s elapsed
│ │ │ │ - -- .739995s elapsed
│ │ │ │ - -- 1.05189s elapsed
│ │ │ │ - -- .696168s elapsed
│ │ │ │ - -- 1.4284s elapsed
│ │ │ │ - -- 1.25123s elapsed
│ │ │ │ - -- 1.4209s elapsed
│ │ │ │ - -- .786654s elapsed
│ │ │ │ - -- .485612s elapsed
│ │ │ │ - -- .40177s elapsed
│ │ │ │ + -- .400714s elapsed
│ │ │ │ + -- .461488s elapsed
│ │ │ │ + -- .820326s elapsed
│ │ │ │ + -- 1.10107s elapsed
│ │ │ │ + -- .660756s elapsed
│ │ │ │ + -- .798543s elapsed
│ │ │ │ + -- .922777s elapsed
│ │ │ │ + -- .991239s elapsed
│ │ │ │ + -- .688378s elapsed
│ │ │ │ + -- .68108s elapsed
│ │ │ │ + -- .350612s elapsed
│ │ │ │ + -- .39335s elapsed
│ │ │ │ + -- .415792s elapsed
│ │ │ │ + -- .608201s elapsed
│ │ │ │ + -- .829812s elapsed
│ │ │ │ + -- 1.10574s elapsed
│ │ │ │ + -- .961423s elapsed
│ │ │ │ + -- .854882s elapsed
│ │ │ │ + -- 1.12791s elapsed
│ │ │ │ + -- .993617s elapsed
│ │ │ │ + -- .753886s elapsed
│ │ │ │ + -- .905183s elapsed
│ │ │ │ + -- 1.21524s elapsed
│ │ │ │ + -- 1.22511s elapsed
│ │ │ │ + -- .508206s elapsed
│ │ │ │ + -- .647445s elapsed
│ │ │ │ + -- 1.20647s elapsed
│ │ │ │ + -- .625125s elapsed
│ │ │ │ + -- .528831s elapsed
│ │ │ │ + -- .757064s elapsed
│ │ │ │ + -- .984286s elapsed
│ │ │ │ + -- .766153s elapsed
│ │ │ │ + -- .523541s elapsed
│ │ │ │ + -- .956656s elapsed
│ │ │ │ + -- .744038s elapsed
│ │ │ │ + -- 1.03215s elapsed
│ │ │ │ + -- .920567s elapsed
│ │ │ │ + -- 1.10685s elapsed
│ │ │ │ + -- 1.1441s elapsed
│ │ │ │ + -- .713198s elapsed
│ │ │ │ + -- .629638s elapsed
│ │ │ │ + -- 1.05435s elapsed
│ │ │ │ + -- 1.33413s elapsed
│ │ │ │ + -- 1.67678s elapsed
│ │ │ │ + -- 1.04044s elapsed
│ │ │ │ + -- 1.02841s elapsed
│ │ │ │ + -- 1.44305s elapsed
│ │ │ │ + -- 1.17036s elapsed
│ │ │ │ + -- .921718s elapsed
│ │ │ │ + -- .977289s elapsed
│ │ │ │ + -- 1.17595s elapsed
│ │ │ │ + -- .796097s elapsed
│ │ │ │ + -- .840068s elapsed
│ │ │ │ + -- .982611s elapsed
│ │ │ │ + -- .5935s elapsed
│ │ │ │ + -- 1.12292s elapsed
│ │ │ │ + -- 1.20133s elapsed
│ │ │ │ + -- 1.26307s elapsed
│ │ │ │ + -- .711245s elapsed
│ │ │ │ + -- .465595s elapsed
│ │ │ │ + -- .366061s elapsed
│ │ │ │  i24 : netList ({{"codimensions", "degrees"}} | allcomps)
│ │ │ │  
│ │ │ │        +------------------------+------------------------+
│ │ │ │  o24 = |codimensions            |degrees                 |
│ │ │ │        +------------------------+------------------------+
│ │ │ │        |{3, 5, 5}               |{2, 4, 6}               |
│ │ │ │        +------------------------+------------------------+
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/___Example_sp4.2_co_spa_sp__K5_spand_sppentagon_spglued_spalong_span_spedge.html
│ │ │ @@ -110,15 +110,15 @@
│ │ │  
│ │ │  o5 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : elapsedTime stablesolsPent = showExoticSolutions Pent
│ │ │ - -- .662s elapsed
│ │ │ + -- .955s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │                                                  1 => {0, 2}
│ │ │                                                  2 => {1, 3}
│ │ │                                                  3 => {2, 4}
│ │ │                                                  4 => {0, 3}
│ │ │  +----+-----+-----+----+-----+-----+-----+-----+
│ │ │  |.309|-.809|-.809|.309|.951 |.588 |-.588|-.951|
│ │ │ @@ -131,15 +131,15 @@
│ │ │  +---+---+---+---+
│ │ │  |72 |144|216|288|
│ │ │  +---+---+---+---+
│ │ │  |288|216|144|72 |
│ │ │  +---+---+---+---+
│ │ │  |0  |0  |0  |0  |
│ │ │  +---+---+---+---+
│ │ │ - -- .721s elapsed
│ │ │ + -- 1s elapsed
│ │ │  
│ │ │  o6 = {{.309, -.809, -.809, .309, .951, .588, -.588, -.951}, {.309, -.809,
│ │ │       ------------------------------------------------------------------------
│ │ │       -.809, .309, -.951, -.588, .588, .951}, {1, 1, 1, 1, 0, 0, 0, 0}}
│ │ │  
│ │ │  o6 : List
    
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -43,15 +43,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       .98, .98, .101, -.98, -.298, .393, .201, .201, .201, -.995, -.201,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       .954}}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : elapsedTime stablesolsPent = showExoticSolutions Pent
│ │ │ │ - -- .662s elapsed
│ │ │ │ + -- .955s elapsed
│ │ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │ │                                                  1 => {0, 2}
│ │ │ │                                                  2 => {1, 3}
│ │ │ │                                                  3 => {2, 4}
│ │ │ │                                                  4 => {0, 3}
│ │ │ │  +----+-----+-----+----+-----+-----+-----+-----+
│ │ │ │  |.309|-.809|-.809|.309|.951 |.588 |-.588|-.951|
│ │ │ │ @@ -64,15 +64,15 @@
│ │ │ │  +---+---+---+---+
│ │ │ │  |72 |144|216|288|
│ │ │ │  +---+---+---+---+
│ │ │ │  |288|216|144|72 |
│ │ │ │  +---+---+---+---+
│ │ │ │  |0  |0  |0  |0  |
│ │ │ │  +---+---+---+---+
│ │ │ │ - -- .721s elapsed
│ │ │ │ + -- 1s elapsed
│ │ │ │  
│ │ │ │  o6 = {{.309, -.809, -.809, .309, .951, .588, -.588, -.951}, {.309, -.809,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       -.809, .309, -.951, -.588, .588, .951}, {1, 1, 1, 1, 0, 0, 0, 0}}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  Computing the (linearly) stable solutions for K5C5 takes a minute or two:
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/___S__C__T_spgraphs_spwith_spexotic_spsolutions.html
│ │ │ @@ -115,19 +115,19 @@
│ │ │  o5 = 3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : for G in Gs list showExoticSolutions G;
│ │ │  warning: some solutions are not regular: {37, 38, 40, 41, 42, 43, 44, 45, 46, 48, 50, 52, 53, 54, 55, 59, 60, 64, 65, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 81, 83, 85, 86, 87, 88, 89}
│ │ │ - -- .763s elapsed
│ │ │ + -- .798s elapsed
│ │ │  warning: some solutions are not regular: {43, 44, 47, 49, 50, 51, 52, 53, 55, 57, 59, 61, 62, 63, 64, 65, 66, 68, 72, 74, 76, 77, 78, 79, 80, 84, 85, 87, 88, 89, 90, 91, 97}
│ │ │ - -- .659s elapsed
│ │ │ - -- .88s elapsed
│ │ │ - -- 1.02s elapsed
│ │ │ + -- .69s elapsed
│ │ │ + -- .907s elapsed
│ │ │ + -- 1.13s 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|
│ │ │ @@ -141,24 +141,24 @@
│ │ │  |144|288|72 |216|
│ │ │  +---+---+---+---+
│ │ │  |0  |0  |0  |0  |
│ │ │  +---+---+---+---+
│ │ │  |216|72 |288|144|
│ │ │  +---+---+---+---+
│ │ │   -- 1.3s elapsed
│ │ │ - -- 1.31s elapsed
│ │ │ + -- 1.37s elapsed
│ │ │  warning: some solutions are not regular: {27, 31, 32, 34, 37, 38, 44, 46, 47, 53, 54, 56, 57, 59, 60}
│ │ │ - -- 1.68s elapsed
│ │ │ + -- 1.75s elapsed
│ │ │  warning: some solutions are not regular: {16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 34}
│ │ │ - -- 1.32s elapsed
│ │ │ - -- 1.4s elapsed
│ │ │ + -- 1.47s elapsed
│ │ │ + -- 1.5s elapsed
│ │ │  warning: some solutions are not regular: {26, 29, 30, 32, 33}
│ │ │ - -- 1.79s elapsed
│ │ │ + -- 1.6s elapsed
│ │ │  warning: some solutions are not regular: {38, 40, 42, 53, 54, 55, 62, 63, 67, 72, 77, 78}
│ │ │ - -- 1.59s elapsed
    
│ │ │ + -- 1.42s elapsed
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    
│ │ │          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,21 +48,21 @@
│ │ │ │  i5 : printingPrecision = 3
│ │ │ │  
│ │ │ │  o5 = 3
│ │ │ │  i6 : for G in Gs list showExoticSolutions G;
│ │ │ │  warning: some solutions are not regular: {37, 38, 40, 41, 42, 43, 44, 45, 46,
│ │ │ │  48, 50, 52, 53, 54, 55, 59, 60, 64, 65, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78,
│ │ │ │  79, 81, 83, 85, 86, 87, 88, 89}
│ │ │ │ - -- .763s elapsed
│ │ │ │ + -- .798s elapsed
│ │ │ │  warning: some solutions are not regular: {43, 44, 47, 49, 50, 51, 52, 53, 55,
│ │ │ │  57, 59, 61, 62, 63, 64, 65, 66, 68, 72, 74, 76, 77, 78, 79, 80, 84, 85, 87, 88,
│ │ │ │  89, 90, 91, 97}
│ │ │ │ - -- .659s elapsed
│ │ │ │ - -- .88s elapsed
│ │ │ │ - -- 1.02s elapsed
│ │ │ │ + -- .69s elapsed
│ │ │ │ + -- .907s elapsed
│ │ │ │ + -- 1.13s 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|
│ │ │ │ @@ -76,23 +76,23 @@
│ │ │ │  |144|288|72 |216|
│ │ │ │  +---+---+---+---+
│ │ │ │  |0  |0  |0  |0  |
│ │ │ │  +---+---+---+---+
│ │ │ │  |216|72 |288|144|
│ │ │ │  +---+---+---+---+
│ │ │ │   -- 1.3s elapsed
│ │ │ │ - -- 1.31s elapsed
│ │ │ │ + -- 1.37s elapsed
│ │ │ │  warning: some solutions are not regular: {27, 31, 32, 34, 37, 38, 44, 46, 47,
│ │ │ │  53, 54, 56, 57, 59, 60}
│ │ │ │ - -- 1.68s elapsed
│ │ │ │ + -- 1.75s elapsed
│ │ │ │  warning: some solutions are not regular: {16, 17, 18, 19, 20, 21, 22, 23, 24,
│ │ │ │  25, 26, 27, 28, 29, 31, 34}
│ │ │ │ - -- 1.32s elapsed
│ │ │ │ - -- 1.4s elapsed
│ │ │ │ + -- 1.47s elapsed
│ │ │ │ + -- 1.5s elapsed
│ │ │ │  warning: some solutions are not regular: {26, 29, 30, 32, 33}
│ │ │ │ - -- 1.79s elapsed
│ │ │ │ + -- 1.6s elapsed
│ │ │ │  warning: some solutions are not regular: {38, 40, 42, 53, 54, 55, 62, 63, 67,
│ │ │ │  72, 77, 78}
│ │ │ │ - -- 1.59s elapsed
│ │ │ │ + -- 1.42s elapsed
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/Oscillators/Documentation.m2:812:0.
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/_get__Linearly__Stable__Solutions.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : getLinearlyStableSolutions(G)
│ │ │  -- warning: experimental computation over inexact field begun
│ │ │  --          results not reliable (one warning given per session)
│ │ │ - -- .250068s elapsed
│ │ │ + -- .256068s elapsed
│ │ │  warning: some solutions are not regular: {4, 5, 7, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21}
│ │ │  
│ │ │  o2 = {{1, 1, 1, 0, 0, 0}}
│ │ │  
│ │ │  o2 : List
    
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,15 +19,15 @@
│ │ │ │  of each oscillator is given by the Kuramoto model. The linear stability of a
│ │ │ │  solution is determined by the eigenvalues of the Jacobian matrix of the system
│ │ │ │  evaluated at the solution.
│ │ │ │  i1 : G = graph({0,1,2,3}, {{0,1},{1,2},{2,3},{0,3}});
│ │ │ │  i2 : getLinearlyStableSolutions(G)
│ │ │ │  -- warning: experimental computation over inexact field begun
│ │ │ │  --          results not reliable (one warning given per session)
│ │ │ │ - -- .250068s elapsed
│ │ │ │ + -- .256068s elapsed
│ │ │ │  warning: some solutions are not regular: {4, 5, 7, 9, 10, 12, 13, 14, 15, 16,
│ │ │ │  17, 18, 19, 20, 21}
│ │ │ │  
│ │ │ │  o2 = {{1, 1, 1, 0, 0, 0}}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/_show__Exotic__Solutions.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : showExoticSolutions G
│ │ │  -- warning: experimental computation over inexact field begun
│ │ │  --          results not reliable (one warning given per session)
│ │ │ - -- .943746s elapsed
│ │ │ + -- 1.05722s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │                                                  1 => {0, 2}
│ │ │                                                  2 => {1, 3}
│ │ │                                                  3 => {2, 4}
│ │ │                                                  4 => {0, 3}
│ │ │  +-------+--------+--------+-------+--------+--------+--------+--------+
│ │ │  |.309017|-.809017|-.809017|.309017|.951057 |.587785 |-.587785|-.951057|
│ │ │ @@ -147,15 +147,15 @@
│ │ │  
│ │ │  o3 : Graph
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : showExoticSolutions G
│ │ │ - -- 1.24035s elapsed
│ │ │ + -- 1.24998s elapsed
│ │ │  
│ │ │  o4 = {{1, 1, 1, 1, 0, 0, 0, 0}}
│ │ │  
│ │ │  o4 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -36,15 +36,15 @@
│ │ │ │             3 => {2, 4}
│ │ │ │             4 => {0, 3}
│ │ │ │  
│ │ │ │  o1 : Graph
│ │ │ │  i2 : showExoticSolutions G
│ │ │ │  -- warning: experimental computation over inexact field begun
│ │ │ │  --          results not reliable (one warning given per session)
│ │ │ │ - -- .943746s elapsed
│ │ │ │ + -- 1.05722s elapsed
│ │ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │ │                                                  1 => {0, 2}
│ │ │ │                                                  2 => {1, 3}
│ │ │ │                                                  3 => {2, 4}
│ │ │ │                                                  4 => {0, 3}
│ │ │ │  +-------+--------+--------+-------+--------+--------+--------+--------+
│ │ │ │  |.309017|-.809017|-.809017|.309017|.951057 |.587785 |-.587785|-.951057|
│ │ │ │ @@ -78,15 +78,15 @@
│ │ │ │             1 => {0, 2}
│ │ │ │             2 => {1, 3, 4}
│ │ │ │             3 => {2, 4}
│ │ │ │             4 => {0, 2, 3}
│ │ │ │  
│ │ │ │  o3 : Graph
│ │ │ │  i4 : showExoticSolutions G
│ │ │ │ - -- 1.24035s elapsed
│ │ │ │ + -- 1.24998s elapsed
│ │ │ │  
│ │ │ │  o4 = {{1, 1, 1, 1, 0, 0, 0, 0}}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _g_e_t_L_i_n_e_a_r_l_y_S_t_a_b_l_e_S_o_l_u_t_i_o_n_s -- Compute linearly stable solutions for the
│ │ │ │        Kuramoto oscillator system associated to a graph
│ │ ├── ./usr/share/doc/Macaulay2/PathSignatures/example-output/___A_spfamily_spof_sppaths_spon_spa_spcone.out
│ │ │ @@ -80,20 +80,20 @@
│ │ │  i19 : needsPackage "MultigradedImplicitization";
│ │ │  
│ │ │  i20 : I = sub(ideal flatten values componentsOfKernel(2, m, Grading => matrix {toList(9:1)}), S);
│ │ │  warning: computation begun over finite field. resulting polynomials may not lie in the ideal
│ │ │  computing total degree: 1
│ │ │  number of monomials = 9
│ │ │  number of distinct multidegrees = 1
│ │ │ - -- .00892428s elapsed
│ │ │ + -- .0108714s elapsed
│ │ │  WARNING: There are linear relations. You may want to reduce the number of variables to speed up the computation.
│ │ │  computing total degree: 2
│ │ │  number of monomials = 45
│ │ │  number of distinct multidegrees = 1
│ │ │ - -- .568854s elapsed
│ │ │ + -- .56807s elapsed
│ │ │  
│ │ │  o20 : Ideal of S
│ │ │  
│ │ │  i21 : dim I
│ │ │  
│ │ │  o21 = 5
│ │ ├── ./usr/share/doc/Macaulay2/PathSignatures/html/___A_spfamily_spof_sppaths_spon_spa_spcone.html
│ │ │ @@ -208,20 +208,20 @@
│ │ │            
│ │ │              
│ │ │                
    i20 : I = sub(ideal flatten values componentsOfKernel(2, m, Grading => matrix {toList(9:1)}), S);
│ │ │  warning: computation begun over finite field. resulting polynomials may not lie in the ideal
│ │ │  computing total degree: 1
│ │ │  number of monomials = 9
│ │ │  number of distinct multidegrees = 1
│ │ │ - -- .00892428s elapsed
│ │ │ + -- .0108714s elapsed
│ │ │  WARNING: There are linear relations. You may want to reduce the number of variables to speed up the computation.
│ │ │  computing total degree: 2
│ │ │  number of monomials = 45
│ │ │  number of distinct multidegrees = 1
│ │ │ - -- .568854s elapsed
│ │ │ + -- .56807s elapsed
│ │ │  
│ │ │  o20 : Ideal of S
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i21 : dim I
│ │ │ ├── html2text {}
│ │ │ │ @@ -77,21 +77,21 @@
│ │ │ │  i20 : I = sub(ideal flatten values componentsOfKernel(2, m, Grading => matrix
│ │ │ │  {toList(9:1)}), S);
│ │ │ │  warning: computation begun over finite field. resulting polynomials may not lie
│ │ │ │  in the ideal
│ │ │ │  computing total degree: 1
│ │ │ │  number of monomials = 9
│ │ │ │  number of distinct multidegrees = 1
│ │ │ │ - -- .00892428s elapsed
│ │ │ │ + -- .0108714s elapsed
│ │ │ │  WARNING: There are linear relations. You may want to reduce the number of
│ │ │ │  variables to speed up the computation.
│ │ │ │  computing total degree: 2
│ │ │ │  number of monomials = 45
│ │ │ │  number of distinct multidegrees = 1
│ │ │ │ - -- .568854s elapsed
│ │ │ │ + -- .56807s elapsed
│ │ │ │  
│ │ │ │  o20 : Ideal of S
│ │ │ │  i21 : dim I
│ │ │ │  
│ │ │ │  o21 = 5
│ │ │ │  i22 : isPrime I
│ │ ├── ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/___Lab__Book__Protocol.out
│ │ │ @@ -41,15 +41,15 @@
│ │ │  i3 : g=3
│ │ │  
│ │ │  o3 = 3
│ │ │  
│ │ │  i4 : kk= ZZ/101;
│ │ │  
│ │ │  i5 : elapsedTime (S,qq,R,u, M1,M2, Mu1, Mu2)=randomNicePencil(kk,g);
│ │ │ - -- 1.19915s elapsed
│ │ │ + -- 1.03951s elapsed
│ │ │  
│ │ │  i6 : M=cliffordModule(Mu1,Mu2,R)
│ │ │  
│ │ │  o6 = CliffordModule{...6...}
│ │ │  
│ │ │  o6 : CliffordModule
│ │ │  
│ │ │ @@ -67,30 +67,30 @@
│ │ │            m12=randomExtension(m1.yAction,m2.yAction);
│ │ │            V = vectorBundleOnE m12;
│ │ │            Ul=tensorProduct(Mor,V);
│ │ │            Ul1=tensorProduct(Mor1,V);
│ │ │            d0=unique degrees target Ul.yAction;
│ │ │            d1=unique degrees target Ul1.yAction;
│ │ │            #d1 >=3 or #d0 >=3) do ();
│ │ │ - -- .463669s elapsed
│ │ │ + -- .345431s elapsed
│ │ │  
│ │ │  i12 : betti Ul.yAction, betti Ul1.yAction
│ │ │  
│ │ │                 0  1          0  1
│ │ │  o12 = (total: 32 32, total: 32 32)
│ │ │            -4: 16  .     -2: 32  .
│ │ │            -3: 16  .     -1:  .  .
│ │ │            -2:  .  .      0:  .  .
│ │ │            -1:  . 16      1:  . 32
│ │ │             0:  . 16
│ │ │  
│ │ │  o12 : Sequence
│ │ │  
│ │ │  i13 : elapsedTime Ul = tensorProduct(M,V); -- the heaviest part computing the actions of generators
│ │ │ - -- 20.5845s elapsed
│ │ │ + -- 13.3419s elapsed
│ │ │  
│ │ │  i14 : M1Ul=sum(#Ul.oddOperators,i->S_i*sub(Ul.oddOperators_i,S));
│ │ │  
│ │ │                32      32
│ │ │  o14 : Matrix S   <-- S
│ │ │  
│ │ │  i15 : r=2
│ │ ├── ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_search__Ulrich.out
│ │ │ @@ -46,30 +46,30 @@
│ │ │  i11 : M=cliffordModule(Mu1,Mu2,R)
│ │ │  
│ │ │  o11 = CliffordModule{...6...}
│ │ │  
│ │ │  o11 : CliffordModule
│ │ │  
│ │ │  i12 : elapsedTime Ulr = searchUlrich(M,S);
│ │ │ - -- .684809s elapsed
│ │ │ + -- .548698s elapsed
│ │ │  
│ │ │  i13 : betti freeResolution Ulr
│ │ │  
│ │ │               0  1 2
│ │ │  o13 = total: 8 16 8
│ │ │            0: 8 16 8
│ │ │  
│ │ │  o13 : BettiTally
│ │ │  
│ │ │  i14 : ann Ulr == ideal qs
│ │ │  
│ │ │  o14 = true
│ │ │  
│ │ │  i15 : elapsedTime Ulr3 = searchUlrich(M,S,3);
│ │ │ - -- 2.41016s elapsed
│ │ │ + -- 1.68676s elapsed
│ │ │  
│ │ │  i16 : betti freeResolution Ulr3
│ │ │  
│ │ │                0  1  2
│ │ │  o16 = total: 12 24 12
│ │ │            0: 12 24 12
│ │ ├── ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/___Lab__Book__Protocol.html
│ │ │ @@ -128,15 +128,15 @@
│ │ │              
│ │ │                
    i4 : kk= ZZ/101;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime (S,qq,R,u, M1,M2, Mu1, Mu2)=randomNicePencil(kk,g);
│ │ │ - -- 1.19915s elapsed
    
│ │ │ + -- 1.03951s elapsed
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : M=cliffordModule(Mu1,Mu2,R)
│ │ │  
│ │ │  o6 = CliffordModule{...6...}
│ │ │ @@ -172,15 +172,15 @@
│ │ │            m12=randomExtension(m1.yAction,m2.yAction);
│ │ │            V = vectorBundleOnE m12;
│ │ │            Ul=tensorProduct(Mor,V);
│ │ │            Ul1=tensorProduct(Mor1,V);
│ │ │            d0=unique degrees target Ul.yAction;
│ │ │            d1=unique degrees target Ul1.yAction;
│ │ │            #d1 >=3 or #d0 >=3) do ();
│ │ │ - -- .463669s elapsed
    
│ │ │ + -- .345431s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : betti Ul.yAction, betti Ul1.yAction
│ │ │  
│ │ │                 0  1          0  1
│ │ │ @@ -193,15 +193,15 @@
│ │ │  
│ │ │  o12 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : elapsedTime Ul = tensorProduct(M,V); -- the heaviest part computing the actions of generators
│ │ │ - -- 20.5845s elapsed
    
│ │ │ + -- 13.3419s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : M1Ul=sum(#Ul.oddOperators,i->S_i*sub(Ul.oddOperators_i,S));
│ │ │  
│ │ │                32      32
│ │ │ ├── html2text {}
│ │ │ │ @@ -55,15 +55,15 @@
│ │ │ │              -- will give an Ulrich bundle, with betti table
│ │ │ │              -- 16 32 16
│ │ │ │  i3 : g=3
│ │ │ │  
│ │ │ │  o3 = 3
│ │ │ │  i4 : kk= ZZ/101;
│ │ │ │  i5 : elapsedTime (S,qq,R,u, M1,M2, Mu1, Mu2)=randomNicePencil(kk,g);
│ │ │ │ - -- 1.19915s elapsed
│ │ │ │ + -- 1.03951s elapsed
│ │ │ │  i6 : M=cliffordModule(Mu1,Mu2,R)
│ │ │ │  
│ │ │ │  o6 = CliffordModule{...6...}
│ │ │ │  
│ │ │ │  o6 : CliffordModule
│ │ │ │  i7 : Mor = vectorBundleOnE M.evenCenter;
│ │ │ │  i8 : Mor1= vectorBundleOnE M.oddCenter;
│ │ │ │ @@ -75,29 +75,29 @@
│ │ │ │            m12=randomExtension(m1.yAction,m2.yAction);
│ │ │ │            V = vectorBundleOnE m12;
│ │ │ │            Ul=tensorProduct(Mor,V);
│ │ │ │            Ul1=tensorProduct(Mor1,V);
│ │ │ │            d0=unique degrees target Ul.yAction;
│ │ │ │            d1=unique degrees target Ul1.yAction;
│ │ │ │            #d1 >=3 or #d0 >=3) do ();
│ │ │ │ - -- .463669s elapsed
│ │ │ │ + -- .345431s elapsed
│ │ │ │  i12 : betti Ul.yAction, betti Ul1.yAction
│ │ │ │  
│ │ │ │                 0  1          0  1
│ │ │ │  o12 = (total: 32 32, total: 32 32)
│ │ │ │            -4: 16  .     -2: 32  .
│ │ │ │            -3: 16  .     -1:  .  .
│ │ │ │            -2:  .  .      0:  .  .
│ │ │ │            -1:  . 16      1:  . 32
│ │ │ │             0:  . 16
│ │ │ │  
│ │ │ │  o12 : Sequence
│ │ │ │  i13 : elapsedTime Ul = tensorProduct(M,V); -- the heaviest part computing the
│ │ │ │  actions of generators
│ │ │ │ - -- 20.5845s elapsed
│ │ │ │ + -- 13.3419s elapsed
│ │ │ │  i14 : M1Ul=sum(#Ul.oddOperators,i->S_i*sub(Ul.oddOperators_i,S));
│ │ │ │  
│ │ │ │                32      32
│ │ │ │  o14 : Matrix S   <-- S
│ │ │ │  i15 : r=2
│ │ │ │  
│ │ │ │  o15 = 2
│ │ ├── ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_search__Ulrich.html
│ │ │ @@ -161,15 +161,15 @@
│ │ │  
│ │ │  o11 : CliffordModule
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : elapsedTime Ulr = searchUlrich(M,S);
│ │ │ - -- .684809s elapsed
    
│ │ │ + -- .548698s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : betti freeResolution Ulr
│ │ │  
│ │ │               0  1 2
│ │ │ @@ -185,15 +185,15 @@
│ │ │  
│ │ │  o14 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : elapsedTime Ulr3 = searchUlrich(M,S,3);
│ │ │ - -- 2.41016s elapsed
    
│ │ │ + -- 1.68676s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i16 : betti freeResolution Ulr3
│ │ │  
│ │ │                0  1  2
│ │ │ ├── html2text {}
│ │ │ │ @@ -64,27 +64,27 @@
│ │ │ │  o10 : Matrix S  <-- S
│ │ │ │  i11 : M=cliffordModule(Mu1,Mu2,R)
│ │ │ │  
│ │ │ │  o11 = CliffordModule{...6...}
│ │ │ │  
│ │ │ │  o11 : CliffordModule
│ │ │ │  i12 : elapsedTime Ulr = searchUlrich(M,S);
│ │ │ │ - -- .684809s elapsed
│ │ │ │ + -- .548698s elapsed
│ │ │ │  i13 : betti freeResolution Ulr
│ │ │ │  
│ │ │ │               0  1 2
│ │ │ │  o13 = total: 8 16 8
│ │ │ │            0: 8 16 8
│ │ │ │  
│ │ │ │  o13 : BettiTally
│ │ │ │  i14 : ann Ulr == ideal qs
│ │ │ │  
│ │ │ │  o14 = true
│ │ │ │  i15 : elapsedTime Ulr3 = searchUlrich(M,S,3);
│ │ │ │ - -- 2.41016s elapsed
│ │ │ │ + -- 1.68676s elapsed
│ │ │ │  i16 : betti freeResolution Ulr3
│ │ │ │  
│ │ │ │                0  1  2
│ │ │ │  o16 = total: 12 24 12
│ │ │ │            0: 12 24 12
│ │ │ │  
│ │ │ │  o16 : BettiTally
│ │ ├── ./usr/share/doc/Macaulay2/Points/example-output/_affine__Fat__Points.out
│ │ │ @@ -66,17 +66,17 @@
│ │ │  i9 : mults = {1,2,3,1,2,3,1,2,3,1,2,3}
│ │ │  
│ │ │  o9 = {1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3}
│ │ │  
│ │ │  o9 : List
│ │ │  
│ │ │  i10 : elapsedTime (Q,inG,G) = affineFatPoints(M,mults,R);
│ │ │ - -- 2.81759s elapsed
│ │ │ + -- 1.71305s elapsed
│ │ │  
│ │ │  i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R);
│ │ │ - -- 4.60723s elapsed
│ │ │ + -- 3.9836s elapsed
│ │ │  
│ │ │  i12 : G==H
│ │ │  
│ │ │  o12 = true
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/Points/html/_affine__Fat__Points.html
│ │ │ @@ -177,21 +177,21 @@
│ │ │  
│ │ │  o9 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : elapsedTime (Q,inG,G) = affineFatPoints(M,mults,R);
│ │ │ - -- 2.81759s elapsed
    
│ │ │ + -- 1.71305s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R);
│ │ │ - -- 4.60723s elapsed
    
│ │ │ + -- 3.9836s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : G==H
│ │ │  
│ │ │  o12 = true
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -81,17 +81,17 @@
│ │ │ │  o8 : Matrix K  <-- K
│ │ │ │  i9 : mults = {1,2,3,1,2,3,1,2,3,1,2,3}
│ │ │ │  
│ │ │ │  o9 = {1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3}
│ │ │ │  
│ │ │ │  o9 : List
│ │ │ │  i10 : elapsedTime (Q,inG,G) = affineFatPoints(M,mults,R);
│ │ │ │ - -- 2.81759s elapsed
│ │ │ │ + -- 1.71305s elapsed
│ │ │ │  i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R);
│ │ │ │ - -- 4.60723s elapsed
│ │ │ │ + -- 3.9836s elapsed
│ │ │ │  i12 : G==H
│ │ │ │  
│ │ │ │  o12 = true
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  For reduced points, this function may be a bit slower than _a_f_f_i_n_e_P_o_i_n_t_s.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _a_f_f_i_n_e_F_a_t_P_o_i_n_t_s_B_y_I_n_t_e_r_s_e_c_t_i_o_n_(_M_a_t_r_i_x_,_L_i_s_t_,_R_i_n_g_) -- computes ideal of fat
│ │ ├── ./usr/share/doc/Macaulay2/Posets/example-output/___Precompute.out
│ │ │ @@ -31,27 +31,27 @@
│ │ │  o5 = CacheTable{name => P}
│ │ │  
│ │ │  i6 : C == P
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : time isDistributive C
│ │ │ - -- used 1.093e-05s (cpu); 7.664e-06s (thread); 0s (gc)
│ │ │ + -- used 1.325e-05s (cpu); 6.367e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = true
│ │ │  
│ │ │  i8 : time isDistributive P
│ │ │ - -- used 6.40308s (cpu); 4.11716s (thread); 0s (gc)
│ │ │ + -- used 6.85424s (cpu); 4.29966s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : C' = dual C;
│ │ │  
│ │ │  i10 : time isDistributive C'
│ │ │ - -- used 6.472e-06s (cpu); 5.841e-06s (thread); 0s (gc)
│ │ │ + -- used 6.635e-06s (cpu); 4.866e-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 0.479688s (cpu); 0.280179s (thread); 0s (gc)
│ │ │ + -- used 0.393378s (cpu); 0.218701s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = Partition{9, 2}
│ │ │  
│ │ │  o4 : Partition
│ │ │  
│ │ │  i5 : time greeneKleitmanPartition(D, Strategy => "chains")
│ │ │ - -- used 1.3956e-05s (cpu); 1.3846e-05s (thread); 0s (gc)
│ │ │ + -- used 1.2207e-05s (cpu); 1.1426e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = Partition{9, 2}
│ │ │  
│ │ │  o5 : Partition
│ │ │  
│ │ │  i6 : greeneKleitmanPartition chain 10
│ │ ├── ./usr/share/doc/Macaulay2/Posets/html/___Precompute.html
│ │ │ @@ -107,23 +107,23 @@
│ │ │  
│ │ │  o6 = true
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time isDistributive C
│ │ │ - -- used 1.093e-05s (cpu); 7.664e-06s (thread); 0s (gc)
│ │ │ + -- used 1.325e-05s (cpu); 6.367e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : time isDistributive P
│ │ │ - -- used 6.40308s (cpu); 4.11716s (thread); 0s (gc)
│ │ │ + -- used 6.85424s (cpu); 4.29966s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    We also know that the dual of a distributive lattice is again a distributive lattice.  Other information is copied when possible.
    
│ │ │ @@ -133,15 +133,15 @@
│ │ │              
│ │ │                
    i9 : C' = dual C;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : time isDistributive C'
│ │ │ - -- used 6.472e-06s (cpu); 5.841e-06s (thread); 0s (gc)
│ │ │ + -- used 6.635e-06s (cpu); 4.866e-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 1.093e-05s (cpu); 7.664e-06s (thread); 0s (gc)
│ │ │ │ + -- used 1.325e-05s (cpu); 6.367e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = true
│ │ │ │  i8 : time isDistributive P
│ │ │ │ - -- used 6.40308s (cpu); 4.11716s (thread); 0s (gc)
│ │ │ │ + -- used 6.85424s (cpu); 4.29966s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  We also know that the dual of a distributive lattice is again a distributive
│ │ │ │  lattice. Other information is copied when possible.
│ │ │ │  i9 : C' = dual C;
│ │ │ │  i10 : time isDistributive C'
│ │ │ │ - -- used 6.472e-06s (cpu); 5.841e-06s (thread); 0s (gc)
│ │ │ │ + -- used 6.635e-06s (cpu); 4.866e-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 0.479688s (cpu); 0.280179s (thread); 0s (gc)
│ │ │ + -- used 0.393378s (cpu); 0.218701s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = Partition{9, 2}
│ │ │  
│ │ │  o4 : Partition
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time greeneKleitmanPartition(D, Strategy => "chains")
│ │ │ - -- used 1.3956e-05s (cpu); 1.3846e-05s (thread); 0s (gc)
│ │ │ + -- used 1.2207e-05s (cpu); 1.1426e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = Partition{9, 2}
│ │ │  
│ │ │  o5 : Partition
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,21 +28,21 @@
│ │ │ │  
│ │ │ │  o2 : Partition
│ │ │ │  The conjugate of $l$ has the same property, but with chains replaced by
│ │ │ │  _a_n_t_i_c_h_a_i_n_s. Because of this, it is often better to count via antichains instead
│ │ │ │  of chains. This can be done by passing "antichains" as the Strategy.
│ │ │ │  i3 : D = dominanceLattice 6;
│ │ │ │  i4 : time greeneKleitmanPartition(D, Strategy => "antichains")
│ │ │ │ - -- used 0.479688s (cpu); 0.280179s (thread); 0s (gc)
│ │ │ │ + -- used 0.393378s (cpu); 0.218701s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = Partition{9, 2}
│ │ │ │  
│ │ │ │  o4 : Partition
│ │ │ │  i5 : time greeneKleitmanPartition(D, Strategy => "chains")
│ │ │ │ - -- used 1.3956e-05s (cpu); 1.3846e-05s (thread); 0s (gc)
│ │ │ │ + -- used 1.2207e-05s (cpu); 1.1426e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = Partition{9, 2}
│ │ │ │  
│ │ │ │  o5 : Partition
│ │ │ │  The Greene-Kleitman partition of the $n$ _c_h_a_i_n is the partition of $n$ with $1$
│ │ │ │  part.
│ │ │ │  i6 : greeneKleitmanPartition chain 10
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_associated__Primes.out
│ │ │ @@ -125,24 +125,24 @@
│ │ │        -----------------------------------------------------------------------
│ │ │        ideal (a, b, c, e), ideal (a, b, d, e), ideal (a, b, c, d, e)}
│ │ │  
│ │ │  o19 : List
│ │ │  
│ │ │  i20 : M1 = set apply(L1, I -> sort flatten entries gens I)
│ │ │  
│ │ │ -o20 = set {{e, d, c, b, a}, {e, a}, {d, c, b, a}, {d, b, a}, {e, d, b, a},
│ │ │ +o20 = set {{e, c, b, a}, {d, b, a}, {d, c, b, a}, {c, b, a}, {e, a}, {e, d,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {c, b, a}, {e, c, b, a}}
│ │ │ +      b, a}, {e, d, c, b, a}}
│ │ │  
│ │ │  o20 : Set
│ │ │  
│ │ │  i21 : M2 = set apply(L2, I -> sort flatten entries gens I)
│ │ │  
│ │ │ -o21 = set {{e, d, c, b, a}, {e, a}, {d, c, b, a}, {d, b, a}, {e, d, b, a},
│ │ │ +o21 = set {{e, c, b, a}, {d, b, a}, {d, c, b, a}, {c, b, a}, {e, a}, {e, d,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {c, b, a}, {e, c, b, a}}
│ │ │ +      b, a}, {e, d, c, b, a}}
│ │ │  
│ │ │  o21 : Set
│ │ │  
│ │ │  i22 : assert(M1 === M2)
│ │ │  
│ │ │  i23 :
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_kernel__Of__Localization.out
│ │ │ @@ -24,35 +24,35 @@
│ │ │                | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                              3
│ │ │  o3 : R-module, quotient of R
│ │ │  
│ │ │  i4 : elapsedTime kernelOfLocalization(M, I1)
│ │ │ - -- .0780135s elapsed
│ │ │ + -- .0955757s elapsed
│ │ │  
│ │ │  o4 = subquotient (| 0 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
│ │ │                    | 1 0 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                    | 0 1 |  | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                                 3
│ │ │  o4 : R-module, subquotient of R
│ │ │  
│ │ │  i5 : elapsedTime kernelOfLocalization(M, I2)
│ │ │ - -- .0175249s elapsed
│ │ │ + -- .022387s elapsed
│ │ │  
│ │ │  o5 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
│ │ │                    | 0 0 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                    | 0 1 |  | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                                 3
│ │ │  o5 : R-module, subquotient of R
│ │ │  
│ │ │  i6 : elapsedTime kernelOfLocalization(M, I3)
│ │ │ - -- .126458s elapsed
│ │ │ + -- .0488983s elapsed
│ │ │  
│ │ │  o6 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
│ │ │                    | 0 1 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                    | 0 0 |  | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                                 3
│ │ │  o6 : R-module, subquotient of R
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_reg__Seq__In__Ideal.out
│ │ │ @@ -13,15 +13,15 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       x x )
│ │ │        0 4
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : elapsedTime regSeqInIdeal I
│ │ │ - -- .157633s elapsed
│ │ │ + -- .0543088s elapsed
│ │ │  
│ │ │  o3 = ideal (x x , x x  + x x , x x  + x x , x x  + x x )
│ │ │               2 7   3 6    0 7   2 5    0 7   1 4    0 7
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : R = QQ[h,l,s,x,y,z]
│ │ │ @@ -41,15 +41,15 @@
│ │ │  o5 : Ideal of R
│ │ │  
│ │ │  i6 : isSubset(I, ideal(s,l,h)) -- implies codim I == 3
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : elapsedTime regSeqInIdeal(I, 3, 3, 1)
│ │ │ - -- .00961892s elapsed
│ │ │ + -- .00777296s elapsed
│ │ │  
│ │ │                     2                3    2     2    8    3    2     2
│ │ │  o7 = ideal (h*l - l  - 4l*s + h*y, h  + l s - h x, s  + h  + l s - h x)
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_associated__Primes.html
│ │ │ @@ -286,28 +286,28 @@
│ │ │  o19 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i20 : M1 = set apply(L1, I -> sort flatten entries gens I)
│ │ │  
│ │ │ -o20 = set {{e, d, c, b, a}, {e, a}, {d, c, b, a}, {d, b, a}, {e, d, b, a},
│ │ │ +o20 = set {{e, c, b, a}, {d, b, a}, {d, c, b, a}, {c, b, a}, {e, a}, {e, d,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {c, b, a}, {e, c, b, a}}
│ │ │ +      b, a}, {e, d, c, b, a}}
│ │ │  
│ │ │  o20 : Set
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i21 : M2 = set apply(L2, I -> sort flatten entries gens I)
│ │ │  
│ │ │ -o21 = set {{e, d, c, b, a}, {e, a}, {d, c, b, a}, {d, b, a}, {e, d, b, a},
│ │ │ +o21 = set {{e, c, b, a}, {d, b, a}, {d, c, b, a}, {c, b, a}, {e, a}, {e, d,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {c, b, a}, {e, c, b, a}}
│ │ │ +      b, a}, {e, d, c, b, a}}
│ │ │  
│ │ │  o21 : Set
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i22 : assert(M1 === M2)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -155,24 +155,24 @@
│ │ │ │  o19 = {ideal (a, e), ideal (a, b, c), ideal (a, b, d), ideal (a, b, c, d),
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        ideal (a, b, c, e), ideal (a, b, d, e), ideal (a, b, c, d, e)}
│ │ │ │  
│ │ │ │  o19 : List
│ │ │ │  i20 : M1 = set apply(L1, I -> sort flatten entries gens I)
│ │ │ │  
│ │ │ │ -o20 = set {{e, d, c, b, a}, {e, a}, {d, c, b, a}, {d, b, a}, {e, d, b, a},
│ │ │ │ +o20 = set {{e, c, b, a}, {d, b, a}, {d, c, b, a}, {c, b, a}, {e, a}, {e, d,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {c, b, a}, {e, c, b, a}}
│ │ │ │ +      b, a}, {e, d, c, b, a}}
│ │ │ │  
│ │ │ │  o20 : Set
│ │ │ │  i21 : M2 = set apply(L2, I -> sort flatten entries gens I)
│ │ │ │  
│ │ │ │ -o21 = set {{e, d, c, b, a}, {e, a}, {d, c, b, a}, {d, b, a}, {e, d, b, a},
│ │ │ │ +o21 = set {{e, c, b, a}, {d, b, a}, {d, c, b, a}, {c, b, a}, {e, a}, {e, d,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {c, b, a}, {e, c, b, a}}
│ │ │ │ +      b, a}, {e, d, c, b, a}}
│ │ │ │  
│ │ │ │  o21 : Set
│ │ │ │  i22 : assert(M1 === M2)
│ │ │ │  The method using Ext modules comes from Eisenbud-Huneke-Vasconcelos, Invent.
│ │ │ │  Math 110 (1992) 207-235.
│ │ │ │  Original author (for ideals): _C_._ _Y_a_c_k_e_l. Updated for modules by J. Chen.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_kernel__Of__Localization.html
│ │ │ @@ -107,41 +107,41 @@
│ │ │                              3
│ │ │  o3 : R-module, quotient of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : elapsedTime kernelOfLocalization(M, I1)
│ │ │ - -- .0780135s elapsed
│ │ │ + -- .0955757s elapsed
│ │ │  
│ │ │  o4 = subquotient (| 0 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
│ │ │                    | 1 0 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                    | 0 1 |  | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                                 3
│ │ │  o4 : R-module, subquotient of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime kernelOfLocalization(M, I2)
│ │ │ - -- .0175249s elapsed
│ │ │ + -- .022387s elapsed
│ │ │  
│ │ │  o5 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
│ │ │                    | 0 0 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                    | 0 1 |  | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                                 3
│ │ │  o5 : R-module, subquotient of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : elapsedTime kernelOfLocalization(M, I3)
│ │ │ - -- .126458s elapsed
│ │ │ + -- .0488983s elapsed
│ │ │  
│ │ │  o6 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
│ │ │                    | 0 1 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                    | 0 0 |  | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                                 3
│ │ │  o6 : R-module, subquotient of R
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,39 +41,39 @@
│ │ │ │  |
│ │ │ │                | 0            0             0            0              x_1^5-
│ │ │ │  x_0x_2^4 |
│ │ │ │  
│ │ │ │                              3
│ │ │ │  o3 : R-module, quotient of R
│ │ │ │  i4 : elapsedTime kernelOfLocalization(M, I1)
│ │ │ │ - -- .0780135s elapsed
│ │ │ │ + -- .0955757s elapsed
│ │ │ │  
│ │ │ │  o4 = subquotient (| 0 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0
│ │ │ │  0              |)
│ │ │ │                    | 1 0 |  | 0            0             0            x_1^3-
│ │ │ │  x_0x_2^2 0              |
│ │ │ │                    | 0 1 |  | 0            0             0            0
│ │ │ │  x_1^5-x_0x_2^4 |
│ │ │ │  
│ │ │ │                                 3
│ │ │ │  o4 : R-module, subquotient of R
│ │ │ │  i5 : elapsedTime kernelOfLocalization(M, I2)
│ │ │ │ - -- .0175249s elapsed
│ │ │ │ + -- .022387s elapsed
│ │ │ │  
│ │ │ │  o5 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0
│ │ │ │  0              |)
│ │ │ │                    | 0 0 |  | 0            0             0            x_1^3-
│ │ │ │  x_0x_2^2 0              |
│ │ │ │                    | 0 1 |  | 0            0             0            0
│ │ │ │  x_1^5-x_0x_2^4 |
│ │ │ │  
│ │ │ │                                 3
│ │ │ │  o5 : R-module, subquotient of R
│ │ │ │  i6 : elapsedTime kernelOfLocalization(M, I3)
│ │ │ │ - -- .126458s elapsed
│ │ │ │ + -- .0488983s elapsed
│ │ │ │  
│ │ │ │  o6 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0
│ │ │ │  0              |)
│ │ │ │                    | 0 1 |  | 0            0             0            x_1^3-
│ │ │ │  x_0x_2^2 0              |
│ │ │ │                    | 0 0 |  | 0            0             0            0
│ │ │ │  x_1^5-x_0x_2^4 |
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_reg__Seq__In__Ideal.html
│ │ │ @@ -102,15 +102,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime regSeqInIdeal I
│ │ │ - -- .157633s elapsed
│ │ │ + -- .0543088s elapsed
│ │ │  
│ │ │  o3 = ideal (x x , x x  + x x , x x  + x x , x x  + x x )
│ │ │               2 7   3 6    0 7   2 5    0 7   1 4    0 7
│ │ │  
│ │ │  o3 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │ @@ -148,15 +148,15 @@
│ │ │  
│ │ │  o6 = true
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : elapsedTime regSeqInIdeal(I, 3, 3, 1)
│ │ │ - -- .00961892s elapsed
│ │ │ + -- .00777296s elapsed
│ │ │  
│ │ │                     2                3    2     2    8    3    2     2
│ │ │  o7 = ideal (h*l - l  - 4l*s + h*y, h  + l s - h x, s  + h  + l s - h x)
│ │ │  
│ │ │  o7 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │               2 7   0 7   3 6   2 6   1 6   0 6   2 5   0 5   3 4   2 4   1 4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       x x )
│ │ │ │        0 4
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : elapsedTime regSeqInIdeal I
│ │ │ │ - -- .157633s elapsed
│ │ │ │ + -- .0543088s elapsed
│ │ │ │  
│ │ │ │  o3 = ideal (x x , x x  + x x , x x  + x x , x x  + x x )
│ │ │ │               2 7   3 6    0 7   2 5    0 7   1 4    0 7
│ │ │ │  
│ │ │ │  o3 : Ideal of R
│ │ │ │  If I is the unit ideal, then an ideal of variables of the ring is returned.
│ │ │ │  If the codimension of I is already known, then one can specify this, along with
│ │ │ │ @@ -70,15 +70,15 @@
│ │ │ │       l , s )
│ │ │ │  
│ │ │ │  o5 : Ideal of R
│ │ │ │  i6 : isSubset(I, ideal(s,l,h)) -- implies codim I == 3
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  i7 : elapsedTime regSeqInIdeal(I, 3, 3, 1)
│ │ │ │ - -- .00961892s elapsed
│ │ │ │ + -- .00777296s elapsed
│ │ │ │  
│ │ │ │                     2                3    2     2    8    3    2     2
│ │ │ │  o7 = ideal (h*l - l  - 4l*s + h*y, h  + l s - h x, s  + h  + l s - h x)
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_a_d_i_c_a_l -- the radical of an ideal
│ │ ├── ./usr/share/doc/Macaulay2/Python/example-output/_iterator_lp__Python__Object_rp.out
│ │ │ @@ -10,12 +10,12 @@
│ │ │  
│ │ │  o2 = range(0, 3)
│ │ │  
│ │ │  o2 : PythonObject of class range
│ │ │  
│ │ │  i3 : i = iterator x
│ │ │  
│ │ │ -o3 = 
│ │ │ +o3 = 
│ │ │  
│ │ │  o3 : PythonObject of class range_iterator
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Python/example-output/_next_lp__Python__Object_rp.out
│ │ │ @@ -10,15 +10,15 @@
│ │ │  
│ │ │  o2 = range(0, 3)
│ │ │  
│ │ │  o2 : PythonObject of class range
│ │ │  
│ │ │  i3 : i = iterator x
│ │ │  
│ │ │ -o3 = 
│ │ │ +o3 = 
│ │ │  
│ │ │  o3 : PythonObject of class range_iterator
│ │ │  
│ │ │  i4 : next i
│ │ │  
│ │ │  o4 = 0
│ │ ├── ./usr/share/doc/Macaulay2/Python/example-output/_python__Run__Script.out
│ │ │ @@ -1,22 +1,22 @@
│ │ │  -- -*- M2-comint -*- hash: 447449196062331972
│ │ │  
│ │ │  i1 : pyfile = temporaryFileName() | ".py"
│ │ │  
│ │ │ -o1 = /tmp/M2-47275-0/0.py
│ │ │ +o1 = /tmp/M2-73748-0/0.py
│ │ │  
│ │ │  i2 : pyfile << "import math" << endl
│ │ │  
│ │ │ -o2 = /tmp/M2-47275-0/0.py
│ │ │ +o2 = /tmp/M2-73748-0/0.py
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : pyfile << "x = math.sin(3.4)" << endl << close
│ │ │  
│ │ │ -o3 = /tmp/M2-47275-0/0.py
│ │ │ +o3 = /tmp/M2-73748-0/0.py
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : get pyfile
│ │ │  
│ │ │  o4 = import math
│ │ │       x = math.sin(3.4)
│ │ ├── ./usr/share/doc/Macaulay2/Python/example-output/_to__Python.out
│ │ │ @@ -72,15 +72,15 @@
│ │ │  
│ │ │  o12 = m2sqrt
│ │ │  
│ │ │  o12 : FunctionClosure
│ │ │  
│ │ │  i13 : pysqrt = toPython m2sqrt
│ │ │  
│ │ │ -o13 = 
│ │ │ +o13 = 
│ │ │  
│ │ │  o13 : PythonObject of class builtin_function_or_method
│ │ │  
│ │ │  i14 : pysqrt 2
│ │ │  calling Macaulay2 code from Python!
│ │ │  
│ │ │  o14 = 1.4142135623730951
│ │ ├── ./usr/share/doc/Macaulay2/Python/example-output/_use_lp__Python__Context_rp.out
│ │ │ @@ -30,15 +30,15 @@
│ │ │  
│ │ │  o7 : Symbol
│ │ │  
│ │ │  i8 : use ctx
│ │ │  
│ │ │  i9 : f
│ │ │  
│ │ │ -o9 =  at 0x7fdae521ccc0>
│ │ │ +o9 =  at 0x7fd4a13dccc0>
│ │ │  
│ │ │  o9 : PythonObject of class function
│ │ │  
│ │ │  i10 : x
│ │ │  
│ │ │  o10 = 5
│ │ ├── ./usr/share/doc/Macaulay2/Python/html/_iterator_lp__Python__Object_rp.html
│ │ │ @@ -90,15 +90,15 @@
│ │ │  o2 : PythonObject of class range
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : i = iterator x
│ │ │  
│ │ │ -o3 = <range_iterator object at 0x7fdae5265860>
│ │ │ +o3 = <range_iterator object at 0x7fd4a14250b0>
│ │ │  
│ │ │  o3 : PythonObject of class range_iterator
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,15 +22,15 @@
│ │ │ │  i2 : x = builtins@@range 3
│ │ │ │  
│ │ │ │  o2 = range(0, 3)
│ │ │ │  
│ │ │ │  o2 : PythonObject of class range
│ │ │ │  i3 : i = iterator x
│ │ │ │  
│ │ │ │ -o3 = 
│ │ │ │ +o3 = 
│ │ │ │  
│ │ │ │  o3 : PythonObject of class range_iterator
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _n_e_x_t_(_P_y_t_h_o_n_O_b_j_e_c_t_) -- retrieve the next item from a python iterator
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _i_t_e_r_a_t_o_r_(_P_y_t_h_o_n_O_b_j_e_c_t_) -- get iterator of iterable python object
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/Python/html/_next_lp__Python__Object_rp.html
│ │ │ @@ -86,15 +86,15 @@
│ │ │  o2 : PythonObject of class range
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : i = iterator x
│ │ │  
│ │ │ -o3 = <range_iterator object at 0x7fdae5259ec0>
│ │ │ +o3 = <range_iterator object at 0x7fd4a14196b0>
│ │ │  
│ │ │  o3 : PythonObject of class range_iterator
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : next i
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,15 +21,15 @@
│ │ │ │  i2 : x = builtins@@range 3
│ │ │ │  
│ │ │ │  o2 = range(0, 3)
│ │ │ │  
│ │ │ │  o2 : PythonObject of class range
│ │ │ │  i3 : i = iterator x
│ │ │ │  
│ │ │ │ -o3 = 
│ │ │ │ +o3 = 
│ │ │ │  
│ │ │ │  o3 : PythonObject of class range_iterator
│ │ │ │  i4 : next i
│ │ │ │  
│ │ │ │  o4 = 0
│ │ │ │  
│ │ │ │  o4 : PythonObject of class int
│ │ ├── ./usr/share/doc/Macaulay2/Python/html/_python__Run__Script.html
│ │ │ @@ -76,31 +76,31 @@
│ │ │            
    The return value is a Python dictionary containing all the variables defined in the global scope.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : pyfile = temporaryFileName() | ".py"
│ │ │  
│ │ │ -o1 = /tmp/M2-47275-0/0.py
    
│ │ │ +o1 = /tmp/M2-73748-0/0.py
│ │ │              
    
│ │ │                
    i2 : pyfile << "import math" << endl
│ │ │  
│ │ │ -o2 = /tmp/M2-47275-0/0.py
│ │ │ +o2 = /tmp/M2-73748-0/0.py
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : pyfile << "x = math.sin(3.4)" << endl << close
│ │ │  
│ │ │ -o3 = /tmp/M2-47275-0/0.py
│ │ │ +o3 = /tmp/M2-73748-0/0.py
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : get pyfile
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,23 +16,23 @@
│ │ │ │  Execute a sequence of statements as if they were read from a Python file. This
│ │ │ │  is for multi-line code that might contain definitions, control structures,
│ │ │ │  imports, etc. It is great for running Python code from a file.
│ │ │ │  The return value is a Python dictionary containing all the variables defined in
│ │ │ │  the global scope.
│ │ │ │  i1 : pyfile = temporaryFileName() | ".py"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-47275-0/0.py
│ │ │ │ +o1 = /tmp/M2-73748-0/0.py
│ │ │ │  i2 : pyfile << "import math" << endl
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-47275-0/0.py
│ │ │ │ +o2 = /tmp/M2-73748-0/0.py
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : pyfile << "x = math.sin(3.4)" << endl << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-47275-0/0.py
│ │ │ │ +o3 = /tmp/M2-73748-0/0.py
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : get pyfile
│ │ │ │  
│ │ │ │  o4 = import math
│ │ │ │       x = math.sin(3.4)
│ │ │ │  i5 : pythonRunScript oo
│ │ ├── ./usr/share/doc/Macaulay2/Python/html/_to__Python.html
│ │ │ @@ -181,15 +181,15 @@
│ │ │  o12 : FunctionClosure
    
│ │ │              
    
│ │ │                
    i13 : pysqrt = toPython m2sqrt
│ │ │  
│ │ │ -o13 = <built-in method m2sqrt of PyCapsule object at 0x7fdae523ac50>
│ │ │ +o13 = <built-in method m2sqrt of PyCapsule object at 0x7fd4a13fec00>
│ │ │  
│ │ │  o13 : PythonObject of class builtin_function_or_method
    
│ │ │              
    
│ │ │                
    i14 : pysqrt 2
│ │ │ ├── html2text {}
│ │ │ │ @@ -72,15 +72,15 @@
│ │ │ │            sqrt x)
│ │ │ │  
│ │ │ │  o12 = m2sqrt
│ │ │ │  
│ │ │ │  o12 : FunctionClosure
│ │ │ │  i13 : pysqrt = toPython m2sqrt
│ │ │ │  
│ │ │ │ -o13 = 
│ │ │ │ +o13 = 
│ │ │ │  
│ │ │ │  o13 : PythonObject of class builtin_function_or_method
│ │ │ │  i14 : pysqrt 2
│ │ │ │  calling Macaulay2 code from Python!
│ │ │ │  
│ │ │ │  o14 = 1.4142135623730951
│ │ ├── ./usr/share/doc/Macaulay2/Python/html/_use_lp__Python__Context_rp.html
│ │ │ @@ -124,15 +124,15 @@
│ │ │                
    i8 : use ctx
    
│ │ │              
    
│ │ │                
    i9 : f
│ │ │  
│ │ │ -o9 = <function <lambda> at 0x7fdae521ccc0>
│ │ │ +o9 = <function <lambda> at 0x7fd4a13dccc0>
│ │ │  
│ │ │  o9 : PythonObject of class function
    
│ │ │              
    
│ │ │                
    i10 : x
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │  
│ │ │ │  o7 = y
│ │ │ │  
│ │ │ │  o7 : Symbol
│ │ │ │  i8 : use ctx
│ │ │ │  i9 : f
│ │ │ │  
│ │ │ │ -o9 =  at 0x7fdae521ccc0>
│ │ │ │ +o9 =  at 0x7fd4a13dccc0>
│ │ │ │  
│ │ │ │  o9 : PythonObject of class function
│ │ │ │  i10 : x
│ │ │ │  
│ │ │ │  o10 = 5
│ │ │ │  
│ │ │ │  o10 : PythonObject of class int
│ │ ├── ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.out
│ │ │ @@ -180,15 +180,15 @@
│ │ │  i21 : L = trim groebnerStratum F;
│ │ │  
│ │ │  o21 : Ideal of T
│ │ │  
│ │ │  i22 : assert(dim L == 18)
│ │ │  
│ │ │  i23 : elapsedTime isPrime L
│ │ │ - -- 2.78979s elapsed
│ │ │ + -- 2.37585s elapsed
│ │ │  
│ │ │  o23 = true
│ │ │  
│ │ │  i24 : I = pointsIdeal randomPoints(S, 6)
│ │ │  
│ │ │                               2                              2   2          
│ │ │  o24 = ideal (a*c - 7b*c - 49c  + 40a*d - 42b*d + 12c*d + 28d , b  - 36b*c -
│ │ │ @@ -302,15 +302,15 @@
│ │ │  o38 = true
│ │ │  
│ │ │  i39 : L441 = trim(L + ideal M1);
│ │ │  
│ │ │  o39 : Ideal of T
│ │ │  
│ │ │  i40 : elapsedTime compsL441 = decompose L441;
│ │ │ - -- 2.4524s elapsed
│ │ │ + -- 1.20257s elapsed
│ │ │  
│ │ │  i41 : #compsL441
│ │ │  
│ │ │  o41 = 2
│ │ │  
│ │ │  i42 : compsL441/dim -- two components, of dimensions 14 and 16.
│ │ │  
│ │ │ @@ -320,37 +320,37 @@
│ │ │  
│ │ │  i43 : compsL441/dim == {16, 14}
│ │ │  
│ │ │  o43 = true
│ │ │  
│ │ │  i44 : pta = randomPointOnRationalVariety compsL441_0
│ │ │  
│ │ │ -o44 = | -27 -13 45 -25 3 38 -20 -30 -41 25 -26 -44 -31 5 14 2 -45 45 21 -27
│ │ │ +o44 = | 32 -41 22 15 22 -46 43 42 -27 -27 -13 10 -24 19 -25 48 31 10 41 49 39
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -23 -29 34 49 32 19 10 26 19 37 15 -28 -50 -10 -32 18 |
│ │ │ +      -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  + 14b*c + 25c  - 44a*d + 38b*d + 45c*d - 27d , a*b + 32b*c +
│ │ │ +o45 = ideal (a  - 25b*c - 27c  + 10a*d - 46b*d + 22c*d + 32d , a*b - 48b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │           2                              2   2              2                
│ │ │ -      21c  - 23a*d - 31b*d - 41c*d - 13d , b  - 32b*c + 26c  - 50a*d - 29b*d
│ │ │ +      41c  + 39a*d - 24b*d - 27c*d - 41d , b  + 26b*c + 18c  - 32a*d - 28b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                  2                   2                            2     2  
│ │ │ -      + 2c*d - 20d , a*c - 28b*c + 19c  + 19a*d - 27b*d + 5c*d + 3d , b*c  +
│ │ │ +                   2                  2                              2     2
│ │ │ +      + 48c*d + 43d , a*c + 37b*c + 5c  + 45a*d + 49b*d + 19c*d + 22d , b*c 
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2         2        2        2      3   3                2 
│ │ │ -      37b*c*d + 34c d + 10a*d  - 45b*d  - 26c*d  - 25d , c  + 18b*c*d + 15c d
│ │ │ +                     2         2        2        2      3   3            
│ │ │ +      + 19b*c*d - 29c d + 15a*d  + 31b*d  - 13c*d  + 15d , c  - 50b*c*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2        2      3
│ │ │ -      - 10a*d  + 49b*d  + 45c*d  - 30d )
│ │ │ +         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 - 37d, a + 4d)                                                                                                                              |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                             2              2                      2   3                2         2        2      3     2                2         2        2 |
│ │ │ -      |ideal (a - 28b + 19c + 48d, b  - 32b*c + 26c  - 15b*d + 43c*d - 44d , c  + 18b*c*d + 15c d - 29b*d  + 33c*d  + 46d , b*c  + 37b*c*d + 34c d + 33b*d  - 14c*d )|
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      +----------------------------------------------------------------------------------------------+
│ │ │ +o47 = |ideal (c + 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 - 37d, a + 4d), ideal (a - 28b + 19c + 48d, b  -
│ │ │ +                                                                            
│ │ │ +o48 = {ideal (c + 45d, b - 34d, a - 35d), ideal (c + 18d, b - 12d, a + 47d),
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2                      2   3                2         2  
│ │ │ -      32b*c + 26c  - 15b*d + 43c*d - 44d , c  + 18b*c*d + 15c d - 29b*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
│ │ │ -      33c*d  + 46d , b*c  + 37b*c*d + 34c d + 33b*d  - 14c*d )}
│ │ │ +                           2
│ │ │ +      - 14b*d + 27c*d - 38d )}
│ │ │  
│ │ │  o48 : List
│ │ │  
│ │ │  i49 : lin = CFa_1_0 -- a linear form, defining a plane.
│ │ │  
│ │ │ -o49 = a - 28b + 19c + 48d
│ │ │ +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 = | 31 42 28 25 19 3 43 -7 -3 -42 -29 -29 14 2 50 5 36 -13 -42 47 13 31
│ │ │ +o53 = | -31 -3 -29 -17 -21 5 -32 33 -24 2 26 -26 -45 -4 16 -22 2 -37 16 -23
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -37 -23 -24 -4 38 -29 -23 21 17 9 0 21 -9 -47 |
│ │ │ +      -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  + 50b*c - 42c  - 29a*d + 3b*d + 28c*d + 31d , a*b - 24b*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                 
│ │ │ -      42c  + 13a*d + 14b*d - 3c*d + 42d , b  - 9b*c - 29c  + 31b*d + 5c*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            
│ │ │ -      43d , a*c + 9b*c - 4c  - 23a*d + 47b*d + 2c*d + 19d , b*c  + 21b*c*d -
│ │ │ +         2                  2                             2     2           
│ │ │ +      32d , a*c - 13b*c + 2c  - 15a*d - 23b*d - 4c*d - 21d , b*c  - 9b*c*d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2         2        2        2      3   3                2         2
│ │ │ -      37c d + 38a*d  + 36b*d  - 29c*d  + 25d , c  - 47b*c*d + 17c d + 21a*d 
│ │ │ +         2         2       2        2      3   3                2         2  
│ │ │ +      29c d + 17a*d  + 2b*d  + 26c*d  - 17d , c  + 21b*c*d - 47c d + 38a*d  +
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2     3
│ │ │ -      - 23b*d  - 13c*d  - 7d )
│ │ │ +           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
│ │ │ @@ -440,114 +442,84 @@
│ │ │            1: . 4 4 1
│ │ │            2: . 2 4 2
│ │ │  
│ │ │  o55 : BettiTally
│ │ │  
│ │ │  i56 : netList decompose Fb --
│ │ │  
│ │ │ -      +-------------------------------------------------------+
│ │ │ -o56 = |ideal (c - 45d, b + 16d, a + 38d)                      |
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |ideal (c + 43d, b + 10d, a + 8d)                       |
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |ideal (c + 34d, b + 15d, a + 28d)                      |
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |ideal (c + 11d, b + 39d, a + 23d)                      |
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |                                      2              2 |
│ │ │ -      |ideal (b - 32c + 42d, a - 19c - 16d, c  - 28c*d - 40d )|
│ │ │ -      +-------------------------------------------------------+
│ │ │ +      +---------------------------------------------------------------+
│ │ │ +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                         2   2                      2                                                                                                             |
│ │ │ -o57 = |ideal (a - 7b + 32c + d, c  + 42b*d + 33c*d - 10d , b*c - b*d + 13c*d + 18d , b  + 28b*d - 32c*d + 16d )                                                                                                            |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                          2                      2                           2   2                      2                                                                                                           |
│ │ │ -      |ideal (a - 7b + 32c + d, c  + 40b*d - 36c*d + 33d , b*c + 45b*d - 16c*d + 39d , b  - 20b*d + 29c*d + 38d )                                                                                                          |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                          2                      2                           2   2                     2                                                                                                            |
│ │ │ -      |ideal (a - 7b + 32c + d, c  - 10b*d + 17c*d - 21d , b*c - 17b*d - 23c*d - 32d , b  - 8b*d - 12c*d - 46d )                                                                                                           |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                                     3      2         2      3                                                                                                                                                      |
│ │ │ -      |ideal (b + 23c - 11d, a - 9c + 25d, c  - 13c d - 14c*d  + 23d )                                                                                                                                                     |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                                     2                      2   2      2                      2   3      2         2        2      3                                                                                |
│ │ │ -      |ideal (a + 48b - 40c - 20d, b*c - 32c  + 43b*d - 21c*d - 12d , b  - 14c  + 14b*d + 18c*d + 36d , c  + 28c d - 20b*d  + 42c*d  - 50d )                                                                               |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                                   2                      2   2      2                      2   3      2         2        2     3                                                                                   |
│ │ │ -      |ideal (a + b + 50c + 26d, b*c - 32c  + 34b*d - 36c*d + 14d , b  - 14c  + 34b*d - 16c*d - 33d , c  + 28c d + 39b*d  - 28c*d  + 4d )                                                                                  |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |        2                            2                                 2                                 2   2                              2                                  2   2                              2 |
│ │ │ -      |ideal (c  - 7a*d - 19b*d + 6c*d - 19d , b*c - 5a*d + 49b*d - 4c*d + 50d , a*c - 6a*d + 35b*d - 39c*d - 2d , b  - 46a*d + 22b*d + 42c*d + 43d , a*b + 3a*d - 12b*d - 49c*d + 40d , a  + 28a*d - 13b*d - 25c*d - 35d )|
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                                     2                     2   2      2                      2   3      2         2        2     3                                                                                  |
│ │ │ -      |ideal (a - 46b + 39c - 29d, b*c - 32c  + 11b*d - 7c*d - 43d , b  - 14c  + 29b*d + 43c*d - 41d , c  + 28c d + 46b*d  - 50c*d  - 5d )                                                                                 |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |        2                              2                                2                                   2   2                              2                                 2   2                           2  |
│ │ │ -      |ideal (c  + 15a*d + 27b*d + 35c*d + 46d , b*c - 6a*d + b*d + 36c*d - 31d , a*c - 10a*d + 45b*d + 20c*d - 23d , b  - 23a*d + 15b*d + 31c*d - 13d , a*b - 6a*d - 40b*d + 8c*d + 18d , a  - 8a*d - 24b*d + c*d - 22d ) |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |        2                              2                                2                                  2   2                      2                                   2   2                              2      |
│ │ │ -      |ideal (c  + 37a*d + 25b*d - 16c*d + 14d , b*c - 7a*d + 47b*d - 3c*d - 2d , a*c - 14a*d + 27b*d - 35c*d - 8d , b  - 33b*d + 19c*d + 27d , a*b - 15a*d - 30b*d - 40c*d - 24d , a  - 44a*d + 16b*d + 11c*d + 12d )     |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      +-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                       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 = | 13 17 -19 -1 -9 -15 -28 -39 -36 1 -47 29 37 -40 35 -31 12 -21 -8 -13
│ │ │ +o58 = | 49 0 -30 -36 -1 0 -9 17 37 29 34 13 19 8 -10 -47 21 -24 -44 42 9 46
│ │ │        -----------------------------------------------------------------------
│ │ │ -      14 15 -23 39 11 8 -24 -13 -42 -2 18 46 -18 -29 -33 -22 |
│ │ │ +      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 = | -45 18 -9 38 21 29 50 -8 -5 45 -47 -26 37 -35 -21 28 27 46 -17 -49
│ │ │ +o59 = | -5 -40 -6 3 -28 -8 -25 15 15 29 26 -37 11 -14 31 14 1 -50 43 37 5 50
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -23 15 -50 37 -39 -14 21 10 -31 3 -18 32 0 3 -15 33 |
│ │ │ +      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  + 35b*c + c  + 29a*d - 15b*d - 19c*d + 13d , a*b + 11b*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                  
│ │ │ -      8c  + 14a*d + 37b*d - 36c*d + 17d , b  - 33b*c - 13c  - 18a*d + 15b*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  
│ │ │ -      31c*d - 28d , a*c + 46b*c + 8c  - 42a*d - 13b*d - 40c*d - 9d , 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   
│ │ │ -      2b*c*d - 23c d - 24a*d  + 12b*d  - 47c*d  - d , c  - 22b*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  + 39b*d  - 21c*d  - 39d )
│ │ │ +             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  - 21b*c + 45c  - 26a*d + 29b*d - 9c*d - 45d , a*b - 39b*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                  
│ │ │ -      17c  - 23a*d + 37b*d - 5c*d + 18d , b  - 15b*c + 10c  + 15b*d + 28c*d +
│ │ │ +                                  2   2              2                  
│ │ │ +      + 5a*d + 11b*d + 15c*d - 40d , b  - 49b*c + 32c  + 50b*d + 14c*d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                   2                              2     2         
│ │ │ -      50d , a*c + 32b*c - 14c  - 31a*d - 49b*d - 35c*d + 21d , b*c  + 3b*c*d
│ │ │ +         2                   2                             2     2          
│ │ │ +      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
│ │ │ -      - 50c d + 21a*d  + 27b*d  - 47c*d  + 38d , c  + 33b*c*d - 18c d + 3a*d 
│ │ │ +           2         2      2        2     3   3               2         2  
│ │ │ +      + 10c d - 18a*d  + b*d  + 26c*d  + 3d , c  + 3b*c*d + 33c d + 21a*d  +
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2     3
│ │ │ -      + 37b*d  + 46c*d  - 8d )
│ │ │ +          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
│ │ │ @@ -565,37 +537,33 @@
│ │ │            1: . 4 4 1
│ │ │            2: . 2 4 2
│ │ │  
│ │ │  o63 : BettiTally
│ │ │  
│ │ │  i64 : netList decompose I0
│ │ │  
│ │ │ -      +---------------------------------------------------------------------------------------------+
│ │ │ -o64 = |ideal (c - 42d, b - 6d, a + 41d)                                                             |
│ │ │ -      +---------------------------------------------------------------------------------------------+
│ │ │ -      |ideal (c + 38d, b - 41d, a - 41d)                                                            |
│ │ │ -      +---------------------------------------------------------------------------------------------+
│ │ │ -      |                                  2                      2   2      2                      2 |
│ │ │ -      |ideal (a + 46b + 8c - 7d, b*c - 7c  + 13b*d + 16c*d - 21d , b  - 42c  - 41b*d + 35c*d - 39d )|
│ │ │ -      +---------------------------------------------------------------------------------------------+
│ │ │ +      +---------------------------------------------------+
│ │ │ +o64 = |ideal (c - 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 + 32d, b + 18d, a - 33d)                     |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |ideal (c + 29d, b - 8d, a + 50d)                      |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |ideal (c + 16d, b + 39d, a - 32d)                     |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |ideal (c + 5d, b - 14d, a + 7d)                       |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |                                     2              2 |
│ │ │ -      |ideal (b - 40c + 5d, a - 47c + 24d, c  - 27c*d + 15d )|
│ │ │ -      +------------------------------------------------------+
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                        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
│ │ │ - -- 2.78979s elapsed
│ │ │ + -- 2.37585s elapsed
│ │ │  
│ │ │  o23 = true
    
│ │ │              
    
│ │ │          
    
│ │ │            
    The Schreyer resolution and minimal Betti numbers
    
│ │ │ @@ -556,15 +556,15 @@
│ │ │  
│ │ │  o39 : Ideal of T    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i40 : elapsedTime compsL441 = decompose L441;
│ │ │ - -- 2.4524s elapsed
    
│ │ │ + -- 1.20257s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i41 : #compsL441
│ │ │  
│ │ │  o41 = 2
    
│ │ │ @@ -591,40 +591,40 @@
│ │ │            
    Both components are rational, and here are random points, one on each component:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i44 : pta = randomPointOnRationalVariety compsL441_0
│ │ │  
│ │ │ -o44 = | -27 -13 45 -25 3 38 -20 -30 -41 25 -26 -44 -31 5 14 2 -45 45 21 -27
│ │ │ +o44 = | 32 -41 22 15 22 -46 43 42 -27 -27 -13 10 -24 19 -25 48 31 10 41 49 39
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -23 -29 34 49 32 19 10 26 19 37 15 -28 -50 -10 -32 18 |
│ │ │ +      -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  + 14b*c + 25c  - 44a*d + 38b*d + 45c*d - 27d , a*b + 32b*c +
│ │ │ +o45 = ideal (a  - 25b*c - 27c  + 10a*d - 46b*d + 22c*d + 32d , a*b - 48b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │           2                              2   2              2                
│ │ │ -      21c  - 23a*d - 31b*d - 41c*d - 13d , b  - 32b*c + 26c  - 50a*d - 29b*d
│ │ │ +      41c  + 39a*d - 24b*d - 27c*d - 41d , b  + 26b*c + 18c  - 32a*d - 28b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                  2                   2                            2     2  
│ │ │ -      + 2c*d - 20d , a*c - 28b*c + 19c  + 19a*d - 27b*d + 5c*d + 3d , b*c  +
│ │ │ +                   2                  2                              2     2
│ │ │ +      + 48c*d + 43d , a*c + 37b*c + 5c  + 45a*d + 49b*d + 19c*d + 22d , b*c 
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2         2        2        2      3   3                2 
│ │ │ -      37b*c*d + 34c d + 10a*d  - 45b*d  - 26c*d  - 25d , c  + 18b*c*d + 15c d
│ │ │ +                     2         2        2        2      3   3            
│ │ │ +      + 19b*c*d - 29c d + 15a*d  + 31b*d  - 13c*d  + 15d , c  - 50b*c*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2        2      3
│ │ │ -      - 10a*d  + 49b*d  + 45c*d  - 30d )
│ │ │ +         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 - 37d, a + 4d)                                                                                                                              |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                             2              2                      2   3                2         2        2      3     2                2         2        2 |
│ │ │ -      |ideal (a - 28b + 19c + 48d, b  - 32b*c + 26c  - 15b*d + 43c*d - 44d , c  + 18b*c*d + 15c d - 29b*d  + 33c*d  + 46d , b*c  + 37b*c*d + 34c d + 33b*d  - 14c*d )|
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------+
    
│ │ │ +      +----------------------------------------------------------------------------------------------+
│ │ │ +o47 = |ideal (c + 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 - 37d, a + 4d), ideal (a - 28b + 19c + 48d, b  -
│ │ │ +                                                                            
│ │ │ +o48 = {ideal (c + 45d, b - 34d, a - 35d), ideal (c + 18d, b - 12d, a + 47d),
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2                      2   3                2         2  
│ │ │ -      32b*c + 26c  - 15b*d + 43c*d - 44d , c  + 18b*c*d + 15c d - 29b*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
│ │ │ -      33c*d  + 46d , b*c  + 37b*c*d + 34c d + 33b*d  - 14c*d )}
│ │ │ +                           2
│ │ │ +      - 14b*d + 27c*d - 38d )}
│ │ │  
│ │ │  o48 : List
    
│ │ │              
    
│ │ │                
    i49 : lin = CFa_1_0 -- a linear form, defining a plane.
│ │ │  
│ │ │ -o49 = a - 28b + 19c + 48d
│ │ │ +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 = | 31 42 28 25 19 3 43 -7 -3 -42 -29 -29 14 2 50 5 36 -13 -42 47 13 31
│ │ │ +o53 = | -31 -3 -29 -17 -21 5 -32 33 -24 2 26 -26 -45 -4 16 -22 2 -37 16 -23
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -37 -23 -24 -4 38 -29 -23 21 17 9 0 21 -9 -47 |
│ │ │ +      -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  + 50b*c - 42c  - 29a*d + 3b*d + 28c*d + 31d , a*b - 24b*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                 
│ │ │ -      42c  + 13a*d + 14b*d - 3c*d + 42d , b  - 9b*c - 29c  + 31b*d + 5c*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            
│ │ │ -      43d , a*c + 9b*c - 4c  - 23a*d + 47b*d + 2c*d + 19d , b*c  + 21b*c*d -
│ │ │ +         2                  2                             2     2           
│ │ │ +      32d , a*c - 13b*c + 2c  - 15a*d - 23b*d - 4c*d - 21d , b*c  - 9b*c*d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2         2        2        2      3   3                2         2
│ │ │ -      37c d + 38a*d  + 36b*d  - 29c*d  + 25d , c  - 47b*c*d + 17c d + 21a*d 
│ │ │ +         2         2       2        2      3   3                2         2  
│ │ │ +      29c d + 17a*d  + 2b*d  + 26c*d  - 17d , c  + 21b*c*d - 47c d + 38a*d  +
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2     3
│ │ │ -      - 23b*d  - 13c*d  - 7d )
│ │ │ +           2        2      3
│ │ │ +      21b*d  - 37c*d  + 33d )
│ │ │  
│ │ │  o54 : Ideal of S
    
│ │ │              
    
│ │ │                
    i55 : betti res Fb
│ │ │ @@ -749,136 +751,106 @@
│ │ │  o55 : BettiTally
    
│ │ │              
    
│ │ │                
    i56 : netList decompose Fb --
│ │ │  
│ │ │ -      +-------------------------------------------------------+
│ │ │ -o56 = |ideal (c - 45d, b + 16d, a + 38d)                      |
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |ideal (c + 43d, b + 10d, a + 8d)                       |
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |ideal (c + 34d, b + 15d, a + 28d)                      |
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |ideal (c + 11d, b + 39d, a + 23d)                      |
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |                                      2              2 |
│ │ │ -      |ideal (b - 32c + 42d, a - 19c - 16d, c  - 28c*d - 40d )|
│ │ │ -      +-------------------------------------------------------+
    
│ │ │ +      +---------------------------------------------------------------+
│ │ │ +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                         2   2                      2                                                                                                             |
│ │ │ -o57 = |ideal (a - 7b + 32c + d, c  + 42b*d + 33c*d - 10d , b*c - b*d + 13c*d + 18d , b  + 28b*d - 32c*d + 16d )                                                                                                            |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                          2                      2                           2   2                      2                                                                                                           |
│ │ │ -      |ideal (a - 7b + 32c + d, c  + 40b*d - 36c*d + 33d , b*c + 45b*d - 16c*d + 39d , b  - 20b*d + 29c*d + 38d )                                                                                                          |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                          2                      2                           2   2                     2                                                                                                            |
│ │ │ -      |ideal (a - 7b + 32c + d, c  - 10b*d + 17c*d - 21d , b*c - 17b*d - 23c*d - 32d , b  - 8b*d - 12c*d - 46d )                                                                                                           |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                                     3      2         2      3                                                                                                                                                      |
│ │ │ -      |ideal (b + 23c - 11d, a - 9c + 25d, c  - 13c d - 14c*d  + 23d )                                                                                                                                                     |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                                     2                      2   2      2                      2   3      2         2        2      3                                                                                |
│ │ │ -      |ideal (a + 48b - 40c - 20d, b*c - 32c  + 43b*d - 21c*d - 12d , b  - 14c  + 14b*d + 18c*d + 36d , c  + 28c d - 20b*d  + 42c*d  - 50d )                                                                               |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                                   2                      2   2      2                      2   3      2         2        2     3                                                                                   |
│ │ │ -      |ideal (a + b + 50c + 26d, b*c - 32c  + 34b*d - 36c*d + 14d , b  - 14c  + 34b*d - 16c*d - 33d , c  + 28c d + 39b*d  - 28c*d  + 4d )                                                                                  |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |        2                            2                                 2                                 2   2                              2                                  2   2                              2 |
│ │ │ -      |ideal (c  - 7a*d - 19b*d + 6c*d - 19d , b*c - 5a*d + 49b*d - 4c*d + 50d , a*c - 6a*d + 35b*d - 39c*d - 2d , b  - 46a*d + 22b*d + 42c*d + 43d , a*b + 3a*d - 12b*d - 49c*d + 40d , a  + 28a*d - 13b*d - 25c*d - 35d )|
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                                     2                     2   2      2                      2   3      2         2        2     3                                                                                  |
│ │ │ -      |ideal (a - 46b + 39c - 29d, b*c - 32c  + 11b*d - 7c*d - 43d , b  - 14c  + 29b*d + 43c*d - 41d , c  + 28c d + 46b*d  - 50c*d  - 5d )                                                                                 |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |        2                              2                                2                                   2   2                              2                                 2   2                           2  |
│ │ │ -      |ideal (c  + 15a*d + 27b*d + 35c*d + 46d , b*c - 6a*d + b*d + 36c*d - 31d , a*c - 10a*d + 45b*d + 20c*d - 23d , b  - 23a*d + 15b*d + 31c*d - 13d , a*b - 6a*d - 40b*d + 8c*d + 18d , a  - 8a*d - 24b*d + c*d - 22d ) |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |        2                              2                                2                                  2   2                      2                                   2   2                              2      |
│ │ │ -      |ideal (c  + 37a*d + 25b*d - 16c*d + 14d , b*c - 7a*d + 47b*d - 3c*d - 2d , a*c - 14a*d + 27b*d - 35c*d - 8d , b  - 33b*d + 19c*d + 27d , a*b - 15a*d - 30b*d - 40c*d - 24d , a  - 44a*d + 16b*d + 11c*d + 12d )     |
│ │ │ -      +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
    
│ │ │ +      +-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                       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 = | 13 17 -19 -1 -9 -15 -28 -39 -36 1 -47 29 37 -40 35 -31 12 -21 -8 -13
│ │ │ +o58 = | 49 0 -30 -36 -1 0 -9 17 37 29 34 13 19 8 -10 -47 21 -24 -44 42 9 46
│ │ │        -----------------------------------------------------------------------
│ │ │ -      14 15 -23 39 11 8 -24 -13 -42 -2 18 46 -18 -29 -33 -22 |
│ │ │ +      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 = | -45 18 -9 38 21 29 50 -8 -5 45 -47 -26 37 -35 -21 28 27 46 -17 -49
│ │ │ +o59 = | -5 -40 -6 3 -28 -8 -25 15 15 29 26 -37 11 -14 31 14 1 -50 43 37 5 50
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -23 15 -50 37 -39 -14 21 10 -31 3 -18 32 0 3 -15 33 |
│ │ │ +      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  + 35b*c + c  + 29a*d - 15b*d - 19c*d + 13d , a*b + 11b*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                  
│ │ │ -      8c  + 14a*d + 37b*d - 36c*d + 17d , b  - 33b*c - 13c  - 18a*d + 15b*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  
│ │ │ -      31c*d - 28d , a*c + 46b*c + 8c  - 42a*d - 13b*d - 40c*d - 9d , 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   
│ │ │ -      2b*c*d - 23c d - 24a*d  + 12b*d  - 47c*d  - d , c  - 22b*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  + 39b*d  - 21c*d  - 39d )
│ │ │ +             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  - 21b*c + 45c  - 26a*d + 29b*d - 9c*d - 45d , a*b - 39b*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                  
│ │ │ -      17c  - 23a*d + 37b*d - 5c*d + 18d , b  - 15b*c + 10c  + 15b*d + 28c*d +
│ │ │ +                                  2   2              2                  
│ │ │ +      + 5a*d + 11b*d + 15c*d - 40d , b  - 49b*c + 32c  + 50b*d + 14c*d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                   2                              2     2         
│ │ │ -      50d , a*c + 32b*c - 14c  - 31a*d - 49b*d - 35c*d + 21d , b*c  + 3b*c*d
│ │ │ +         2                   2                             2     2          
│ │ │ +      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
│ │ │ -      - 50c d + 21a*d  + 27b*d  - 47c*d  + 38d , c  + 33b*c*d - 18c d + 3a*d 
│ │ │ +           2         2      2        2     3   3               2         2  
│ │ │ +      + 10c d - 18a*d  + b*d  + 26c*d  + 3d , c  + 3b*c*d + 33c d + 21a*d  +
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2     3
│ │ │ -      + 37b*d  + 46c*d  - 8d )
│ │ │ +          2        2      3
│ │ │ +      3b*d  - 50c*d  + 15d )
│ │ │  
│ │ │  o61 : Ideal of S
    
│ │ │              
    
│ │ │                
    i62 : betti res I0
│ │ │ @@ -907,40 +879,36 @@
│ │ │            
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i64 : netList decompose I0
│ │ │  
│ │ │ -      +---------------------------------------------------------------------------------------------+
│ │ │ -o64 = |ideal (c - 42d, b - 6d, a + 41d)                                                             |
│ │ │ -      +---------------------------------------------------------------------------------------------+
│ │ │ -      |ideal (c + 38d, b - 41d, a - 41d)                                                            |
│ │ │ -      +---------------------------------------------------------------------------------------------+
│ │ │ -      |                                  2                      2   2      2                      2 |
│ │ │ -      |ideal (a + 46b + 8c - 7d, b*c - 7c  + 13b*d + 16c*d - 21d , b  - 42c  - 41b*d + 35c*d - 39d )|
│ │ │ -      +---------------------------------------------------------------------------------------------+
    
│ │ │ +      +---------------------------------------------------+
│ │ │ +o64 = |ideal (c - 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 + 32d, b + 18d, a - 33d)                     |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |ideal (c + 29d, b - 8d, a + 50d)                      |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |ideal (c + 16d, b + 39d, a - 32d)                     |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |ideal (c + 5d, b - 14d, a + 7d)                       |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |                                     2              2 |
│ │ │ -      |ideal (b - 40c + 5d, a - 47c + 24d, c  - 27c*d + 15d )|
│ │ │ -      +------------------------------------------------------+
    
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                        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
│ │ │ │ - -- 2.78979s elapsed
│ │ │ │ + -- 2.37585s elapsed
│ │ │ │  
│ │ │ │  o23 = true
│ │ │ │  ********** TThhee SScchhrreeyyeerr rreessoolluuttiioonn aanndd mmiinniimmaall BBeettttii nnuummbbeerrss **********
│ │ │ │  Schreyer's construction of a nonminimal free resolution starts with a Groebner
│ │ │ │  basis. First, one constructs the SScchhrreeyyeerr ffrraammee (see La Scala, Stillman). This
│ │ │ │  is determined solely from the initial ideal $J$ and its minimal generators (but
│ │ │ │  depends on some choices of ordering, but otherwise is combinatorial). This
│ │ │ │ @@ -415,15 +415,15 @@
│ │ │ │  We now compute the locus in $V(L)$ where the Betti diagram has no cancellation.
│ │ │ │  This is a closed subscheme of $V(L)$, which is a closed subscheme of the
│ │ │ │  Hilbert scheme. Notice that there are two components.
│ │ │ │  i39 : L441 = trim(L + ideal M1);
│ │ │ │  
│ │ │ │  o39 : Ideal of T
│ │ │ │  i40 : elapsedTime compsL441 = decompose L441;
│ │ │ │ - -- 2.4524s elapsed
│ │ │ │ + -- 1.20257s elapsed
│ │ │ │  i41 : #compsL441
│ │ │ │  
│ │ │ │  o41 = 2
│ │ │ │  i42 : compsL441/dim -- two components, of dimensions 14 and 16.
│ │ │ │  
│ │ │ │  o42 = {16, 14}
│ │ │ │  
│ │ │ │ @@ -431,36 +431,36 @@
│ │ │ │  i43 : compsL441/dim == {16, 14}
│ │ │ │  
│ │ │ │  o43 = true
│ │ │ │  Both components are rational, and here are random points, one on each
│ │ │ │  component:
│ │ │ │  i44 : pta = randomPointOnRationalVariety compsL441_0
│ │ │ │  
│ │ │ │ -o44 = | -27 -13 45 -25 3 38 -20 -30 -41 25 -26 -44 -31 5 14 2 -45 45 21 -27
│ │ │ │ +o44 = | 32 -41 22 15 22 -46 43 42 -27 -27 -13 10 -24 19 -25 48 31 10 41 49 39
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -23 -29 34 49 32 19 10 26 19 37 15 -28 -50 -10 -32 18 |
│ │ │ │ +      -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  + 14b*c + 25c  - 44a*d + 38b*d + 45c*d - 27d , a*b + 32b*c +
│ │ │ │ +o45 = ideal (a  - 25b*c - 27c  + 10a*d - 46b*d + 22c*d + 32d , a*b - 48b*c +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │           2                              2   2              2
│ │ │ │ -      21c  - 23a*d - 31b*d - 41c*d - 13d , b  - 32b*c + 26c  - 50a*d - 29b*d
│ │ │ │ +      41c  + 39a*d - 24b*d - 27c*d - 41d , b  + 26b*c + 18c  - 32a*d - 28b*d
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                  2                   2                            2     2
│ │ │ │ -      + 2c*d - 20d , a*c - 28b*c + 19c  + 19a*d - 27b*d + 5c*d + 3d , b*c  +
│ │ │ │ +                   2                  2                              2     2
│ │ │ │ +      + 48c*d + 43d , a*c + 37b*c + 5c  + 45a*d + 49b*d + 19c*d + 22d , b*c
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                   2         2        2        2      3   3                2
│ │ │ │ -      37b*c*d + 34c d + 10a*d  - 45b*d  - 26c*d  - 25d , c  + 18b*c*d + 15c d
│ │ │ │ +                     2         2        2        2      3   3
│ │ │ │ +      + 19b*c*d - 29c d + 15a*d  + 31b*d  - 13c*d  + 15d , c  - 50b*c*d +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -             2        2        2      3
│ │ │ │ -      - 10a*d  + 49b*d  + 45c*d  - 30d )
│ │ │ │ +         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 . . .
│ │ │ │ @@ -468,254 +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 - 37d, a + 4d)
│ │ │ │ +----------------------+
│ │ │ │ +o47 = |ideal (c + 45d, b - 34d, a - 35d)
│ │ │ │  |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------------------------
│ │ │ │ --------+
│ │ │ │ -      |                             2              2                      2   3
│ │ │ │ -2         2        2      3     2                2         2        2 |
│ │ │ │ -      |ideal (a - 28b + 19c + 48d, b  - 32b*c + 26c  - 15b*d + 43c*d - 44d , c
│ │ │ │ -+ 18b*c*d + 15c d - 29b*d  + 33c*d  + 46d , b*c  + 37b*c*d + 34c d + 33b*d  -
│ │ │ │ -14c*d )|
│ │ │ │ +----------------------+
│ │ │ │ +      |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 - 37d, a + 4d), ideal (a - 28b + 19c + 48d, b  -
│ │ │ │ +
│ │ │ │ +o48 = {ideal (c + 45d, b - 34d, a - 35d), ideal (c + 18d, b - 12d, a + 47d),
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                 2                      2   3                2         2
│ │ │ │ -      32b*c + 26c  - 15b*d + 43c*d - 44d , c  + 18b*c*d + 15c d - 29b*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
│ │ │ │ -      33c*d  + 46d , b*c  + 37b*c*d + 34c d + 33b*d  - 14c*d )}
│ │ │ │ +                           2
│ │ │ │ +      - 14b*d + 27c*d - 38d )}
│ │ │ │  
│ │ │ │  o48 : List
│ │ │ │  i49 : lin = CFa_1_0 -- a linear form, defining a plane.
│ │ │ │  
│ │ │ │ -o49 = a - 28b + 19c + 48d
│ │ │ │ +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 = | 31 42 28 25 19 3 43 -7 -3 -42 -29 -29 14 2 50 5 36 -13 -42 47 13 31
│ │ │ │ +o53 = | -31 -3 -29 -17 -21 5 -32 33 -24 2 26 -26 -45 -4 16 -22 2 -37 16 -23
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -37 -23 -24 -4 38 -29 -23 21 17 9 0 21 -9 -47 |
│ │ │ │ +      -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  + 50b*c - 42c  - 29a*d + 3b*d + 28c*d + 31d , a*b - 24b*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
│ │ │ │ -      42c  + 13a*d + 14b*d - 3c*d + 42d , b  - 9b*c - 29c  + 31b*d + 5c*d +
│ │ │ │ +      16c  - 42a*d - 45b*d - 24c*d - 3d , b  + 47b*c + 9c  + 19b*d - 22c*d -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -         2                 2                             2     2
│ │ │ │ -      43d , a*c + 9b*c - 4c  - 23a*d + 47b*d + 2c*d + 19d , b*c  + 21b*c*d -
│ │ │ │ +         2                  2                             2     2
│ │ │ │ +      32d , a*c - 13b*c + 2c  - 15a*d - 23b*d - 4c*d - 21d , b*c  - 9b*c*d -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -         2         2        2        2      3   3                2         2
│ │ │ │ -      37c d + 38a*d  + 36b*d  - 29c*d  + 25d , c  - 47b*c*d + 17c d + 21a*d
│ │ │ │ +         2         2       2        2      3   3                2         2
│ │ │ │ +      29c d + 17a*d  + 2b*d  + 26c*d  - 17d , c  + 21b*c*d - 47c d + 38a*d  +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -             2        2     3
│ │ │ │ -      - 23b*d  - 13c*d  - 7d )
│ │ │ │ +           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 - 45d, b + 16d, a + 38d)                      |
│ │ │ │ -      +-------------------------------------------------------+
│ │ │ │ -      |ideal (c + 43d, b + 10d, a + 8d)                       |
│ │ │ │ -      +-------------------------------------------------------+
│ │ │ │ -      |ideal (c + 34d, b + 15d, a + 28d)                      |
│ │ │ │ -      +-------------------------------------------------------+
│ │ │ │ -      |ideal (c + 11d, b + 39d, a + 23d)                      |
│ │ │ │ -      +-------------------------------------------------------+
│ │ │ │ -      |                                      2              2 |
│ │ │ │ -      |ideal (b - 32c + 42d, a - 19c - 16d, c  - 28c*d - 40d )|
│ │ │ │ -      +-------------------------------------------------------+
│ │ │ │ +      +---------------------------------------------------------------+
│ │ │ │ +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
│ │ │ │ -2   2                      2
│ │ │ │ -|
│ │ │ │ -o57 = |ideal (a - 7b + 32c + d, c  + 42b*d + 33c*d - 10d , b*c - b*d + 13c*d +
│ │ │ │ -18d , b  + 28b*d - 32c*d + 16d )
│ │ │ │ -|
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------+
│ │ │ │ -      |                          2                      2
│ │ │ │ -2   2                      2
│ │ │ │ -|
│ │ │ │ -      |ideal (a - 7b + 32c + d, c  + 40b*d - 36c*d + 33d , b*c + 45b*d - 16c*d
│ │ │ │ -+ 39d , b  - 20b*d + 29c*d + 38d )
│ │ │ │ -|
│ │ │ │ +---------------------------------------------------------------------------+
│ │ │ │ +      |                       2                             2   2             2
│ │ │ │ +2                  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
│ │ │ │ +)|
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------+
│ │ │ │ -      |                          2                      2
│ │ │ │ -2   2                     2
│ │ │ │ -|
│ │ │ │ -      |ideal (a - 7b + 32c + d, c  - 10b*d + 17c*d - 21d , b*c - 17b*d - 23c*d
│ │ │ │ -- 32d , b  - 8b*d - 12c*d - 46d )
│ │ │ │ -|
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------+
│ │ │ │ -      |                                     3      2         2      3
│ │ │ │ -|
│ │ │ │ -      |ideal (b + 23c - 11d, a - 9c + 25d, c  - 13c d - 14c*d  + 23d )
│ │ │ │ -|
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------+
│ │ │ │ -      |                                     2                      2   2      2
│ │ │ │ -2   3      2         2        2      3
│ │ │ │ -|
│ │ │ │ -      |ideal (a + 48b - 40c - 20d, b*c - 32c  + 43b*d - 21c*d - 12d , b  - 14c
│ │ │ │ -+ 14b*d + 18c*d + 36d , c  + 28c d - 20b*d  + 42c*d  - 50d )
│ │ │ │ -|
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------+
│ │ │ │ -      |                                   2                      2   2      2
│ │ │ │ -2   3      2         2        2     3
│ │ │ │ -|
│ │ │ │ -      |ideal (a + b + 50c + 26d, b*c - 32c  + 34b*d - 36c*d + 14d , b  - 14c  +
│ │ │ │ -34b*d - 16c*d - 33d , c  + 28c d + 39b*d  - 28c*d  + 4d )
│ │ │ │ -|
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------+
│ │ │ │ -      |        2                            2                                 2
│ │ │ │ -2   2                              2                                  2   2
│ │ │ │ -2 |
│ │ │ │ -      |ideal (c  - 7a*d - 19b*d + 6c*d - 19d , b*c - 5a*d + 49b*d - 4c*d + 50d
│ │ │ │ -, a*c - 6a*d + 35b*d - 39c*d - 2d , b  - 46a*d + 22b*d + 42c*d + 43d , a*b +
│ │ │ │ -3a*d - 12b*d - 49c*d + 40d , a  + 28a*d - 13b*d - 25c*d - 35d )|
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------+
│ │ │ │ -      |                                     2                     2   2      2
│ │ │ │ -2   3      2         2        2     3
│ │ │ │ -|
│ │ │ │ -      |ideal (a - 46b + 39c - 29d, b*c - 32c  + 11b*d - 7c*d - 43d , b  - 14c
│ │ │ │ -+ 29b*d + 43c*d - 41d , c  + 28c d + 46b*d  - 50c*d  - 5d )
│ │ │ │ -|
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------+
│ │ │ │ -      |        2                              2
│ │ │ │ -2                                   2   2                              2
│ │ │ │ -2   2                           2  |
│ │ │ │ -      |ideal (c  + 15a*d + 27b*d + 35c*d + 46d , b*c - 6a*d + b*d + 36c*d - 31d
│ │ │ │ -, a*c - 10a*d + 45b*d + 20c*d - 23d , b  - 23a*d + 15b*d + 31c*d - 13d , a*b -
│ │ │ │ -6a*d - 40b*d + 8c*d + 18d , a  - 8a*d - 24b*d + c*d - 22d ) |
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------+
│ │ │ │ -      |        2                              2
│ │ │ │ -2                                  2   2                      2
│ │ │ │ -2   2                              2      |
│ │ │ │ -      |ideal (c  + 37a*d + 25b*d - 16c*d + 14d , b*c - 7a*d + 47b*d - 3c*d - 2d
│ │ │ │ -, a*c - 14a*d + 27b*d - 35c*d - 8d , b  - 33b*d + 19c*d + 27d , a*b - 15a*d -
│ │ │ │ -30b*d - 40c*d - 24d , a  - 44a*d + 16b*d + 11c*d + 12d )     |
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------+
│ │ │ │ +---------------------------------------------------------------------------+
│ │ │ │  i58 : pt0 = randomPointOnRationalVariety(compsL441_0)
│ │ │ │  
│ │ │ │ -o58 = | 13 17 -19 -1 -9 -15 -28 -39 -36 1 -47 29 37 -40 35 -31 12 -21 -8 -13
│ │ │ │ +o58 = | 49 0 -30 -36 -1 0 -9 17 37 29 34 13 19 8 -10 -47 21 -24 -44 42 9 46
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      14 15 -23 39 11 8 -24 -13 -42 -2 18 46 -18 -29 -33 -22 |
│ │ │ │ +      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 = | -45 18 -9 38 21 29 50 -8 -5 45 -47 -26 37 -35 -21 28 27 46 -17 -49
│ │ │ │ +o59 = | -5 -40 -6 3 -28 -8 -25 15 15 29 26 -37 11 -14 31 14 1 -50 43 37 5 50
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -23 15 -50 37 -39 -14 21 10 -31 3 -18 32 0 3 -15 33 |
│ │ │ │ +      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  + 35b*c + c  + 29a*d - 15b*d - 19c*d + 13d , a*b + 11b*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
│ │ │ │ -      8c  + 14a*d + 37b*d - 36c*d + 17d , b  - 33b*c - 13c  - 18a*d + 15b*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
│ │ │ │ -      31c*d - 28d , a*c + 46b*c + 8c  - 42a*d - 13b*d - 40c*d - 9d , 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
│ │ │ │ -      2b*c*d - 23c d - 24a*d  + 12b*d  - 47c*d  - d , c  - 22b*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  + 39b*d  - 21c*d  - 39d )
│ │ │ │ +             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  - 21b*c + 45c  - 26a*d + 29b*d - 9c*d - 45d , a*b - 39b*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
│ │ │ │ -      17c  - 23a*d + 37b*d - 5c*d + 18d , b  - 15b*c + 10c  + 15b*d + 28c*d +
│ │ │ │ +                                  2   2              2
│ │ │ │ +      + 5a*d + 11b*d + 15c*d - 40d , b  - 49b*c + 32c  + 50b*d + 14c*d -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -         2                   2                              2     2
│ │ │ │ -      50d , a*c + 32b*c - 14c  - 31a*d - 49b*d - 35c*d + 21d , b*c  + 3b*c*d
│ │ │ │ +         2                   2                             2     2
│ │ │ │ +      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
│ │ │ │ -      - 50c d + 21a*d  + 27b*d  - 47c*d  + 38d , c  + 33b*c*d - 18c d + 3a*d
│ │ │ │ +           2         2      2        2     3   3               2         2
│ │ │ │ +      + 10c d - 18a*d  + b*d  + 26c*d  + 3d , c  + 3b*c*d + 33c d + 21a*d  +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -             2        2     3
│ │ │ │ -      + 37b*d  + 46c*d  - 8d )
│ │ │ │ +          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 . . .
│ │ │ │ @@ -730,44 +653,44 @@
│ │ │ │            0: 1 . . .
│ │ │ │            1: . 4 4 1
│ │ │ │            2: . 2 4 2
│ │ │ │  
│ │ │ │  o63 : BettiTally
│ │ │ │  i64 : netList decompose I0
│ │ │ │  
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ ----------------------+
│ │ │ │ -o64 = |ideal (c - 42d, b - 6d, a + 41d)
│ │ │ │ -|
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ ----------------------+
│ │ │ │ -      |ideal (c + 38d, b - 41d, a - 41d)
│ │ │ │ -|
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ ----------------------+
│ │ │ │ -      |                                  2                      2   2      2
│ │ │ │ -2 |
│ │ │ │ -      |ideal (a + 46b + 8c - 7d, b*c - 7c  + 13b*d + 16c*d - 21d , b  - 42c  -
│ │ │ │ -41b*d + 35c*d - 39d )|
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ ----------------------+
│ │ │ │ +      +---------------------------------------------------+
│ │ │ │ +o64 = |ideal (c - 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 + 32d, b + 18d, a - 33d)                     |
│ │ │ │ -      +------------------------------------------------------+
│ │ │ │ -      |ideal (c + 29d, b - 8d, a + 50d)                      |
│ │ │ │ -      +------------------------------------------------------+
│ │ │ │ -      |ideal (c + 16d, b + 39d, a - 32d)                     |
│ │ │ │ -      +------------------------------------------------------+
│ │ │ │ -      |ideal (c + 5d, b - 14d, a + 7d)                       |
│ │ │ │ -      +------------------------------------------------------+
│ │ │ │ -      |                                     2              2 |
│ │ │ │ -      |ideal (b - 40c + 5d, a - 47c + 24d, c  - 27c*d + 15d )|
│ │ │ │ -      +------------------------------------------------------+
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------------------+
│ │ │ │ +      |                        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)
│ │ │ │  
│ │ │ │         1      4      5      2
│ │ │ │  o67 = S  <-- S  <-- S  <-- S  <-- 0
│ │ ├── ./usr/share/doc/Macaulay2/RandomCanonicalCurves/example-output/_canonical__Curve.out
│ │ │ @@ -6,15 +6,15 @@
│ │ │  i2 : g=14;
│ │ │  
│ │ │  i3 : FF=ZZ/10007;
│ │ │  
│ │ │  i4 : R=FF[x_0..x_(g-1)];
│ │ │  
│ │ │  i5 : time betti(I=(random canonicalCurve)(g,R))
│ │ │ - -- used 8.79941s (cpu); 6.0168s (thread); 0s (gc)
│ │ │ + -- used 7.06347s (cpu); 5.60269s (thread); 0s (gc)
│ │ │  
│ │ │              0  1
│ │ │  o5 = total: 1 66
│ │ │           0: 1  .
│ │ │           1: . 66
│ │ │  
│ │ │  o5 : BettiTally
│ │ ├── ./usr/share/doc/Macaulay2/RandomCanonicalCurves/html/_canonical__Curve.html
│ │ │ @@ -92,15 +92,15 @@
│ │ │              
    		
│ │ │                
    i4 : R=FF[x_0..x_(g-1)];
    
│ │ │              
    
│ │ │                
    i5 : time betti(I=(random canonicalCurve)(g,R))
│ │ │ - -- used 8.79941s (cpu); 6.0168s (thread); 0s (gc)
│ │ │ + -- used 7.06347s (cpu); 5.60269s (thread); 0s (gc)
│ │ │  
│ │ │              0  1
│ │ │  o5 = total: 1 66
│ │ │           0: 1  .
│ │ │           1: . 66
│ │ │  
│ │ │  o5 : BettiTally
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,15 +17,15 @@
│ │ │ │  unirationality of $M_g$ by Severi, Sernesi, Chang-Ran and Verra.
│ │ │ │  i1 : setRandomSeed "alpha";
│ │ │ │   -- setting random seed to 10206284518
│ │ │ │  i2 : g=14;
│ │ │ │  i3 : FF=ZZ/10007;
│ │ │ │  i4 : R=FF[x_0..x_(g-1)];
│ │ │ │  i5 : time betti(I=(random canonicalCurve)(g,R))
│ │ │ │ - -- used 8.79941s (cpu); 6.0168s (thread); 0s (gc)
│ │ │ │ + -- used 7.06347s (cpu); 5.60269s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              0  1
│ │ │ │  o5 = total: 1 66
│ │ │ │           0: 1  .
│ │ │ │           1: . 66
│ │ │ │  
│ │ │ │  o5 : BettiTally
│ │ ├── ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_test__Time__For__L__L__Lon__Syzygies.out
│ │ │ @@ -7,42 +7,42 @@
│ │ │  
│ │ │  o2 = (10, 20)
│ │ │  
│ │ │  o2 : Sequence
│ │ │  
│ │ │  i3 : (m,t1,t2)=testTimeForLLLonSyzygies(r,n,Height=>11)
│ │ │  
│ │ │ -o3 = ({5, 2.91596e52, 9}, .00182097, .000868731)
│ │ │ +o3 = ({5, 2.91596e52, 9}, .00192931, .000933583)
│ │ │  
│ │ │  o3 : Sequence
│ │ │  
│ │ │  i4 : (m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100)
│ │ │  
│ │ │ -o4 = ({50, 2.30853e454, 98}, .0051242, .0353355)
│ │ │ +o4 = ({50, 2.30853e454, 98}, .00500121, .0392925)
│ │ │  
│ │ │  o4 : Sequence
│ │ │  
│ │ │  i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})
│ │ │  
│ │ │ -o5 = {{.00505599, .0122226}, {.00470716, .00413436}, {.00612262, .00687245},
│ │ │ +o5 = {{.00507434, .0137547}, {.00479582, .00426515}, {.00584127, .00769176},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {.00521976, .00990609}, {.00508104, .013251}, {.100364, .012526},
│ │ │ +     {.00636604, .0106617}, {.00568013, .0143842}, {.0859937, .014311},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {.00513575, .00813997}, {.00513784, .00750516}, {.00491794, .00544569},
│ │ │ +     {.00453487, .00927446}, {.00426814, .0084229}, {.0034686, .00596937},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {.00753335, .00820152}}
│ │ │ +     {.00547447, .0091573}}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : 1/10*sum(L,t->t_0)
│ │ │  
│ │ │ -o6 = .01492759439999993
│ │ │ +o6 = .01314973480000003
│ │ │  
│ │ │  o6 : RR (of precision 53)
│ │ │  
│ │ │  i7 : 1/10*sum(L,t->t_1)
│ │ │  
│ │ │ -o7 = .008820477599999954
│ │ │ +o7 = .009789265800000013
│ │ │  
│ │ │  o7 : RR (of precision 53)
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomComplexes/html/_test__Time__For__L__L__Lon__Syzygies.html
│ │ │ @@ -93,57 +93,57 @@
│ │ │  o2 : Sequence
│ │ │              
    
│ │ │                
    i3 : (m,t1,t2)=testTimeForLLLonSyzygies(r,n,Height=>11)
│ │ │  
│ │ │ -o3 = ({5, 2.91596e52, 9}, .00182097, .000868731)
│ │ │ +o3 = ({5, 2.91596e52, 9}, .00192931, .000933583)
│ │ │  
│ │ │  o3 : Sequence
    
│ │ │              
    
│ │ │                
    i4 : (m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100)
│ │ │  
│ │ │ -o4 = ({50, 2.30853e454, 98}, .0051242, .0353355)
│ │ │ +o4 = ({50, 2.30853e454, 98}, .00500121, .0392925)
│ │ │  
│ │ │  o4 : Sequence
    
│ │ │              
    
│ │ │                
    i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})
│ │ │  
│ │ │ -o5 = {{.00505599, .0122226}, {.00470716, .00413436}, {.00612262, .00687245},
│ │ │ +o5 = {{.00507434, .0137547}, {.00479582, .00426515}, {.00584127, .00769176},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {.00521976, .00990609}, {.00508104, .013251}, {.100364, .012526},
│ │ │ +     {.00636604, .0106617}, {.00568013, .0143842}, {.0859937, .014311},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {.00513575, .00813997}, {.00513784, .00750516}, {.00491794, .00544569},
│ │ │ +     {.00453487, .00927446}, {.00426814, .0084229}, {.0034686, .00596937},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {.00753335, .00820152}}
│ │ │ +     {.00547447, .0091573}}
│ │ │  
│ │ │  o5 : List
    
│ │ │              
    
│ │ │                
    i6 : 1/10*sum(L,t->t_0)
│ │ │  
│ │ │ -o6 = .01492759439999993
│ │ │ +o6 = .01314973480000003
│ │ │  
│ │ │  o6 : RR (of precision 53)
    
│ │ │              
    
│ │ │                
    i7 : 1/10*sum(L,t->t_1)
│ │ │  
│ │ │ -o7 = .008820477599999954
│ │ │ +o7 = .009789265800000013
│ │ │  
│ │ │  o7 : RR (of precision 53)
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,41 +25,41 @@
│ │ │ │  i2 : r=10,n=20
│ │ │ │  
│ │ │ │  o2 = (10, 20)
│ │ │ │  
│ │ │ │  o2 : Sequence
│ │ │ │  i3 : (m,t1,t2)=testTimeForLLLonSyzygies(r,n,Height=>11)
│ │ │ │  
│ │ │ │ -o3 = ({5, 2.91596e52, 9}, .00182097, .000868731)
│ │ │ │ +o3 = ({5, 2.91596e52, 9}, .00192931, .000933583)
│ │ │ │  
│ │ │ │  o3 : Sequence
│ │ │ │  i4 : (m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100)
│ │ │ │  
│ │ │ │ -o4 = ({50, 2.30853e454, 98}, .0051242, .0353355)
│ │ │ │ +o4 = ({50, 2.30853e454, 98}, .00500121, .0392925)
│ │ │ │  
│ │ │ │  o4 : Sequence
│ │ │ │  i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})
│ │ │ │  
│ │ │ │ -o5 = {{.00505599, .0122226}, {.00470716, .00413436}, {.00612262, .00687245},
│ │ │ │ +o5 = {{.00507434, .0137547}, {.00479582, .00426515}, {.00584127, .00769176},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {.00521976, .00990609}, {.00508104, .013251}, {.100364, .012526},
│ │ │ │ +     {.00636604, .0106617}, {.00568013, .0143842}, {.0859937, .014311},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {.00513575, .00813997}, {.00513784, .00750516}, {.00491794, .00544569},
│ │ │ │ +     {.00453487, .00927446}, {.00426814, .0084229}, {.0034686, .00596937},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {.00753335, .00820152}}
│ │ │ │ +     {.00547447, .0091573}}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : 1/10*sum(L,t->t_0)
│ │ │ │  
│ │ │ │ -o6 = .01492759439999993
│ │ │ │ +o6 = .01314973480000003
│ │ │ │  
│ │ │ │  o6 : RR (of precision 53)
│ │ │ │  i7 : 1/10*sum(L,t->t_1)
│ │ │ │  
│ │ │ │ -o7 = .008820477599999954
│ │ │ │ +o7 = .009789265800000013
│ │ │ │  
│ │ │ │  o7 : RR (of precision 53)
│ │ │ │  ********** WWaayyss ttoo uussee tteessttTTiimmeeFFoorrLLLLLLoonnSSyyzzyyggiieess:: **********
│ │ │ │      * testTimeForLLLonSyzygies(ZZ,ZZ)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _t_e_s_t_T_i_m_e_F_o_r_L_L_L_o_n_S_y_z_y_g_i_e_s is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s.
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/example-output/_smooth__Canonical__Curve.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 11549527689790345152
│ │ │  
│ │ │  i1 : time ICan = smoothCanonicalCurve(11,5);
│ │ │ - -- used 1.56147s (cpu); 1.25457s (thread); 0s (gc)
│ │ │ + -- used 1.47082s (cpu); 1.18233s (thread); 0s (gc)
│ │ │  
│ │ │                ZZ
│ │ │  o1 : Ideal of --[t ..t  ]
│ │ │                 5  0   10
│ │ │  
│ │ │  i2 : (dim ICan, genus ICan, degree ICan)
│ │ ├── ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/_smooth__Canonical__Curve.html
│ │ │ @@ -82,15 +82,15 @@
│ │ │            
    If the option Printing is set to true then printings about the current step in the construction are displayed.
    
│ │ │            
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  For g<=10 the curves are constructed via plane models.
│ │ │ │  For g<=13 the curves are constructed via space models.
│ │ │ │  For g=14 the curves are constructed by Verra's method.
│ │ │ │  For g=15 the curves are constructed via matrix factorizations.
│ │ │ │  If the option Printing is set to true then printings about the current step in
│ │ │ │  the construction are displayed.
│ │ │ │  i1 : time ICan = smoothCanonicalCurve(11,5);
│ │ │ │ - -- used 1.56147s (cpu); 1.25457s (thread); 0s (gc)
│ │ │ │ + -- used 1.47082s (cpu); 1.18233s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                ZZ
│ │ │ │  o1 : Ideal of --[t ..t  ]
│ │ │ │                 5  0   10
│ │ │ │  i2 : (dim ICan, genus ICan, degree ICan)
│ │ │ │  
│ │ │ │  o2 = (2, 11, 20)
│ │ ├── ./usr/share/doc/Macaulay2/RandomGenus14Curves/example-output/_random__Curve__Genus14__Degree18in__P6.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │   -- setting random seed to 10206284518
│ │ │  
│ │ │  i2 : FF=ZZ/10007;
│ │ │  
│ │ │  i3 : S=FF[x_0..x_6];
│ │ │  
│ │ │  i4 : time I=randomCurveGenus14Degree18inP6(S);
│ │ │ - -- used 1.58351s (cpu); 1.29415s (thread); 0s (gc)
│ │ │ + -- used 1.7615s (cpu); 1.52378s (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 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : time ICan = smoothCanonicalCurve(11,5);
│ │ │ - -- used 1.56147s (cpu); 1.25457s (thread); 0s (gc)
│ │ │ + -- used 1.47082s (cpu); 1.18233s (thread); 0s (gc)
│ │ │  
│ │ │                ZZ
│ │ │  o1 : Ideal of --[t ..t  ]
│ │ │                 5  0   10
    
│ │ │              
    
│ │ │                
    i3 : S=FF[x_0..x_6];
    
│ │ │              
    
│ │ │                
    i4 : time I=randomCurveGenus14Degree18inP6(S);
│ │ │ - -- used 1.58351s (cpu); 1.29415s (thread); 0s (gc)
│ │ │ + -- used 1.7615s (cpu); 1.52378s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of S
    
│ │ │              
    
│ │ │                
    i5 : betti res I
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,15 +28,15 @@
│ │ │ │  fields of the chosen finite characteristic 10007, for fields of characteristic
│ │ │ │  0 by semi-continuity, and, hence, for all but finitely many primes $p$.
│ │ │ │  i1 : setRandomSeed("alpha");
│ │ │ │   -- setting random seed to 10206284518
│ │ │ │  i2 : FF=ZZ/10007;
│ │ │ │  i3 : S=FF[x_0..x_6];
│ │ │ │  i4 : time I=randomCurveGenus14Degree18inP6(S);
│ │ │ │ - -- used 1.58351s (cpu); 1.29415s (thread); 0s (gc)
│ │ │ │ + -- used 1.7615s (cpu); 1.52378s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : Ideal of S
│ │ │ │  i5 : betti res I
│ │ │ │  
│ │ │ │              0  1  2  3  4 5
│ │ │ │  o5 = total: 1 13 45 56 25 2
│ │ │ │           0: 1  .  .  .  . .
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Random__Ideals.out
│ │ │ @@ -1,24 +1,23 @@
│ │ │  -- -*- M2-comint -*- hash: 9542801742429495161
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1765726966
│ │ │ + -- setting random seed to 1767266088
│ │ │  
│ │ │ -o1 = 1765726966
│ │ │ +o1 = 1767266088
│ │ │  
│ │ │  i2 : kk=ZZ/101;
│ │ │  
│ │ │  i3 : S=kk[vars(0..5)];
│ │ │  
│ │ │  i4 : time tally for n from 1 to 500 list regularity randomMonomialIdeal(10:3,S)
│ │ │ - -- used 2.93525s (cpu); 1.5644s (thread); 0s (gc)
│ │ │ + -- used 2.9617s (cpu); 1.52285s (thread); 0s (gc)
│ │ │  
│ │ │ -o4 = Tally{4 => 50 }
│ │ │ -           5 => 198
│ │ │ -           6 => 177
│ │ │ -           7 => 62
│ │ │ -           8 => 12
│ │ │ -           9 => 1
│ │ │ +o4 = Tally{4 => 46 }
│ │ │ +           5 => 217
│ │ │ +           6 => 174
│ │ │ +           7 => 52
│ │ │ +           8 => 11
│ │ │  
│ │ │  o4 : Tally
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 5959465567197821046
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1765726971
│ │ │ + -- setting random seed to 1767266092
│ │ │  
│ │ │ -o1 = 1765726971
│ │ │ +o1 = 1767266092
│ │ │  
│ │ │  i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │  
│ │ │  o2 : QuotientRing
│ │ │  
│ │ │ @@ -15,13 +15,13 @@
│ │ │  
│ │ │  o3 = S
│ │ │  
│ │ │  o3 : PolynomialRing
│ │ │  
│ │ │  i4 : randomMonomial(3,S)
│ │ │  
│ │ │ -      2
│ │ │ -o4 = a c
│ │ │ +      3
│ │ │ +o4 = b
│ │ │  
│ │ │  o4 : S
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Monomial__Ideal.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 8876340562021865447
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1765726980
│ │ │ + -- setting random seed to 1767266098
│ │ │  
│ │ │ -o1 = 1765726980
│ │ │ +o1 = 1767266098
│ │ │  
│ │ │  i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │  
│ │ │  o2 : QuotientRing
│ │ │  
│ │ │ @@ -28,12 +28,12 @@
│ │ │  
│ │ │  o5 = ideal(a*c*d)
│ │ │  
│ │ │  o5 : Ideal of S
│ │ │  
│ │ │  i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │  
│ │ │ -o6 = ideal (c*e, a*d, d*e, b*e, c*d)
│ │ │ +o6 = ideal (c*e, a*c, c*d, a*e, b*d)
│ │ │  
│ │ │  o6 : Ideal of S
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Step.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 10504911213508281315
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1765726975
│ │ │ + -- setting random seed to 1767266095
│ │ │  
│ │ │ -o1 = 1765726975
│ │ │ +o1 = 1767266095
│ │ │  
│ │ │  i2 : S=ZZ/2[vars(0..3)]
│ │ │  
│ │ │  o2 = S
│ │ │  
│ │ │  o2 : PolynomialRing
│ │ │  
│ │ │ @@ -15,17 +15,15 @@
│ │ │  
│ │ │  o3 = monomialIdeal (a*b, a*d, b*c*d)
│ │ │  
│ │ │  o3 : MonomialIdeal of S
│ │ │  
│ │ │  i4 : randomSquareFreeStep J
│ │ │  
│ │ │ -o4 = {monomialIdeal (a*b, a*c*d, b*c*d), {a*b, a*c*d, b*c*d}, {c*d, b*d, a*d,
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     b*c, a*c}}
│ │ │ +o4 = {monomialIdeal (a*b, a*d, b*d), {a*b, a*d, b*d}, {c*d, b*c, a*c}}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : setRandomSeed(1)
│ │ │   -- setting random seed to 1
│ │ │  
│ │ │  o5 = 1
│ │ │ @@ -39,15 +37,15 @@
│ │ │  i7 : J = monomialIdeal 0_S
│ │ │  
│ │ │  o7 = monomialIdeal ()
│ │ │  
│ │ │  o7 : MonomialIdeal of S
│ │ │  
│ │ │  i8 : time T=tally for t from 1 to 5000 list first (J=rsfs(J,AlexanderProbability => .01));
│ │ │ - -- used 4.75542s (cpu); 3.39881s (thread); 0s (gc)
│ │ │ + -- used 4.46924s (cpu); 3.07173s (thread); 0s (gc)
│ │ │  
│ │ │  i9 : #T
│ │ │  
│ │ │  o9 = 168
│ │ │  
│ │ │  i10 : T
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Monomial.html
│ │ │ @@ -71,17 +71,17 @@
│ │ │          
    
│ │ │            
    Chooses a random monomial.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1765726971
│ │ │ + -- setting random seed to 1767266092
│ │ │  
│ │ │ -o1 = 1765726971
    
│ │ │ +o1 = 1767266092
│ │ │              
    
│ │ │                
    i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │ @@ -98,16 +98,16 @@
│ │ │  o3 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i4 : randomMonomial(3,S)
│ │ │  
│ │ │ -      2
│ │ │ -o4 = a c
│ │ │ +      3
│ │ │ +o4 = b
│ │ │  
│ │ │  o4 : S
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,31 +11,31 @@
│ │ │ │            o d, an _i_n_t_e_g_e_r, non-negative
│ │ │ │            o S, a _r_i_n_g, polynomial ring
│ │ │ │      * Outputs:
│ │ │ │            o m, a _r_i_n_g_ _e_l_e_m_e_n_t, monomial of S
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Chooses a random monomial.
│ │ │ │  i1 : setRandomSeed(currentTime())
│ │ │ │ - -- setting random seed to 1765726971
│ │ │ │ + -- setting random seed to 1767266092
│ │ │ │  
│ │ │ │ -o1 = 1765726971
│ │ │ │ +o1 = 1767266092
│ │ │ │  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
│ │ │ │ +      3
│ │ │ │ +o4 = b
│ │ │ │  
│ │ │ │  o4 : S
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_a_n_d_o_m_M_o_n_o_m_i_a_l_I_d_e_a_l -- random monomial ideal with given degree generators
│ │ │ │      * _r_a_n_d_o_m_S_q_u_a_r_e_F_r_e_e_M_o_n_o_m_i_a_l_I_d_e_a_l -- random square-free monomial ideal with
│ │ │ │        given degree generators
│ │ │ │  ********** WWaayyss ttoo uussee rraannddoommMMoonnoommiiaall:: **********
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Square__Free__Monomial__Ideal.html
│ │ │ @@ -71,17 +71,17 @@
│ │ │          
    
│ │ │            
    Choose a random square-free monomial ideal whose generators have the degrees specified by the list or sequence L.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1765726980
│ │ │ + -- setting random seed to 1767266098
│ │ │  
│ │ │ -o1 = 1765726980
    
│ │ │ +o1 = 1767266098
│ │ │              
    
│ │ │                
    i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │ @@ -117,15 +117,15 @@
│ │ │  o5 : Ideal of S
    
│ │ │              
    
│ │ │                
    i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │  
│ │ │ -o6 = ideal (c*e, a*d, d*e, b*e, c*d)
│ │ │ +o6 = ideal (c*e, a*c, c*d, a*e, b*d)
│ │ │  
│ │ │  o6 : Ideal of S
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,17 +13,17 @@
│ │ │ │      * Outputs:
│ │ │ │            o I, an _i_d_e_a_l, square-free monomial ideal with generators of
│ │ │ │              specified degrees
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Choose a random square-free monomial ideal whose generators have the degrees
│ │ │ │  specified by the list or sequence L.
│ │ │ │  i1 : setRandomSeed(currentTime())
│ │ │ │ - -- setting random seed to 1765726980
│ │ │ │ + -- setting random seed to 1767266098
│ │ │ │  
│ │ │ │ -o1 = 1765726980
│ │ │ │ +o1 = 1767266098
│ │ │ │  i2 : kk=ZZ/101
│ │ │ │  
│ │ │ │  o2 = kk
│ │ │ │  
│ │ │ │  o2 : QuotientRing
│ │ │ │  i3 : S=kk[a..e]
│ │ │ │  
│ │ │ │ @@ -39,15 +39,15 @@
│ │ │ │  low degree gens generated everything
│ │ │ │  
│ │ │ │  o5 = ideal(a*c*d)
│ │ │ │  
│ │ │ │  o5 : Ideal of S
│ │ │ │  i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │ │  
│ │ │ │ -o6 = ideal (c*e, a*d, d*e, b*e, c*d)
│ │ │ │ +o6 = ideal (c*e, a*c, c*d, a*e, b*d)
│ │ │ │  
│ │ │ │  o6 : Ideal of S
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  The ideal is constructed degree by degree, starting from the lowest degree
│ │ │ │  specified. If there are not enough monomials of the next specified degree that
│ │ │ │  are not already in the ideal, the function prints a warning and returns an
│ │ │ │  ideal containing all the generators of that degree.
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Square__Free__Step.html
│ │ │ @@ -79,17 +79,17 @@
│ │ │            
    With probability p the routine takes the Alexander dual of I; the default value of p is .05, and it can be set with the option AlexanderProbility.
    
│ │ │            
    Otherwise uses the Metropolis algorithm to produce a random walk on the space of square-free ideals. Note that there are a LOT of square-free ideals; these are the Dedekind numbers, and the sequence (with 1,2,3,4,5,6,7,8 variables) begins 3,6,20,168,7581, 7828354, 2414682040998, 56130437228687557907788. (see the Online Encyclopedia of Integer Sequences for more information). Given I in a polynomial ring S, we make a list ISocgens of the square-free minimal monomial generators of the socle of S/(squares+I) and a list of minimal generators Igens of I. A candidate "next" ideal J is formed as follows: We choose randomly from the union of these lists; if a socle element is chosen, it's added to I; if a minimal generator is chosen, it's replaced by the square-free part of the maximal ideal times it. the chance of making the given move is then 1/(#ISocgens+#Igens), and the chance of making the move back would be the similar quantity for J, so we make the move or not depending on whether random RR < (nJ+nSocJ)/(nI+nSocI) or not; here random RR is a random number in [0,1].
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1765726975
│ │ │ + -- setting random seed to 1767266095
│ │ │  
│ │ │ -o1 = 1765726975
    
│ │ │ +o1 = 1767266095
│ │ │              
    
│ │ │                
    i2 : S=ZZ/2[vars(0..3)]
│ │ │  
│ │ │  o2 = S
│ │ │ @@ -106,17 +106,15 @@
│ │ │  o3 : MonomialIdeal of S
    
│ │ │              
    
│ │ │                
    i4 : randomSquareFreeStep J
│ │ │  
│ │ │ -o4 = {monomialIdeal (a*b, a*c*d, b*c*d), {a*b, a*c*d, b*c*d}, {c*d, b*d, a*d,
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     b*c, a*c}}
│ │ │ +o4 = {monomialIdeal (a*b, a*d, b*d), {a*b, a*d, b*d}, {c*d, b*c, a*c}}
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │          
    
│ │ │            
    With 4 variables and 168 possible monomial ideals, a run of 5000 takes less than 6 seconds on a reasonably fast machine. With 10 variables a run of 1000 takes about 2 seconds.
    
│ │ │ @@ -147,15 +145,15 @@
│ │ │  
│ │ │  o7 : MonomialIdeal of S
    
│ │ │              
    
│ │ │                
    i8 : time T=tally for t from 1 to 5000 list first (J=rsfs(J,AlexanderProbability => .01));
│ │ │ - -- used 4.75542s (cpu); 3.39881s (thread); 0s (gc)
    
│ │ │ + -- used 4.46924s (cpu); 3.07173s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i9 : #T
│ │ │  
│ │ │  o9 = 168
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,32 +35,30 @@
│ │ │ │  choose randomly from the union of these lists; if a socle element is chosen,
│ │ │ │  it's added to I; if a minimal generator is chosen, it's replaced by the square-
│ │ │ │  free part of the maximal ideal times it. the chance of making the given move is
│ │ │ │  then 1/(#ISocgens+#Igens), and the chance of making the move back would be the
│ │ │ │  similar quantity for J, so we make the move or not depending on whether random
│ │ │ │  RR < (nJ+nSocJ)/(nI+nSocI) or not; here random RR is a random number in [0,1].
│ │ │ │  i1 : setRandomSeed(currentTime())
│ │ │ │ - -- setting random seed to 1765726975
│ │ │ │ + -- setting random seed to 1767266095
│ │ │ │  
│ │ │ │ -o1 = 1765726975
│ │ │ │ +o1 = 1767266095
│ │ │ │  i2 : S=ZZ/2[vars(0..3)]
│ │ │ │  
│ │ │ │  o2 = S
│ │ │ │  
│ │ │ │  o2 : PolynomialRing
│ │ │ │  i3 : J = monomialIdeal"ab,ad, bcd"
│ │ │ │  
│ │ │ │  o3 = monomialIdeal (a*b, a*d, b*c*d)
│ │ │ │  
│ │ │ │  o3 : MonomialIdeal of S
│ │ │ │  i4 : randomSquareFreeStep J
│ │ │ │  
│ │ │ │ -o4 = {monomialIdeal (a*b, a*c*d, b*c*d), {a*b, a*c*d, b*c*d}, {c*d, b*d, a*d,
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     b*c, a*c}}
│ │ │ │ +o4 = {monomialIdeal (a*b, a*d, b*d), {a*b, a*d, b*d}, {c*d, b*c, a*c}}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  With 4 variables and 168 possible monomial ideals, a run of 5000 takes less
│ │ │ │  than 6 seconds on a reasonably fast machine. With 10 variables a run of 1000
│ │ │ │  takes about 2 seconds.
│ │ │ │  i5 : setRandomSeed(1)
│ │ │ │   -- setting random seed to 1
│ │ │ │ @@ -74,15 +72,15 @@
│ │ │ │  i7 : J = monomialIdeal 0_S
│ │ │ │  
│ │ │ │  o7 = monomialIdeal ()
│ │ │ │  
│ │ │ │  o7 : MonomialIdeal of S
│ │ │ │  i8 : time T=tally for t from 1 to 5000 list first (J=rsfs
│ │ │ │  (J,AlexanderProbability => .01));
│ │ │ │ - -- used 4.75542s (cpu); 3.39881s (thread); 0s (gc)
│ │ │ │ + -- used 4.46924s (cpu); 3.07173s (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.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -72,22 +72,21 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1765726966
│ │ │ + -- setting random seed to 1767266088
│ │ │  
│ │ │ -o1 = 1765726966
    
│ │ │ +o1 = 1767266088
│ │ │              
    
│ │ │                
    i2 : kk=ZZ/101;
    
│ │ │              
    
│ │ │                
    i3 : S=kk[vars(0..5)];
    
│ │ │              
    
│ │ │                
    i4 : time tally for n from 1 to 500 list regularity randomMonomialIdeal(10:3,S)
│ │ │ - -- used 2.93525s (cpu); 1.5644s (thread); 0s (gc)
│ │ │ + -- used 2.9617s (cpu); 1.52285s (thread); 0s (gc)
│ │ │  
│ │ │ -o4 = Tally{4 => 50 }
│ │ │ -           5 => 198
│ │ │ -           6 => 177
│ │ │ -           7 => 62
│ │ │ -           8 => 12
│ │ │ -           9 => 1
│ │ │ +o4 = Tally{4 => 46 }
│ │ │ +           5 => 217
│ │ │ +           6 => 174
│ │ │ +           7 => 52
│ │ │ +           8 => 11
│ │ │  
│ │ │  o4 : Tally
    
│ │ │              
    
│ │ │          
    
│ │ │            
    How does this compare with the case of binomial ideals? or pure binomial ideals? We invite the reader to experiment, replacing "randomMonomialIdeal" above with "randomBinomialIdeal" or "randomPureBinomialIdeal", or taking larger numbers of examples. Click the link "Finding Extreme Examples" below to see some other, more elaborate ways to search.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,28 +9,27 @@
│ │ │ │  This package can be used to make experiments, trying many ideals, perhaps over
│ │ │ │  small fields. For example...what would you expect the regularities of "typical"
│ │ │ │  monomial ideals with 10 generators of degree 3 in 6 variables to be? Try a
│ │ │ │  bunch of examples -- it's fast. Here we do only 500 -- this takes about a
│ │ │ │  second on a fast machine -- but with a little patience, thousands can be done
│ │ │ │  conveniently.
│ │ │ │  i1 : setRandomSeed(currentTime())
│ │ │ │ - -- setting random seed to 1765726966
│ │ │ │ + -- setting random seed to 1767266088
│ │ │ │  
│ │ │ │ -o1 = 1765726966
│ │ │ │ +o1 = 1767266088
│ │ │ │  i2 : kk=ZZ/101;
│ │ │ │  i3 : S=kk[vars(0..5)];
│ │ │ │  i4 : time tally for n from 1 to 500 list regularity randomMonomialIdeal(10:3,S)
│ │ │ │ - -- used 2.93525s (cpu); 1.5644s (thread); 0s (gc)
│ │ │ │ + -- used 2.9617s (cpu); 1.52285s (thread); 0s (gc)
│ │ │ │  
│ │ │ │ -o4 = Tally{4 => 50 }
│ │ │ │ -           5 => 198
│ │ │ │ -           6 => 177
│ │ │ │ -           7 => 62
│ │ │ │ -           8 => 12
│ │ │ │ -           9 => 1
│ │ │ │ +o4 = Tally{4 => 46 }
│ │ │ │ +           5 => 217
│ │ │ │ +           6 => 174
│ │ │ │ +           7 => 52
│ │ │ │ +           8 => 11
│ │ │ │  
│ │ │ │  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)
│ │ │ - -- 1.63992s elapsed
│ │ │ + -- 1.33023s elapsed
│ │ │  
│ │ │  o4 = 4
│ │ │  
│ │ │  i5 : elapsedTime dim I
│ │ │ - -- 3.02509s elapsed
│ │ │ + -- 3.20197s elapsed
│ │ │  
│ │ │  o5 = 4
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/example-output/_extend__Ideal__By__Non__Zero__Minor.out
│ │ │ @@ -35,15 +35,15 @@
│ │ │  i8 : i = 0;
│ │ │  
│ │ │  i9 : J = I;
│ │ │  
│ │ │  o9 : Ideal of T
│ │ │  
│ │ │  i10 : elapsedTime(while (i < 10) and dim J > 1 do (i = i+1; J = extendIdealByNonZeroMinor(4, M, J)) );
│ │ │ - -- 2.0239s elapsed
│ │ │ + -- 1.51607s elapsed
│ │ │  
│ │ │  i11 : dim J
│ │ │  
│ │ │  o11 = 1
│ │ │  
│ │ │  i12 : i
│ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/example-output/_random__Points.out
│ │ │ @@ -27,24 +27,24 @@
│ │ │  i6 : S=ZZ/103[y_0..y_30];
│ │ │  
│ │ │  i7 : I=minors(2,random(S^3,S^{3:-1}));
│ │ │  
│ │ │  o7 : Ideal of S
│ │ │  
│ │ │  i8 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, DecompositionStrategy=>MultiplicationTable)
│ │ │ - -- 3.12043s elapsed
│ │ │ + -- 2.6456s elapsed
│ │ │  
│ │ │  o8 = {{-4, -35, -7, 0, 0, 1, 5, -13, 0, -47, 0, 41, 0, -51, -46, 35, 0, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │       -47, 14, -30, 42, 30, 4, -41, 24, 0, 0, 15, 20, 1}}
│ │ │  
│ │ │  o8 : List
│ │ │  
│ │ │  i9 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, DecompositionStrategy=>Decompose)
│ │ │ - -- 2.62392s elapsed
│ │ │ + -- 1.95895s elapsed
│ │ │  
│ │ │  o9 = {{11, 9, -9, -15, -7, 27, 19, -36, 48, 26, -4, 3, 29, -8, 7, -32, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │       11, 7, 7, 25, -14, -39, 17, -16, 4, -50, -12, 21, -50, 51}}
│ │ │  
│ │ │  o9 : List
│ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/html/_dim__Via__Bezout.html
│ │ │ @@ -95,23 +95,23 @@
│ │ │  
│ │ │  o3 : Ideal of S
│ │ │              
    
│ │ │                
    i4 : elapsedTime dimViaBezout(I)
│ │ │ - -- 1.63992s elapsed
│ │ │ + -- 1.33023s elapsed
│ │ │  
│ │ │  o4 = 4
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime dim I
│ │ │ - -- 3.02509s elapsed
│ │ │ + -- 3.20197s elapsed
│ │ │  
│ │ │  o5 = 4
    
│ │ │              
    
│ │ │          
    
│ │ │            
    The user may set the MinimumFieldSize to ensure that the field being worked over is big enough.  For instance, there are relatively few linear spaces over a field of characteristic 2, and this can cause incorrect results to be provided.  If no size is provided, the function tries to guess a good size based on ambient ring.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -32,19 +32,19 @@
│ │ │ │  examples, the built in dim function is much faster.
│ │ │ │  i1 : kk=ZZ/101;
│ │ │ │  i2 : S=kk[y_0..y_8];
│ │ │ │  i3 : I=ideal random(S^1,S^{-2,-2,-2,-2})+(ideal random(2,S))^2;
│ │ │ │  
│ │ │ │  o3 : Ideal of S
│ │ │ │  i4 : elapsedTime dimViaBezout(I)
│ │ │ │ - -- 1.63992s elapsed
│ │ │ │ + -- 1.33023s elapsed
│ │ │ │  
│ │ │ │  o4 = 4
│ │ │ │  i5 : elapsedTime dim I
│ │ │ │ - -- 3.02509s elapsed
│ │ │ │ + -- 3.20197s elapsed
│ │ │ │  
│ │ │ │  o5 = 4
│ │ │ │  The user may set the MinimumFieldSize to ensure that the field being worked
│ │ │ │  over is big enough. For instance, there are relatively few linear spaces over a
│ │ │ │  field of characteristic 2, and this can cause incorrect results to be provided.
│ │ │ │  If no size is provided, the function tries to guess a good size based on
│ │ │ │  ambient ring.
│ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/html/_extend__Ideal__By__Non__Zero__Minor.html
│ │ │ @@ -155,15 +155,15 @@
│ │ │  
│ │ │  o9 : Ideal of T
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : elapsedTime(while (i < 10) and dim J > 1 do (i = i+1; J = extendIdealByNonZeroMinor(4, M, J)) );
│ │ │ - -- 2.0239s elapsed
    
│ │ │ + -- 1.51607s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : dim J
│ │ │  
│ │ │  o11 = 1
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -79,15 +79,15 @@
│ │ │ │  o7 : Matrix T  <-- T
│ │ │ │  i8 : i = 0;
│ │ │ │  i9 : J = I;
│ │ │ │  
│ │ │ │  o9 : Ideal of T
│ │ │ │  i10 : elapsedTime(while (i < 10) and dim J > 1 do (i = i+1; J =
│ │ │ │  extendIdealByNonZeroMinor(4, M, J)) );
│ │ │ │ - -- 2.0239s elapsed
│ │ │ │ + -- 1.51607s elapsed
│ │ │ │  i11 : dim J
│ │ │ │  
│ │ │ │  o11 = 1
│ │ │ │  i12 : i
│ │ │ │  
│ │ │ │  o12 = 4
│ │ │ │  In this particular example, there tend to be about 5 associated primes when
│ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/html/_random__Points.html
│ │ │ @@ -144,27 +144,27 @@
│ │ │  
│ │ │  o7 : Ideal of S
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, DecompositionStrategy=>MultiplicationTable)
│ │ │ - -- 3.12043s elapsed
│ │ │ + -- 2.6456s elapsed
│ │ │  
│ │ │  o8 = {{-4, -35, -7, 0, 0, 1, 5, -13, 0, -47, 0, 41, 0, -51, -46, 35, 0, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │       -47, 14, -30, 42, 30, 4, -41, 24, 0, 0, 15, 20, 1}}
│ │ │  
│ │ │  o8 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, DecompositionStrategy=>Decompose)
│ │ │ - -- 2.62392s elapsed
│ │ │ + -- 1.95895s elapsed
│ │ │  
│ │ │  o9 = {{11, 9, -9, -15, -7, 27, 19, -36, 48, 26, -4, 3, 29, -8, 7, -32, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │       11, 7, 7, 25, -14, -39, 17, -16, 4, -50, -12, 21, -50, 51}}
│ │ │  
│ │ │  o9 : List
    
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -66,24 +66,24 @@
│ │ │ │  first in rings with more variables.
│ │ │ │  i6 : S=ZZ/103[y_0..y_30];
│ │ │ │  i7 : I=minors(2,random(S^3,S^{3:-1}));
│ │ │ │  
│ │ │ │  o7 : Ideal of S
│ │ │ │  i8 : elapsedTime randomPoints(I, Strategy=>LinearIntersection,
│ │ │ │  DecompositionStrategy=>MultiplicationTable)
│ │ │ │ - -- 3.12043s elapsed
│ │ │ │ + -- 2.6456s elapsed
│ │ │ │  
│ │ │ │  o8 = {{-4, -35, -7, 0, 0, 1, 5, -13, 0, -47, 0, 41, 0, -51, -46, 35, 0, 0,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       -47, 14, -30, 42, 30, 4, -41, 24, 0, 0, 15, 20, 1}}
│ │ │ │  
│ │ │ │  o8 : List
│ │ │ │  i9 : elapsedTime randomPoints(I, Strategy=>LinearIntersection,
│ │ │ │  DecompositionStrategy=>Decompose)
│ │ │ │ - -- 2.62392s elapsed
│ │ │ │ + -- 1.95895s elapsed
│ │ │ │  
│ │ │ │  o9 = {{11, 9, -9, -15, -7, 27, 19, -36, 48, 26, -4, 3, 29, -8, 7, -32, 16,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       11, 7, 7, 25, -14, -39, 17, -16, 4, -50, -12, 21, -50, 51}}
│ │ │ │  
│ │ │ │  o9 : List
│ │ │ │  ********** WWaayyss ttoo uussee rraannddoommPPooiinnttss:: **********
│ │ ├── ./usr/share/doc/Macaulay2/RationalMaps/example-output/_inverse__Of__Map.out
│ │ │ @@ -49,15 +49,15 @@
│ │ │  i12 : Q=QQ[x,y,z,t,u];
│ │ │  
│ │ │  i13 : phi=map(Q,Q,matrix{{x^5,y*x^4,z*x^4+y^5,t*x^4+z^5,u*x^4+t^5}});
│ │ │  
│ │ │  o13 : RingMap Q <-- Q
│ │ │  
│ │ │  i14 : time inverseOfMap(phi,CheckBirational=>false, Verbosity=>0)
│ │ │ - -- used 0.612163s (cpu); 0.416347s (thread); 0s (gc)
│ │ │ + -- used 0.80984s (cpu); 0.455423s (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
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : time inverseOfMap(phi,CheckBirational=>false, Verbosity=>0)
│ │ │ - -- used 0.612163s (cpu); 0.416347s (thread); 0s (gc)
│ │ │ + -- used 0.80984s (cpu); 0.455423s (thread); 0s (gc)
│ │ │  
│ │ │                                  125   124      120 5    124    100 25     104 20       108 15 2      112 10 3     116 5 4    120 5    124      125      4 120        8 115 2        12 110 3         16 105 4         20 100 5          24 95 6          28 90 7           32 85 8           36 80 9           40 75 10           44 70 11           48 65 12           52 60 13           56 55 14           60 50 15           64 45 16           68 40 17          72 35 18          76 30 19         80 25 20         84 20 21        88 15 22       92 10 23      96 5 24    100 25     24 100        28 95          32 90 2         36 85 3          40 80 4          44 75 5           48 70 6           52 65 7           56 60 8           60 55 9           64 50 10           68 45 11           72 40 12           76 35 13           80 30 14          84 25 15          88 20 16         92 15 17        96 10 18        100 5 19      104 20       48 75 2       52 70   2        56 65 2 2        60 60 3 2         64 55 4 2         68 50 5 2         72 45 6 2         76 40 7 2         80 35 8 2         84 30 9 2         88 25 10 2         92 20 11 2        96 15 12 2        100 10 13 2       104 5 14 2      108 15 2      72 50 3       76 45   3       80 40 2 3        84 35 3 3        88 30 4 3        92 25 5 3        96 20 6 3        100 15 7 3       104 10 8 3       108 5 9 3      112 10 3     96 25 4      100 20   4      104 15 2 4      108 10 3 4      112 5 4 4     116 5 4    120 5    124
│ │ │  o14 = Proj Q - - - > Proj Q   {x   , x   y, - x   y  + x   z, x   y   - 5x   y  z + 10x   y  z  - 10x   y  z  + 5x   y z  - x   z  + x   t, - y    + 25x y   z - 300x y   z  + 2300x  y   z  - 12650x  y   z  + 53130x  y   z  - 177100x  y  z  + 480700x  y  z  - 1081575x  y  z  + 2042975x  y  z  - 3268760x  y  z   + 4457400x  y  z   - 5200300x  y  z   + 5200300x  y  z   - 4457400x  y  z   + 3268760x  y  z   - 2042975x  y  z   + 1081575x  y  z   - 480700x  y  z   + 177100x  y  z   - 53130x  y  z   + 12650x  y  z   - 2300x  y  z   + 300x  y  z   - 25x  y z   + x   z   - 5x  y   t + 100x  y  z*t - 950x  y  z t + 5700x  y  z t - 24225x  y  z t + 77520x  y  z t - 193800x  y  z t + 387600x  y  z t - 629850x  y  z t + 839800x  y  z t - 923780x  y  z  t + 839800x  y  z  t - 629850x  y  z  t + 387600x  y  z  t - 193800x  y  z  t + 77520x  y  z  t - 24225x  y  z  t + 5700x  y  z  t - 950x  y  z  t + 100x   y z  t - 5x   z  t - 10x  y  t  + 150x  y  z*t  - 1050x  y  z t  + 4550x  y  z t  - 13650x  y  z t  + 30030x  y  z t  - 50050x  y  z t  + 64350x  y  z t  - 64350x  y  z t  + 50050x  y  z t  - 30030x  y  z  t  + 13650x  y  z  t  - 4550x  y  z  t  + 1050x   y  z  t  - 150x   y z  t  + 10x   z  t  - 10x  y  t  + 100x  y  z*t  - 450x  y  z t  + 1200x  y  z t  - 2100x  y  z t  + 2520x  y  z t  - 2100x  y  z t  + 1200x   y  z t  - 450x   y  z t  + 100x   y z t  - 10x   z  t  - 5x  y  t  + 25x   y  z*t  - 50x   y  z t  + 50x   y  z t  - 25x   y z t  + 5x   z t  - x   t  + x   u}
│ │ │  
│ │ │  o14 : RationalMapping
    
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -94,15 +94,15 @@
│ │ │ │  o11 : Ideal of blowUpSubvar
│ │ │ │  The next example is a birational map on $\mathbb{P}^4$.
│ │ │ │  i12 : Q=QQ[x,y,z,t,u];
│ │ │ │  i13 : phi=map(Q,Q,matrix{{x^5,y*x^4,z*x^4+y^5,t*x^4+z^5,u*x^4+t^5}});
│ │ │ │  
│ │ │ │  o13 : RingMap Q <-- Q
│ │ │ │  i14 : time inverseOfMap(phi,CheckBirational=>false, Verbosity=>0)
│ │ │ │ - -- used 0.612163s (cpu); 0.416347s (thread); 0s (gc)
│ │ │ │ + -- used 0.80984s (cpu); 0.455423s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                                  125   124      120 5    124    100 25     104
│ │ │ │  20       108 15 2      112 10 3     116 5 4    120 5    124      125      4 120
│ │ │ │  8 115 2        12 110 3         16 105 4         20 100 5          24 95 6
│ │ │ │  28 90 7           32 85 8           36 80 9           40 75 10           44 70
│ │ │ │  11           48 65 12           52 60 13           56 55 14           60 50 15
│ │ │ │  64 45 16           68 40 17          72 35 18          76 30 19         80 25
│ │ ├── ./usr/share/doc/Macaulay2/RationalPoints2/example-output/_rational__Points.out
│ │ │ @@ -48,15 +48,15 @@
│ │ │               0    1    2    3    4    5    6    7    8    9    10
│ │ │  
│ │ │                  ZZ
│ │ │  o13 : Ideal of ---[u ..u  ]
│ │ │                 101  0   10
│ │ │  
│ │ │  i14 : time rationalPoints(I, Amount => true)
│ │ │ - -- used 0.00322822s (cpu); 0.00322771s (thread); 0s (gc)
│ │ │ + -- used 0.00457055s (cpu); 0.0045688s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 110462212541120451001
│ │ │  
│ │ │  i15 : QQ[x,y,z]; I = homogenize(ideal(y^2-x*(x-1)*(x-2)*(x-5)*(x-6)), z);
│ │ │  
│ │ │  o16 : Ideal of QQ[x..z]
│ │ │  
│ │ │ @@ -142,23 +142,23 @@
│ │ │  
│ │ │  i31 : nodes = I + ideal jacobian I;
│ │ │  
│ │ │  o31 : Ideal of R
│ │ │  
│ │ │  i32 : time rationalPoints(variety nodes, Split=>true, Verbose=>true);
│ │ │  -- base change to the field QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25)
│ │ │ - -- used 0.996963s (cpu); 0.801505s (thread); 0s (gc)
│ │ │ + -- used 1.04689s (cpu); 0.856754s (thread); 0s (gc)
│ │ │  
│ │ │  i33 : #oo
│ │ │  
│ │ │  o33 = 31
│ │ │  
│ │ │  i34 : nodes' = baseChange_32003 nodes;
│ │ │  
│ │ │  o34 : Ideal of GF 1048969271299456081[x..z, w]
│ │ │  
│ │ │  i35 : time #rationalPoints(variety nodes', Split=>true, Verbose=>true)
│ │ │ - -- used 0.273936s (cpu); 0.210148s (thread); 0s (gc)
│ │ │ + -- used 0.289067s (cpu); 0.219977s (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
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : time rationalPoints(I, Amount => true)
│ │ │ - -- used 0.00322822s (cpu); 0.00322771s (thread); 0s (gc)
│ │ │ + -- used 0.00457055s (cpu); 0.0045688s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 110462212541120451001
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    Over number fields
    
│ │ │ @@ -348,15 +348,15 @@
│ │ │  o31 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i32 : time rationalPoints(variety nodes, Split=>true, Verbose=>true);
│ │ │  -- base change to the field QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25)
│ │ │ - -- used 0.996963s (cpu); 0.801505s (thread); 0s (gc)
    
│ │ │ + -- used 1.04689s (cpu); 0.856754s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i33 : #oo
│ │ │  
│ │ │  o33 = 31
    
│ │ │ @@ -373,15 +373,15 @@
│ │ │  
│ │ │  o34 : Ideal of GF 1048969271299456081[x..z, w]
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i35 : time #rationalPoints(variety nodes', Split=>true, Verbose=>true)
│ │ │ - -- used 0.273936s (cpu); 0.210148s (thread); 0s (gc)
│ │ │ + -- used 0.289067s (cpu); 0.219977s (thread); 0s (gc)
│ │ │  
│ │ │  o35 = 31
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -89,15 +89,15 @@
│ │ │ │  o13 = ideal(u  + u  + u  + u  + u  + u  + u  + u  + u  + u  + u  )
│ │ │ │               0    1    2    3    4    5    6    7    8    9    10
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o13 : Ideal of ---[u ..u  ]
│ │ │ │                 101  0   10
│ │ │ │  i14 : time rationalPoints(I, Amount => true)
│ │ │ │ - -- used 0.00322822s (cpu); 0.00322771s (thread); 0s (gc)
│ │ │ │ + -- used 0.00457055s (cpu); 0.0045688s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 = 110462212541120451001
│ │ │ │  ****** OOvveerr nnuummbbeerr ffiieellddss ******
│ │ │ │  Over a number field one can use the option Bound to specify a maximal
│ │ │ │  multiplicative height given by $(x_0:\dots:x_n)\mapsto \prod_{v}\max_i|x_i|_v ^
│ │ │ │  {d_v/d}$ (this is also available as a method _g_l_o_b_a_l_H_e_i_g_h_t).
│ │ │ │  i15 : QQ[x,y,z]; I = homogenize(ideal(y^2-x*(x-1)*(x-2)*(x-5)*(x-6)), z);
│ │ │ │ @@ -197,24 +197,24 @@
│ │ │ │  
│ │ │ │  o30 : Ideal of R
│ │ │ │  i31 : nodes = I + ideal jacobian I;
│ │ │ │  
│ │ │ │  o31 : Ideal of R
│ │ │ │  i32 : time rationalPoints(variety nodes, Split=>true, Verbose=>true);
│ │ │ │  -- base change to the field QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25)
│ │ │ │ - -- used 0.996963s (cpu); 0.801505s (thread); 0s (gc)
│ │ │ │ + -- used 1.04689s (cpu); 0.856754s (thread); 0s (gc)
│ │ │ │  i33 : #oo
│ │ │ │  
│ │ │ │  o33 = 31
│ │ │ │  Still it runs a lot faster when reduced to a positive characteristic.
│ │ │ │  i34 : nodes' = baseChange_32003 nodes;
│ │ │ │  
│ │ │ │  o34 : Ideal of GF 1048969271299456081[x..z, w]
│ │ │ │  i35 : time #rationalPoints(variety nodes', Split=>true, Verbose=>true)
│ │ │ │ - -- used 0.273936s (cpu); 0.210148s (thread); 0s (gc)
│ │ │ │ + -- used 0.289067s (cpu); 0.219977s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o35 = 31
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  For a number field other than QQ, the enumeration of elements with bounded
│ │ │ │  height depends on an algorithm by Doyle–Krumm, which is currently only
│ │ │ │  implemented in Sage.
│ │ │ │  ******** MMeennuu ********
│ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/___Plane__Curve__Singularities.out
│ │ │ @@ -331,15 +331,15 @@
│ │ │                 2 2    2   2             2 2    2
│ │ │        - p w , p y  - p , p w y - p p , p w  - p )
│ │ │           2 1   0      1   0 0     1 2   0 0    2
│ │ │  
│ │ │  o47 : Ideal of B2
│ │ │  
│ │ │  i48 : time sing2 = ideal singularLocus strictTransform2;
│ │ │ - -- used 0.826819s (cpu); 0.679734s (thread); 0s (gc)
│ │ │ + -- used 0.959232s (cpu); 0.804259s (thread); 0s (gc)
│ │ │  
│ │ │                   ZZ
│ │ │  o48 : Ideal of -----[p ..p , w ..w , x..y]
│ │ │                 32003  0   2   0   1
│ │ │  
│ │ │  i49 : saturate(sing2, sub(irrelTot, ring sing2))
│ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_expected__Rees__Ideal.out
│ │ │ @@ -58,15 +58,15 @@
│ │ │  o5 : Matrix S  <-- S
│ │ │  
│ │ │  i6 : I = minors(n-1, M);
│ │ │  
│ │ │  o6 : Ideal of S
│ │ │  
│ │ │  i7 : time rI = expectedReesIdeal I; -- n= 5 case takes < 1 sec.
│ │ │ - -- used 1.03s (cpu); 0.739203s (thread); 0s (gc)
│ │ │ + -- used 1.02858s (cpu); 0.80405s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of S[w ..w ]
│ │ │                   0   4
│ │ │  
│ │ │  i8 : kk = ZZ/101;
│ │ │  
│ │ │  i9 : S = kk[x,y,z];
│ │ │ @@ -77,19 +77,19 @@
│ │ │  o10 : Matrix S  <-- S
│ │ │  
│ │ │  i11 : I = minors(3,m);
│ │ │  
│ │ │  o11 : Ideal of S
│ │ │  
│ │ │  i12 : time reesIdeal (I, I_0);
│ │ │ - -- used 1.73256s (cpu); 1.40832s (thread); 0s (gc)
│ │ │ + -- used 1.78483s (cpu); 1.46876s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of S[w ..w ]
│ │ │                    0   3
│ │ │  
│ │ │  i13 : time reesIdeal (I, I_0, Jacobian =>false);
│ │ │ - -- used 1.5974s (cpu); 1.21866s (thread); 0s (gc)
│ │ │ + -- used 1.60907s (cpu); 1.39787s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of S[w ..w ]
│ │ │                    0   3
│ │ │  
│ │ │  i14 :
│ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_rees__Ideal.out
│ │ │ @@ -13,21 +13,21 @@
│ │ │                  3    2
│ │ │       - x x x , x  - x x )
│ │ │          0 2 4   1    0 4
│ │ │  
│ │ │  o3 : Ideal of S
│ │ │  
│ │ │  i4 : time V1 = reesIdeal i;
│ │ │ - -- used 0.0275853s (cpu); 0.0256772s (thread); 0s (gc)
│ │ │ + -- used 0.0793139s (cpu); 0.0320054s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of S[w ..w ]
│ │ │                   0   6
│ │ │  
│ │ │  i5 : time V2 = reesIdeal(i,i_0);
│ │ │ - -- used 0.113109s (cpu); 0.112422s (thread); 0s (gc)
│ │ │ + -- used 0.327239s (cpu); 0.164424s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : Ideal of S[w ..w ]
│ │ │                   0   6
│ │ │  
│ │ │  i6 : S=kk[a,b,c]
│ │ │  
│ │ │  o6 = S
│ │ │ @@ -47,21 +47,21 @@
│ │ │  
│ │ │               2        2
│ │ │  o8 = ideal (a , a*b, b )
│ │ │  
│ │ │  o8 : Ideal of S
│ │ │  
│ │ │  i9 : time I1 = reesIdeal i;
│ │ │ - -- used 0.0195203s (cpu); 0.0182383s (thread); 0s (gc)
│ │ │ + -- used 0.137162s (cpu); 0.03244s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of S[w ..w ]
│ │ │                   0   2
│ │ │  
│ │ │  i10 : time I2 = reesIdeal(i,i_0);
│ │ │ - -- used 0.00761992s (cpu); 0.00733771s (thread); 0s (gc)
│ │ │ + -- used 0.0203954s (cpu); 0.00915512s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of S[w ..w ]
│ │ │                    0   2
│ │ │  
│ │ │  i11 : transpose gens I1
│ │ │  
│ │ │  o11 = {-1, -3} | aw_1-bw_2    |
│ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/html/___Plane__Curve__Singularities.html
│ │ │ @@ -587,15 +587,15 @@
│ │ │          
    
│ │ │            
    We compute the singular locus once again:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -325,15 +325,15 @@
│ │ │ │                 2 2    2   2             2 2    2
│ │ │ │        - p w , p y  - p , p w y - p p , p w  - p )
│ │ │ │           2 1   0      1   0 0     1 2   0 0    2
│ │ │ │  
│ │ │ │  o47 : Ideal of B2
│ │ │ │  We compute the singular locus once again:
│ │ │ │  i48 : time sing2 = ideal singularLocus strictTransform2;
│ │ │ │ - -- used 0.826819s (cpu); 0.679734s (thread); 0s (gc)
│ │ │ │ + -- used 0.959232s (cpu); 0.804259s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                   ZZ
│ │ │ │  o48 : Ideal of -----[p ..p , w ..w , x..y]
│ │ │ │                 32003  0   2   0   1
│ │ │ │  i49 : saturate(sing2, sub(irrelTot, ring sing2))
│ │ │ │  
│ │ │ │  o49 = ideal 1
│ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/html/_expected__Rees__Ideal.html
│ │ │ @@ -151,15 +151,15 @@
│ │ │  
│ │ │  o6 : Ideal of S
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i48 : time sing2 = ideal singularLocus strictTransform2;
│ │ │ - -- used 0.826819s (cpu); 0.679734s (thread); 0s (gc)
│ │ │ + -- used 0.959232s (cpu); 0.804259s (thread); 0s (gc)
│ │ │  
│ │ │                   ZZ
│ │ │  o48 : Ideal of -----[p ..p , w ..w , x..y]
│ │ │                 32003  0   2   0   1
    
│ │ │              
    
│ │ │                
    i7 : time rI = expectedReesIdeal I; -- n= 5 case takes < 1 sec.
│ │ │ - -- used 1.03s (cpu); 0.739203s (thread); 0s (gc)
│ │ │ + -- used 1.02858s (cpu); 0.80405s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of S[w ..w ]
│ │ │                   0   4
    
│ │ │              
    
│ │ │ @@ -185,24 +185,24 @@
│ │ │  
│ │ │  o11 : Ideal of S
│ │ │              
    
│ │ │                
    i12 : time reesIdeal (I, I_0);
│ │ │ - -- used 1.73256s (cpu); 1.40832s (thread); 0s (gc)
│ │ │ + -- used 1.78483s (cpu); 1.46876s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of S[w ..w ]
│ │ │                    0   3
    
│ │ │              
    
│ │ │                
    i13 : time reesIdeal (I, I_0, Jacobian =>false);
│ │ │ - -- used 1.5974s (cpu); 1.21866s (thread); 0s (gc)
│ │ │ + -- used 1.60907s (cpu); 1.39787s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of S[w ..w ]
│ │ │                    0   3
    
│ │ │              
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -86,34 +86,34 @@
│ │ │ │  
│ │ │ │               5      4
│ │ │ │  o5 : Matrix S  <-- S
│ │ │ │  i6 : I = minors(n-1, M);
│ │ │ │  
│ │ │ │  o6 : Ideal of S
│ │ │ │  i7 : time rI = expectedReesIdeal I; -- n= 5 case takes < 1 sec.
│ │ │ │ - -- used 1.03s (cpu); 0.739203s (thread); 0s (gc)
│ │ │ │ + -- used 1.02858s (cpu); 0.80405s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : Ideal of S[w ..w ]
│ │ │ │                   0   4
│ │ │ │  i8 : kk = ZZ/101;
│ │ │ │  i9 : S = kk[x,y,z];
│ │ │ │  i10 : m = random(S^3, S^{4:-2});
│ │ │ │  
│ │ │ │                3      4
│ │ │ │  o10 : Matrix S  <-- S
│ │ │ │  i11 : I = minors(3,m);
│ │ │ │  
│ │ │ │  o11 : Ideal of S
│ │ │ │  i12 : time reesIdeal (I, I_0);
│ │ │ │ - -- used 1.73256s (cpu); 1.40832s (thread); 0s (gc)
│ │ │ │ + -- used 1.78483s (cpu); 1.46876s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o12 : Ideal of S[w ..w ]
│ │ │ │                    0   3
│ │ │ │  i13 : time reesIdeal (I, I_0, Jacobian =>false);
│ │ │ │ - -- used 1.5974s (cpu); 1.21866s (thread); 0s (gc)
│ │ │ │ + -- used 1.60907s (cpu); 1.39787s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 : Ideal of S[w ..w ]
│ │ │ │                    0   3
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _s_y_m_m_e_t_r_i_c_A_l_g_e_b_r_a_I_d_e_a_l -- Ideal of the symmetric algebra of an ideal or
│ │ │ │        module
│ │ │ │      * _j_a_c_o_b_i_a_n_D_u_a_l -- Computes the 'jacobian dual', part of a method of finding
│ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/html/_rees__Ideal.html
│ │ │ @@ -110,24 +110,24 @@
│ │ │  
│ │ │  o3 : Ideal of S
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time V1 = reesIdeal i;
│ │ │ - -- used 0.0275853s (cpu); 0.0256772s (thread); 0s (gc)
│ │ │ + -- used 0.0793139s (cpu); 0.0320054s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of S[w ..w ]
│ │ │                   0   6
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time V2 = reesIdeal(i,i_0);
│ │ │ - -- used 0.113109s (cpu); 0.112422s (thread); 0s (gc)
│ │ │ + -- used 0.327239s (cpu); 0.164424s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : Ideal of S[w ..w ]
│ │ │                   0   6
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │ @@ -164,24 +164,24 @@
│ │ │  
│ │ │  o8 : Ideal of S
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : time I1 = reesIdeal i;
│ │ │ - -- used 0.0195203s (cpu); 0.0182383s (thread); 0s (gc)
│ │ │ + -- used 0.137162s (cpu); 0.03244s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of S[w ..w ]
│ │ │                   0   2
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : time I2 = reesIdeal(i,i_0);
│ │ │ - -- used 0.00761992s (cpu); 0.00733771s (thread); 0s (gc)
│ │ │ + -- used 0.0203954s (cpu); 0.00915512s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of S[w ..w ]
│ │ │                    0   2
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -51,20 +51,20 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                  3    2
│ │ │ │       - x x x , x  - x x )
│ │ │ │          0 2 4   1    0 4
│ │ │ │  
│ │ │ │  o3 : Ideal of S
│ │ │ │  i4 : time V1 = reesIdeal i;
│ │ │ │ - -- used 0.0275853s (cpu); 0.0256772s (thread); 0s (gc)
│ │ │ │ + -- used 0.0793139s (cpu); 0.0320054s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : Ideal of S[w ..w ]
│ │ │ │                   0   6
│ │ │ │  i5 : time V2 = reesIdeal(i,i_0);
│ │ │ │ - -- used 0.113109s (cpu); 0.112422s (thread); 0s (gc)
│ │ │ │ + -- used 0.327239s (cpu); 0.164424s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : Ideal of S[w ..w ]
│ │ │ │                   0   6
│ │ │ │  The following example shows how we handle degrees
│ │ │ │  i6 : S=kk[a,b,c]
│ │ │ │  
│ │ │ │  o6 = S
│ │ │ │ @@ -81,20 +81,20 @@
│ │ │ │  i8 : i=minors(2,m)
│ │ │ │  
│ │ │ │               2        2
│ │ │ │  o8 = ideal (a , a*b, b )
│ │ │ │  
│ │ │ │  o8 : Ideal of S
│ │ │ │  i9 : time I1 = reesIdeal i;
│ │ │ │ - -- used 0.0195203s (cpu); 0.0182383s (thread); 0s (gc)
│ │ │ │ + -- used 0.137162s (cpu); 0.03244s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 : Ideal of S[w ..w ]
│ │ │ │                   0   2
│ │ │ │  i10 : time I2 = reesIdeal(i,i_0);
│ │ │ │ - -- used 0.00761992s (cpu); 0.00733771s (thread); 0s (gc)
│ │ │ │ + -- used 0.0203954s (cpu); 0.00915512s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 : Ideal of S[w ..w ]
│ │ │ │                    0   2
│ │ │ │  i11 : transpose gens I1
│ │ │ │  
│ │ │ │  o11 = {-1, -3} | aw_1-bw_2    |
│ │ │ │        {-1, -3} | aw_0-bw_1    |
│ │ ├── ./usr/share/doc/Macaulay2/Regularity/example-output/_m__Regularity.out
│ │ │ @@ -71,15 +71,15 @@
│ │ │       x x x , x  + x x  - x x  - x x x , x  + x  - x x )
│ │ │        0 1 3   0    0 1    1 2    0 2 5   0    2    0 5
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 : benchmark "mRegularity I1"
│ │ │  
│ │ │ -o8 = .2590714710000002
│ │ │ +o8 = .2568463147272727
│ │ │  
│ │ │  o8 : RR (of precision 53)
│ │ │  
│ │ │  i9 : R = QQ[x_0..x_5]
│ │ │  
│ │ │  o9 = R
│ │ │  
│ │ │ @@ -87,17 +87,17 @@
│ │ │  
│ │ │  i10 : I2 = ideal ( x_0^2+x_5^2, x_0^2+x_0*x_3+x_4^2, x_0^2+x_0*x_5+x_2*x_5, x_0^2-x_0*x_3-x_3*x_5, x_0^2-x_3*x_4, x_0*x_3);
│ │ │  
│ │ │  o10 : Ideal of R
│ │ │  
│ │ │  i11 : benchmark " mRegularity I2"
│ │ │  
│ │ │ -o11 = .07735652088059701
│ │ │ +o11 = .07241933712244897
│ │ │  
│ │ │  o11 : RR (of precision 53)
│ │ │  
│ │ │  i12 : time regularity I2  
│ │ │ - -- used 0.00237225s (cpu); 0.00237209s (thread); 0s (gc)
│ │ │ + -- used 0.00216214s (cpu); 0.00216562s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = 4
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/Regularity/html/_m__Regularity.html
│ │ │ @@ -176,15 +176,15 @@
│ │ │  o7 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : benchmark "mRegularity I1"
│ │ │  
│ │ │ -o8 = .2590714710000002
│ │ │ +o8 = .2568463147272727
│ │ │  
│ │ │  o8 : RR (of precision 53)
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    This is an example where regularity is faster than mRegularity.
    
│ │ │          
│ │ │ @@ -204,23 +204,23 @@
│ │ │  o10 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i11 : benchmark " mRegularity I2"
│ │ │  
│ │ │ -o11 = .07735652088059701
│ │ │ +o11 = .07241933712244897
│ │ │  
│ │ │  o11 : RR (of precision 53)
    
│ │ │              
    
│ │ │                
    i12 : time regularity I2  
│ │ │ - -- used 0.00237225s (cpu); 0.00237209s (thread); 0s (gc)
│ │ │ + -- used 0.00216214s (cpu); 0.00216562s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = 4
    
│ │ │              
    
│ │ │          
    This symbol is provided by the package Regularity.
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -94,34 +94,34 @@
│ │ │ │                3    2        2            3    3      2
│ │ │ │       x x x , x  + x x  - x x  - x x x , x  + x  - x x )
│ │ │ │        0 1 3   0    0 1    1 2    0 2 5   0    2    0 5
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  i8 : benchmark "mRegularity I1"
│ │ │ │  
│ │ │ │ -o8 = .2590714710000002
│ │ │ │ +o8 = .2568463147272727
│ │ │ │  
│ │ │ │  o8 : RR (of precision 53)
│ │ │ │  This is an example where regularity is faster than mRegularity.
│ │ │ │  i9 : R = QQ[x_0..x_5]
│ │ │ │  
│ │ │ │  o9 = R
│ │ │ │  
│ │ │ │  o9 : PolynomialRing
│ │ │ │  i10 : I2 = ideal ( x_0^2+x_5^2, x_0^2+x_0*x_3+x_4^2, x_0^2+x_0*x_5+x_2*x_5,
│ │ │ │  x_0^2-x_0*x_3-x_3*x_5, x_0^2-x_3*x_4, x_0*x_3);
│ │ │ │  
│ │ │ │  o10 : Ideal of R
│ │ │ │  i11 : benchmark " mRegularity I2"
│ │ │ │  
│ │ │ │ -o11 = .07735652088059701
│ │ │ │ +o11 = .07241933712244897
│ │ │ │  
│ │ │ │  o11 : RR (of precision 53)
│ │ │ │  i12 : time regularity I2
│ │ │ │ - -- used 0.00237225s (cpu); 0.00237209s (thread); 0s (gc)
│ │ │ │ + -- used 0.00216214s (cpu); 0.00216562s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o12 = 4
│ │ │ │  This symbol is provided by the package Regularity.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_g_u_l_a_r_i_t_y -- compute the Castelnuovo-Mumford regularity
│ │ │ │  ********** WWaayyss ttoo uussee mmRReegguullaarriittyy:: **********
│ │ │ │      * mRegularity(Ideal)
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_cayley__Trick.out
│ │ │ @@ -5,18 +5,18 @@
│ │ │  o2 = ideal(x x  - x x )
│ │ │              0 1    2 3
│ │ │  
│ │ │  o2 : Ideal of QQ[x ..x ]
│ │ │                    0   3
│ │ │  
│ │ │  i3 : time (P1xP1xP2,P1xP1xP2') = cayleyTrick(P1xP1,2);
│ │ │ - -- used 0.127171s (cpu); 0.070403s (thread); 0s (gc)
│ │ │ + -- used 0.131859s (cpu); 0.0745575s (thread); 0s (gc)
│ │ │  
│ │ │  i4 : time (P1xP1xP1,P1xP1xP1') = cayleyTrick(P1xP1,1)
│ │ │ - -- used 0.124664s (cpu); 0.0700762s (thread); 0s (gc)
│ │ │ + -- used 0.139932s (cpu); 0.0797331s (thread); 0s (gc)
│ │ │  
│ │ │                                                                             
│ │ │  o4 = (ideal (x   x    - x   x   , x   x    - x   x   , x   x    - x   x   ,
│ │ │                0,3 1,2    0,2 1,3   1,0 1,1    1,2 1,3   0,3 1,1    0,1 1,3 
│ │ │       ------------------------------------------------------------------------
│ │ │                                                                              
│ │ │       x   x    - x   x   , x   x    - x   x   , x   x    - x   x   , x   x   
│ │ │ @@ -37,17 +37,17 @@
│ │ │                                                2   2
│ │ │       4x   x   x   x    - 2x   x   x   x    + x   x   ))
│ │ │         0,0 0,1 1,2 1,3     0,2 0,3 1,2 1,3    0,2 1,3
│ │ │  
│ │ │  o4 : Sequence
│ │ │  
│ │ │  i5 : time cayleyTrick(P1xP1,1,Duality=>true);
│ │ │ - -- used 0.153844s (cpu); 0.0993609s (thread); 0s (gc)
│ │ │ + -- used 0.17014s (cpu); 0.114168s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : assert(oo == (P1xP1xP1,P1xP1xP1'))
│ │ │  
│ │ │  i7 : time cayleyTrick(P1xP1,2,Duality=>true);
│ │ │ - -- used 0.197861s (cpu); 0.13749s (thread); 0s (gc)
│ │ │ + -- used 0.177206s (cpu); 0.115764s (thread); 0s (gc)
│ │ │  
│ │ │  i8 : assert(oo == (P1xP1xP2,P1xP1xP2'))
│ │ │  
│ │ │  i9 :
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_chow__Equations.out
│ │ │ @@ -9,15 +9,15 @@
│ │ │  o2 = ideal (x  + x  + x  + x , x x  + x x  + x x )
│ │ │               0    1    2    3   0 1    1 2    2 3
│ │ │  
│ │ │  o2 : Ideal of P3
│ │ │  
│ │ │  i3 : -- Chow equations of C
│ │ │       time eqsC = chowEquations chowForm C
│ │ │ - -- used 0.127688s (cpu); 0.062s (thread); 0s (gc)
│ │ │ + -- used 0.135471s (cpu); 0.0630562s (thread); 0s (gc)
│ │ │  
│ │ │               2 2    2 2    2 2    4                2      2 2   2      
│ │ │  o3 = ideal (x x  + x x  + x x  + x , x x x x  + x x x  + x x , x x x  +
│ │ │               0 3    1 3    2 3    3   0 1 2 3    1 2 3    2 3   0 2 3  
│ │ │       ------------------------------------------------------------------------
│ │ │        2        3           2         2      3   3         2          2    2 2
│ │ │       x x x  + x x  - 2x x x  - 2x x x  - x x , x x  + 2x x x  - x x x  + x x 
│ │ │ @@ -72,15 +72,15 @@
│ │ │  o5 = ideal (x  - x x , x  - x x x , x x  - x x )
│ │ │               1    0 2   2    0 1 3   1 2    0 3
│ │ │  
│ │ │  o5 : Ideal of P3
│ │ │  
│ │ │  i6 : -- Chow equations of D
│ │ │       time eqsD = chowEquations chowForm D
│ │ │ - -- used 0.113843s (cpu); 0.0571957s (thread); 0s (gc)
│ │ │ + -- used 0.13548s (cpu); 0.0604597s (thread); 0s (gc)
│ │ │  
│ │ │               4      3 2     3        2 2     3      2   2   2 2      2   2 
│ │ │  o6 = ideal (x x  - x x , x x x  - x x x , x x x  - x x x , x x x  - x x x ,
│ │ │               2 3    1 3   1 2 3    0 1 3   0 2 3    0 1 3   1 2 3    0 1 3 
│ │ │       ------------------------------------------------------------------------
│ │ │            2      3 2   3        3 2   4         2         2 2       3    
│ │ │       x x x x  - x x , x x x  - x x , x x  - 4x x x x  + 3x x x , x x x  -
│ │ │ @@ -117,24 +117,24 @@
│ │ │  o9 = ideal(x x  + x x )
│ │ │              0 1    2 3
│ │ │  
│ │ │  o9 : Ideal of P3
│ │ │  
│ │ │  i10 : -- tangential Chow forms of Q
│ │ │        time (W0,W1,W2) = (tangentialChowForm(Q,0),tangentialChowForm(Q,1),tangentialChowForm(Q,2))
│ │ │ - -- used 0.158592s (cpu); 0.109834s (thread); 0s (gc)
│ │ │ + -- used 0.263346s (cpu); 0.119511s (thread); 0s (gc)
│ │ │  
│ │ │                       2                              2
│ │ │  o10 = (x x  + x x , x    - 4x   x    + 2x   x    + x   , x     x      +
│ │ │          0 1    2 3   0,1     0,2 1,3     0,1 2,3    2,3   0,1,2 0,1,3  
│ │ │        -----------------------------------------------------------------------
│ │ │        x     x     )
│ │ │         0,2,3 1,2,3
│ │ │  
│ │ │  o10 : Sequence
│ │ │  
│ │ │  i11 : time (Q,Q,Q) == (chowEquations(W0,0),chowEquations(W1,1),chowEquations(W2,2))
│ │ │ - -- used 0.150222s (cpu); 0.0893773s (thread); 0s (gc)
│ │ │ + -- used 0.158772s (cpu); 0.0913561s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = true
│ │ │  
│ │ │  i12 :
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_chow__Form.out
│ │ │ @@ -16,15 +16,15 @@
│ │ │  
│ │ │                 ZZ
│ │ │  o2 : Ideal of ----[x ..x ]
│ │ │                3331  0   5
│ │ │  
│ │ │  i3 : -- Chow form of V in Grass(2,5) (performing internal computations on an affine chart of the Grassmannian)
│ │ │       time ChowV = chowForm(V,AffineChartGrass=>{1,2,3})
│ │ │ - -- used 5.31048s (cpu); 4.82715s (thread); 0s (gc)
│ │ │ + -- used 5.21966s (cpu); 4.85951s (thread); 0s (gc)
│ │ │  
│ │ │        4               2              2     2               2            
│ │ │  o3 = x      + 2x     x     x      + x     x      - 2x     x     x      +
│ │ │        1,2,4     0,2,4 1,2,4 2,3,4    0,2,4 2,3,4     1,2,3 1,2,4 1,2,5  
│ │ │       ------------------------------------------------------------------------
│ │ │        2     2              2                                       
│ │ │       x     x      - x     x     x      + x     x     x     x      +
│ │ │ @@ -143,19 +143,19 @@
│ │ │                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            3331  0,1,2   0,1,3   0,2,3   1,2,3   0,1,4   0,2,4   1,2,4   0,3,4   1,3,4   2,3,4   0,1,5   0,2,5   1,2,5   0,3,5   1,3,5   2,3,5   0,4,5   1,4,5   2,4,5   3,4,5
│ │ │  o3 : -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
│ │ │       (x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x      - x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x      + x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x      - x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     )
│ │ │         2,3,5 1,4,5    1,3,5 2,4,5    1,2,5 3,4,5   2,3,4 1,4,5    1,3,4 2,4,5    1,2,4 3,4,5   2,3,5 0,4,5    0,3,5 2,4,5    0,2,5 3,4,5   1,3,5 0,4,5    0,3,5 1,4,5    0,1,5 3,4,5   1,2,5 0,4,5    0,2,5 1,4,5    0,1,5 2,4,5   2,3,4 0,4,5    0,3,4 2,4,5    0,2,4 3,4,5   1,3,4 0,4,5    0,3,4 1,4,5    0,1,4 3,4,5   1,2,4 0,4,5    0,2,4 1,4,5    0,1,4 2,4,5   1,2,3 0,4,5    0,2,3 1,4,5    0,1,3 2,4,5    0,1,2 3,4,5   2,3,4 1,3,5    1,3,4 2,3,5    1,2,3 3,4,5   1,2,5 0,3,5    0,2,5 1,3,5    0,1,5 2,3,5   2,3,4 0,3,5    0,3,4 2,3,5    0,2,3 3,4,5   1,3,4 0,3,5    0,3,4 1,3,5    0,1,3 3,4,5   1,2,4 0,3,5    0,2,4 1,3,5    0,1,4 2,3,5    0,1,2 3,4,5   1,2,3 0,3,5    0,2,3 1,3,5    0,1,3 2,3,5   2,3,4 1,2,5    1,2,4 2,3,5    1,2,3 2,4,5   1,3,4 1,2,5    1,2,4 1,3,5    1,2,3 1,4,5   0,3,4 1,2,5    0,2,4 1,3,5    0,1,4 2,3,5    0,2,3 1,4,5    0,1,3 2,4,5    0,1,2 3,4,5   2,3,4 0,2,5    0,2,4 2,3,5    0,2,3 2,4,5   1,3,4 0,2,5    0,2,4 1,3,5    0,2,3 1,4,5    0,1,2 3,4,5   0,3,4 0,2,5    0,2,4 0,3,5    0,2,3 0,4,5   1,2,4 0,2,5    0,2,4 1,2,5    0,1,2 2,4,5   1,2,3 0,2,5    0,2,3 1,2,5    0,1,2 2,3,5   2,3,4 0,1,5    0,1,4 2,3,5    0,1,3 2,4,5    0,1,2 3,4,5   1,3,4 0,1,5    0,1,4 1,3,5    0,1,3 1,4,5   0,3,4 0,1,5    0,1,4 0,3,5    0,1,3 0,4,5   1,2,4 0,1,5    0,1,4 1,2,5    0,1,2 1,4,5   0,2,4 0,1,5    0,1,4 0,2,5    0,1,2 0,4,5   1,2,3 0,1,5    0,1,3 1,2,5    0,1,2 1,3,5   0,2,3 0,1,5    0,1,3 0,2,5    0,1,2 0,3,5   1,2,4 0,3,4    0,2,4 1,3,4    0,1,4 2,3,4   1,2,3 0,3,4    0,2,3 1,3,4    0,1,3 2,3,4   1,2,3 0,2,4    0,2,3 1,2,4    0,1,2 2,3,4   1,2,3 0,1,4    0,1,3 1,2,4    0,1,2 1,3,4   0,2,3 0,1,4    0,1,3 0,2,4    0,1,2 0,3,4
│ │ │  
│ │ │  i4 : -- equivalently (but faster)...
│ │ │       time assert(ChowV === chowForm f)
│ │ │ - -- used 1.12682s (cpu); 0.99711s (thread); 0s (gc)
│ │ │ + -- used 1.15009s (cpu); 1.09337s (thread); 0s (gc)
│ │ │  
│ │ │  i5 : -- X-resultant of V
│ │ │       time Xres = fromPluckerToStiefel dualize ChowV;
│ │ │ - -- used 0.235111s (cpu); 0.179065s (thread); 0s (gc)
│ │ │ + -- used 0.284789s (cpu); 0.220968s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : -- three generic ternary quadrics
│ │ │       F = genericPolynomials({2,2,2},ZZ/3331)
│ │ │  
│ │ │           2               2                        2     2               2  
│ │ │  o6 = {a x  + a x x  + a x  + a x x  + a x x  + a x , b x  + b x x  + b x  +
│ │ │         0 0    1 0 1    3 1    2 0 2    4 1 2    5 2   0 0    1 0 1    3 1  
│ │ │ @@ -164,12 +164,12 @@
│ │ │       b x x  + b x x  + b x , c x  + c x x  + c x  + c x x  + c x x  + c x }
│ │ │        2 0 2    4 1 2    5 2   0 0    1 0 1    3 1    2 0 2    4 1 2    5 2
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : -- resultant of the three forms
│ │ │       time resF = resultant F;
│ │ │ - -- used 0.285058s (cpu); 0.174585s (thread); 0s (gc)
│ │ │ + -- used 0.287939s (cpu); 0.220002s (thread); 0s (gc)
│ │ │  
│ │ │  i8 : assert(resF === sub(Xres,vars ring resF) and Xres === sub(resF,vars ring Xres))
│ │ │  
│ │ │  i9 :
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_discriminant_lp__Ring__Element_rp.out
│ │ │ @@ -4,30 +4,30 @@
│ │ │  
│ │ │          2              2
│ │ │  o2 = a*x  + b*x*y + c*y
│ │ │  
│ │ │  o2 : ZZ[a..c][x..y]
│ │ │  
│ │ │  i3 : time discriminant F
│ │ │ - -- used 0.00807841s (cpu); 0.00807736s (thread); 0s (gc)
│ │ │ + -- used 0.00983715s (cpu); 0.00983637s (thread); 0s (gc)
│ │ │  
│ │ │          2
│ │ │  o3 = - b  + 4a*c
│ │ │  
│ │ │  o3 : ZZ[a..c]
│ │ │  
│ │ │  i4 : ZZ[a,b,c,d][x,y]; F = a*x^3+b*x^2*y+c*x*y^2+d*y^3
│ │ │  
│ │ │          3      2         2      3
│ │ │  o5 = a*x  + b*x y + c*x*y  + d*y
│ │ │  
│ │ │  o5 : ZZ[a..d][x..y]
│ │ │  
│ │ │  i6 : time discriminant F
│ │ │ - -- used 0.0087506s (cpu); 0.00875122s (thread); 0s (gc)
│ │ │ + -- used 0.0101472s (cpu); 0.010148s (thread); 0s (gc)
│ │ │  
│ │ │          2 2       3     3                   2 2
│ │ │  o6 = - b c  + 4a*c  + 4b d - 18a*b*c*d + 27a d
│ │ │  
│ │ │  o6 : ZZ[a..d]
│ │ │  
│ │ │  i7 : x=symbol x; R=ZZ/331[x_0..x_3]
│ │ │ @@ -59,15 +59,15 @@
│ │ │                  4        3      4             4        3      4
│ │ │  o12 = (t  + t )x  - t x x  + t x  + (t  - t )x  + t x x  + t x
│ │ │          0    1  0    1 0 1    0 1     0    1  2    1 2 3    0 3
│ │ │  
│ │ │  o12 : R'
│ │ │  
│ │ │  i13 : time D=discriminant pencil
│ │ │ - -- used 0.461801s (cpu); 0.427218s (thread); 0s (gc)
│ │ │ + -- used 0.480519s (cpu); 0.420479s (thread); 0s (gc)
│ │ │  
│ │ │             108      106 2       102 6      100 8       98 10       96 12  
│ │ │  o13 = - 62t    + 19t   t  + 160t   t  + 91t   t  + 129t  t   + 117t  t   +
│ │ │             0        0   1       0   1      0   1       0  1        0  1   
│ │ │        -----------------------------------------------------------------------
│ │ │            94 14       92 16      90 18      88 20      86 22       84 24  
│ │ │        161t  t   + 124t  t   - 82t  t   - 21t  t   - 49t  t   - 123t  t   +
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_dual__Variety.out
│ │ │ @@ -9,25 +9,25 @@
│ │ │       x x )
│ │ │        0 3
│ │ │  
│ │ │  o1 : Ideal of QQ[x ..x ]
│ │ │                    0   5
│ │ │  
│ │ │  i2 : time V' = dualVariety V
│ │ │ - -- used 0.170711s (cpu); 0.119239s (thread); 0s (gc)
│ │ │ + -- used 0.198598s (cpu); 0.132067s (thread); 0s (gc)
│ │ │  
│ │ │              2                 2    2
│ │ │  o2 = ideal(x x  - x x x  + x x  + x x  - 4x x x )
│ │ │              2 3    1 2 4    0 4    1 5     0 3 5
│ │ │  
│ │ │  o2 : Ideal of QQ[x ..x ]
│ │ │                    0   5
│ │ │  
│ │ │  i3 : time V == dualVariety V'
│ │ │ - -- used 0.204333s (cpu); 0.146742s (thread); 0s (gc)
│ │ │ + -- used 0.230838s (cpu); 0.168205s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 : F = first genericPolynomials({3,-1,-1},ZZ/3331)
│ │ │  
│ │ │          3      2          2      3      2                   2          2  
│ │ │  o4 = a x  + a x x  + a x x  + a x  + a x x  + a x x x  + a x x  + a x x  +
│ │ │ @@ -38,22 +38,22 @@
│ │ │        8 1 2    9 2
│ │ │  
│ │ │        ZZ
│ │ │  o4 : ----[a ..a ][x ..x ]
│ │ │       3331  0   9   0   2
│ │ │  
│ │ │  i5 : time discF = ideal discriminant F;
│ │ │ - -- used 0.0566354s (cpu); 0.0566377s (thread); 0s (gc)
│ │ │ + -- used 0.0657693s (cpu); 0.065771s (thread); 0s (gc)
│ │ │  
│ │ │                 ZZ
│ │ │  o5 : Ideal of ----[a ..a ]
│ │ │                3331  0   9
│ │ │  
│ │ │  i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true);
│ │ │ - -- used 0.611958s (cpu); 0.563695s (thread); 0s (gc)
│ │ │ + -- used 0.730716s (cpu); 0.662534s (thread); 0s (gc)
│ │ │  
│ │ │                 ZZ
│ │ │  o6 : Ideal of ----[x ..x ]
│ │ │                3331  0   9
│ │ │  
│ │ │  i7 : discF == sub(Z,vars ring discF) and Z == sub(discF,vars ring Z)
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_from__Plucker__To__Stiefel.out
│ │ │ @@ -6,15 +6,15 @@
│ │ │  o1 = ideal (x  - x x , x x  - x x , x  - x x )
│ │ │               2    1 3   1 2    0 3   1    0 2
│ │ │  
│ │ │  o1 : Ideal of QQ[x ..x ]
│ │ │                    0   3
│ │ │  
│ │ │  i2 : time fromPluckerToStiefel dualize chowForm C
│ │ │ - -- used 0.110649s (cpu); 0.0580797s (thread); 0s (gc)
│ │ │ + -- used 0.134421s (cpu); 0.0678798s (thread); 0s (gc)
│ │ │  
│ │ │          3   3          2   2              2       2          2   3    
│ │ │  o2 = - x   x    + x   x   x   x    - x   x   x   x    + x   x   x    -
│ │ │          0,3 1,0    0,2 0,3 1,0 1,1    0,1 0,3 1,0 1,1    0,0 0,3 1,1  
│ │ │       ------------------------------------------------------------------------
│ │ │        2       2               2   2                                   
│ │ │       x   x   x   x    + 2x   x   x   x    + x   x   x   x   x   x    -
│ │ │ @@ -56,15 +56,15 @@
│ │ │       x   x   x   x    - 2x   x   x   x    - x   x   x   x    + x   x
│ │ │        0,0 0,1 1,1 1,3     0,0 0,2 1,1 1,3    0,0 0,1 1,2 1,3    0,0 1,3
│ │ │  
│ │ │  o2 : QQ[x   ..x   ]
│ │ │           0,0   1,3
│ │ │  
│ │ │  i3 : time fromPluckerToStiefel(dualize chowForm C,AffineChartGrass=>{0,1})
│ │ │ - -- used 0.040301s (cpu); 0.0403043s (thread); 0s (gc)
│ │ │ + -- used 0.0498309s (cpu); 0.0498325s (thread); 0s (gc)
│ │ │  
│ │ │              3          2                         2                        
│ │ │  o3 = - x   x    + x   x   x    - x   x   x    + x   x    + 3x   x   x    -
│ │ │          0,3 1,2    0,2 1,2 1,3    0,2 0,3 1,2    0,2 1,3     0,3 1,2 1,3  
│ │ │       ------------------------------------------------------------------------
│ │ │             2      3      2
│ │ │       2x   x    + x    + x
│ │ │ @@ -85,15 +85,15 @@
│ │ │  
│ │ │  o4 : QQ[a   ..a   ]
│ │ │           0,0   1,1
│ │ │  
│ │ │  i5 : w = chowForm C;
│ │ │  
│ │ │  i6 : time U = apply(subsets(4,2),s->ideal fromPluckerToStiefel(w,AffineChartGrass=>s))
│ │ │ - -- used 0.0190728s (cpu); 0.0190737s (thread); 0s (gc)
│ │ │ + -- used 0.0249667s (cpu); 0.0249684s (thread); 0s (gc)
│ │ │  
│ │ │                     3          2          3                       2        
│ │ │  o6 = {ideal(- x   x    + x   x   x    - x    - 3x   x   x    + 2x   x    +
│ │ │                 0,3 1,2    0,2 1,2 1,3    0,2     0,2 0,3 1,2     0,2 1,3  
│ │ │       ------------------------------------------------------------------------
│ │ │                           2      2            2   3               2        
│ │ │       x   x   x    - x   x    + x   ), ideal(x   x    - 2x   x   x   x    +
│ │ │ @@ -130,14 +130,14 @@
│ │ │             2      3      2
│ │ │       2x   x    - x    + x   )}
│ │ │         0,0 1,1    1,1    1,0
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : time apply(U,u->dim singularLocus u)
│ │ │ - -- used 0.0170671s (cpu); 0.0170726s (thread); 0s (gc)
│ │ │ + -- used 0.0211123s (cpu); 0.0211149s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {2, 2, 2, 2, 2, 2}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_hurwitz__Form.out
│ │ │ @@ -10,15 +10,15 @@
│ │ │       + -p  + -p p  + 7p p  + 6p p  + -p p  + --p )
│ │ │         4 3   9 0 4     1 4     2 4   9 3 4   10 4
│ │ │  
│ │ │  o1 : Ideal of QQ[p ..p ]
│ │ │                    0   4
│ │ │  
│ │ │  i2 : time hurwitzForm Q
│ │ │ - -- used 0.0393717s (cpu); 0.0393716s (thread); 0s (gc)
│ │ │ + -- used 0.0449735s (cpu); 0.0449738s (thread); 0s (gc)
│ │ │  
│ │ │                2                                 2                      
│ │ │  o2 = 11966535p    + 14645610p   p    + 11354175p    + 1666980p   p    +
│ │ │                0,1            0,1 0,2            0,2           0,1 1,2  
│ │ │       ------------------------------------------------------------------------
│ │ │                                 2                                          
│ │ │       4456620p   p    + 1127196p    + 54176850p   p    + 20326950p   p    +
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_is__Coisotropic.out
│ │ │ @@ -26,15 +26,15 @@
│ │ │       QQ[p   ..p   , p   , p   , p   , p   ]
│ │ │           0,1   0,2   1,2   0,3   1,3   2,3
│ │ │  o1 : --------------------------------------
│ │ │           p   p    - p   p    + p   p
│ │ │            1,2 0,3    0,2 1,3    0,1 2,3
│ │ │  
│ │ │  i2 : time isCoisotropic w
│ │ │ - -- used 0.00787565s (cpu); 0.00787232s (thread); 0s (gc)
│ │ │ + -- used 0.00855771s (cpu); 0.00855747s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = true
│ │ │  
│ │ │  i3 : -- random quadric in G(1,3)
│ │ │       w' = random(2,Grass(1,3))
│ │ │  
│ │ │         2     5           10 2     2                          2      3        
│ │ │ @@ -56,12 +56,12 @@
│ │ │       QQ[p   ..p   , p   , p   , p   , p   ]
│ │ │           0,1   0,2   1,2   0,3   1,3   2,3
│ │ │  o3 : --------------------------------------
│ │ │           p   p    - p   p    + p   p
│ │ │            1,2 0,3    0,2 1,3    0,1 2,3
│ │ │  
│ │ │  i4 : time isCoisotropic w'
│ │ │ - -- used 0.00654309s (cpu); 0.00654341s (thread); 0s (gc)
│ │ │ + -- used 0.00707054s (cpu); 0.00707066s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = false
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_is__In__Coisotropic.out
│ │ │ @@ -31,12 +31,12 @@
│ │ │            4        5
│ │ │  
│ │ │                  ZZ
│ │ │  o3 : Ideal of -----[x ..x ]
│ │ │                33331  0   5
│ │ │  
│ │ │  i4 : time isInCoisotropic(L,I) -- whether L belongs to Z_1(V(I))
│ │ │ - -- used 0.0197451s (cpu); 0.0197449s (thread); 0s (gc)
│ │ │ + -- used 0.0208729s (cpu); 0.0208724s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_macaulay__Formula.out
│ │ │ @@ -13,15 +13,15 @@
│ │ │                     2          2        2      3
│ │ │       c x x x  + c x x  + c x x  + c x x  + c x }
│ │ │        4 0 1 2    7 1 2    5 0 2    8 1 2    9 2
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : time (D,D') = macaulayFormula F
│ │ │ - -- used 0.00462514s (cpu); 0.00462065s (thread); 0s (gc)
│ │ │ + -- used 0.00419877s (cpu); 0.00419752s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = (| a_0 a_1 a_2 a_3 a_4 a_5 0   0   0   0   0   0   0   0   0   0   0  
│ │ │        | 0   a_0 0   a_1 a_2 0   a_3 a_4 a_5 0   0   0   0   0   0   0   0  
│ │ │        | 0   0   a_0 0   a_1 a_2 0   a_3 a_4 a_5 0   0   0   0   0   0   0  
│ │ │        | 0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0   0   0  
│ │ │        | 0   0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0   0  
│ │ │        | 0   0   0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0  
│ │ │ @@ -78,15 +78,15 @@
│ │ │       10   2   7   2   5 3
│ │ │       --p p  + -p p  + -p }
│ │ │        9 0 2   8 1 2   6 2
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : time (D,D') = macaulayFormula F
│ │ │ - -- used 0.00237921s (cpu); 0.00238132s (thread); 0s (gc)
│ │ │ + -- used 0.00263209s (cpu); 0.00263066s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = (| 9/2 9/4 3/4 7/4  7/9  7/10 0    0    0    0    0   0   0    0    0   
│ │ │        | 0   9/2 0   9/4  3/4  0    7/4  7/9  7/10 0    0   0   0    0    0   
│ │ │        | 0   0   9/2 0    9/4  3/4  0    7/4  7/9  7/10 0   0   0    0    0   
│ │ │        | 0   0   0   9/2  0    0    9/4  3/4  0    0    7/4 7/9 7/10 0    0   
│ │ │        | 0   0   0   0    9/2  0    0    9/4  3/4  0    0   7/4 7/9  7/10 0   
│ │ │        | 0   0   0   0    0    9/2  0    0    9/4  3/4  0   0   7/4  7/9  7/10
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_plucker.out
│ │ │ @@ -9,29 +9,29 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       664x )
│ │ │           4
│ │ │  
│ │ │  o3 : Ideal of P4
│ │ │  
│ │ │  i4 : time p = plucker L
│ │ │ - -- used 0.00452289s (cpu); 0.00452084s (thread); 0s (gc)
│ │ │ + -- used 0.00547559s (cpu); 0.00547367s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = ideal (x    + 8480x   , x    - 6727x   , x    + 15777x   , x    +
│ │ │               2,4        3,4   1,4        3,4   0,4         3,4   2,3  
│ │ │       ------------------------------------------------------------------------
│ │ │       11656x   , x    - 14853x   , x    + 664x   , x    + 13522x   , x    +
│ │ │             3,4   1,3         3,4   0,3       3,4   1,2         3,4   0,2  
│ │ │       ------------------------------------------------------------------------
│ │ │       11804x   , x    + 14854x   )
│ │ │             3,4   0,1         3,4
│ │ │  
│ │ │  o4 : Ideal of G'1'4
│ │ │  
│ │ │  i5 : time L' = plucker p
│ │ │ - -- used 0.111022s (cpu); 0.0535001s (thread); 0s (gc)
│ │ │ + -- used 0.12088s (cpu); 0.0581025s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = ideal (x  + 8480x  - 11656x , x  - 6727x  + 14853x , x  + 15777x  -
│ │ │               2        3         4   1        3         4   0         3  
│ │ │       ------------------------------------------------------------------------
│ │ │       664x )
│ │ │           4
│ │ │  
│ │ │ @@ -40,25 +40,25 @@
│ │ │  i6 : assert(L' == L)
│ │ │  
│ │ │  i7 : Y = ideal apply(5,i->random(1,G'1'4)); -- an elliptic curve
│ │ │  
│ │ │  o7 : Ideal of G'1'4
│ │ │  
│ │ │  i8 : time W = plucker Y; -- surface swept out by the lines of Y
│ │ │ - -- used 0.0361963s (cpu); 0.0361951s (thread); 0s (gc)
│ │ │ + -- used 0.0373625s (cpu); 0.0373403s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of P4
│ │ │  
│ │ │  i9 : (codim W,degree W)
│ │ │  
│ │ │  o9 = (2, 5)
│ │ │  
│ │ │  o9 : Sequence
│ │ │  
│ │ │  i10 : time Y' = plucker(W,1); -- variety of lines contained in W
│ │ │ - -- used 0.142146s (cpu); 0.142149s (thread); 0s (gc)
│ │ │ + -- used 0.174625s (cpu); 0.17463s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of G'1'4
│ │ │  
│ │ │  i11 : assert(Y' == Y)
│ │ │  
│ │ │  i12 :
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_resultant_lp..._cm__Algorithm_eq_gt..._rp.out
│ │ │ @@ -35,15 +35,15 @@
│ │ │       3     2    9    7     2    9        3       1    8    4
│ │ │       -b)y*w  + (-a + -b)z*w  + (-a + 2b)w , 2x + -y + -z + -w}
│ │ │       4          8    8          7                4    3    5
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : time resultant(F,Algorithm=>"Poisson2")
│ │ │ - -- used 0.316083s (cpu); 0.220122s (thread); 0s (gc)
│ │ │ + -- used 0.333645s (cpu); 0.20846s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o3 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -56,15 +56,15 @@
│ │ │       1146977327343523453866040839029   4   194441910898734675845094443 5
│ │ │       -------------------------------a*b  - ---------------------------b
│ │ │            1119954511872000000000                895963609497600000
│ │ │  
│ │ │  o3 : QQ[a..b]
│ │ │  
│ │ │  i4 : time resultant(F,Algorithm=>"Macaulay2")
│ │ │ - -- used 0.199752s (cpu); 0.138876s (thread); 0s (gc)
│ │ │ + -- used 0.160974s (cpu); 0.0991529s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o4 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -77,15 +77,15 @@
│ │ │       1146977327343523453866040839029   4   194441910898734675845094443 5
│ │ │       -------------------------------a*b  - ---------------------------b
│ │ │            1119954511872000000000                895963609497600000
│ │ │  
│ │ │  o4 : QQ[a..b]
│ │ │  
│ │ │  i5 : time resultant(F,Algorithm=>"Poisson")
│ │ │ - -- used 0.514703s (cpu); 0.461162s (thread); 0s (gc)
│ │ │ + -- used 0.401593s (cpu); 0.342364s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o5 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -98,15 +98,15 @@
│ │ │       1146977327343523453866040839029   4   194441910898734675845094443 5
│ │ │       -------------------------------a*b  - ---------------------------b
│ │ │            1119954511872000000000                895963609497600000
│ │ │  
│ │ │  o5 : QQ[a..b]
│ │ │  
│ │ │  i6 : time resultant(F,Algorithm=>"Macaulay")
│ │ │ - -- used 0.590923s (cpu); 0.537731s (thread); 0s (gc)
│ │ │ + -- used 0.71147s (cpu); 0.643863s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o6 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_resultant_lp__Matrix_rp.out
│ │ │ @@ -10,15 +10,15 @@
│ │ │  
│ │ │         2               2                            3      2         4
│ │ │  o2 = {x  + 3t*y*z - u*z , (t + 3u - 1)x - y, - t*x*y  + t*x y*z + u*z }
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : time resultant F
│ │ │ - -- used 0.0229913s (cpu); 0.0229909s (thread); 0s (gc)
│ │ │ + -- used 0.0283289s (cpu); 0.0283286s (thread); 0s (gc)
│ │ │  
│ │ │            12         11 2         10 3         9 4          8 5          7 6
│ │ │  o3 = - 81t  u - 1701t  u  - 15309t  u  - 76545t u  - 229635t u  - 413343t u 
│ │ │       ------------------------------------------------------------------------
│ │ │                6 7          5 8       11          10 2         9 3  
│ │ │       - 413343t u  - 177147t u  + 567t  u + 10206t  u  + 76545t u  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -64,15 +64,15 @@
│ │ │            3
│ │ │       + c x }
│ │ │          9 2
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : time resultant F
│ │ │ - -- used 2.72711s (cpu); 2.06378s (thread); 0s (gc)
│ │ │ + -- used 2.44305s (cpu); 1.91459s (thread); 0s (gc)
│ │ │  
│ │ │        6 3 2       5 2   2     2 4   2 2    3 3 3 2     2 4 2   2  
│ │ │  o5 = a b c  - 3a a b b c  + 3a a b b c  - a a b c  + 3a a b b c  -
│ │ │        2 3 0     1 2 3 4 0     1 2 3 4 0    1 2 4 0     1 2 3 5 0  
│ │ │       ------------------------------------------------------------------------
│ │ │         3 3       2     4 2 2   2     4 2   2 2     5     2 2    6 3 2  
│ │ │       6a a b b b c  + 3a a b b c  + 3a a b b c  - 3a a b b c  + a b c  -
│ │ │ @@ -1690,12 +1690,12 @@
│ │ │                            2     2               2                        2
│ │ │       b x x  + b x x  + b x , c x  + c x x  + c x  + c x x  + c x x  + c x }
│ │ │        2 0 2    4 1 2    5 2   0 0    1 0 1    3 1    2 0 2    4 1 2    5 2
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : time # terms resultant F
│ │ │ - -- used 0.58204s (cpu); 0.40137s (thread); 0s (gc)
│ │ │ + -- used 0.436586s (cpu); 0.362757s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = 21894
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_tangential__Chow__Form.out
│ │ │ @@ -8,15 +8,15 @@
│ │ │                 1 2    0 3     1 3    0 4     3    2 4
│ │ │  
│ │ │  o2 : Ideal of QQ[p ..p ]
│ │ │                    0   4
│ │ │  
│ │ │  i3 : -- 0-th associated hypersurface of S in G(1,4) (Chow form)
│ │ │       time tangentialChowForm(S,0)
│ │ │ - -- used 0.028783s (cpu); 0.0287839s (thread); 0s (gc)
│ │ │ + -- used 0.0340372s (cpu); 0.0340373s (thread); 0s (gc)
│ │ │  
│ │ │        2                                                       2        
│ │ │  o3 = p   p    - p   p   p    - p   p   p    + p   p   p    + p   p    +
│ │ │        1,3 2,3    1,2 1,3 2,4    0,3 1,3 2,4    0,2 1,4 2,4    1,2 3,4  
│ │ │       ------------------------------------------------------------------------
│ │ │        2
│ │ │       p   p    - 2p   p   p    - p   p   p
│ │ │ @@ -26,15 +26,15 @@
│ │ │                                                            0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
│ │ │  o3 : ----------------------------------------------------------------------------------------------------------------------------------------------------------------
│ │ │       (p   p    - p   p    + p   p   , p   p    - p   p    + p   p   , p   p    - p   p    + p   p   , p   p    - p   p    + p   p   , p   p    - p   p    + p   p   )
│ │ │         2,3 1,4    1,3 2,4    1,2 3,4   2,3 0,4    0,3 2,4    0,2 3,4   1,3 0,4    0,3 1,4    0,1 3,4   1,2 0,4    0,2 1,4    0,1 2,4   1,2 0,3    0,2 1,3    0,1 2,3
│ │ │  
│ │ │  i4 : -- 1-th associated hypersurface of S in G(2,4)
│ │ │       time tangentialChowForm(S,1)
│ │ │ - -- used 0.120285s (cpu); 0.0736976s (thread); 0s (gc)
│ │ │ + -- used 0.133792s (cpu); 0.0760563s (thread); 0s (gc)
│ │ │  
│ │ │        2     2        2     2               3        2     2      
│ │ │  o4 = p     p      + p     p      - 2p     p      + p     p      -
│ │ │        1,2,3 1,2,4    0,2,4 1,2,4     0,2,3 1,2,4    0,2,4 0,3,4  
│ │ │       ------------------------------------------------------------------------
│ │ │               3         3               3            
│ │ │       4p     p      - 4p     p      - 2p     p      +
│ │ │ @@ -68,32 +68,32 @@
│ │ │                                                                                0,1,2   0,1,3   0,2,3   1,2,3   0,1,4   0,2,4   1,2,4   0,3,4   1,3,4   2,3,4
│ │ │  o4 : ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
│ │ │       (p     p      - p     p      + p     p     , p     p      - p     p      + p     p     , p     p      - p     p      + p     p     , p     p      - p     p      + p     p     , p     p      - p     p      + p     p     )
│ │ │         1,2,4 0,3,4    0,2,4 1,3,4    0,1,4 2,3,4   1,2,3 0,3,4    0,2,3 1,3,4    0,1,3 2,3,4   1,2,3 0,2,4    0,2,3 1,2,4    0,1,2 2,3,4   1,2,3 0,1,4    0,1,3 1,2,4    0,1,2 1,3,4   0,2,3 0,1,4    0,1,3 0,2,4    0,1,2 0,3,4
│ │ │  
│ │ │  i5 : -- 2-th associated hypersurface of S in G(3,4) (parameterizing tangent hyperplanes to S)
│ │ │       time tangentialChowForm(S,2)
│ │ │ - -- used 0.0319774s (cpu); 0.0319788s (thread); 0s (gc)
│ │ │ + -- used 0.0393206s (cpu); 0.0393146s (thread); 0s (gc)
│ │ │  
│ │ │                2                                             2
│ │ │  o5 = p       p        - p       p       p        + p       p
│ │ │        0,1,3,4 0,2,3,4    0,1,2,4 0,2,3,4 1,2,3,4    0,1,2,3 1,2,3,4
│ │ │  
│ │ │  o5 : QQ[p       ..p       , p       , p       , p       ]
│ │ │           0,1,2,3   0,1,2,4   0,1,3,4   0,2,3,4   1,2,3,4
│ │ │  
│ │ │  i6 : -- we get the dual hypersurface of S in G(0,4) by dualizing
│ │ │       time S' = ideal dualize tangentialChowForm(S,2)
│ │ │ - -- used 0.11267s (cpu); 0.0697575s (thread); 0s (gc)
│ │ │ + -- used 0.121272s (cpu); 0.057695s (thread); 0s (gc)
│ │ │  
│ │ │              2               2
│ │ │  o6 = ideal(p p  - p p p  + p p )
│ │ │              1 2    0 1 3    0 4
│ │ │  
│ │ │  o6 : Ideal of QQ[p ..p ]
│ │ │                    0   4
│ │ │  
│ │ │  i7 : -- we then can recover S
│ │ │       time assert(dualize tangentialChowForm(S',3) == S)
│ │ │ - -- used 0.148995s (cpu); 0.103293s (thread); 0s (gc)
│ │ │ + -- used 0.171014s (cpu); 0.10791s (thread); 0s (gc)
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_cayley__Trick.html
│ │ │ @@ -86,24 +86,24 @@
│ │ │  o2 : Ideal of QQ[x ..x ]
│ │ │                    0   3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time (P1xP1xP2,P1xP1xP2') = cayleyTrick(P1xP1,2);
│ │ │ - -- used 0.127171s (cpu); 0.070403s (thread); 0s (gc)
    
│ │ │ + -- used 0.131859s (cpu); 0.0745575s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    In the next example, we calculate the defining ideal of $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\subset\mathbb{P}^7$ and that of its dual variety.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │          
    
│ │ │                
    i4 : time (P1xP1xP1,P1xP1xP1') = cayleyTrick(P1xP1,1)
│ │ │ - -- used 0.124664s (cpu); 0.0700762s (thread); 0s (gc)
│ │ │ + -- used 0.139932s (cpu); 0.0797331s (thread); 0s (gc)
│ │ │  
│ │ │                                                                             
│ │ │  o4 = (ideal (x   x    - x   x   , x   x    - x   x   , x   x    - x   x   ,
│ │ │                0,3 1,2    0,2 1,3   1,0 1,1    1,2 1,3   0,3 1,1    0,1 1,3 
│ │ │       ------------------------------------------------------------------------
│ │ │                                                                              
│ │ │       x   x    - x   x   , x   x    - x   x   , x   x    - x   x   , x   x   
│ │ │ @@ -130,26 +130,26 @@
│ │ │            
    
│ │ │          
    If the option Duality is set to true, then the method applies the so-called "dual Cayley trick".
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -38,20 +38,20 @@
│ │ │ │  
│ │ │ │  o2 = ideal(x x  - x x )
│ │ │ │              0 1    2 3
│ │ │ │  
│ │ │ │  o2 : Ideal of QQ[x ..x ]
│ │ │ │                    0   3
│ │ │ │  i3 : time (P1xP1xP2,P1xP1xP2') = cayleyTrick(P1xP1,2);
│ │ │ │ - -- used 0.127171s (cpu); 0.070403s (thread); 0s (gc)
│ │ │ │ + -- used 0.131859s (cpu); 0.0745575s (thread); 0s (gc)
│ │ │ │  In the next example, we calculate the defining ideal of $\mathbb
│ │ │ │  {P}^1\times\mathbb{P}^1\times\mathbb{P}^1\subset\mathbb{P}^7$ and that of its
│ │ │ │  dual variety.
│ │ │ │  i4 : time (P1xP1xP1,P1xP1xP1') = cayleyTrick(P1xP1,1)
│ │ │ │ - -- used 0.124664s (cpu); 0.0700762s (thread); 0s (gc)
│ │ │ │ + -- used 0.139932s (cpu); 0.0797331s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  
│ │ │ │  o4 = (ideal (x   x    - x   x   , x   x    - x   x   , x   x    - x   x   ,
│ │ │ │                0,3 1,2    0,2 1,3   1,0 1,1    1,2 1,3   0,3 1,1    0,1 1,3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │  
│ │ │ │       x   x    - x   x   , x   x    - x   x   , x   x    - x   x   , x   x
│ │ │ │ @@ -73,18 +73,18 @@
│ │ │ │       4x   x   x   x    - 2x   x   x   x    + x   x   ))
│ │ │ │         0,0 0,1 1,2 1,3     0,2 0,3 1,2 1,3    0,2 1,3
│ │ │ │  
│ │ │ │  o4 : Sequence
│ │ │ │  If the option Duality is set to true, then the method applies the so-called
│ │ │ │  "dual Cayley trick".
│ │ │ │  i5 : time cayleyTrick(P1xP1,1,Duality=>true);
│ │ │ │ - -- used 0.153844s (cpu); 0.0993609s (thread); 0s (gc)
│ │ │ │ + -- used 0.17014s (cpu); 0.114168s (thread); 0s (gc)
│ │ │ │  i6 : assert(oo == (P1xP1xP1,P1xP1xP1'))
│ │ │ │  i7 : time cayleyTrick(P1xP1,2,Duality=>true);
│ │ │ │ - -- used 0.197861s (cpu); 0.13749s (thread); 0s (gc)
│ │ │ │ + -- used 0.177206s (cpu); 0.115764s (thread); 0s (gc)
│ │ │ │  i8 : assert(oo == (P1xP1xP2,P1xP1xP2'))
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_u_a_l_V_a_r_i_e_t_y -- projective dual variety
│ │ │ │  ********** WWaayyss ttoo uussee ccaayylleeyyTTrriicckk:: **********
│ │ │ │      * cayleyTrick(Ideal,ZZ)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _c_a_y_l_e_y_T_r_i_c_k is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s.
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_chow__Equations.html
│ │ │ @@ -90,15 +90,15 @@
│ │ │  o2 : Ideal of P3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : time cayleyTrick(P1xP1,1,Duality=>true);
│ │ │ - -- used 0.153844s (cpu); 0.0993609s (thread); 0s (gc)
    
│ │ │ + -- used 0.17014s (cpu); 0.114168s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i6 : assert(oo == (P1xP1xP1,P1xP1xP1'))
    
│ │ │              
    
│ │ │                
    i7 : time cayleyTrick(P1xP1,2,Duality=>true);
│ │ │ - -- used 0.197861s (cpu); 0.13749s (thread); 0s (gc)
    
│ │ │ + -- used 0.177206s (cpu); 0.115764s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i8 : assert(oo == (P1xP1xP2,P1xP1xP2'))
    
│ │ │              
    
│ │ │                
    i3 : -- Chow equations of C
│ │ │       time eqsC = chowEquations chowForm C
│ │ │ - -- used 0.127688s (cpu); 0.062s (thread); 0s (gc)
│ │ │ + -- used 0.135471s (cpu); 0.0630562s (thread); 0s (gc)
│ │ │  
│ │ │               2 2    2 2    2 2    4                2      2 2   2      
│ │ │  o3 = ideal (x x  + x x  + x x  + x , x x x x  + x x x  + x x , x x x  +
│ │ │               0 3    1 3    2 3    3   0 1 2 3    1 2 3    2 3   0 2 3  
│ │ │       ------------------------------------------------------------------------
│ │ │        2        3           2         2      3   3         2          2    2 2
│ │ │       x x x  + x x  - 2x x x  - 2x x x  - x x , x x  + 2x x x  - x x x  + x x 
│ │ │ @@ -162,15 +162,15 @@
│ │ │  o5 : Ideal of P3
    
│ │ │              
    
│ │ │                
    i6 : -- Chow equations of D
│ │ │       time eqsD = chowEquations chowForm D
│ │ │ - -- used 0.113843s (cpu); 0.0571957s (thread); 0s (gc)
│ │ │ + -- used 0.13548s (cpu); 0.0604597s (thread); 0s (gc)
│ │ │  
│ │ │               4      3 2     3        2 2     3      2   2   2 2      2   2 
│ │ │  o6 = ideal (x x  - x x , x x x  - x x x , x x x  - x x x , x x x  - x x x ,
│ │ │               2 3    1 3   1 2 3    0 1 3   0 2 3    0 1 3   1 2 3    0 1 3 
│ │ │       ------------------------------------------------------------------------
│ │ │            2      3 2   3        3 2   4         2         2 2       3    
│ │ │       x x x x  - x x , x x x  - x x , x x  - 4x x x x  + 3x x x , x x x  -
│ │ │ @@ -222,30 +222,30 @@
│ │ │  o9 : Ideal of P3
    
│ │ │              
    
│ │ │                
    i10 : -- tangential Chow forms of Q
│ │ │        time (W0,W1,W2) = (tangentialChowForm(Q,0),tangentialChowForm(Q,1),tangentialChowForm(Q,2))
│ │ │ - -- used 0.158592s (cpu); 0.109834s (thread); 0s (gc)
│ │ │ + -- used 0.263346s (cpu); 0.119511s (thread); 0s (gc)
│ │ │  
│ │ │                       2                              2
│ │ │  o10 = (x x  + x x , x    - 4x   x    + 2x   x    + x   , x     x      +
│ │ │          0 1    2 3   0,1     0,2 1,3     0,1 2,3    2,3   0,1,2 0,1,3  
│ │ │        -----------------------------------------------------------------------
│ │ │        x     x     )
│ │ │         0,2,3 1,2,3
│ │ │  
│ │ │  o10 : Sequence
    
│ │ │              
    
│ │ │                
    i11 : time (Q,Q,Q) == (chowEquations(W0,0),chowEquations(W1,1),chowEquations(W2,2))
│ │ │ - -- used 0.150222s (cpu); 0.0893773s (thread); 0s (gc)
│ │ │ + -- used 0.158772s (cpu); 0.0913561s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = true
    
│ │ │              
    
│ │ │          
    Note that chowEquations(W,0) is not the same as chowEquations W.
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,15 +28,15 @@
│ │ │ │               2    2    2    2
│ │ │ │  o2 = ideal (x  + x  + x  + x , x x  + x x  + x x )
│ │ │ │               0    1    2    3   0 1    1 2    2 3
│ │ │ │  
│ │ │ │  o2 : Ideal of P3
│ │ │ │  i3 : -- Chow equations of C
│ │ │ │       time eqsC = chowEquations chowForm C
│ │ │ │ - -- used 0.127688s (cpu); 0.062s (thread); 0s (gc)
│ │ │ │ + -- used 0.135471s (cpu); 0.0630562s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               2 2    2 2    2 2    4                2      2 2   2
│ │ │ │  o3 = ideal (x x  + x x  + x x  + x , x x x x  + x x x  + x x , x x x  +
│ │ │ │               0 3    1 3    2 3    3   0 1 2 3    1 2 3    2 3   0 2 3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        2        3           2         2      3   3         2          2    2 2
│ │ │ │       x x x  + x x  - 2x x x  - 2x x x  - x x , x x  + 2x x x  - x x x  + x x
│ │ │ │ @@ -88,15 +88,15 @@
│ │ │ │               2          3              2    2
│ │ │ │  o5 = ideal (x  - x x , x  - x x x , x x  - x x )
│ │ │ │               1    0 2   2    0 1 3   1 2    0 3
│ │ │ │  
│ │ │ │  o5 : Ideal of P3
│ │ │ │  i6 : -- Chow equations of D
│ │ │ │       time eqsD = chowEquations chowForm D
│ │ │ │ - -- used 0.113843s (cpu); 0.0571957s (thread); 0s (gc)
│ │ │ │ + -- used 0.13548s (cpu); 0.0604597s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               4      3 2     3        2 2     3      2   2   2 2      2   2
│ │ │ │  o6 = ideal (x x  - x x , x x x  - x x x , x x x  - x x x , x x x  - x x x ,
│ │ │ │               2 3    1 3   1 2 3    0 1 3   0 2 3    0 1 3   1 2 3    0 1 3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │            2      3 2   3        3 2   4         2         2 2       3
│ │ │ │       x x x x  - x x , x x x  - x x , x x  - 4x x x x  + 3x x x , x x x  -
│ │ │ │ @@ -135,27 +135,27 @@
│ │ │ │  o9 = ideal(x x  + x x )
│ │ │ │              0 1    2 3
│ │ │ │  
│ │ │ │  o9 : Ideal of P3
│ │ │ │  i10 : -- tangential Chow forms of Q
│ │ │ │        time (W0,W1,W2) = (tangentialChowForm(Q,0),tangentialChowForm
│ │ │ │  (Q,1),tangentialChowForm(Q,2))
│ │ │ │ - -- used 0.158592s (cpu); 0.109834s (thread); 0s (gc)
│ │ │ │ + -- used 0.263346s (cpu); 0.119511s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                       2                              2
│ │ │ │  o10 = (x x  + x x , x    - 4x   x    + 2x   x    + x   , x     x      +
│ │ │ │          0 1    2 3   0,1     0,2 1,3     0,1 2,3    2,3   0,1,2 0,1,3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        x     x     )
│ │ │ │         0,2,3 1,2,3
│ │ │ │  
│ │ │ │  o10 : Sequence
│ │ │ │  i11 : time (Q,Q,Q) == (chowEquations(W0,0),chowEquations(W1,1),chowEquations
│ │ │ │  (W2,2))
│ │ │ │ - -- used 0.150222s (cpu); 0.0893773s (thread); 0s (gc)
│ │ │ │ + -- used 0.158772s (cpu); 0.0913561s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = true
│ │ │ │  Note that chowEquations(W,0) is not the same as chowEquations W.
│ │ │ │  ********** WWaayyss ttoo uussee cchhoowwEEqquuaattiioonnss:: **********
│ │ │ │      * chowEquations(RingElement)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _c_h_o_w_E_q_u_a_t_i_o_n_s is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s.
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_chow__Form.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │                3331  0   5
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : -- Chow form of V in Grass(2,5) (performing internal computations on an affine chart of the Grassmannian)
│ │ │       time ChowV = chowForm(V,AffineChartGrass=>{1,2,3})
│ │ │ - -- used 5.31048s (cpu); 4.82715s (thread); 0s (gc)
│ │ │ + -- used 5.21966s (cpu); 4.85951s (thread); 0s (gc)
│ │ │  
│ │ │        4               2              2     2               2            
│ │ │  o3 = x      + 2x     x     x      + x     x      - 2x     x     x      +
│ │ │        1,2,4     0,2,4 1,2,4 2,3,4    0,2,4 2,3,4     1,2,3 1,2,4 1,2,5  
│ │ │       ------------------------------------------------------------------------
│ │ │        2     2              2                                       
│ │ │       x     x      - x     x     x      + x     x     x     x      +
│ │ │ @@ -227,22 +227,22 @@
│ │ │         2,3,5 1,4,5    1,3,5 2,4,5    1,2,5 3,4,5   2,3,4 1,4,5    1,3,4 2,4,5    1,2,4 3,4,5   2,3,5 0,4,5    0,3,5 2,4,5    0,2,5 3,4,5   1,3,5 0,4,5    0,3,5 1,4,5    0,1,5 3,4,5   1,2,5 0,4,5    0,2,5 1,4,5    0,1,5 2,4,5   2,3,4 0,4,5    0,3,4 2,4,5    0,2,4 3,4,5   1,3,4 0,4,5    0,3,4 1,4,5    0,1,4 3,4,5   1,2,4 0,4,5    0,2,4 1,4,5    0,1,4 2,4,5   1,2,3 0,4,5    0,2,3 1,4,5    0,1,3 2,4,5    0,1,2 3,4,5   2,3,4 1,3,5    1,3,4 2,3,5    1,2,3 3,4,5   1,2,5 0,3,5    0,2,5 1,3,5    0,1,5 2,3,5   2,3,4 0,3,5    0,3,4 2,3,5    0,2,3 3,4,5   1,3,4 0,3,5    0,3,4 1,3,5    0,1,3 3,4,5   1,2,4 0,3,5    0,2,4 1,3,5    0,1,4 2,3,5    0,1,2 3,4,5   1,2,3 0,3,5    0,2,3 1,3,5    0,1,3 2,3,5   2,3,4 1,2,5    1,2,4 2,3,5    1,2,3 2,4,5   1,3,4 1,2,5    1,2,4 1,3,5    1,2,3 1,4,5   0,3,4 1,2,5    0,2,4 1,3,5    0,1,4 2,3,5    0,2,3 1,4,5    0,1,3 2,4,5    0,1,2 3,4,5   2,3,4 0,2,5    0,2,4 2,3,5    0,2,3 2,4,5   1,3,4 0,2,5    0,2,4 1,3,5    0,2,3 1,4,5    0,1,2 3,4,5   0,3,4 0,2,5    0,2,4 0,3,5    0,2,3 0,4,5   1,2,4 0,2,5    0,2,4 1,2,5    0,1,2 2,4,5   1,2,3 0,2,5    0,2,3 1,2,5    0,1,2 2,3,5   2,3,4 0,1,5    0,1,4 2,3,5    0,1,3 2,4,5    0,1,2 3,4,5   1,3,4 0,1,5    0,1,4 1,3,5    0,1,3 1,4,5   0,3,4 0,1,5    0,1,4 0,3,5    0,1,3 0,4,5   1,2,4 0,1,5    0,1,4 1,2,5    0,1,2 1,4,5   0,2,4 0,1,5    0,1,4 0,2,5    0,1,2 0,4,5   1,2,3 0,1,5    0,1,3 1,2,5    0,1,2 1,3,5   0,2,3 0,1,5    0,1,3 0,2,5    0,1,2 0,3,5   1,2,4 0,3,4    0,2,4 1,3,4    0,1,4 2,3,4   1,2,3 0,3,4    0,2,3 1,3,4    0,1,3 2,3,4   1,2,3 0,2,4    0,2,3 1,2,4    0,1,2 2,3,4   1,2,3 0,1,4    0,1,3 1,2,4    0,1,2 1,3,4   0,2,3 0,1,4    0,1,3 0,2,4    0,1,2 0,3,4
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : -- equivalently (but faster)...
│ │ │       time assert(ChowV === chowForm f)
│ │ │ - -- used 1.12682s (cpu); 0.99711s (thread); 0s (gc)
    
│ │ │ + -- used 1.15009s (cpu); 1.09337s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : -- X-resultant of V
│ │ │       time Xres = fromPluckerToStiefel dualize ChowV;
│ │ │ - -- used 0.235111s (cpu); 0.179065s (thread); 0s (gc)
    
│ │ │ + -- used 0.284789s (cpu); 0.220968s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : -- three generic ternary quadrics
│ │ │       F = genericPolynomials({2,2,2},ZZ/3331)
│ │ │  
│ │ │ @@ -257,15 +257,15 @@
│ │ │  o6 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : -- resultant of the three forms
│ │ │       time resF = resultant F;
│ │ │ - -- used 0.285058s (cpu); 0.174585s (thread); 0s (gc)
    
│ │ │ + -- used 0.287939s (cpu); 0.220002s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : assert(resF === sub(Xres,vars ring resF) and Xres === sub(resF,vars ring Xres))
    
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,15 +41,15 @@
│ │ │ │  
│ │ │ │                 ZZ
│ │ │ │  o2 : Ideal of ----[x ..x ]
│ │ │ │                3331  0   5
│ │ │ │  i3 : -- Chow form of V in Grass(2,5) (performing internal computations on an
│ │ │ │  affine chart of the Grassmannian)
│ │ │ │       time ChowV = chowForm(V,AffineChartGrass=>{1,2,3})
│ │ │ │ - -- used 5.31048s (cpu); 4.82715s (thread); 0s (gc)
│ │ │ │ + -- used 5.21966s (cpu); 4.85951s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        4               2              2     2               2
│ │ │ │  o3 = x      + 2x     x     x      + x     x      - 2x     x     x      +
│ │ │ │        1,2,4     0,2,4 1,2,4 2,3,4    0,2,4 2,3,4     1,2,3 1,2,4 1,2,5
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        2     2              2
│ │ │ │       x     x      - x     x     x      + x     x     x     x      +
│ │ │ │ @@ -234,33 +234,33 @@
│ │ │ │  1,4,5   0,2,4 0,1,5    0,1,4 0,2,5    0,1,2 0,4,5   1,2,3 0,1,5    0,1,3 1,2,5
│ │ │ │  0,1,2 1,3,5   0,2,3 0,1,5    0,1,3 0,2,5    0,1,2 0,3,5   1,2,4 0,3,4    0,2,4
│ │ │ │  1,3,4    0,1,4 2,3,4   1,2,3 0,3,4    0,2,3 1,3,4    0,1,3 2,3,4   1,2,3 0,2,4
│ │ │ │  0,2,3 1,2,4    0,1,2 2,3,4   1,2,3 0,1,4    0,1,3 1,2,4    0,1,2 1,3,4   0,2,3
│ │ │ │  0,1,4    0,1,3 0,2,4    0,1,2 0,3,4
│ │ │ │  i4 : -- equivalently (but faster)...
│ │ │ │       time assert(ChowV === chowForm f)
│ │ │ │ - -- used 1.12682s (cpu); 0.99711s (thread); 0s (gc)
│ │ │ │ + -- used 1.15009s (cpu); 1.09337s (thread); 0s (gc)
│ │ │ │  i5 : -- X-resultant of V
│ │ │ │       time Xres = fromPluckerToStiefel dualize ChowV;
│ │ │ │ - -- used 0.235111s (cpu); 0.179065s (thread); 0s (gc)
│ │ │ │ + -- used 0.284789s (cpu); 0.220968s (thread); 0s (gc)
│ │ │ │  i6 : -- three generic ternary quadrics
│ │ │ │       F = genericPolynomials({2,2,2},ZZ/3331)
│ │ │ │  
│ │ │ │           2               2                        2     2               2
│ │ │ │  o6 = {a x  + a x x  + a x  + a x x  + a x x  + a x , b x  + b x x  + b x  +
│ │ │ │         0 0    1 0 1    3 1    2 0 2    4 1 2    5 2   0 0    1 0 1    3 1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                            2     2               2                        2
│ │ │ │       b x x  + b x x  + b x , c x  + c x x  + c x  + c x x  + c x x  + c x }
│ │ │ │        2 0 2    4 1 2    5 2   0 0    1 0 1    3 1    2 0 2    4 1 2    5 2
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : -- resultant of the three forms
│ │ │ │       time resF = resultant F;
│ │ │ │ - -- used 0.285058s (cpu); 0.174585s (thread); 0s (gc)
│ │ │ │ + -- used 0.287939s (cpu); 0.220002s (thread); 0s (gc)
│ │ │ │  i8 : assert(resF === sub(Xres,vars ring resF) and Xres === sub(resF,vars ring
│ │ │ │  Xres))
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_a_n_g_e_n_t_i_a_l_C_h_o_w_F_o_r_m -- higher Chow forms of a projective variety
│ │ │ │      * _h_u_r_w_i_t_z_F_o_r_m -- Hurwitz form of a projective variety
│ │ │ │  ********** WWaayyss ttoo uussee cchhoowwFFoorrmm:: **********
│ │ │ │      * chowForm(Ideal)
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_discriminant_lp__Ring__Element_rp.html
│ │ │ @@ -83,15 +83,15 @@
│ │ │  
│ │ │  o2 : ZZ[a..c][x..y]
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time discriminant F
│ │ │ - -- used 0.00807841s (cpu); 0.00807736s (thread); 0s (gc)
│ │ │ + -- used 0.00983715s (cpu); 0.00983637s (thread); 0s (gc)
│ │ │  
│ │ │          2
│ │ │  o3 = - b  + 4a*c
│ │ │  
│ │ │  o3 : ZZ[a..c]
    
│ │ │              
│ │ │            
│ │ │ @@ -104,15 +104,15 @@
│ │ │  
│ │ │  o5 : ZZ[a..d][x..y]
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time discriminant F
│ │ │ - -- used 0.0087506s (cpu); 0.00875122s (thread); 0s (gc)
│ │ │ + -- used 0.0101472s (cpu); 0.010148s (thread); 0s (gc)
│ │ │  
│ │ │          2 2       3     3                   2 2
│ │ │  o6 = - b c  + 4a*c  + 4b d - 18a*b*c*d + 27a d
│ │ │  
│ │ │  o6 : ZZ[a..d]
    
│ │ │              
│ │ │            
│ │ │ @@ -165,15 +165,15 @@
│ │ │  
│ │ │  o12 : R'
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : time D=discriminant pencil
│ │ │ - -- used 0.461801s (cpu); 0.427218s (thread); 0s (gc)
│ │ │ + -- used 0.480519s (cpu); 0.420479s (thread); 0s (gc)
│ │ │  
│ │ │             108      106 2       102 6      100 8       98 10       96 12  
│ │ │  o13 = - 62t    + 19t   t  + 160t   t  + 91t   t  + 129t  t   + 117t  t   +
│ │ │             0        0   1       0   1      0   1       0  1        0  1   
│ │ │        -----------------------------------------------------------------------
│ │ │            94 14       92 16      90 18      88 20      86 22       84 24  
│ │ │        161t  t   + 124t  t   - 82t  t   - 21t  t   - 49t  t   - 123t  t   +
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,28 +23,28 @@
│ │ │ │  i1 : ZZ[a,b,c][x,y]; F = a*x^2+b*x*y+c*y^2
│ │ │ │  
│ │ │ │          2              2
│ │ │ │  o2 = a*x  + b*x*y + c*y
│ │ │ │  
│ │ │ │  o2 : ZZ[a..c][x..y]
│ │ │ │  i3 : time discriminant F
│ │ │ │ - -- used 0.00807841s (cpu); 0.00807736s (thread); 0s (gc)
│ │ │ │ + -- used 0.00983715s (cpu); 0.00983637s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          2
│ │ │ │  o3 = - b  + 4a*c
│ │ │ │  
│ │ │ │  o3 : ZZ[a..c]
│ │ │ │  i4 : ZZ[a,b,c,d][x,y]; F = a*x^3+b*x^2*y+c*x*y^2+d*y^3
│ │ │ │  
│ │ │ │          3      2         2      3
│ │ │ │  o5 = a*x  + b*x y + c*x*y  + d*y
│ │ │ │  
│ │ │ │  o5 : ZZ[a..d][x..y]
│ │ │ │  i6 : time discriminant F
│ │ │ │ - -- used 0.0087506s (cpu); 0.00875122s (thread); 0s (gc)
│ │ │ │ + -- used 0.0101472s (cpu); 0.010148s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          2 2       3     3                   2 2
│ │ │ │  o6 = - b c  + 4a*c  + 4b d - 18a*b*c*d + 27a d
│ │ │ │  
│ │ │ │  o6 : ZZ[a..d]
│ │ │ │  The next example illustrates how computing the intersection of a pencil
│ │ │ │  generated by two degree $d$ forms $F(x_0,\ldots,x_n), G(x_0,\ldots,x_n)$ with
│ │ │ │ @@ -74,15 +74,15 @@
│ │ │ │  
│ │ │ │                  4        3      4             4        3      4
│ │ │ │  o12 = (t  + t )x  - t x x  + t x  + (t  - t )x  + t x x  + t x
│ │ │ │          0    1  0    1 0 1    0 1     0    1  2    1 2 3    0 3
│ │ │ │  
│ │ │ │  o12 : R'
│ │ │ │  i13 : time D=discriminant pencil
│ │ │ │ - -- used 0.461801s (cpu); 0.427218s (thread); 0s (gc)
│ │ │ │ + -- used 0.480519s (cpu); 0.420479s (thread); 0s (gc)
│ │ │ │  
│ │ │ │             108      106 2       102 6      100 8       98 10       96 12
│ │ │ │  o13 = - 62t    + 19t   t  + 160t   t  + 91t   t  + 129t  t   + 117t  t   +
│ │ │ │             0        0   1       0   1      0   1       0  1        0  1
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │            94 14       92 16      90 18      88 20      86 22       84 24
│ │ │ │        161t  t   + 124t  t   - 82t  t   - 21t  t   - 49t  t   - 123t  t   +
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_dual__Variety.html
│ │ │ @@ -90,28 +90,28 @@
│ │ │  o1 : Ideal of QQ[x ..x ]
│ │ │                    0   5
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time V' = dualVariety V
│ │ │ - -- used 0.170711s (cpu); 0.119239s (thread); 0s (gc)
│ │ │ + -- used 0.198598s (cpu); 0.132067s (thread); 0s (gc)
│ │ │  
│ │ │              2                 2    2
│ │ │  o2 = ideal(x x  - x x x  + x x  + x x  - 4x x x )
│ │ │              2 3    1 2 4    0 4    1 5     0 3 5
│ │ │  
│ │ │  o2 : Ideal of QQ[x ..x ]
│ │ │                    0   5
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time V == dualVariety V'
│ │ │ - -- used 0.204333s (cpu); 0.146742s (thread); 0s (gc)
│ │ │ + -- used 0.230838s (cpu); 0.168205s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    In the next example, we verify that the discriminant of a generic ternary cubic form coincides with the dual variety of the 3-th Veronese embedding of the plane, which is a hypersurface of degree 12 in $\mathbb{P}^9$
    
│ │ │          
│ │ │ @@ -131,25 +131,25 @@
│ │ │  o4 : ----[a ..a ][x ..x ]
│ │ │       3331  0   9   0   2
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -31,24 +31,24 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       x x )
│ │ │ │        0 3
│ │ │ │  
│ │ │ │  o1 : Ideal of QQ[x ..x ]
│ │ │ │                    0   5
│ │ │ │  i2 : time V' = dualVariety V
│ │ │ │ - -- used 0.170711s (cpu); 0.119239s (thread); 0s (gc)
│ │ │ │ + -- used 0.198598s (cpu); 0.132067s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              2                 2    2
│ │ │ │  o2 = ideal(x x  - x x x  + x x  + x x  - 4x x x )
│ │ │ │              2 3    1 2 4    0 4    1 5     0 3 5
│ │ │ │  
│ │ │ │  o2 : Ideal of QQ[x ..x ]
│ │ │ │                    0   5
│ │ │ │  i3 : time V == dualVariety V'
│ │ │ │ - -- used 0.204333s (cpu); 0.146742s (thread); 0s (gc)
│ │ │ │ + -- used 0.230838s (cpu); 0.168205s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  In the next example, we verify that the discriminant of a generic ternary cubic
│ │ │ │  form coincides with the dual variety of the 3-th Veronese embedding of the
│ │ │ │  plane, which is a hypersurface of degree 12 in $\mathbb{P}^9$
│ │ │ │  i4 : F = first genericPolynomials({3,-1,-1},ZZ/3331)
│ │ │ │  
│ │ │ │ @@ -60,21 +60,21 @@
│ │ │ │       a x x  + a x
│ │ │ │        8 1 2    9 2
│ │ │ │  
│ │ │ │        ZZ
│ │ │ │  o4 : ----[a ..a ][x ..x ]
│ │ │ │       3331  0   9   0   2
│ │ │ │  i5 : time discF = ideal discriminant F;
│ │ │ │ - -- used 0.0566354s (cpu); 0.0566377s (thread); 0s (gc)
│ │ │ │ + -- used 0.0657693s (cpu); 0.065771s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                 ZZ
│ │ │ │  o5 : Ideal of ----[a ..a ]
│ │ │ │                3331  0   9
│ │ │ │  i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true);
│ │ │ │ - -- used 0.611958s (cpu); 0.563695s (thread); 0s (gc)
│ │ │ │ + -- used 0.730716s (cpu); 0.662534s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                 ZZ
│ │ │ │  o6 : Ideal of ----[x ..x ]
│ │ │ │                3331  0   9
│ │ │ │  i7 : discF == sub(Z,vars ring discF) and Z == sub(discF,vars ring Z)
│ │ │ │  
│ │ │ │  o7 = true
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_from__Plucker__To__Stiefel.html
│ │ │ @@ -85,15 +85,15 @@
│ │ │  o1 : Ideal of QQ[x ..x ]
│ │ │                    0   3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : time discF = ideal discriminant F;
│ │ │ - -- used 0.0566354s (cpu); 0.0566377s (thread); 0s (gc)
│ │ │ + -- used 0.0657693s (cpu); 0.065771s (thread); 0s (gc)
│ │ │  
│ │ │                 ZZ
│ │ │  o5 : Ideal of ----[a ..a ]
│ │ │                3331  0   9
    
│ │ │              
    
│ │ │                
    i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true);
│ │ │ - -- used 0.611958s (cpu); 0.563695s (thread); 0s (gc)
│ │ │ + -- used 0.730716s (cpu); 0.662534s (thread); 0s (gc)
│ │ │  
│ │ │                 ZZ
│ │ │  o6 : Ideal of ----[x ..x ]
│ │ │                3331  0   9
    
│ │ │              
    
│ │ │                
    i2 : time fromPluckerToStiefel dualize chowForm C
│ │ │ - -- used 0.110649s (cpu); 0.0580797s (thread); 0s (gc)
│ │ │ + -- used 0.134421s (cpu); 0.0678798s (thread); 0s (gc)
│ │ │  
│ │ │          3   3          2   2              2       2          2   3    
│ │ │  o2 = - x   x    + x   x   x   x    - x   x   x   x    + x   x   x    -
│ │ │          0,3 1,0    0,2 0,3 1,0 1,1    0,1 0,3 1,0 1,1    0,0 0,3 1,1  
│ │ │       ------------------------------------------------------------------------
│ │ │        2       2               2   2                                   
│ │ │       x   x   x   x    + 2x   x   x   x    + x   x   x   x   x   x    -
│ │ │ @@ -138,15 +138,15 @@
│ │ │  o2 : QQ[x   ..x   ]
│ │ │           0,0   1,3
    
│ │ │              
    
│ │ │                
    i3 : time fromPluckerToStiefel(dualize chowForm C,AffineChartGrass=>{0,1})
│ │ │ - -- used 0.040301s (cpu); 0.0403043s (thread); 0s (gc)
│ │ │ + -- used 0.0498309s (cpu); 0.0498325s (thread); 0s (gc)
│ │ │  
│ │ │              3          2                         2                        
│ │ │  o3 = - x   x    + x   x   x    - x   x   x    + x   x    + 3x   x   x    -
│ │ │          0,3 1,2    0,2 1,2 1,3    0,2 0,3 1,2    0,2 1,3     0,3 1,2 1,3  
│ │ │       ------------------------------------------------------------------------
│ │ │             2      3      2
│ │ │       2x   x    + x    + x
│ │ │ @@ -179,15 +179,15 @@
│ │ │              
    		
│ │ │                
    i5 : w = chowForm C;
    
│ │ │              
    
│ │ │                
    i6 : time U = apply(subsets(4,2),s->ideal fromPluckerToStiefel(w,AffineChartGrass=>s))
│ │ │ - -- used 0.0190728s (cpu); 0.0190737s (thread); 0s (gc)
│ │ │ + -- used 0.0249667s (cpu); 0.0249684s (thread); 0s (gc)
│ │ │  
│ │ │                     3          2          3                       2        
│ │ │  o6 = {ideal(- x   x    + x   x   x    - x    - 3x   x   x    + 2x   x    +
│ │ │                 0,3 1,2    0,2 1,2 1,3    0,2     0,2 0,3 1,2     0,2 1,3  
│ │ │       ------------------------------------------------------------------------
│ │ │                           2      2            2   3               2        
│ │ │       x   x   x    - x   x    + x   ), ideal(x   x    - 2x   x   x   x    +
│ │ │ @@ -227,15 +227,15 @@
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : time apply(U,u->dim singularLocus u)
│ │ │ - -- used 0.0170671s (cpu); 0.0170726s (thread); 0s (gc)
│ │ │ + -- used 0.0211123s (cpu); 0.0211149s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {2, 2, 2, 2, 2, 2}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │               2                       2
│ │ │ │  o1 = ideal (x  - x x , x x  - x x , x  - x x )
│ │ │ │               2    1 3   1 2    0 3   1    0 2
│ │ │ │  
│ │ │ │  o1 : Ideal of QQ[x ..x ]
│ │ │ │                    0   3
│ │ │ │  i2 : time fromPluckerToStiefel dualize chowForm C
│ │ │ │ - -- used 0.110649s (cpu); 0.0580797s (thread); 0s (gc)
│ │ │ │ + -- used 0.134421s (cpu); 0.0678798s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          3   3          2   2              2       2          2   3
│ │ │ │  o2 = - x   x    + x   x   x   x    - x   x   x   x    + x   x   x    -
│ │ │ │          0,3 1,0    0,2 0,3 1,0 1,1    0,1 0,3 1,0 1,1    0,0 0,3 1,1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        2       2               2   2
│ │ │ │       x   x   x   x    + 2x   x   x   x    + x   x   x   x   x   x    -
│ │ │ │ @@ -75,15 +75,15 @@
│ │ │ │            2       2       2           2      2           2      3   3
│ │ │ │       x   x   x   x    - 2x   x   x   x    - x   x   x   x    + x   x
│ │ │ │        0,0 0,1 1,1 1,3     0,0 0,2 1,1 1,3    0,0 0,1 1,2 1,3    0,0 1,3
│ │ │ │  
│ │ │ │  o2 : QQ[x   ..x   ]
│ │ │ │           0,0   1,3
│ │ │ │  i3 : time fromPluckerToStiefel(dualize chowForm C,AffineChartGrass=>{0,1})
│ │ │ │ - -- used 0.040301s (cpu); 0.0403043s (thread); 0s (gc)
│ │ │ │ + -- used 0.0498309s (cpu); 0.0498325s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              3          2                         2
│ │ │ │  o3 = - x   x    + x   x   x    - x   x   x    + x   x    + 3x   x   x    -
│ │ │ │          0,3 1,2    0,2 1,2 1,3    0,2 0,3 1,2    0,2 1,3     0,3 1,2 1,3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │             2      3      2
│ │ │ │       2x   x    + x    + x
│ │ │ │ @@ -105,15 +105,15 @@
│ │ │ │  o4 : QQ[a   ..a   ]
│ │ │ │           0,0   1,1
│ │ │ │  As another application, we check that the singular locus of the Chow form of
│ │ │ │  the twisted cubic has dimension 2 (on each standard chart).
│ │ │ │  i5 : w = chowForm C;
│ │ │ │  i6 : time U = apply(subsets(4,2),s->ideal fromPluckerToStiefel
│ │ │ │  (w,AffineChartGrass=>s))
│ │ │ │ - -- used 0.0190728s (cpu); 0.0190737s (thread); 0s (gc)
│ │ │ │ + -- used 0.0249667s (cpu); 0.0249684s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                     3          2          3                       2
│ │ │ │  o6 = {ideal(- x   x    + x   x   x    - x    - 3x   x   x    + 2x   x    +
│ │ │ │                 0,3 1,2    0,2 1,2 1,3    0,2     0,2 0,3 1,2     0,2 1,3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                           2      2            2   3               2
│ │ │ │       x   x   x    - x   x    + x   ), ideal(x   x    - 2x   x   x   x    +
│ │ │ │ @@ -149,15 +149,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │             2      3      2
│ │ │ │       2x   x    - x    + x   )}
│ │ │ │         0,0 1,1    1,1    1,0
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : time apply(U,u->dim singularLocus u)
│ │ │ │ - -- used 0.0170671s (cpu); 0.0170726s (thread); 0s (gc)
│ │ │ │ + -- used 0.0211123s (cpu); 0.0211149s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = {2, 2, 2, 2, 2, 2}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  ********** WWaayyss ttoo uussee ffrroommPPlluucckkeerrTTooSSttiieeffeell:: **********
│ │ │ │      * fromPluckerToStiefel(Ideal)
│ │ │ │      * fromPluckerToStiefel(Matrix)
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_hurwitz__Form.html
│ │ │ @@ -92,15 +92,15 @@
│ │ │  o1 : Ideal of QQ[p ..p ]
│ │ │                    0   4
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time hurwitzForm Q
│ │ │ - -- used 0.0393717s (cpu); 0.0393716s (thread); 0s (gc)
│ │ │ + -- used 0.0449735s (cpu); 0.0449738s (thread); 0s (gc)
│ │ │  
│ │ │                2                                 2                      
│ │ │  o2 = 11966535p    + 14645610p   p    + 11354175p    + 1666980p   p    +
│ │ │                0,1            0,1 0,2            0,2           0,1 1,2  
│ │ │       ------------------------------------------------------------------------
│ │ │                                 2                                          
│ │ │       4456620p   p    + 1127196p    + 54176850p   p    + 20326950p   p    +
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │         5 2   7                       2        3 2
│ │ │ │       + -p  + -p p  + 7p p  + 6p p  + -p p  + --p )
│ │ │ │         4 3   9 0 4     1 4     2 4   9 3 4   10 4
│ │ │ │  
│ │ │ │  o1 : Ideal of QQ[p ..p ]
│ │ │ │                    0   4
│ │ │ │  i2 : time hurwitzForm Q
│ │ │ │ - -- used 0.0393717s (cpu); 0.0393716s (thread); 0s (gc)
│ │ │ │ + -- used 0.0449735s (cpu); 0.0449738s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                2                                 2
│ │ │ │  o2 = 11966535p    + 14645610p   p    + 11354175p    + 1666980p   p    +
│ │ │ │                0,1            0,1 0,2            0,2           0,1 1,2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                                 2
│ │ │ │       4456620p   p    + 1127196p    + 54176850p   p    + 20326950p   p    +
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_is__Coisotropic.html
│ │ │ @@ -104,15 +104,15 @@
│ │ │           p   p    - p   p    + p   p
│ │ │            1,2 0,3    0,2 1,3    0,1 2,3
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time isCoisotropic w
│ │ │ - -- used 0.00787565s (cpu); 0.00787232s (thread); 0s (gc)
│ │ │ + -- used 0.00855771s (cpu); 0.00855747s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : -- random quadric in G(1,3)
│ │ │ @@ -140,15 +140,15 @@
│ │ │           p   p    - p   p    + p   p
│ │ │            1,2 0,3    0,2 1,3    0,1 2,3
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time isCoisotropic w'
│ │ │ - -- used 0.00654309s (cpu); 0.00654341s (thread); 0s (gc)
│ │ │ + -- used 0.00707054s (cpu); 0.00707066s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = false
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -44,15 +44,15 @@
│ │ │ │  
│ │ │ │       QQ[p   ..p   , p   , p   , p   , p   ]
│ │ │ │           0,1   0,2   1,2   0,3   1,3   2,3
│ │ │ │  o1 : --------------------------------------
│ │ │ │           p   p    - p   p    + p   p
│ │ │ │            1,2 0,3    0,2 1,3    0,1 2,3
│ │ │ │  i2 : time isCoisotropic w
│ │ │ │ - -- used 0.00787565s (cpu); 0.00787232s (thread); 0s (gc)
│ │ │ │ + -- used 0.00855771s (cpu); 0.00855747s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = true
│ │ │ │  i3 : -- random quadric in G(1,3)
│ │ │ │       w' = random(2,Grass(1,3))
│ │ │ │  
│ │ │ │         2     5           10 2     2                          2      3
│ │ │ │  o3 = 6p    + -p   p    + --p    + -p   p    + 10p   p    + 5p    + --p   p
│ │ │ │ @@ -72,15 +72,15 @@
│ │ │ │  
│ │ │ │       QQ[p   ..p   , p   , p   , p   , p   ]
│ │ │ │           0,1   0,2   1,2   0,3   1,3   2,3
│ │ │ │  o3 : --------------------------------------
│ │ │ │           p   p    - p   p    + p   p
│ │ │ │            1,2 0,3    0,2 1,3    0,1 2,3
│ │ │ │  i4 : time isCoisotropic w'
│ │ │ │ - -- used 0.00654309s (cpu); 0.00654341s (thread); 0s (gc)
│ │ │ │ + -- used 0.00707054s (cpu); 0.00707066s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = false
│ │ │ │  ********** WWaayyss ttoo uussee iissCCooiissoottrrooppiicc:: **********
│ │ │ │      * isCoisotropic(RingElement)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _i_s_C_o_i_s_o_t_r_o_p_i_c is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s.
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_is__In__Coisotropic.html
│ │ │ @@ -117,15 +117,15 @@
│ │ │  o3 : Ideal of -----[x ..x ]
│ │ │                33331  0   5
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time isInCoisotropic(L,I) -- whether L belongs to Z_1(V(I))
│ │ │ - -- used 0.0197451s (cpu); 0.0197449s (thread); 0s (gc)
│ │ │ + -- used 0.0208729s (cpu); 0.0208724s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -54,15 +54,15 @@
│ │ │ │       2380x  + 9482x )
│ │ │ │            4        5
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o3 : Ideal of -----[x ..x ]
│ │ │ │                33331  0   5
│ │ │ │  i4 : time isInCoisotropic(L,I) -- whether L belongs to Z_1(V(I))
│ │ │ │ - -- used 0.0197451s (cpu); 0.0197449s (thread); 0s (gc)
│ │ │ │ + -- used 0.0208729s (cpu); 0.0208724s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_a_n_g_e_n_t_i_a_l_C_h_o_w_F_o_r_m -- higher Chow forms of a projective variety
│ │ │ │      * _p_l_u_c_k_e_r -- get the Plücker coordinates of a linear subspace
│ │ │ │  ********** WWaayyss ttoo uussee iissIInnCCooiissoottrrooppiicc:: **********
│ │ │ │      * isInCoisotropic(Ideal,Ideal)
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_macaulay__Formula.html
│ │ │ @@ -87,15 +87,15 @@
│ │ │  
│ │ │  o1 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time (D,D') = macaulayFormula F
│ │ │ - -- used 0.00462514s (cpu); 0.00462065s (thread); 0s (gc)
│ │ │ + -- used 0.00419877s (cpu); 0.00419752s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = (| a_0 a_1 a_2 a_3 a_4 a_5 0   0   0   0   0   0   0   0   0   0   0  
│ │ │        | 0   a_0 0   a_1 a_2 0   a_3 a_4 a_5 0   0   0   0   0   0   0   0  
│ │ │        | 0   0   a_0 0   a_1 a_2 0   a_3 a_4 a_5 0   0   0   0   0   0   0  
│ │ │        | 0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0   0   0  
│ │ │        | 0   0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0   0  
│ │ │        | 0   0   0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0  
│ │ │ @@ -158,15 +158,15 @@
│ │ │  
│ │ │  o3 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time (D,D') = macaulayFormula F
│ │ │ - -- used 0.00237921s (cpu); 0.00238132s (thread); 0s (gc)
│ │ │ + -- used 0.00263209s (cpu); 0.00263066s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = (| 9/2 9/4 3/4 7/4  7/9  7/10 0    0    0    0    0   0   0    0    0   
│ │ │        | 0   9/2 0   9/4  3/4  0    7/4  7/9  7/10 0    0   0   0    0    0   
│ │ │        | 0   0   9/2 0    9/4  3/4  0    7/4  7/9  7/10 0   0   0    0    0   
│ │ │        | 0   0   0   9/2  0    0    9/4  3/4  0    0    7/4 7/9 7/10 0    0   
│ │ │        | 0   0   0   0    9/2  0    0    9/4  3/4  0    0   7/4 7/9  7/10 0   
│ │ │        | 0   0   0   0    0    9/2  0    0    9/4  3/4  0   0   7/4  7/9  7/10
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,15 +28,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                     2          2        2      3
│ │ │ │       c x x x  + c x x  + c x x  + c x x  + c x }
│ │ │ │        4 0 1 2    7 1 2    5 0 2    8 1 2    9 2
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : time (D,D') = macaulayFormula F
│ │ │ │ - -- used 0.00462514s (cpu); 0.00462065s (thread); 0s (gc)
│ │ │ │ + -- used 0.00419877s (cpu); 0.00419752s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = (| a_0 a_1 a_2 a_3 a_4 a_5 0   0   0   0   0   0   0   0   0   0   0
│ │ │ │        | 0   a_0 0   a_1 a_2 0   a_3 a_4 a_5 0   0   0   0   0   0   0   0
│ │ │ │        | 0   0   a_0 0   a_1 a_2 0   a_3 a_4 a_5 0   0   0   0   0   0   0
│ │ │ │        | 0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0   0   0
│ │ │ │        | 0   0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0   0
│ │ │ │        | 0   0   0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0
│ │ │ │ @@ -91,15 +91,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       10   2   7   2   5 3
│ │ │ │       --p p  + -p p  + -p }
│ │ │ │        9 0 2   8 1 2   6 2
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : time (D,D') = macaulayFormula F
│ │ │ │ - -- used 0.00237921s (cpu); 0.00238132s (thread); 0s (gc)
│ │ │ │ + -- used 0.00263209s (cpu); 0.00263066s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = (| 9/2 9/4 3/4 7/4  7/9  7/10 0    0    0    0    0   0   0    0    0
│ │ │ │        | 0   9/2 0   9/4  3/4  0    7/4  7/9  7/10 0    0   0   0    0    0
│ │ │ │        | 0   0   9/2 0    9/4  3/4  0    7/4  7/9  7/10 0   0   0    0    0
│ │ │ │        | 0   0   0   9/2  0    0    9/4  3/4  0    0    7/4 7/9 7/10 0    0
│ │ │ │        | 0   0   0   0    9/2  0    0    9/4  3/4  0    0   7/4 7/9  7/10 0
│ │ │ │        | 0   0   0   0    0    9/2  0    0    9/4  3/4  0   0   7/4  7/9  7/10
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_plucker.html
│ │ │ @@ -92,15 +92,15 @@
│ │ │  
│ │ │  o3 : Ideal of P4
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time p = plucker L
│ │ │ - -- used 0.00452289s (cpu); 0.00452084s (thread); 0s (gc)
│ │ │ + -- used 0.00547559s (cpu); 0.00547367s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = ideal (x    + 8480x   , x    - 6727x   , x    + 15777x   , x    +
│ │ │               2,4        3,4   1,4        3,4   0,4         3,4   2,3  
│ │ │       ------------------------------------------------------------------------
│ │ │       11656x   , x    - 14853x   , x    + 664x   , x    + 13522x   , x    +
│ │ │             3,4   1,3         3,4   0,3       3,4   1,2         3,4   0,2  
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -109,15 +109,15 @@
│ │ │  
│ │ │  o4 : Ideal of G'1'4
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time L' = plucker p
│ │ │ - -- used 0.111022s (cpu); 0.0535001s (thread); 0s (gc)
│ │ │ + -- used 0.12088s (cpu); 0.0581025s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = ideal (x  + 8480x  - 11656x , x  - 6727x  + 14853x , x  + 15777x  -
│ │ │               2        3         4   1        3         4   0         3  
│ │ │       ------------------------------------------------------------------------
│ │ │       664x )
│ │ │           4
│ │ │  
│ │ │ @@ -138,15 +138,15 @@
│ │ │  
│ │ │  o7 : Ideal of G'1'4
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : time W = plucker Y; -- surface swept out by the lines of Y
│ │ │ - -- used 0.0361963s (cpu); 0.0361951s (thread); 0s (gc)
│ │ │ + -- used 0.0373625s (cpu); 0.0373403s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of P4
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : (codim W,degree W)
│ │ │ @@ -158,15 +158,15 @@
│ │ │            
│ │ │          
│ │ │          
    In this example, we can recover the subvariety $Y\subset\mathbb{G}(k,\mathbb{P}^n)$ by computing the Fano variety of $k$-planes contained in $W$.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : time Y' = plucker(W,1); -- variety of lines contained in W
│ │ │ - -- used 0.142146s (cpu); 0.142149s (thread); 0s (gc)
│ │ │ + -- used 0.174625s (cpu); 0.17463s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of G'1'4
    
│ │ │              
    
│ │ │                
    i11 : assert(Y' == Y)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,28 +28,28 @@
│ │ │ │               2        3         4   1        3         4   0         3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       664x )
│ │ │ │           4
│ │ │ │  
│ │ │ │  o3 : Ideal of P4
│ │ │ │  i4 : time p = plucker L
│ │ │ │ - -- used 0.00452289s (cpu); 0.00452084s (thread); 0s (gc)
│ │ │ │ + -- used 0.00547559s (cpu); 0.00547367s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = ideal (x    + 8480x   , x    - 6727x   , x    + 15777x   , x    +
│ │ │ │               2,4        3,4   1,4        3,4   0,4         3,4   2,3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       11656x   , x    - 14853x   , x    + 664x   , x    + 13522x   , x    +
│ │ │ │             3,4   1,3         3,4   0,3       3,4   1,2         3,4   0,2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       11804x   , x    + 14854x   )
│ │ │ │             3,4   0,1         3,4
│ │ │ │  
│ │ │ │  o4 : Ideal of G'1'4
│ │ │ │  i5 : time L' = plucker p
│ │ │ │ - -- used 0.111022s (cpu); 0.0535001s (thread); 0s (gc)
│ │ │ │ + -- used 0.12088s (cpu); 0.0581025s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = ideal (x  + 8480x  - 11656x , x  - 6727x  + 14853x , x  + 15777x  -
│ │ │ │               2        3         4   1        3         4   0         3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       664x )
│ │ │ │           4
│ │ │ │  
│ │ │ │ @@ -60,26 +60,26 @@
│ │ │ │  $W\subset\mathbb{P}^n$ swept out by the linear spaces corresponding to points
│ │ │ │  of $Y$. As an example, we now compute a surface scroll $W\subset\mathbb{P}^4$
│ │ │ │  over an elliptic curve $Y\subset\mathbb{G}(1,\mathbb{P}^4)$.
│ │ │ │  i7 : Y = ideal apply(5,i->random(1,G'1'4)); -- an elliptic curve
│ │ │ │  
│ │ │ │  o7 : Ideal of G'1'4
│ │ │ │  i8 : time W = plucker Y; -- surface swept out by the lines of Y
│ │ │ │ - -- used 0.0361963s (cpu); 0.0361951s (thread); 0s (gc)
│ │ │ │ + -- used 0.0373625s (cpu); 0.0373403s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 : Ideal of P4
│ │ │ │  i9 : (codim W,degree W)
│ │ │ │  
│ │ │ │  o9 = (2, 5)
│ │ │ │  
│ │ │ │  o9 : Sequence
│ │ │ │  In this example, we can recover the subvariety $Y\subset\mathbb{G}(k,\mathbb
│ │ │ │  {P}^n)$ by computing the Fano variety of $k$-planes contained in $W$.
│ │ │ │  i10 : time Y' = plucker(W,1); -- variety of lines contained in W
│ │ │ │ - -- used 0.142146s (cpu); 0.142149s (thread); 0s (gc)
│ │ │ │ + -- used 0.174625s (cpu); 0.17463s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 : Ideal of G'1'4
│ │ │ │  i11 : assert(Y' == Y)
│ │ │ │  WWaarrnniinngg: Notice that, by default, the computation is done on a randomly chosen
│ │ │ │  affine chart on the Grassmannian. To change this behavior, you can use the
│ │ │ │  _A_f_f_i_n_e_C_h_a_r_t_G_r_a_s_s option.
│ │ │ │  ********** WWaayyss ttoo uussee pplluucckkeerr:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_resultant_lp..._cm__Algorithm_eq_gt..._rp.html
│ │ │ @@ -108,15 +108,15 @@
│ │ │  
│ │ │  o2 : List
│ │ │              
    
│ │ │                
    i3 : time resultant(F,Algorithm=>"Poisson2")
│ │ │ - -- used 0.316083s (cpu); 0.220122s (thread); 0s (gc)
│ │ │ + -- used 0.333645s (cpu); 0.20846s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o3 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -132,15 +132,15 @@
│ │ │  
│ │ │  o3 : QQ[a..b]
    
│ │ │              
    
│ │ │                
    i4 : time resultant(F,Algorithm=>"Macaulay2")
│ │ │ - -- used 0.199752s (cpu); 0.138876s (thread); 0s (gc)
│ │ │ + -- used 0.160974s (cpu); 0.0991529s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o4 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -156,15 +156,15 @@
│ │ │  
│ │ │  o4 : QQ[a..b]
    
│ │ │              
    
│ │ │                
    i5 : time resultant(F,Algorithm=>"Poisson")
│ │ │ - -- used 0.514703s (cpu); 0.461162s (thread); 0s (gc)
│ │ │ + -- used 0.401593s (cpu); 0.342364s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o5 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -180,15 +180,15 @@
│ │ │  
│ │ │  o5 : QQ[a..b]
    
│ │ │              
    
│ │ │                
    i6 : time resultant(F,Algorithm=>"Macaulay")
│ │ │ - -- used 0.590923s (cpu); 0.537731s (thread); 0s (gc)
│ │ │ + -- used 0.71147s (cpu); 0.643863s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o6 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ ├── html2text {}
│ │ │ │ @@ -58,15 +58,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       3     2    9    7     2    9        3       1    8    4
│ │ │ │       -b)y*w  + (-a + -b)z*w  + (-a + 2b)w , 2x + -y + -z + -w}
│ │ │ │       4          8    8          7                4    3    5
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  i3 : time resultant(F,Algorithm=>"Poisson2")
│ │ │ │ - -- used 0.316083s (cpu); 0.220122s (thread); 0s (gc)
│ │ │ │ + -- used 0.333645s (cpu); 0.20846s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4
│ │ │ │  o3 = - -----------------------------a  - -------------------------------a b -
│ │ │ │             2222549728809984000000            12700284164628480000000
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       348237304382147063838108483692249 3 2
│ │ │ │       ---------------------------------a b  -
│ │ │ │ @@ -78,15 +78,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       1146977327343523453866040839029   4   194441910898734675845094443 5
│ │ │ │       -------------------------------a*b  - ---------------------------b
│ │ │ │            1119954511872000000000                895963609497600000
│ │ │ │  
│ │ │ │  o3 : QQ[a..b]
│ │ │ │  i4 : time resultant(F,Algorithm=>"Macaulay2")
│ │ │ │ - -- used 0.199752s (cpu); 0.138876s (thread); 0s (gc)
│ │ │ │ + -- used 0.160974s (cpu); 0.0991529s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4
│ │ │ │  o4 = - -----------------------------a  - -------------------------------a b -
│ │ │ │             2222549728809984000000            12700284164628480000000
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       348237304382147063838108483692249 3 2
│ │ │ │       ---------------------------------a b  -
│ │ │ │ @@ -98,15 +98,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       1146977327343523453866040839029   4   194441910898734675845094443 5
│ │ │ │       -------------------------------a*b  - ---------------------------b
│ │ │ │            1119954511872000000000                895963609497600000
│ │ │ │  
│ │ │ │  o4 : QQ[a..b]
│ │ │ │  i5 : time resultant(F,Algorithm=>"Poisson")
│ │ │ │ - -- used 0.514703s (cpu); 0.461162s (thread); 0s (gc)
│ │ │ │ + -- used 0.401593s (cpu); 0.342364s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4
│ │ │ │  o5 = - -----------------------------a  - -------------------------------a b -
│ │ │ │             2222549728809984000000            12700284164628480000000
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       348237304382147063838108483692249 3 2
│ │ │ │       ---------------------------------a b  -
│ │ │ │ @@ -118,15 +118,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       1146977327343523453866040839029   4   194441910898734675845094443 5
│ │ │ │       -------------------------------a*b  - ---------------------------b
│ │ │ │            1119954511872000000000                895963609497600000
│ │ │ │  
│ │ │ │  o5 : QQ[a..b]
│ │ │ │  i6 : time resultant(F,Algorithm=>"Macaulay")
│ │ │ │ - -- used 0.590923s (cpu); 0.537731s (thread); 0s (gc)
│ │ │ │ + -- used 0.71147s (cpu); 0.643863s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4
│ │ │ │  o6 = - -----------------------------a  - -------------------------------a b -
│ │ │ │             2222549728809984000000            12700284164628480000000
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       348237304382147063838108483692249 3 2
│ │ │ │       ---------------------------------a b  -
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_resultant_lp__Matrix_rp.html
│ │ │ @@ -92,15 +92,15 @@
│ │ │  
│ │ │  o2 : List
    
│ │ │              
    
│ │ │                
    i3 : time resultant F
│ │ │ - -- used 0.0229913s (cpu); 0.0229909s (thread); 0s (gc)
│ │ │ + -- used 0.0283289s (cpu); 0.0283286s (thread); 0s (gc)
│ │ │  
│ │ │            12         11 2         10 3         9 4          8 5          7 6
│ │ │  o3 = - 81t  u - 1701t  u  - 15309t  u  - 76545t u  - 229635t u  - 413343t u 
│ │ │       ------------------------------------------------------------------------
│ │ │                6 7          5 8       11          10 2         9 3  
│ │ │       - 413343t u  - 177147t u  + 567t  u + 10206t  u  + 76545t u  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -155,15 +155,15 @@
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │                
    i5 : time resultant F
│ │ │ - -- used 2.72711s (cpu); 2.06378s (thread); 0s (gc)
│ │ │ + -- used 2.44305s (cpu); 1.91459s (thread); 0s (gc)
│ │ │  
│ │ │        6 3 2       5 2   2     2 4   2 2    3 3 3 2     2 4 2   2  
│ │ │  o5 = a b c  - 3a a b b c  + 3a a b b c  - a a b c  + 3a a b b c  -
│ │ │        2 3 0     1 2 3 4 0     1 2 3 4 0    1 2 4 0     1 2 3 5 0  
│ │ │       ------------------------------------------------------------------------
│ │ │         3 3       2     4 2 2   2     4 2   2 2     5     2 2    6 3 2  
│ │ │       6a a b b b c  + 3a a b b c  + 3a a b b c  - 3a a b b c  + a b c  -
│ │ │ @@ -1790,15 +1790,15 @@
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : time # terms resultant F
│ │ │ - -- used 0.58204s (cpu); 0.40137s (thread); 0s (gc)
│ │ │ + -- used 0.436586s (cpu); 0.362757s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = 21894
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -32,15 +32,15 @@
│ │ │ │  i2 : F = {x^2+3*t*y*z-u*z^2,(t+3*u-1)*x-y,-t*x*y^3+t*x^2*y*z+u*z^4}
│ │ │ │  
│ │ │ │         2               2                            3      2         4
│ │ │ │  o2 = {x  + 3t*y*z - u*z , (t + 3u - 1)x - y, - t*x*y  + t*x y*z + u*z }
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  i3 : time resultant F
│ │ │ │ - -- used 0.0229913s (cpu); 0.0229909s (thread); 0s (gc)
│ │ │ │ + -- used 0.0283289s (cpu); 0.0283286s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            12         11 2         10 3         9 4          8 5          7 6
│ │ │ │  o3 = - 81t  u - 1701t  u  - 15309t  u  - 76545t u  - 229635t u  - 413343t u
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                6 7          5 8       11          10 2         9 3
│ │ │ │       - 413343t u  - 177147t u  + 567t  u + 10206t  u  + 76545t u  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -86,15 +86,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │            3
│ │ │ │       + c x }
│ │ │ │          9 2
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : time resultant F
│ │ │ │ - -- used 2.72711s (cpu); 2.06378s (thread); 0s (gc)
│ │ │ │ + -- used 2.44305s (cpu); 1.91459s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        6 3 2       5 2   2     2 4   2 2    3 3 3 2     2 4 2   2
│ │ │ │  o5 = a b c  - 3a a b b c  + 3a a b b c  - a a b c  + 3a a b b c  -
│ │ │ │        2 3 0     1 2 3 4 0     1 2 3 4 0    1 2 4 0     1 2 3 5 0
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │         3 3       2     4 2 2   2     4 2   2 2     5     2 2    6 3 2
│ │ │ │       6a a b b b c  + 3a a b b c  + 3a a b b c  - 3a a b b c  + a b c  -
│ │ │ │ @@ -1712,15 +1712,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                            2     2               2                        2
│ │ │ │       b x x  + b x x  + b x , c x  + c x x  + c x  + c x x  + c x x  + c x }
│ │ │ │        2 0 2    4 1 2    5 2   0 0    1 0 1    3 1    2 0 2    4 1 2    5 2
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : time # terms resultant F
│ │ │ │ - -- used 0.58204s (cpu); 0.40137s (thread); 0s (gc)
│ │ │ │ + -- used 0.436586s (cpu); 0.362757s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = 21894
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_h_o_w_F_o_r_m -- Chow form of a projective variety
│ │ │ │      * _d_i_s_c_r_i_m_i_n_a_n_t_(_R_i_n_g_E_l_e_m_e_n_t_)
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * resultant(List)
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_tangential__Chow__Form.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │                    0   4
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : -- 0-th associated hypersurface of S in G(1,4) (Chow form)
│ │ │       time tangentialChowForm(S,0)
│ │ │ - -- used 0.028783s (cpu); 0.0287839s (thread); 0s (gc)
│ │ │ + -- used 0.0340372s (cpu); 0.0340373s (thread); 0s (gc)
│ │ │  
│ │ │        2                                                       2        
│ │ │  o3 = p   p    - p   p   p    - p   p   p    + p   p   p    + p   p    +
│ │ │        1,3 2,3    1,2 1,3 2,4    0,3 1,3 2,4    0,2 1,4 2,4    1,2 3,4  
│ │ │       ------------------------------------------------------------------------
│ │ │        2
│ │ │       p   p    - 2p   p   p    - p   p   p
│ │ │ @@ -118,15 +118,15 @@
│ │ │         2,3 1,4    1,3 2,4    1,2 3,4   2,3 0,4    0,3 2,4    0,2 3,4   1,3 0,4    0,3 1,4    0,1 3,4   1,2 0,4    0,2 1,4    0,1 2,4   1,2 0,3    0,2 1,3    0,1 2,3
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : -- 1-th associated hypersurface of S in G(2,4)
│ │ │       time tangentialChowForm(S,1)
│ │ │ - -- used 0.120285s (cpu); 0.0736976s (thread); 0s (gc)
│ │ │ + -- used 0.133792s (cpu); 0.0760563s (thread); 0s (gc)
│ │ │  
│ │ │        2     2        2     2               3        2     2      
│ │ │  o4 = p     p      + p     p      - 2p     p      + p     p      -
│ │ │        1,2,3 1,2,4    0,2,4 1,2,4     0,2,3 1,2,4    0,2,4 0,3,4  
│ │ │       ------------------------------------------------------------------------
│ │ │               3         3               3            
│ │ │       4p     p      - 4p     p      - 2p     p      +
│ │ │ @@ -163,43 +163,43 @@
│ │ │         1,2,4 0,3,4    0,2,4 1,3,4    0,1,4 2,3,4   1,2,3 0,3,4    0,2,3 1,3,4    0,1,3 2,3,4   1,2,3 0,2,4    0,2,3 1,2,4    0,1,2 2,3,4   1,2,3 0,1,4    0,1,3 1,2,4    0,1,2 1,3,4   0,2,3 0,1,4    0,1,3 0,2,4    0,1,2 0,3,4
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : -- 2-th associated hypersurface of S in G(3,4) (parameterizing tangent hyperplanes to S)
│ │ │       time tangentialChowForm(S,2)
│ │ │ - -- used 0.0319774s (cpu); 0.0319788s (thread); 0s (gc)
│ │ │ + -- used 0.0393206s (cpu); 0.0393146s (thread); 0s (gc)
│ │ │  
│ │ │                2                                             2
│ │ │  o5 = p       p        - p       p       p        + p       p
│ │ │        0,1,3,4 0,2,3,4    0,1,2,4 0,2,3,4 1,2,3,4    0,1,2,3 1,2,3,4
│ │ │  
│ │ │  o5 : QQ[p       ..p       , p       , p       , p       ]
│ │ │           0,1,2,3   0,1,2,4   0,1,3,4   0,2,3,4   1,2,3,4
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : -- we get the dual hypersurface of S in G(0,4) by dualizing
│ │ │       time S' = ideal dualize tangentialChowForm(S,2)
│ │ │ - -- used 0.11267s (cpu); 0.0697575s (thread); 0s (gc)
│ │ │ + -- used 0.121272s (cpu); 0.057695s (thread); 0s (gc)
│ │ │  
│ │ │              2               2
│ │ │  o6 = ideal(p p  - p p p  + p p )
│ │ │              1 2    0 1 3    0 4
│ │ │  
│ │ │  o6 : Ideal of QQ[p ..p ]
│ │ │                    0   4
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : -- we then can recover S
│ │ │       time assert(dualize tangentialChowForm(S',3) == S)
│ │ │ - -- used 0.148995s (cpu); 0.103293s (thread); 0s (gc)
    
│ │ │ + -- used 0.171014s (cpu); 0.10791s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -63,15 +63,15 @@
│ │ │ │  o2 = ideal (- p p  + p p , - p p  + p p , - p  + p p )
│ │ │ │                 1 2    0 3     1 3    0 4     3    2 4
│ │ │ │  
│ │ │ │  o2 : Ideal of QQ[p ..p ]
│ │ │ │                    0   4
│ │ │ │  i3 : -- 0-th associated hypersurface of S in G(1,4) (Chow form)
│ │ │ │       time tangentialChowForm(S,0)
│ │ │ │ - -- used 0.028783s (cpu); 0.0287839s (thread); 0s (gc)
│ │ │ │ + -- used 0.0340372s (cpu); 0.0340373s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        2                                                       2
│ │ │ │  o3 = p   p    - p   p   p    - p   p   p    + p   p   p    + p   p    +
│ │ │ │        1,3 2,3    1,2 1,3 2,4    0,3 1,3 2,4    0,2 1,4 2,4    1,2 3,4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        2
│ │ │ │       p   p    - 2p   p   p    - p   p   p
│ │ │ │ @@ -88,15 +88,15 @@
│ │ │ │  - p   p    + p   p   , p   p    - p   p    + p   p   , p   p    - p   p    + p
│ │ │ │  p   )
│ │ │ │         2,3 1,4    1,3 2,4    1,2 3,4   2,3 0,4    0,3 2,4    0,2 3,4   1,3 0,4
│ │ │ │  0,3 1,4    0,1 3,4   1,2 0,4    0,2 1,4    0,1 2,4   1,2 0,3    0,2 1,3    0,1
│ │ │ │  2,3
│ │ │ │  i4 : -- 1-th associated hypersurface of S in G(2,4)
│ │ │ │       time tangentialChowForm(S,1)
│ │ │ │ - -- used 0.120285s (cpu); 0.0736976s (thread); 0s (gc)
│ │ │ │ + -- used 0.133792s (cpu); 0.0760563s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        2     2        2     2               3        2     2
│ │ │ │  o4 = p     p      + p     p      - 2p     p      + p     p      -
│ │ │ │        1,2,3 1,2,4    0,2,4 1,2,4     0,2,3 1,2,4    0,2,4 0,3,4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │               3         3               3
│ │ │ │       4p     p      - 4p     p      - 2p     p      +
│ │ │ │ @@ -138,35 +138,35 @@
│ │ │ │  p      + p     p     , p     p      - p     p      + p     p     )
│ │ │ │         1,2,4 0,3,4    0,2,4 1,3,4    0,1,4 2,3,4   1,2,3 0,3,4    0,2,3 1,3,4
│ │ │ │  0,1,3 2,3,4   1,2,3 0,2,4    0,2,3 1,2,4    0,1,2 2,3,4   1,2,3 0,1,4    0,1,3
│ │ │ │  1,2,4    0,1,2 1,3,4   0,2,3 0,1,4    0,1,3 0,2,4    0,1,2 0,3,4
│ │ │ │  i5 : -- 2-th associated hypersurface of S in G(3,4) (parameterizing tangent
│ │ │ │  hyperplanes to S)
│ │ │ │       time tangentialChowForm(S,2)
│ │ │ │ - -- used 0.0319774s (cpu); 0.0319788s (thread); 0s (gc)
│ │ │ │ + -- used 0.0393206s (cpu); 0.0393146s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                2                                             2
│ │ │ │  o5 = p       p        - p       p       p        + p       p
│ │ │ │        0,1,3,4 0,2,3,4    0,1,2,4 0,2,3,4 1,2,3,4    0,1,2,3 1,2,3,4
│ │ │ │  
│ │ │ │  o5 : QQ[p       ..p       , p       , p       , p       ]
│ │ │ │           0,1,2,3   0,1,2,4   0,1,3,4   0,2,3,4   1,2,3,4
│ │ │ │  i6 : -- we get the dual hypersurface of S in G(0,4) by dualizing
│ │ │ │       time S' = ideal dualize tangentialChowForm(S,2)
│ │ │ │ - -- used 0.11267s (cpu); 0.0697575s (thread); 0s (gc)
│ │ │ │ + -- used 0.121272s (cpu); 0.057695s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              2               2
│ │ │ │  o6 = ideal(p p  - p p p  + p p )
│ │ │ │              1 2    0 1 3    0 4
│ │ │ │  
│ │ │ │  o6 : Ideal of QQ[p ..p ]
│ │ │ │                    0   4
│ │ │ │  i7 : -- we then can recover S
│ │ │ │       time assert(dualize tangentialChowForm(S',3) == S)
│ │ │ │ - -- used 0.148995s (cpu); 0.103293s (thread); 0s (gc)
│ │ │ │ + -- used 0.171014s (cpu); 0.10791s (thread); 0s (gc)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_C_o_i_s_o_t_r_o_p_i_c -- whether a hypersurface of a Grassmannian is a tangential
│ │ │ │        Chow form
│ │ │ │      * _c_h_o_w_F_o_r_m -- Chow form of a projective variety
│ │ │ │  ********** WWaayyss ttoo uussee ttaannggeennttiiaallCChhoowwFFoorrmm:: **********
│ │ │ │      * tangentialChowForm(Ideal,ZZ)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_resource_splimits.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  time(seconds)        700
│ │ │  file(blocks)         unlimited
│ │ │  data(kbytes)         unlimited
│ │ │  stack(kbytes)        8192
│ │ │  coredump(blocks)     unlimited
│ │ │  memory(kbytes)       850000
│ │ │  locked memory(kbytes) 8192
│ │ │ -process              63811
│ │ │ +process              63521
│ │ │  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/M2-29954-0/0.m2
│ │ │ +o1 = /tmp/M2-43124-0/0.m2
│ │ │  
│ │ │  i2 : fn< (stderr<<"Running"< ( exit(27); ); ///< (stderr<<"Spinning!!"<"/tmp/M2-29954-0/1.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/1.m2" >"/tmp/M2-43124-0/1.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  i7 : h
│ │ │  
│ │ │  o7 = HashTable{"answer file" => null}
│ │ │                 "exit code" => 0
│ │ │                 "output file" => null
│ │ │                 "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/M2-29954-0/2.m2" >"/tmp/M2-29954-0/2.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/2.m2" >"/tmp/M2-43124-0/2.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
│ │ │  
│ │ │  i11 : h
│ │ │  
│ │ │ -o11 = HashTable{"answer file" => /tmp/M2-29954-0/2.ans}
│ │ │ +o11 = HashTable{"answer file" => /tmp/M2-43124-0/2.ans}
│ │ │                  "exit code" => 27
│ │ │ -                "output file" => /tmp/M2-29954-0/2.out
│ │ │ +                "output file" => /tmp/M2-43124-0/2.out
│ │ │                  "return code" => 6912
│ │ │                  "statistics" => null
│ │ │ -                "time used" => 1
│ │ │ +                "time used" => 2
│ │ │                  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/M2-29954-0/3.m2" >"/tmp/M2-29954-0/3.out" 2>&1 ))
│ │ │ +Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/3.m2" >"/tmp/M2-43124-0/3.out" 2>&1 ))
│ │ │  Killed
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
│ │ │  
│ │ │  i15 : h
│ │ │  
│ │ │ -o15 = HashTable{"answer file" => /tmp/M2-29954-0/3.ans}
│ │ │ +o15 = HashTable{"answer file" => /tmp/M2-43124-0/3.ans}
│ │ │                  "exit code" => 0
│ │ │ -                "output file" => /tmp/M2-29954-0/3.out
│ │ │ +                "output file" => /tmp/M2-43124-0/3.out
│ │ │                  "return code" => 9
│ │ │                  "statistics" => null
│ │ │ -                "time used" => 2
│ │ │ +                "time used" => 1
│ │ │                  value => null
│ │ │  
│ │ │  o15 : HashTable
│ │ │  
│ │ │  i16 : if h#"output file" =!= null and fileExists(h#"output file") then get(h#"output file")
│ │ │  
│ │ │  o16 = 
│ │ │ -      i1 : -- Script /tmp/M2-29954-0/3.m2 automatically generated by RunExternalM2
│ │ │ +      i1 : -- Script /tmp/M2-43124-0/3.m2 automatically generated by RunExternalM2
│ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │  
│ │ │ -      i2 : load "/tmp/M2-29954-0/0.m2";
│ │ │ +      i2 : load "/tmp/M2-43124-0/0.m2";
│ │ │  
│ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-29954-0/3.ans",spin (10));
│ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-43124-0/3.ans",spin (10));
│ │ │        Spinning!!
│ │ │  
│ │ │  
│ │ │  i17 : if h#"answer file" =!= null and fileExists(h#"answer file") then get(h#"answer file")
│ │ │  
│ │ │  i18 : h=runExternalM2(fn,"spin",3,KeepStatistics=>true);
│ │ │ -Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-29954-0/4.m2" >"/tmp/M2-29954-0/4.out" 2>&1') >"/tmp/M2-29954-0/4.stat" 2>&1 ))
│ │ │ +Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/4.m2" >"/tmp/M2-43124-0/4.out" 2>&1') >"/tmp/M2-43124-0/4.stat" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  i19 : h#"statistics"
│ │ │  
│ │ │ -o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-29954-0/4.m2" >"/tmp/M2-29954-0/4.out" 2>&1"
│ │ │ -              User time (seconds): 5.25
│ │ │ -              System time (seconds): 0.12
│ │ │ -              Percent of CPU this job got: 77%
│ │ │ -              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:06.97
│ │ │ +o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/4.m2" >"/tmp/M2-43124-0/4.out" 2>&1"
│ │ │ +              User time (seconds): 4.34
│ │ │ +              System time (seconds): 0.28
│ │ │ +              Percent of CPU this job got: 119%
│ │ │ +              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:03.86
│ │ │                Average shared text size (kbytes): 0
│ │ │                Average unshared data size (kbytes): 0
│ │ │                Average stack size (kbytes): 0
│ │ │                Average total size (kbytes): 0
│ │ │ -              Maximum resident set size (kbytes): 251780
│ │ │ +              Maximum resident set size (kbytes): 338928
│ │ │                Average resident set size (kbytes): 0
│ │ │                Major (requiring I/O) page faults: 0
│ │ │ -              Minor (reclaiming a frame) page faults: 8554
│ │ │ -              Voluntary context switches: 1820
│ │ │ -              Involuntary context switches: 2015
│ │ │ +              Minor (reclaiming a frame) page faults: 11480
│ │ │ +              Voluntary context switches: 5569
│ │ │ +              Involuntary context switches: 899
│ │ │                Swaps: 0
│ │ │                File system inputs: 0
│ │ │ -              File system outputs: 0
│ │ │ +              File system outputs: 16
│ │ │                Socket messages sent: 0
│ │ │                Socket messages received: 0
│ │ │                Signals delivered: 0
│ │ │                Page size (bytes): 4096
│ │ │                Exit status: 0
│ │ │  
│ │ │  
│ │ │  i20 : v=/// A complicated string^%&C@#CERQVASDFQ#BQBSDH"' ewrjwklsf///;
│ │ │  
│ │ │  i21 : (runExternalM2(fn,identity,v))#value===v
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-29954-0/6.m2" >"/tmp/M2-29954-0/6.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/6.m2" >"/tmp/M2-43124-0/6.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  o21 = true
│ │ │  
│ │ │  i22 : R=QQ[x,y];
│ │ │  
│ │ │  i23 : v=coker random(R^2,R^{3:-1})
│ │ │ @@ -139,54 +139,54 @@
│ │ │  o23 = cokernel | 9/2x+9/4y 7/9x+7/10y 7x+3/7y |
│ │ │                 | 3/4x+7/4y 7/10x+7/3y 6/7x+6y |
│ │ │  
│ │ │                               2
│ │ │  o23 : R-module, quotient of R
│ │ │  
│ │ │  i24 : h=runExternalM2(fn,identity,v)
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-29954-0/7.m2" >"/tmp/M2-29954-0/7.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/7.m2" >"/tmp/M2-43124-0/7.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
│ │ │  
│ │ │ -o24 = HashTable{"answer file" => /tmp/M2-29954-0/7.ans}
│ │ │ +o24 = HashTable{"answer file" => /tmp/M2-43124-0/7.ans}
│ │ │                  "exit code" => 1
│ │ │ -                "output file" => /tmp/M2-29954-0/7.out
│ │ │ +                "output file" => /tmp/M2-43124-0/7.out
│ │ │                  "return code" => 256
│ │ │                  "statistics" => null
│ │ │ -                "time used" => 1
│ │ │ +                "time used" => 2
│ │ │                  value => null
│ │ │  
│ │ │  o24 : HashTable
│ │ │  
│ │ │  i25 : get(h#"output file")
│ │ │  
│ │ │  o25 = 
│ │ │ -      i1 : -- Script /tmp/M2-29954-0/7.m2 automatically generated by RunExternalM2
│ │ │ +      i1 : -- Script /tmp/M2-43124-0/7.m2 automatically generated by RunExternalM2
│ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │  
│ │ │ -      i2 : load "/tmp/M2-29954-0/0.m2";
│ │ │ +      i2 : load "/tmp/M2-43124-0/0.m2";
│ │ │  
│ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-29954-0/7.ans",identity (cokernel(map(R^2,R^{3:{-1}},{{(9/2)*x+(9/4)*y, (7/9)*x+(7/10)*y, 7*x+(3/7)*y}, {(3/4)*x+(7/4)*y, (7/10)*x+(7/3)*y, (6/7)*x+6*y}}))));
│ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-43124-0/7.ans",identity (cokernel(map(R^2,R^{3:{-1}},{{(9/2)*x+(9/4)*y, (7/9)*x+(7/10)*y, 7*x+(3/7)*y}, {(3/4)*x+(7/4)*y, (7/10)*x+(7/3)*y, (6/7)*x+6*y}}))));
│ │ │        stdio:4:74:(3):[1]: error: no method for binary operator ^ applied to objects:
│ │ │                    R (of class Symbol)
│ │ │              ^     2 (of class ZZ)
│ │ │  
│ │ │  
│ │ │  i26 : fn<"/tmp/M2-29954-0/8.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/8.m2" >"/tmp/M2-43124-0/8.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  o27 = true
│ │ │  
│ │ │  i28 : v=R;
│ │ │  
│ │ │  i29 : h=runExternalM2(fn,identity,v);
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-29954-0/9.m2" >"/tmp/M2-29954-0/9.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/9.m2" >"/tmp/M2-43124-0/9.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  i30 : h#value
│ │ │  
│ │ │  o30 = QQ[x..y]
│ │ │  
│ │ │  o30 : PolynomialRing
│ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/html/_resource_splimits.html
│ │ │ @@ -75,15 +75,15 @@
│ │ │  time(seconds)        700
│ │ │  file(blocks)         unlimited
│ │ │  data(kbytes)         unlimited
│ │ │  stack(kbytes)        8192
│ │ │  coredump(blocks)     unlimited
│ │ │  memory(kbytes)       850000
│ │ │  locked memory(kbytes) 8192
│ │ │ -process              63811
│ │ │ +process              63521
│ │ │  nofiles              512
│ │ │  vmemory(kbytes)      unlimited
│ │ │  locks                unlimited
│ │ │  rtprio               0
│ │ │  
│ │ │  o1 = 0
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,15 +35,15 @@
│ │ │ │  time(seconds)        700
│ │ │ │  file(blocks)         unlimited
│ │ │ │  data(kbytes)         unlimited
│ │ │ │  stack(kbytes)        8192
│ │ │ │  coredump(blocks)     unlimited
│ │ │ │  memory(kbytes)       850000
│ │ │ │  locked memory(kbytes) 8192
│ │ │ │ -process              63811
│ │ │ │ +process              63521
│ │ │ │  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/M2-29954-0/0.m2
    
│ │ │ +o1 = /tmp/M2-43124-0/0.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/M2-29954-0/1.m2" >"/tmp/M2-29954-0/1.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/1.m2" >"/tmp/M2-43124-0/1.out" 2>&1 ))
│ │ │  Finished running.
    
│ │ │              
    
│ │ │                
    i7 : h
│ │ │  
│ │ │  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/M2-29954-0/2.m2" >"/tmp/M2-29954-0/2.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/2.m2" >"/tmp/M2-43124-0/2.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
    
│ │ │              
    
│ │ │                
    i11 : h
│ │ │  
│ │ │ -o11 = HashTable{"answer file" => /tmp/M2-29954-0/2.ans}
│ │ │ +o11 = HashTable{"answer file" => /tmp/M2-43124-0/2.ans}
│ │ │                  "exit code" => 27
│ │ │ -                "output file" => /tmp/M2-29954-0/2.out
│ │ │ +                "output file" => /tmp/M2-43124-0/2.out
│ │ │                  "return code" => 6912
│ │ │                  "statistics" => null
│ │ │ -                "time used" => 1
│ │ │ +                "time used" => 2
│ │ │                  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/M2-29954-0/3.m2" >"/tmp/M2-29954-0/3.out" 2>&1 ))
│ │ │ +Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/3.m2" >"/tmp/M2-43124-0/3.out" 2>&1 ))
│ │ │  Killed
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
    
│ │ │              
    
│ │ │                
    i15 : h
│ │ │  
│ │ │ -o15 = HashTable{"answer file" => /tmp/M2-29954-0/3.ans}
│ │ │ +o15 = HashTable{"answer file" => /tmp/M2-43124-0/3.ans}
│ │ │                  "exit code" => 0
│ │ │ -                "output file" => /tmp/M2-29954-0/3.out
│ │ │ +                "output file" => /tmp/M2-43124-0/3.out
│ │ │                  "return code" => 9
│ │ │                  "statistics" => null
│ │ │ -                "time used" => 2
│ │ │ +                "time used" => 1
│ │ │                  value => null
│ │ │  
│ │ │  o15 : HashTable
    
│ │ │              
    
│ │ │                
    i16 : if h#"output file" =!= null and fileExists(h#"output file") then get(h#"output file")
│ │ │  
│ │ │  o16 = 
│ │ │ -      i1 : -- Script /tmp/M2-29954-0/3.m2 automatically generated by RunExternalM2
│ │ │ +      i1 : -- Script /tmp/M2-43124-0/3.m2 automatically generated by RunExternalM2
│ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │  
│ │ │ -      i2 : load "/tmp/M2-29954-0/0.m2";
│ │ │ +      i2 : load "/tmp/M2-43124-0/0.m2";
│ │ │  
│ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-29954-0/3.ans",spin (10));
│ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-43124-0/3.ans",spin (10));
│ │ │        Spinning!!
    
│ │ │              
    
│ │ │                
    i17 : if h#"answer file" =!= null and fileExists(h#"answer file") then get(h#"answer file")
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -295,15 +295,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i18 : h=runExternalM2(fn,"spin",3,KeepStatistics=>true);
│ │ │ -Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-29954-0/4.m2" >"/tmp/M2-29954-0/4.out" 2>&1') >"/tmp/M2-29954-0/4.stat" 2>&1 ))
│ │ │ +Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/4.m2" >"/tmp/M2-43124-0/4.out" 2>&1') >"/tmp/M2-43124-0/4.stat" 2>&1 ))
│ │ │  Finished running.
    
│ │ │              
    
│ │ │                
    i19 : h#"statistics"
│ │ │  
│ │ │ -o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-29954-0/4.m2" >"/tmp/M2-29954-0/4.out" 2>&1"
│ │ │ -              User time (seconds): 5.25
│ │ │ -              System time (seconds): 0.12
│ │ │ -              Percent of CPU this job got: 77%
│ │ │ -              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:06.97
│ │ │ +o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/4.m2" >"/tmp/M2-43124-0/4.out" 2>&1"
│ │ │ +              User time (seconds): 4.34
│ │ │ +              System time (seconds): 0.28
│ │ │ +              Percent of CPU this job got: 119%
│ │ │ +              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:03.86
│ │ │                Average shared text size (kbytes): 0
│ │ │                Average unshared data size (kbytes): 0
│ │ │                Average stack size (kbytes): 0
│ │ │                Average total size (kbytes): 0
│ │ │ -              Maximum resident set size (kbytes): 251780
│ │ │ +              Maximum resident set size (kbytes): 338928
│ │ │                Average resident set size (kbytes): 0
│ │ │                Major (requiring I/O) page faults: 0
│ │ │ -              Minor (reclaiming a frame) page faults: 8554
│ │ │ -              Voluntary context switches: 1820
│ │ │ -              Involuntary context switches: 2015
│ │ │ +              Minor (reclaiming a frame) page faults: 11480
│ │ │ +              Voluntary context switches: 5569
│ │ │ +              Involuntary context switches: 899
│ │ │                Swaps: 0
│ │ │                File system inputs: 0
│ │ │ -              File system outputs: 0
│ │ │ +              File system outputs: 16
│ │ │                Socket messages sent: 0
│ │ │                Socket messages received: 0
│ │ │                Signals delivered: 0
│ │ │                Page size (bytes): 4096
│ │ │                Exit status: 0
    
│ │ │              
    
│ │ │                
    i20 : v=/// A complicated string^%&C@#CERQVASDFQ#BQBSDH"' ewrjwklsf///;
    
│ │ │              
    
│ │ │                
    i21 : (runExternalM2(fn,identity,v))#value===v
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-29954-0/6.m2" >"/tmp/M2-29954-0/6.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/6.m2" >"/tmp/M2-43124-0/6.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  o21 = true
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -325,24 +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/M2-29954-0/7.m2" >"/tmp/M2-29954-0/7.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/7.m2" >"/tmp/M2-43124-0/7.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
│ │ │  
│ │ │ -o24 = HashTable{"answer file" => /tmp/M2-29954-0/7.ans}
│ │ │ +o24 = HashTable{"answer file" => /tmp/M2-43124-0/7.ans}
│ │ │                  "exit code" => 1
│ │ │ -                "output file" => /tmp/M2-29954-0/7.out
│ │ │ +                "output file" => /tmp/M2-43124-0/7.out
│ │ │                  "return code" => 256
│ │ │                  "statistics" => null
│ │ │ -                "time used" => 1
│ │ │ +                "time used" => 2
│ │ │                  value => null
│ │ │  
│ │ │  o24 : HashTable
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -350,20 +350,20 @@
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i25 : get(h#"output file")
│ │ │  
│ │ │  o25 = 
│ │ │ -      i1 : -- Script /tmp/M2-29954-0/7.m2 automatically generated by RunExternalM2
│ │ │ +      i1 : -- Script /tmp/M2-43124-0/7.m2 automatically generated by RunExternalM2
│ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │  
│ │ │ -      i2 : load "/tmp/M2-29954-0/0.m2";
│ │ │ +      i2 : load "/tmp/M2-43124-0/0.m2";
│ │ │  
│ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-29954-0/7.ans",identity (cokernel(map(R^2,R^{3:{-1}},{{(9/2)*x+(9/4)*y, (7/9)*x+(7/10)*y, 7*x+(3/7)*y}, {(3/4)*x+(7/4)*y, (7/10)*x+(7/3)*y, (6/7)*x+6*y}}))));
│ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-43124-0/7.ans",identity (cokernel(map(R^2,R^{3:{-1}},{{(9/2)*x+(9/4)*y, (7/9)*x+(7/10)*y, 7*x+(3/7)*y}, {(3/4)*x+(7/4)*y, (7/10)*x+(7/3)*y, (6/7)*x+6*y}}))));
│ │ │        stdio:4:74:(3):[1]: error: no method for binary operator ^ applied to objects:
│ │ │                    R (of class Symbol)
│ │ │              ^     2 (of class ZZ)
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -374,15 +374,15 @@
│ │ │              
    		
│ │ │                
    i26 : fn<<///R=QQ[x,y];///<<endl<<flush;
    
│ │ │              
    
│ │ │                
    i27 : (runExternalM2(fn,identity,v))#value===v
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-29954-0/8.m2" >"/tmp/M2-29954-0/8.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/8.m2" >"/tmp/M2-43124-0/8.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  o27 = true
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -394,15 +394,15 @@
│ │ │              
│ │ │                
    i28 : v=R;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i29 : h=runExternalM2(fn,identity,v);
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-29954-0/9.m2" >"/tmp/M2-29954-0/9.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43124-0/9.m2" >"/tmp/M2-43124-0/9.out" 2>&1 ))
│ │ │  Finished running.
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i30 : h#value
│ │ │ ├── html2text {}
│ │ │ │ @@ -45,201 +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/M2-29954-0/0.m2
│ │ │ │ +o1 = /tmp/M2-43124-0/0.m2
│ │ │ │  i2 : fn< (stderr<<"Running"< ( exit(27); ); ///< (stderr<<"Spinning!!"<"/tmp/M2-29954-0/1.out" 2>&1 ))
│ │ │ │ +M2-43124-0/1.m2" >"/tmp/M2-43124-0/1.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  i7 : h
│ │ │ │  
│ │ │ │  o7 = HashTable{"answer file" => null}
│ │ │ │                 "exit code" => 0
│ │ │ │                 "output file" => null
│ │ │ │                 "return code" => 0
│ │ │ │                 "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/
│ │ │ │ -M2-29954-0/2.m2" >"/tmp/M2-29954-0/2.out" 2>&1 ))
│ │ │ │ +M2-43124-0/2.m2" >"/tmp/M2-43124-0/2.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  RunExternalM2: expected answer file does not exist
│ │ │ │  i11 : h
│ │ │ │  
│ │ │ │ -o11 = HashTable{"answer file" => /tmp/M2-29954-0/2.ans}
│ │ │ │ +o11 = HashTable{"answer file" => /tmp/M2-43124-0/2.ans}
│ │ │ │                  "exit code" => 27
│ │ │ │ -                "output file" => /tmp/M2-29954-0/2.out
│ │ │ │ +                "output file" => /tmp/M2-43124-0/2.out
│ │ │ │                  "return code" => 6912
│ │ │ │                  "statistics" => null
│ │ │ │ -                "time used" => 1
│ │ │ │ +                "time used" => 2
│ │ │ │                  value => null
│ │ │ │  
│ │ │ │  o11 : HashTable
│ │ │ │  i12 : fileExists(h#"output file")
│ │ │ │  
│ │ │ │  o12 = true
│ │ │ │  i13 : fileExists(h#"answer file")
│ │ │ │  
│ │ │ │  o13 = false
│ │ │ │  Here, we use _r_e_s_o_u_r_c_e_ _l_i_m_i_t_s to limit the routine to 2 seconds of computational
│ │ │ │  time, while the system is asked to use 10 seconds of computational time:
│ │ │ │  i14 : h=runExternalM2(fn,"spin",10,PreRunScript=>"ulimit -t 2");
│ │ │ │  Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -
│ │ │ │ -q  <"/tmp/M2-29954-0/3.m2" >"/tmp/M2-29954-0/3.out" 2>&1 ))
│ │ │ │ +q  <"/tmp/M2-43124-0/3.m2" >"/tmp/M2-43124-0/3.out" 2>&1 ))
│ │ │ │  Killed
│ │ │ │  Finished running.
│ │ │ │  RunExternalM2: expected answer file does not exist
│ │ │ │  i15 : h
│ │ │ │  
│ │ │ │ -o15 = HashTable{"answer file" => /tmp/M2-29954-0/3.ans}
│ │ │ │ +o15 = HashTable{"answer file" => /tmp/M2-43124-0/3.ans}
│ │ │ │                  "exit code" => 0
│ │ │ │ -                "output file" => /tmp/M2-29954-0/3.out
│ │ │ │ +                "output file" => /tmp/M2-43124-0/3.out
│ │ │ │                  "return code" => 9
│ │ │ │                  "statistics" => null
│ │ │ │ -                "time used" => 2
│ │ │ │ +                "time used" => 1
│ │ │ │                  value => null
│ │ │ │  
│ │ │ │  o15 : HashTable
│ │ │ │  i16 : if h#"output file" =!= null and fileExists(h#"output file") then get
│ │ │ │  (h#"output file")
│ │ │ │  
│ │ │ │  o16 =
│ │ │ │ -      i1 : -- Script /tmp/M2-29954-0/3.m2 automatically generated by
│ │ │ │ +      i1 : -- Script /tmp/M2-43124-0/3.m2 automatically generated by
│ │ │ │  RunExternalM2
│ │ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │ │  
│ │ │ │ -      i2 : load "/tmp/M2-29954-0/0.m2";
│ │ │ │ +      i2 : load "/tmp/M2-43124-0/0.m2";
│ │ │ │  
│ │ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-29954-0/3.ans",spin (10));
│ │ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-43124-0/3.ans",spin (10));
│ │ │ │        Spinning!!
│ │ │ │  i17 : if h#"answer file" =!= null and fileExists(h#"answer file") then get
│ │ │ │  (h#"answer file")
│ │ │ │  We can get quite a lot of detail on the resources used with the _K_e_e_p_S_t_a_t_i_s_t_i_c_s
│ │ │ │  command:
│ │ │ │  i18 : h=runExternalM2(fn,"spin",3,KeepStatistics=>true);
│ │ │ │  Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary  --stop -
│ │ │ │ --no-debug --silent  -q  <"/tmp/M2-29954-0/4.m2" >"/tmp/M2-29954-0/4.out" 2>&1')
│ │ │ │ ->"/tmp/M2-29954-0/4.stat" 2>&1 ))
│ │ │ │ +-no-debug --silent  -q  <"/tmp/M2-43124-0/4.m2" >"/tmp/M2-43124-0/4.out" 2>&1')
│ │ │ │ +>"/tmp/M2-43124-0/4.stat" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  i19 : h#"statistics"
│ │ │ │  
│ │ │ │  o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug
│ │ │ │ ---silent  -q  <"/tmp/M2-29954-0/4.m2" >"/tmp/M2-29954-0/4.out" 2>&1"
│ │ │ │ -              User time (seconds): 5.25
│ │ │ │ -              System time (seconds): 0.12
│ │ │ │ -              Percent of CPU this job got: 77%
│ │ │ │ -              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:06.97
│ │ │ │ +--silent  -q  <"/tmp/M2-43124-0/4.m2" >"/tmp/M2-43124-0/4.out" 2>&1"
│ │ │ │ +              User time (seconds): 4.34
│ │ │ │ +              System time (seconds): 0.28
│ │ │ │ +              Percent of CPU this job got: 119%
│ │ │ │ +              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:03.86
│ │ │ │                Average shared text size (kbytes): 0
│ │ │ │                Average unshared data size (kbytes): 0
│ │ │ │                Average stack size (kbytes): 0
│ │ │ │                Average total size (kbytes): 0
│ │ │ │ -              Maximum resident set size (kbytes): 251780
│ │ │ │ +              Maximum resident set size (kbytes): 338928
│ │ │ │                Average resident set size (kbytes): 0
│ │ │ │                Major (requiring I/O) page faults: 0
│ │ │ │ -              Minor (reclaiming a frame) page faults: 8554
│ │ │ │ -              Voluntary context switches: 1820
│ │ │ │ -              Involuntary context switches: 2015
│ │ │ │ +              Minor (reclaiming a frame) page faults: 11480
│ │ │ │ +              Voluntary context switches: 5569
│ │ │ │ +              Involuntary context switches: 899
│ │ │ │                Swaps: 0
│ │ │ │                File system inputs: 0
│ │ │ │ -              File system outputs: 0
│ │ │ │ +              File system outputs: 16
│ │ │ │                Socket messages sent: 0
│ │ │ │                Socket messages received: 0
│ │ │ │                Signals delivered: 0
│ │ │ │                Page size (bytes): 4096
│ │ │ │                Exit status: 0
│ │ │ │  We can handle most kinds of objects as return values, although _M_u_t_a_b_l_e_M_a_t_r_i_x
│ │ │ │  does not work. Here, we use the built-in _i_d_e_n_t_i_t_y function:
│ │ │ │  i20 : v=/// A complicated string^%&C@#CERQVASDFQ#BQBSDH"' ewrjwklsf///;
│ │ │ │  i21 : (runExternalM2(fn,identity,v))#value===v
│ │ │ │  Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/
│ │ │ │ -M2-29954-0/6.m2" >"/tmp/M2-29954-0/6.out" 2>&1 ))
│ │ │ │ +M2-43124-0/6.m2" >"/tmp/M2-43124-0/6.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  
│ │ │ │  o21 = true
│ │ │ │  Some care is required, however:
│ │ │ │  i22 : R=QQ[x,y];
│ │ │ │  i23 : v=coker random(R^2,R^{3:-1})
│ │ │ │  
│ │ │ │  o23 = cokernel | 9/2x+9/4y 7/9x+7/10y 7x+3/7y |
│ │ │ │                 | 3/4x+7/4y 7/10x+7/3y 6/7x+6y |
│ │ │ │  
│ │ │ │                               2
│ │ │ │  o23 : R-module, quotient of R
│ │ │ │  i24 : h=runExternalM2(fn,identity,v)
│ │ │ │  Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/
│ │ │ │ -M2-29954-0/7.m2" >"/tmp/M2-29954-0/7.out" 2>&1 ))
│ │ │ │ +M2-43124-0/7.m2" >"/tmp/M2-43124-0/7.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  RunExternalM2: expected answer file does not exist
│ │ │ │  
│ │ │ │ -o24 = HashTable{"answer file" => /tmp/M2-29954-0/7.ans}
│ │ │ │ +o24 = HashTable{"answer file" => /tmp/M2-43124-0/7.ans}
│ │ │ │                  "exit code" => 1
│ │ │ │ -                "output file" => /tmp/M2-29954-0/7.out
│ │ │ │ +                "output file" => /tmp/M2-43124-0/7.out
│ │ │ │                  "return code" => 256
│ │ │ │                  "statistics" => null
│ │ │ │ -                "time used" => 1
│ │ │ │ +                "time used" => 2
│ │ │ │                  value => null
│ │ │ │  
│ │ │ │  o24 : HashTable
│ │ │ │  To view the error message:
│ │ │ │  i25 : get(h#"output file")
│ │ │ │  
│ │ │ │  o25 =
│ │ │ │ -      i1 : -- Script /tmp/M2-29954-0/7.m2 automatically generated by
│ │ │ │ +      i1 : -- Script /tmp/M2-43124-0/7.m2 automatically generated by
│ │ │ │  RunExternalM2
│ │ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │ │  
│ │ │ │ -      i2 : load "/tmp/M2-29954-0/0.m2";
│ │ │ │ +      i2 : load "/tmp/M2-43124-0/0.m2";
│ │ │ │  
│ │ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-29954-0/7.ans",identity (cokernel
│ │ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-43124-0/7.ans",identity (cokernel
│ │ │ │  (map(R^2,R^{3:{-1}},{{(9/2)*x+(9/4)*y, (7/9)*x+(7/10)*y, 7*x+(3/7)*y}, {(3/
│ │ │ │  4)*x+(7/4)*y, (7/10)*x+(7/3)*y, (6/7)*x+6*y}}))));
│ │ │ │        stdio:4:74:(3):[1]: error: no method for binary operator ^ applied to
│ │ │ │  objects:
│ │ │ │                    R (of class Symbol)
│ │ │ │              ^     2 (of class ZZ)
│ │ │ │  Keep in mind that the object you are passing must make sense in the context of
│ │ │ │  the file containing your function! For instance, here we need to define the
│ │ │ │  ring:
│ │ │ │  i26 : fn<"/tmp/M2-29954-0/8.out" 2>&1 ))
│ │ │ │ +M2-43124-0/8.m2" >"/tmp/M2-43124-0/8.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  
│ │ │ │  o27 = true
│ │ │ │  This problem can be avoided by following some _s_u_g_g_e_s_t_i_o_n_s_ _f_o_r_ _u_s_i_n_g
│ │ │ │  _R_u_n_E_x_t_e_r_n_a_l_M_2.
│ │ │ │  The objects may unavoidably lose some internal references, though:
│ │ │ │  i28 : v=R;
│ │ │ │  i29 : h=runExternalM2(fn,identity,v);
│ │ │ │  Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/
│ │ │ │ -M2-29954-0/9.m2" >"/tmp/M2-29954-0/9.out" 2>&1 ))
│ │ │ │ +M2-43124-0/9.m2" >"/tmp/M2-43124-0/9.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  i30 : h#value
│ │ │ │  
│ │ │ │  o30 = QQ[x..y]
│ │ │ │  
│ │ │ │  o30 : PolynomialRing
│ │ │ │  i31 : v===h#value
│ │ ├── ./usr/share/doc/Macaulay2/SLPexpressions/example-output/___S__L__Pexpressions.out
│ │ │ @@ -30,23 +30,23 @@
│ │ │                                              )
│ │ │  
│ │ │                            "variable positions" => {-1}
│ │ │  
│ │ │  o5 : InterpretedSLProgram
│ │ │  
│ │ │  i6 : time A = evaluate(slp,matrix{{1}});
│ │ │ - -- used 0.000111089s (cpu); 0.000225292s (thread); 0s (gc)
│ │ │ + -- used 0.00117205s (cpu); 0.000228059s (thread); 0s (gc)
│ │ │  
│ │ │                1       1
│ │ │  o6 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i7 : ZZ[y];
│ │ │  
│ │ │  i8 : time B = sub((y+1)^(2^n),{y=>1})
│ │ │ - -- used 5.14794s (cpu); 3.65518s (thread); 0s (gc)
│ │ │ + -- used 4.27824s (cpu); 3.0255s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = 104438888141315250669175271071662438257996424904738378038423348328395390
│ │ │       797155745684882681193499755834089010671443926283798757343818579360726323
│ │ │       608785136527794595697654370999834036159013438371831442807001185594622637
│ │ │       631883939771274567233468434458661749680790870580370407128404874011860911
│ │ │       446797778359802900668693897688178778594690563019026094059957945343282346
│ │ │       930302669644305902501597239986771421554169383555988529148631823791443449
│ │ ├── ./usr/share/doc/Macaulay2/SLPexpressions/html/index.html
│ │ │ @@ -104,29 +104,29 @@
│ │ │  
│ │ │  o5 : InterpretedSLProgram
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time A = evaluate(slp,matrix{{1}});
│ │ │ - -- used 0.000111089s (cpu); 0.000225292s (thread); 0s (gc)
│ │ │ + -- used 0.00117205s (cpu); 0.000228059s (thread); 0s (gc)
│ │ │  
│ │ │                1       1
│ │ │  o6 : Matrix ZZ  <-- ZZ
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : ZZ[y];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : time B = sub((y+1)^(2^n),{y=>1})
│ │ │ - -- used 5.14794s (cpu); 3.65518s (thread); 0s (gc)
│ │ │ + -- used 4.27824s (cpu); 3.0255s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = 104438888141315250669175271071662438257996424904738378038423348328395390
│ │ │       797155745684882681193499755834089010671443926283798757343818579360726323
│ │ │       608785136527794595697654370999834036159013438371831442807001185594622637
│ │ │       631883939771274567233468434458661749680790870580370407128404874011860911
│ │ │       446797778359802900668693897688178778594690563019026094059957945343282346
│ │ │       930302669644305902501597239986771421554169383555988529148631823791443449
│ │ │ ├── html2text {}
│ │ │ │ @@ -38,21 +38,21 @@
│ │ │ │                                              output nodes: 1
│ │ │ │                                              )
│ │ │ │  
│ │ │ │                            "variable positions" => {-1}
│ │ │ │  
│ │ │ │  o5 : InterpretedSLProgram
│ │ │ │  i6 : time A = evaluate(slp,matrix{{1}});
│ │ │ │ - -- used 0.000111089s (cpu); 0.000225292s (thread); 0s (gc)
│ │ │ │ + -- used 0.00117205s (cpu); 0.000228059s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                1       1
│ │ │ │  o6 : Matrix ZZ  <-- ZZ
│ │ │ │  i7 : ZZ[y];
│ │ │ │  i8 : time B = sub((y+1)^(2^n),{y=>1})
│ │ │ │ - -- used 5.14794s (cpu); 3.65518s (thread); 0s (gc)
│ │ │ │ + -- used 4.27824s (cpu); 3.0255s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = 104438888141315250669175271071662438257996424904738378038423348328395390
│ │ │ │       797155745684882681193499755834089010671443926283798757343818579360726323
│ │ │ │       608785136527794595697654370999834036159013438371831442807001185594622637
│ │ │ │       631883939771274567233468434458661749680790870580370407128404874011860911
│ │ │ │       446797778359802900668693897688178778594690563019026094059957945343282346
│ │ │ │       930302669644305902501597239986771421554169383555988529148631823791443449
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Complex.out
│ │ │ @@ -15,15 +15,15 @@
│ │ │  i3 : r={5,11,3,2}
│ │ │  
│ │ │  o3 = {5, 11, 3, 2}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : elapsedTime C=randomChainComplex(h,r,Height=>4)
│ │ │ - -- .00628004s elapsed
│ │ │ + -- .00749109s elapsed
│ │ │  
│ │ │         6       19       19       7       3
│ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ  <-- ZZ
│ │ │                                          
│ │ │       0       1        2        3       4
│ │ │  
│ │ │  o4 : ChainComplex
│ │ │ @@ -51,15 +51,15 @@
│ │ │         53        53         53         53        53
│ │ │                                                  
│ │ │       -1        0          1          2         3
│ │ │  
│ │ │  o6 : ChainComplex
│ │ │  
│ │ │  i7 : elapsedTime (h,U)=SVDComplex CR;
│ │ │ - -- .00224378s elapsed
│ │ │ + -- .00294503s elapsed
│ │ │  
│ │ │  i8 : h
│ │ │  
│ │ │  o8 = HashTable{-1 => 1}
│ │ │                 0 => 3
│ │ │                 1 => 5
│ │ │                 2 => 2
│ │ │ @@ -95,15 +95,15 @@
│ │ │  i12 : maximalEntry chainComplex errors
│ │ │  
│ │ │  o12 = {8.43769e-15, 6.39488e-14, 1.06581e-13, 9.76996e-15}
│ │ │  
│ │ │  o12 : List
│ │ │  
│ │ │  i13 : elapsedTime (hL,U)=SVDComplex(CR,Strategy=>Laplacian);
│ │ │ - -- .00497913s elapsed
│ │ │ + -- .00611395s elapsed
│ │ │  
│ │ │  i14 : hL === h
│ │ │  
│ │ │  o14 = true
│ │ │  
│ │ │  i15 : SigmaL =source U;
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Homology.out
│ │ │ @@ -15,15 +15,15 @@
│ │ │  i3 : r={4,3,3}
│ │ │  
│ │ │  o3 = {4, 3, 3}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .00283463s elapsed
│ │ │ + -- .00300798s elapsed
│ │ │  
│ │ │         5       10       11       5
│ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o4 : ChainComplex
│ │ │ @@ -47,25 +47,25 @@
│ │ │         53        53         53         53
│ │ │                                        
│ │ │       0         1          2          3
│ │ │  
│ │ │  o6 : ChainComplex
│ │ │  
│ │ │  i7 : elapsedTime (h,h1)=SVDHomology CR
│ │ │ - -- .000620476s elapsed
│ │ │ + -- .000634316s elapsed
│ │ │  
│ │ │  o7 = (HashTable{0 => 1}, HashTable{1 => (7.87842, 1.31052, )           })
│ │ │                  1 => 3             2 => (37.9214, 30.3707, 1.61954e-14)
│ │ │                  2 => 5             3 => (14.972, 8.57847, 3.90646e-15)
│ │ │                  3 => 2
│ │ │  
│ │ │  o7 : Sequence
│ │ │  
│ │ │  i8 : elapsedTime (hL,hL1)=SVDHomology(CR,Strategy=>Laplacian)
│ │ │ - -- .00139661s elapsed
│ │ │ + -- .00146477s elapsed
│ │ │  
│ │ │  o8 = (HashTable{0 => 1}, HashTable{0 => (, 1.71747, -1.72291e-14)      })
│ │ │                  1 => 3             1 => (1.71747, 922.381, 2.51496e-13)
│ │ │                  2 => 5             2 => (922.381, 73.5901, 1.81323e-13)
│ │ │                  3 => 2             3 => (73.5901, , 2.82914e-13)
│ │ │  
│ │ │  o8 : Sequence
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_common__Entries.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │  i4 : r={4,3,5}
│ │ │  
│ │ │  o4 = {4, 3, 5}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : elapsedTime C=randomChainComplex(h,r,Height=>100,ZeroMean=>true)
│ │ │ - -- .00359076s elapsed
│ │ │ + -- .00400183s elapsed
│ │ │  
│ │ │         6       10       13       8
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : ChainComplex
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_euclidean__Distance.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │  i4 : r={4,3,3}
│ │ │  
│ │ │  o4 = {4, 3, 3}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .0027493s elapsed
│ │ │ + -- .00296269s elapsed
│ │ │  
│ │ │         6       10       11       5
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : ChainComplex
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_project__To__Complex.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │  i4 : r={4,3,3}
│ │ │  
│ │ │  o4 = {4, 3, 3}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .00303026s elapsed
│ │ │ + -- .00309145s elapsed
│ │ │  
│ │ │         6       10       11       5
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : ChainComplex
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Complex.html
│ │ │ @@ -105,15 +105,15 @@
│ │ │  
│ │ │  o3 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : elapsedTime C=randomChainComplex(h,r,Height=>4)
│ │ │ - -- .00628004s elapsed
│ │ │ + -- .00749109s elapsed
│ │ │  
│ │ │         6       19       19       7       3
│ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ  <-- ZZ
│ │ │                                          
│ │ │       0       1        2        3       4
│ │ │  
│ │ │  o4 : ChainComplex
    
│ │ │ @@ -150,15 +150,15 @@
│ │ │  
│ │ │  o6 : ChainComplex
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : elapsedTime (h,U)=SVDComplex CR;
│ │ │ - -- .00224378s elapsed
    
│ │ │ + -- .00294503s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : h
│ │ │  
│ │ │  o8 = HashTable{-1 => 1}
│ │ │ @@ -212,15 +212,15 @@
│ │ │  
│ │ │  o12 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : elapsedTime (hL,U)=SVDComplex(CR,Strategy=>Laplacian);
│ │ │ - -- .00497913s elapsed
    
│ │ │ + -- .00611395s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : hL === h
│ │ │  
│ │ │  o14 = true
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -37,15 +37,15 @@
│ │ │ │  o2 : List
│ │ │ │  i3 : r={5,11,3,2}
│ │ │ │  
│ │ │ │  o3 = {5, 11, 3, 2}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : elapsedTime C=randomChainComplex(h,r,Height=>4)
│ │ │ │ - -- .00628004s elapsed
│ │ │ │ + -- .00749109s elapsed
│ │ │ │  
│ │ │ │         6       19       19       7       3
│ │ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ  <-- ZZ
│ │ │ │  
│ │ │ │       0       1        2        3       4
│ │ │ │  
│ │ │ │  o4 : ChainComplex
│ │ │ │ @@ -70,15 +70,15 @@
│ │ │ │  o6 = RR    <-- RR     <-- RR     <-- RR    <-- RR
│ │ │ │         53        53         53         53        53
│ │ │ │  
│ │ │ │       -1        0          1          2         3
│ │ │ │  
│ │ │ │  o6 : ChainComplex
│ │ │ │  i7 : elapsedTime (h,U)=SVDComplex CR;
│ │ │ │ - -- .00224378s elapsed
│ │ │ │ + -- .00294503s elapsed
│ │ │ │  i8 : h
│ │ │ │  
│ │ │ │  o8 = HashTable{-1 => 1}
│ │ │ │                 0 => 3
│ │ │ │                 1 => 5
│ │ │ │                 2 => 2
│ │ │ │                 3 => 1
│ │ │ │ @@ -109,15 +109,15 @@
│ │ │ │  1)*Sigma.dd_ell*transpose U_ell);
│ │ │ │  i12 : maximalEntry chainComplex errors
│ │ │ │  
│ │ │ │  o12 = {8.43769e-15, 6.39488e-14, 1.06581e-13, 9.76996e-15}
│ │ │ │  
│ │ │ │  o12 : List
│ │ │ │  i13 : elapsedTime (hL,U)=SVDComplex(CR,Strategy=>Laplacian);
│ │ │ │ - -- .00497913s elapsed
│ │ │ │ + -- .00611395s elapsed
│ │ │ │  i14 : hL === h
│ │ │ │  
│ │ │ │  o14 = true
│ │ │ │  i15 : SigmaL =source U;
│ │ │ │  i16 : for i from min CR+1 to max CR list maximalEntry(SigmaL.dd_i -Sigma.dd_i)
│ │ │ │  
│ │ │ │  o16 = {1.77636e-14, 6.39488e-14, 8.52651e-14, 3.55271e-15}
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Homology.html
│ │ │ @@ -107,15 +107,15 @@
│ │ │  
│ │ │  o3 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .00283463s elapsed
│ │ │ + -- .00300798s elapsed
│ │ │  
│ │ │         5       10       11       5
│ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o4 : ChainComplex
    
│ │ │ @@ -148,28 +148,28 @@
│ │ │  
│ │ │  o6 : ChainComplex
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : elapsedTime (h,h1)=SVDHomology CR
│ │ │ - -- .000620476s elapsed
│ │ │ + -- .000634316s elapsed
│ │ │  
│ │ │  o7 = (HashTable{0 => 1}, HashTable{1 => (7.87842, 1.31052, )           })
│ │ │                  1 => 3             2 => (37.9214, 30.3707, 1.61954e-14)
│ │ │                  2 => 5             3 => (14.972, 8.57847, 3.90646e-15)
│ │ │                  3 => 2
│ │ │  
│ │ │  o7 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : elapsedTime (hL,hL1)=SVDHomology(CR,Strategy=>Laplacian)
│ │ │ - -- .00139661s elapsed
│ │ │ + -- .00146477s elapsed
│ │ │  
│ │ │  o8 = (HashTable{0 => 1}, HashTable{0 => (, 1.71747, -1.72291e-14)      })
│ │ │                  1 => 3             1 => (1.71747, 922.381, 2.51496e-13)
│ │ │                  2 => 5             2 => (922.381, 73.5901, 1.81323e-13)
│ │ │                  3 => 2             3 => (73.5901, , 2.82914e-13)
│ │ │  
│ │ │  o8 : Sequence
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │  o2 : List
│ │ │ │  i3 : r={4,3,3}
│ │ │ │  
│ │ │ │  o3 = {4, 3, 3}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ │ - -- .00283463s elapsed
│ │ │ │ + -- .00300798s elapsed
│ │ │ │  
│ │ │ │         5       10       11       5
│ │ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │ │  
│ │ │ │       0       1        2        3
│ │ │ │  
│ │ │ │  o4 : ChainComplex
│ │ │ │ @@ -69,24 +69,24 @@
│ │ │ │  o6 = RR    <-- RR     <-- RR     <-- RR
│ │ │ │         53        53         53         53
│ │ │ │  
│ │ │ │       0         1          2          3
│ │ │ │  
│ │ │ │  o6 : ChainComplex
│ │ │ │  i7 : elapsedTime (h,h1)=SVDHomology CR
│ │ │ │ - -- .000620476s elapsed
│ │ │ │ + -- .000634316s elapsed
│ │ │ │  
│ │ │ │  o7 = (HashTable{0 => 1}, HashTable{1 => (7.87842, 1.31052, )           })
│ │ │ │                  1 => 3             2 => (37.9214, 30.3707, 1.61954e-14)
│ │ │ │                  2 => 5             3 => (14.972, 8.57847, 3.90646e-15)
│ │ │ │                  3 => 2
│ │ │ │  
│ │ │ │  o7 : Sequence
│ │ │ │  i8 : elapsedTime (hL,hL1)=SVDHomology(CR,Strategy=>Laplacian)
│ │ │ │ - -- .00139661s elapsed
│ │ │ │ + -- .00146477s elapsed
│ │ │ │  
│ │ │ │  o8 = (HashTable{0 => 1}, HashTable{0 => (, 1.71747, -1.72291e-14)      })
│ │ │ │                  1 => 3             1 => (1.71747, 922.381, 2.51496e-13)
│ │ │ │                  2 => 5             2 => (922.381, 73.5901, 1.81323e-13)
│ │ │ │                  3 => 2             3 => (73.5901, , 2.82914e-13)
│ │ │ │  
│ │ │ │  o8 : Sequence
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/_common__Entries.html
│ │ │ @@ -110,15 +110,15 @@
│ │ │  
│ │ │  o4 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime C=randomChainComplex(h,r,Height=>100,ZeroMean=>true)
│ │ │ - -- .00359076s elapsed
│ │ │ + -- .00400183s elapsed
│ │ │  
│ │ │         6       10       13       8
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : ChainComplex
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │  o3 : List
│ │ │ │  i4 : r={4,3,5}
│ │ │ │  
│ │ │ │  o4 = {4, 3, 5}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : elapsedTime C=randomChainComplex(h,r,Height=>100,ZeroMean=>true)
│ │ │ │ - -- .00359076s elapsed
│ │ │ │ + -- .00400183s elapsed
│ │ │ │  
│ │ │ │         6       10       13       8
│ │ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │ │  
│ │ │ │       0       1        2        3
│ │ │ │  
│ │ │ │  o5 : ChainComplex
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/_euclidean__Distance.html
│ │ │ @@ -104,15 +104,15 @@
│ │ │  
│ │ │  o4 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .0027493s elapsed
│ │ │ + -- .00296269s elapsed
│ │ │  
│ │ │         6       10       11       5
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : ChainComplex
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  o3 : List
│ │ │ │  i4 : r={4,3,3}
│ │ │ │  
│ │ │ │  o4 = {4, 3, 3}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ │ - -- .0027493s elapsed
│ │ │ │ + -- .00296269s elapsed
│ │ │ │  
│ │ │ │         6       10       11       5
│ │ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │ │  
│ │ │ │       0       1        2        3
│ │ │ │  
│ │ │ │  o5 : ChainComplex
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/_project__To__Complex.html
│ │ │ @@ -104,15 +104,15 @@
│ │ │  
│ │ │  o4 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .00303026s elapsed
│ │ │ + -- .00309145s elapsed
│ │ │  
│ │ │         6       10       11       5
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : ChainComplex
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  o3 : List
│ │ │ │  i4 : r={4,3,3}
│ │ │ │  
│ │ │ │  o4 = {4, 3, 3}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ │ - -- .00303026s elapsed
│ │ │ │ + -- .00309145s 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 0.386864s (cpu); 0.31337s (thread); 0s (gc)
│ │ │ + -- used 0.375116s (cpu); 0.375118s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of S
│ │ │  
│ │ │  i8 : time quotient(I^3, J^2, Strategy => Quotient);
│ │ │ - -- used 0.49095s (cpu); 0.490915s (thread); 0s (gc)
│ │ │ + -- used 0.693251s (cpu); 0.611114s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of S
│ │ │  
│ │ │  i9 : S = ZZ/101[vars(0..4)];
│ │ │  
│ │ │  i10 : I = ideal vars S;
│ │ │  
│ │ │  o10 : Ideal of S
│ │ │  
│ │ │  i11 : time quotient(I^5, I^3, Strategy => Iterate);
│ │ │ - -- used 0.0268459s (cpu); 0.0268439s (thread); 0s (gc)
│ │ │ + -- used 0.0265589s (cpu); 0.0265634s (thread); 0s (gc)
│ │ │  
│ │ │  o11 : Ideal of S
│ │ │  
│ │ │  i12 : time quotient(I^5, I^3, Strategy => Quotient);
│ │ │ - -- used 0.00767377s (cpu); 0.00767479s (thread); 0s (gc)
│ │ │ + -- used 0.0089403s (cpu); 0.00894681s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of S
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp..._cm__Strategy_eq_gt..._rp.html
│ │ │ @@ -125,23 +125,23 @@
│ │ │  
│ │ │  o6 : Ideal of S
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time quotient(I^3, J^2, Strategy => Iterate);
│ │ │ - -- used 0.386864s (cpu); 0.31337s (thread); 0s (gc)
│ │ │ + -- used 0.375116s (cpu); 0.375118s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of S
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : time quotient(I^3, J^2, Strategy => Quotient);
│ │ │ - -- used 0.49095s (cpu); 0.490915s (thread); 0s (gc)
│ │ │ + -- used 0.693251s (cpu); 0.611114s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of S
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    Strategy => Quotient is faster in other cases:
    
│ │ │ @@ -158,23 +158,23 @@
│ │ │  
│ │ │  o10 : Ideal of S
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : time quotient(I^5, I^3, Strategy => Iterate);
│ │ │ - -- used 0.0268459s (cpu); 0.0268439s (thread); 0s (gc)
│ │ │ + -- used 0.0265589s (cpu); 0.0265634s (thread); 0s (gc)
│ │ │  
│ │ │  o11 : Ideal of S
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : time quotient(I^5, I^3, Strategy => Quotient);
│ │ │ - -- used 0.00767377s (cpu); 0.00767479s (thread); 0s (gc)
│ │ │ + -- used 0.0089403s (cpu); 0.00894681s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of S
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -56,32 +56,32 @@
│ │ │ │  i5 : I = monomialCurveIdeal(S, 1..n-1);
│ │ │ │  
│ │ │ │  o5 : Ideal of S
│ │ │ │  i6 : J = ideal(map(S^1, S^n, (p, q) -> S_q^5));
│ │ │ │  
│ │ │ │  o6 : Ideal of S
│ │ │ │  i7 : time quotient(I^3, J^2, Strategy => Iterate);
│ │ │ │ - -- used 0.386864s (cpu); 0.31337s (thread); 0s (gc)
│ │ │ │ + -- used 0.375116s (cpu); 0.375118s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : Ideal of S
│ │ │ │  i8 : time quotient(I^3, J^2, Strategy => Quotient);
│ │ │ │ - -- used 0.49095s (cpu); 0.490915s (thread); 0s (gc)
│ │ │ │ + -- used 0.693251s (cpu); 0.611114s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 : Ideal of S
│ │ │ │  Strategy => Quotient is faster in other cases:
│ │ │ │  i9 : S = ZZ/101[vars(0..4)];
│ │ │ │  i10 : I = ideal vars S;
│ │ │ │  
│ │ │ │  o10 : Ideal of S
│ │ │ │  i11 : time quotient(I^5, I^3, Strategy => Iterate);
│ │ │ │ - -- used 0.0268459s (cpu); 0.0268439s (thread); 0s (gc)
│ │ │ │ + -- used 0.0265589s (cpu); 0.0265634s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 : Ideal of S
│ │ │ │  i12 : time quotient(I^5, I^3, Strategy => Quotient);
│ │ │ │ - -- used 0.00767377s (cpu); 0.00767479s (thread); 0s (gc)
│ │ │ │ + -- used 0.0089403s (cpu); 0.00894681s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o12 : Ideal of S
│ │ │ │  ********** RReeffeerreenncceess **********
│ │ │ │  For further information see for example Exercise 15.41 in Eisenbud's
│ │ │ │  Commutative Algebra with a View Towards Algebraic Geometry.
│ │ │ │  ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd SSttrraatteeggyy:: **********
│ │ │ │      * addHook(...,Strategy=>...) -- see _a_d_d_H_o_o_k -- add a hook function to an
│ │ ├── ./usr/share/doc/Macaulay2/Schubert2/example-output/___Lines_spon_sphypersurfaces.out
│ │ │ @@ -40,23 +40,23 @@
│ │ │            )
│ │ │  
│ │ │  o6 = f
│ │ │  
│ │ │  o6 : FunctionClosure
│ │ │  
│ │ │  i7 : for n from 2 to 10 list time f n
│ │ │ - -- used 0.00503697s (cpu); 0.00503317s (thread); 0s (gc)
│ │ │ - -- used 0.00641821s (cpu); 0.0064191s (thread); 0s (gc)
│ │ │ - -- used 0.00958149s (cpu); 0.00958239s (thread); 0s (gc)
│ │ │ - -- used 0.016275s (cpu); 0.0162761s (thread); 0s (gc)
│ │ │ - -- used 0.03087s (cpu); 0.0308718s (thread); 0s (gc)
│ │ │ - -- used 0.0536082s (cpu); 0.0536137s (thread); 0s (gc)
│ │ │ - -- used 0.0913388s (cpu); 0.0913434s (thread); 0s (gc)
│ │ │ - -- used 0.278118s (cpu); 0.17744s (thread); 0s (gc)
│ │ │ - -- used 0.223118s (cpu); 0.223096s (thread); 0s (gc)
│ │ │ + -- used 0.00640037s (cpu); 0.0063991s (thread); 0s (gc)
│ │ │ + -- used 0.00766984s (cpu); 0.00767694s (thread); 0s (gc)
│ │ │ + -- used 0.0104622s (cpu); 0.0104677s (thread); 0s (gc)
│ │ │ + -- used 0.0175219s (cpu); 0.0175313s (thread); 0s (gc)
│ │ │ + -- used 0.0437386s (cpu); 0.0437534s (thread); 0s (gc)
│ │ │ + -- used 0.0600615s (cpu); 0.0600733s (thread); 0s (gc)
│ │ │ + -- used 0.103727s (cpu); 0.103737s (thread); 0s (gc)
│ │ │ + -- used 0.167023s (cpu); 0.167033s (thread); 0s (gc)
│ │ │ + -- used 0.416645s (cpu); 0.303049s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775,
│ │ │       ------------------------------------------------------------------------
│ │ │       289139638632755625, 520764738758073845321}
│ │ │  
│ │ │  o7 : List
│ │ ├── ./usr/share/doc/Macaulay2/Schubert2/html/___Lines_spon_sphypersurfaces.html
│ │ │ @@ -126,23 +126,23 @@
│ │ │  
│ │ │  o6 : FunctionClosure
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : for n from 2 to 10 list time f n
│ │ │ - -- used 0.00503697s (cpu); 0.00503317s (thread); 0s (gc)
│ │ │ - -- used 0.00641821s (cpu); 0.0064191s (thread); 0s (gc)
│ │ │ - -- used 0.00958149s (cpu); 0.00958239s (thread); 0s (gc)
│ │ │ - -- used 0.016275s (cpu); 0.0162761s (thread); 0s (gc)
│ │ │ - -- used 0.03087s (cpu); 0.0308718s (thread); 0s (gc)
│ │ │ - -- used 0.0536082s (cpu); 0.0536137s (thread); 0s (gc)
│ │ │ - -- used 0.0913388s (cpu); 0.0913434s (thread); 0s (gc)
│ │ │ - -- used 0.278118s (cpu); 0.17744s (thread); 0s (gc)
│ │ │ - -- used 0.223118s (cpu); 0.223096s (thread); 0s (gc)
│ │ │ + -- used 0.00640037s (cpu); 0.0063991s (thread); 0s (gc)
│ │ │ + -- used 0.00766984s (cpu); 0.00767694s (thread); 0s (gc)
│ │ │ + -- used 0.0104622s (cpu); 0.0104677s (thread); 0s (gc)
│ │ │ + -- used 0.0175219s (cpu); 0.0175313s (thread); 0s (gc)
│ │ │ + -- used 0.0437386s (cpu); 0.0437534s (thread); 0s (gc)
│ │ │ + -- used 0.0600615s (cpu); 0.0600733s (thread); 0s (gc)
│ │ │ + -- used 0.103727s (cpu); 0.103737s (thread); 0s (gc)
│ │ │ + -- used 0.167023s (cpu); 0.167033s (thread); 0s (gc)
│ │ │ + -- used 0.416645s (cpu); 0.303049s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775,
│ │ │       ------------------------------------------------------------------------
│ │ │       289139638632755625, 520764738758073845321}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -56,23 +56,23 @@
│ │ │ │            integral chern symmetricPower_(2*n-3) last bundles G
│ │ │ │            )
│ │ │ │  
│ │ │ │  o6 = f
│ │ │ │  
│ │ │ │  o6 : FunctionClosure
│ │ │ │  i7 : for n from 2 to 10 list time f n
│ │ │ │ - -- used 0.00503697s (cpu); 0.00503317s (thread); 0s (gc)
│ │ │ │ - -- used 0.00641821s (cpu); 0.0064191s (thread); 0s (gc)
│ │ │ │ - -- used 0.00958149s (cpu); 0.00958239s (thread); 0s (gc)
│ │ │ │ - -- used 0.016275s (cpu); 0.0162761s (thread); 0s (gc)
│ │ │ │ - -- used 0.03087s (cpu); 0.0308718s (thread); 0s (gc)
│ │ │ │ - -- used 0.0536082s (cpu); 0.0536137s (thread); 0s (gc)
│ │ │ │ - -- used 0.0913388s (cpu); 0.0913434s (thread); 0s (gc)
│ │ │ │ - -- used 0.278118s (cpu); 0.17744s (thread); 0s (gc)
│ │ │ │ - -- used 0.223118s (cpu); 0.223096s (thread); 0s (gc)
│ │ │ │ + -- used 0.00640037s (cpu); 0.0063991s (thread); 0s (gc)
│ │ │ │ + -- used 0.00766984s (cpu); 0.00767694s (thread); 0s (gc)
│ │ │ │ + -- used 0.0104622s (cpu); 0.0104677s (thread); 0s (gc)
│ │ │ │ + -- used 0.0175219s (cpu); 0.0175313s (thread); 0s (gc)
│ │ │ │ + -- used 0.0437386s (cpu); 0.0437534s (thread); 0s (gc)
│ │ │ │ + -- used 0.0600615s (cpu); 0.0600733s (thread); 0s (gc)
│ │ │ │ + -- used 0.103727s (cpu); 0.103737s (thread); 0s (gc)
│ │ │ │ + -- used 0.167023s (cpu); 0.167033s (thread); 0s (gc)
│ │ │ │ + -- used 0.416645s (cpu); 0.303049s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       289139638632755625, 520764738758073845321}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  Note: in characteristic zero, using Bertini's theorem, the numbers computed can
│ │ ├── ./usr/share/doc/Macaulay2/SchurFunctors/dump/rawdocumentation.dump
│ │ │ @@ -1,8 +1,8 @@
│ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sun Dec 14 14:09:54 2025
│ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sun Dec 14 14:09:53 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │  #:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  c3BsaXRDaGFyYWN0ZXI=
│ │ ├── ./usr/share/doc/Macaulay2/SegreClasses/example-output/_is__Component__Contained.out
│ │ │ @@ -53,15 +53,15 @@
│ │ │  o9 : Ideal of R
│ │ │  
│ │ │  i10 : X=((W)*ideal(y)+ideal(f));
│ │ │  
│ │ │  o10 : Ideal of R
│ │ │  
│ │ │  i11 : time isComponentContained(X,Y)
│ │ │ - -- used 4.58407s (cpu); 3.45047s (thread); 0s (gc)
│ │ │ + -- used 7.45695s (cpu); 3.86118s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = true
│ │ │  
│ │ │  i12 : print "we could confirm this with the computation:"
│ │ │  we could confirm this with the computation:
│ │ │  
│ │ │  i13 : B=ideal(x)*ideal(y)*ideal(z)
│ │ │ @@ -71,12 +71,12 @@
│ │ │        b*d*g, b*d*h, b*d*i, b*e*g, b*e*h, b*e*i, b*f*g, b*f*h, b*f*i, c*d*g,
│ │ │        -----------------------------------------------------------------------
│ │ │        c*d*h, c*d*i, c*e*g, c*e*h, c*e*i, c*f*g, c*f*h, c*f*i)
│ │ │  
│ │ │  o13 : Ideal of R
│ │ │  
│ │ │  i14 : time isSubset(saturate(Y,B),saturate(X,B))
│ │ │ - -- used 50.6991s (cpu); 46.8604s (thread); 0s (gc)
│ │ │ + -- used 61.3289s (cpu); 55.8154s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = true
│ │ │  
│ │ │  i15 :
│ │ ├── ./usr/share/doc/Macaulay2/SegreClasses/example-output/_segre__Dim__X.out
│ │ │ @@ -23,24 +23,24 @@
│ │ │  i5 : A = makeChowRing(R)
│ │ │  
│ │ │  o5 = A
│ │ │  
│ │ │  o5 : QuotientRing
│ │ │  
│ │ │  i6 : time s = segreDimX(X,Y,A)
│ │ │ - -- used 0.374749s (cpu); 0.242963s (thread); 0s (gc)
│ │ │ + -- used 0.539407s (cpu); 0.185342s (thread); 0s (gc)
│ │ │  
│ │ │         2             2
│ │ │  o6 = 2H  + 4H H  + 2H
│ │ │         1     1 2     2
│ │ │  
│ │ │  o6 : A
│ │ │  
│ │ │  i7 : time segre(X,Y,A)
│ │ │ - -- used 0.16565s (cpu); 0.105524s (thread); 0s (gc)
│ │ │ + -- used 0.300199s (cpu); 0.127528s (thread); 0s (gc)
│ │ │  
│ │ │          2 2     2         2     2             2
│ │ │  o7 = 12H H  - 6H H  - 6H H  + 2H  + 4H H  + 2H
│ │ │          1 2     1 2     1 2     1     1 2     2
│ │ │  
│ │ │  o7 : A
│ │ ├── ./usr/share/doc/Macaulay2/SegreClasses/html/_is__Component__Contained.html
│ │ │ @@ -162,15 +162,15 @@
│ │ │  
│ │ │  o10 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : time isComponentContained(X,Y)
│ │ │ - -- used 4.58407s (cpu); 3.45047s (thread); 0s (gc)
│ │ │ + -- used 7.45695s (cpu); 3.86118s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : print "we could confirm this with the computation:"
│ │ │ @@ -189,15 +189,15 @@
│ │ │  
│ │ │  o13 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : time isSubset(saturate(Y,B),saturate(X,B))
│ │ │ - -- used 50.6991s (cpu); 46.8604s (thread); 0s (gc)
│ │ │ + -- used 61.3289s (cpu); 55.8154s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -68,30 +68,30 @@
│ │ │ │  i9 : Y=ideal (z_0*W_0-z_1*W_1)+ideal(f);
│ │ │ │  
│ │ │ │  o9 : Ideal of R
│ │ │ │  i10 : X=((W)*ideal(y)+ideal(f));
│ │ │ │  
│ │ │ │  o10 : Ideal of R
│ │ │ │  i11 : time isComponentContained(X,Y)
│ │ │ │ - -- used 4.58407s (cpu); 3.45047s (thread); 0s (gc)
│ │ │ │ + -- used 7.45695s (cpu); 3.86118s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = true
│ │ │ │  i12 : print "we could confirm this with the computation:"
│ │ │ │  we could confirm this with the computation:
│ │ │ │  i13 : B=ideal(x)*ideal(y)*ideal(z)
│ │ │ │  
│ │ │ │  o13 = ideal (a*d*g, a*d*h, a*d*i, a*e*g, a*e*h, a*e*i, a*f*g, a*f*h, a*f*i,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        b*d*g, b*d*h, b*d*i, b*e*g, b*e*h, b*e*i, b*f*g, b*f*h, b*f*i, c*d*g,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        c*d*h, c*d*i, c*e*g, c*e*h, c*e*i, c*f*g, c*f*h, c*f*i)
│ │ │ │  
│ │ │ │  o13 : Ideal of R
│ │ │ │  i14 : time isSubset(saturate(Y,B),saturate(X,B))
│ │ │ │ - -- used 50.6991s (cpu); 46.8604s (thread); 0s (gc)
│ │ │ │ + -- used 61.3289s (cpu); 55.8154s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 = true
│ │ │ │  ********** WWaayyss ttoo uussee iissCCoommppoonneennttCCoonnttaaiinneedd:: **********
│ │ │ │      * isComponentContained(Ideal,Ideal)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _i_s_C_o_m_p_o_n_e_n_t_C_o_n_t_a_i_n_e_d is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s.
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/SegreClasses/html/_segre__Dim__X.html
│ │ │ @@ -118,27 +118,27 @@
│ │ │  
│ │ │  o5 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time s = segreDimX(X,Y,A)
│ │ │ - -- used 0.374749s (cpu); 0.242963s (thread); 0s (gc)
│ │ │ + -- used 0.539407s (cpu); 0.185342s (thread); 0s (gc)
│ │ │  
│ │ │         2             2
│ │ │  o6 = 2H  + 4H H  + 2H
│ │ │         1     1 2     2
│ │ │  
│ │ │  o6 : A
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time segre(X,Y,A)
│ │ │ - -- used 0.16565s (cpu); 0.105524s (thread); 0s (gc)
│ │ │ + -- used 0.300199s (cpu); 0.127528s (thread); 0s (gc)
│ │ │  
│ │ │          2 2     2         2     2             2
│ │ │  o7 = 12H H  - 6H H  - 6H H  + 2H  + 4H H  + 2H
│ │ │          1 2     1 2     1 2     1     1 2     2
│ │ │  
│ │ │  o7 : A
    
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,23 +48,23 @@
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : A = makeChowRing(R)
│ │ │ │  
│ │ │ │  o5 = A
│ │ │ │  
│ │ │ │  o5 : QuotientRing
│ │ │ │  i6 : time s = segreDimX(X,Y,A)
│ │ │ │ - -- used 0.374749s (cpu); 0.242963s (thread); 0s (gc)
│ │ │ │ + -- used 0.539407s (cpu); 0.185342s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         2             2
│ │ │ │  o6 = 2H  + 4H H  + 2H
│ │ │ │         1     1 2     2
│ │ │ │  
│ │ │ │  o6 : A
│ │ │ │  i7 : time segre(X,Y,A)
│ │ │ │ - -- used 0.16565s (cpu); 0.105524s (thread); 0s (gc)
│ │ │ │ + -- used 0.300199s (cpu); 0.127528s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          2 2     2         2     2             2
│ │ │ │  o7 = 12H H  - 6H H  - 6H H  + 2H  + 4H H  + 2H
│ │ │ │          1 2     1 2     1 2     1     1 2     2
│ │ │ │  
│ │ │ │  o7 : A
│ │ │ │  ********** WWaayyss ttoo uussee sseeggrreeDDiimmXX:: **********
│ │ ├── ./usr/share/doc/Macaulay2/SimpleDoc/example-output/_test__Example.out
│ │ │ @@ -1,6 +1,6 @@
│ │ │  -- -*- M2-comint -*- hash: 1331702921222
│ │ │  
│ │ │  i1 : check SimpleDoc
│ │ │ - -- capturing check(0, "SimpleDoc")           -- .226545s elapsed
│ │ │ + -- capturing check(0, "SimpleDoc")           -- .145078s elapsed
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/SimpleDoc/html/_test__Example.html
│ │ │ @@ -74,15 +74,15 @@
│ │ │          
    
│ │ │            
    The check method executes all package tests defined this way.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : check SimpleDoc
│ │ │ - -- capturing check(0, "SimpleDoc")           -- .226545s elapsed
    
│ │ │ + -- capturing check(0, "SimpleDoc")           -- .145078s elapsed
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,15 +10,15 @@
│ │ │ │  The variable testExample is a _S_t_r_i_n_g containing an example of the use of _T_E_S_T
│ │ │ │  to write a test case.
│ │ │ │  TEST /// -* test for simpleDocFrob *-
│ │ │ │    assert(simpleDocFrob(2,matrix{{1,2}}) == matrix{{1,2,0,0},{0,0,1,2}})
│ │ │ │  ///
│ │ │ │  The _c_h_e_c_k method executes all package tests defined this way.
│ │ │ │  i1 : check SimpleDoc
│ │ │ │ - -- capturing check(0, "SimpleDoc")           -- .226545s elapsed
│ │ │ │ + -- capturing check(0, "SimpleDoc")           -- .145078s elapsed
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _T_E_S_T -- add a test for a package
│ │ │ │      * _c_h_e_c_k -- perform tests of a package
│ │ │ │      * _p_a_c_k_a_g_e_T_e_m_p_l_a_t_e -- a template for a package
│ │ │ │      * _d_o_c_E_x_a_m_p_l_e -- an example of a documentation string
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _t_e_s_t_E_x_a_m_p_l_e is a _s_t_r_i_n_g.
│ │ ├── ./usr/share/doc/Macaulay2/SlackIdeals/example-output/_rehomogenize__Polynomial.out
│ │ │ @@ -9,14 +9,14 @@
│ │ │  
│ │ │  i3 : (Y, T) = setOnesForest X;
│ │ │  
│ │ │  i4 : remVars := flatten entries Y - set{0_(ring Y), 1_(ring Y)};
│ │ │  
│ │ │  i5 : h = rehomogenizePolynomial(X, Y, T, remVars_0^2+remVars_0*remVars_1-1)
│ │ │  
│ │ │ -        2 2 2 2          2 2 2 2          2 2
│ │ │ -o5 = - x x x x x  x   + x x x x x  x   + x x x x x x x x
│ │ │ -        1 4 6 7 10 11    2 3 5 8 10 11    2 3 5 6 7 8 9 12
│ │ │ +      2 2 2 2          2 2 2 2                  2 2
│ │ │ +o5 = x x x x x  x   - x x x x x  x   + x x x x x x x x
│ │ │ +      1 4 6 7 10 11    2 3 5 8 10 11    1 2 3 4 6 7 9 12
│ │ │  
│ │ │  o5 : R
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/SlackIdeals/html/_rehomogenize__Polynomial.html
│ │ │ @@ -99,17 +99,17 @@
│ │ │                
      i4 : remVars := flatten entries Y - set{0_(ring Y), 1_(ring Y)};
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i5 : h = rehomogenizePolynomial(X, Y, T, remVars_0^2+remVars_0*remVars_1-1)
│ │ │  
│ │ │ -        2 2 2 2          2 2 2 2          2 2
│ │ │ -o5 = - x x x x x  x   + x x x x x  x   + x x x x x x x x
│ │ │ -        1 4 6 7 10 11    2 3 5 8 10 11    2 3 5 6 7 8 9 12
│ │ │ +      2 2 2 2          2 2 2 2                  2 2
│ │ │ +o5 = x x x x x  x   - x x x x x  x   + x x x x x x x x
│ │ │ +      1 4 6 7 10 11    2 3 5 8 10 11    1 2 3 4 6 7 9 12
│ │ │  
│ │ │  o5 : R
      
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -31,17 +31,17 @@
│ │ │ │  
│ │ │ │               6      5
│ │ │ │  o2 : Matrix R  <-- R
│ │ │ │  i3 : (Y, T) = setOnesForest X;
│ │ │ │  i4 : remVars := flatten entries Y - set{0_(ring Y), 1_(ring Y)};
│ │ │ │  i5 : h = rehomogenizePolynomial(X, Y, T, remVars_0^2+remVars_0*remVars_1-1)
│ │ │ │  
│ │ │ │ -        2 2 2 2          2 2 2 2          2 2
│ │ │ │ -o5 = - x x x x x  x   + x x x x x  x   + x x x x x x x x
│ │ │ │ -        1 4 6 7 10 11    2 3 5 8 10 11    2 3 5 6 7 8 9 12
│ │ │ │ +      2 2 2 2          2 2 2 2                  2 2
│ │ │ │ +o5 = x x x x x  x   - x x x x x  x   + x x x x x x x x
│ │ │ │ +      1 4 6 7 10 11    2 3 5 8 10 11    1 2 3 4 6 7 9 12
│ │ │ │  
│ │ │ │  o5 : R
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _s_e_t_O_n_e_s_F_o_r_e_s_t -- sets to 1 variables in a symbolic slack matrix which
│ │ │ │        corresponding to edges of a spanning forest
│ │ │ │      * _s_l_a_c_k_I_d_e_a_l -- computes the slack ideal
│ │ │ │      * _s_y_m_b_o_l_i_c_S_l_a_c_k_M_a_t_r_i_x -- computes the symbolic slack matrix
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_degree__Determinant.out
│ │ │ @@ -3,15 +3,15 @@
│ │ │  i1 : n = {2,3,2}
│ │ │  
│ │ │  o1 = {2, 3, 2}
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : time degreeDeterminant n
│ │ │ - -- used 0.000112722s (cpu); 0.000101711s (thread); 0s (gc)
│ │ │ + -- used 7.8019e-05s (cpu); 7.0604e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 6
│ │ │  
│ │ │  i3 : M = genericMultidimensionalMatrix n;
│ │ │  warning: clearing value of symbol x2 to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 1368   or with command line option   --debug 1368
│ │ │  warning: clearing value of symbol x1 to allow access to subscripted variables based on it
│ │ │ @@ -19,14 +19,14 @@
│ │ │  warning: clearing value of symbol x0 to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 6010   or with command line option   --debug 6010
│ │ │  
│ │ │  o3 : 3-dimensional matrix of shape 2 x 3 x 2 over ZZ[a     ..a     ]
│ │ │                                                        0,0,0   1,2,1
│ │ │  
│ │ │  i4 : time degree determinant M
│ │ │ - -- used 0.0350456s (cpu); 0.0342305s (thread); 0s (gc)
│ │ │ + -- used 0.16978s (cpu); 0.0529805s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = {6}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dense__Discriminant.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 17130321902108223178
│ │ │  
│ │ │  i1 : (d,n) := (2,3);
│ │ │  
│ │ │  i2 : time Disc = denseDiscriminant(d,n)
│ │ │ - -- used 0.408829s (cpu); 0.223024s (thread); 0s (gc)
│ │ │ + -- used 0.430955s (cpu); 0.241127s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = Disc
│ │ │  
│ │ │  o2 : SparseDiscriminant (sparse discriminant associated to | 0 0 0 0 0 0 1 1 1 2 |)
│ │ │                                                             | 0 0 0 1 1 2 0 0 1 0 |
│ │ │                                                             | 0 1 2 0 1 0 0 1 0 0 |
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dense__Resultant.out
│ │ │ @@ -9,18 +9,18 @@
│ │ │                   2
│ │ │       c x x  + c x  + c x  + c x  + c )
│ │ │        4 1 2    2 2    3 1    1 2    0
│ │ │  
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : time denseResultant(f0,f1,f2); -- using Poisson formula
│ │ │ - -- used 0.163836s (cpu); 0.123645s (thread); 0s (gc)
│ │ │ + -- used 0.209212s (cpu); 0.151545s (thread); 0s (gc)
│ │ │  
│ │ │  i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay formula
│ │ │ - -- used 0.292245s (cpu); 0.237931s (thread); 0s (gc)
│ │ │ + -- used 0.344805s (cpu); 0.28602s (thread); 0s (gc)
│ │ │  
│ │ │  i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant
│ │ │ - -- used 0.365133s (cpu); 0.330814s (thread); 0s (gc)
│ │ │ + -- used 0.328512s (cpu); 0.29171s (thread); 0s (gc)
│ │ │  
│ │ │  i5 : assert(o2 == o3 and o3 == o4)
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_determinant_lp__Multidimensional__Matrix_rp.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  o1 = {{{{8, 1}, {3, 7}}, {{8, 3}, {3, 7}}}, {{{8, 8}, {5, 7}}, {{8, 5}, {2,
│ │ │       ------------------------------------------------------------------------
│ │ │       3}}}}
│ │ │  
│ │ │  o1 : 4-dimensional matrix of shape 2 x 2 x 2 x 2 over ZZ
│ │ │  
│ │ │  i2 : time det M
│ │ │ - -- used 0.0870547s (cpu); 0.0854223s (thread); 0s (gc)
│ │ │ + -- used 0.148646s (cpu); 0.10084s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 9698337990421512192
│ │ │  
│ │ │  i3 : M = randomMultidimensionalMatrix(2,2,2,2,5)
│ │ │  
│ │ │  o3 = {{{{{6, 3, 6, 8, 6}, {9, 3, 7, 6, 9}}, {{6, 2, 6, 0, 2}, {6, 9, 3, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -24,13 +24,13 @@
│ │ │       7, 4, 5}}}, {{{4, 0, 1, 4, 4}, {2, 6, 1, 1, 4}}, {{5, 4, 9, 7, 4}, {6,
│ │ │       ------------------------------------------------------------------------
│ │ │       4, 8, 4, 2}}}}}
│ │ │  
│ │ │  o3 : 5-dimensional matrix of shape 2 x 2 x 2 x 2 x 5 over ZZ
│ │ │  
│ │ │  i4 : time det M
│ │ │ - -- used 0.514606s (cpu); 0.440355s (thread); 0s (gc)
│ │ │ + -- used 0.46196s (cpu); 0.461782s (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 2.62265s (cpu); 2.22282s (thread); 0s (gc)
│ │ │ + -- used 2.76719s (cpu); 2.48392s (thread); 0s (gc)
│ │ │  
│ │ │                                                     2                        
│ │ │  o2 = a     a     a     a     a     a      - a     a     a     a     a      -
│ │ │        0,1,1 0,2,0 0,2,1 1,0,0 1,0,1 1,1,0    0,1,0 0,2,1 1,0,0 1,0,1 1,1,0  
│ │ │       ------------------------------------------------------------------------
│ │ │              2     2                                2            
│ │ │       a     a     a     a      + a     a     a     a     a      -
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_sparse__Resultant.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 16228363821945730064
│ │ │  
│ │ │  i1 : time Res = sparseResultant(matrix{{0,1,1,2},{0,0,1,1}},matrix{{0,1,2,2},{1,0,1,2}},matrix{{0,0,1,1},{0,1,0,1}})
│ │ │ - -- used 0.507781s (cpu); 0.44829s (thread); 0s (gc)
│ │ │ + -- used 0.437655s (cpu); 0.417926s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = Res
│ │ │  
│ │ │  o1 : SparseResultant (sparse mixed resultant associated to {| 0 1 1 2 |, | 0 1 2 2 |, | 0 0 1 1 |})
│ │ │                                                              | 0 0 1 1 |  | 1 0 1 2 |  | 0 1 0 1 |
│ │ │  
│ │ │  i2 : QQ[c_(1,1)..c_(3,4)][x,y];
│ │ │ @@ -18,15 +18,15 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       c   x*y + c   x + c   y + c   )
│ │ │        3,3       3,4     3,2     3,1
│ │ │  
│ │ │  o3 : Sequence
│ │ │  
│ │ │  i4 : time Res(f,g,h)
│ │ │ - -- used 0.00905261s (cpu); 0.00905263s (thread); 0s (gc)
│ │ │ + -- used 0.0105012s (cpu); 0.0105005s (thread); 0s (gc)
│ │ │  
│ │ │          2                       4      2   2               4    
│ │ │  o4 = - c   c   c   c   c   c   c    + c   c   c   c   c   c    +
│ │ │          1,2 1,3 1,4 2,1 2,2 2,3 3,1    1,2 1,3 2,1 2,2 2,4 3,1  
│ │ │       ------------------------------------------------------------------------
│ │ │        3       2       3               2                   3        
│ │ │       c   c   c   c   c   c    - 2c   c   c   c   c   c   c   c    +
│ │ │ @@ -730,30 +730,30 @@
│ │ │  
│ │ │  o4 : QQ[c   ..c   ]
│ │ │           1,1   3,4
│ │ │  
│ │ │  i5 : assert(Res(f,g,h) == sparseResultant(f,g,h))
│ │ │  
│ │ │  i6 : time Res = sparseResultant(matrix{{0,0,1,1},{0,1,0,1}},CoefficientRing=>ZZ/3331);
│ │ │ - -- used 0.0304897s (cpu); 0.0292273s (thread); 0s (gc)
│ │ │ + -- used 0.174291s (cpu); 0.0527137s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : SparseResultant (sparse unmixed resultant associated to | 0 0 1 1 | over ZZ/3331)
│ │ │                                                               | 0 1 0 1 |
│ │ │  
│ │ │  i7 : ZZ/3331[a_0..a_3,b_0..b_3,c_0..c_3][x,y];
│ │ │  
│ │ │  i8 : (f,g,h) = (a_0 + a_1*x + a_2*y + a_3*x*y, b_0 + b_1*x + b_2*y + b_3*x*y, c_0 + c_1*x + c_2*y + c_3*x*y)
│ │ │  
│ │ │  o8 = (a x*y + a x + a y + a , b x*y + b x + b y + b , c x*y + c x + c y + c )
│ │ │         3       1     2     0   3       1     2     0   3       1     2     0
│ │ │  
│ │ │  o8 : Sequence
│ │ │  
│ │ │  i9 : time Res(f,g,h)
│ │ │ - -- used 0.00316112s (cpu); 0.00316119s (thread); 0s (gc)
│ │ │ + -- used 0.00426482s (cpu); 0.00426298s (thread); 0s (gc)
│ │ │  
│ │ │        2     2            2            2        2 2    2          
│ │ │  o9 = a b b c  - a a b b c  - a a b b c  + a a b c  - a b b c c  -
│ │ │        3 1 2 0    2 3 1 3 0    1 3 2 3 0    1 2 3 0    3 0 2 0 1  
│ │ │       ------------------------------------------------------------------------
│ │ │                           2                       2                         
│ │ │       a a b b c c  + a a b c c  + a a b b c c  + a b b c c  - a a b b c c  +
│ │ │ @@ -822,15 +822,15 @@
│ │ │                    2
│ │ │        c x x  + c x  + c x  + c x  + c )
│ │ │         4 1 2    2 2    3 1    1 2    0
│ │ │  
│ │ │  o11 : Sequence
│ │ │  
│ │ │  i12 : time (MixedRes,UnmixedRes) = (sparseResultant(f,g,h),sparseResultant(f,g,h,Unmixed=>true));
│ │ │ - -- used 0.260966s (cpu); 0.209622s (thread); 0s (gc)
│ │ │ + -- used 0.386656s (cpu); 0.211447s (thread); 0s (gc)
│ │ │  
│ │ │  i13 : quotientRemainder(UnmixedRes,MixedRes)
│ │ │  
│ │ │          2 2                   2    2                               2 2
│ │ │  o13 = (b c  - b b c c  + b b c  + b c c  - 2b b c c  - b b c c  + b c , 0)
│ │ │          5 2    4 5 2 4    2 5 4    4 2 5     2 5 2 5    2 4 4 5    2 5
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_degree__Determinant.html
│ │ │ @@ -76,15 +76,15 @@
│ │ │  
│ │ │  o1 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time degreeDeterminant n
│ │ │ - -- used 0.000112722s (cpu); 0.000101711s (thread); 0s (gc)
│ │ │ + -- used 7.8019e-05s (cpu); 7.0604e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 6
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : M = genericMultidimensionalMatrix n;
│ │ │ @@ -98,15 +98,15 @@
│ │ │  o3 : 3-dimensional matrix of shape 2 x 3 x 2 over ZZ[a     ..a     ]
│ │ │                                                        0,0,0   1,2,1
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time degree determinant M
│ │ │ - -- used 0.0350456s (cpu); 0.0342305s (thread); 0s (gc)
│ │ │ + -- used 0.16978s (cpu); 0.0529805s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = {6}
│ │ │  
│ │ │  o4 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : n = {2,3,2}
│ │ │ │  
│ │ │ │  o1 = {2, 3, 2}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : time degreeDeterminant n
│ │ │ │ - -- used 0.000112722s (cpu); 0.000101711s (thread); 0s (gc)
│ │ │ │ + -- used 7.8019e-05s (cpu); 7.0604e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = 6
│ │ │ │  i3 : M = genericMultidimensionalMatrix n;
│ │ │ │  warning: clearing value of symbol x2 to allow access to subscripted variables
│ │ │ │  based on it
│ │ │ │         : debug with expression   debug 1368   or with command line option   --
│ │ │ │  debug 1368
│ │ │ │ @@ -35,15 +35,15 @@
│ │ │ │  based on it
│ │ │ │         : debug with expression   debug 6010   or with command line option   --
│ │ │ │  debug 6010
│ │ │ │  
│ │ │ │  o3 : 3-dimensional matrix of shape 2 x 3 x 2 over ZZ[a     ..a     ]
│ │ │ │                                                        0,0,0   1,2,1
│ │ │ │  i4 : time degree determinant M
│ │ │ │ - -- used 0.0350456s (cpu); 0.0342305s (thread); 0s (gc)
│ │ │ │ + -- used 0.16978s (cpu); 0.0529805s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = {6}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_e_t_e_r_m_i_n_a_n_t_(_M_u_l_t_i_d_i_m_e_n_s_i_o_n_a_l_M_a_t_r_i_x_) -- hyperdeterminant of a
│ │ │ │        multidimensional matrix
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_dense__Discriminant.html
│ │ │ @@ -80,15 +80,15 @@
│ │ │              
│ │ │                
    i1 : (d,n) := (2,3);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time Disc = denseDiscriminant(d,n)
│ │ │ - -- used 0.408829s (cpu); 0.223024s (thread); 0s (gc)
│ │ │ + -- used 0.430955s (cpu); 0.241127s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = Disc
│ │ │  
│ │ │  o2 : SparseDiscriminant (sparse discriminant associated to | 0 0 0 0 0 0 1 1 1 2 |)
│ │ │                                                             | 0 0 0 1 1 2 0 0 1 0 |
│ │ │                                                             | 0 1 2 0 1 0 0 1 0 0 |
    
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,15 +18,15 @@
│ │ │ │      * Outputs:
│ │ │ │            o for (d,n), this is the same as _s_p_a_r_s_e_D_i_s_c_r_i_m_i_n_a_n_t _e_x_p_o_n_e_n_t_s_M_a_t_r_i_x
│ │ │ │              ""ggeenneerriicc ppoollyynnoommiiaall ooff ddeeggrreeee dd iinn nn vvaarriiaabblleess"";;
│ │ │ │            o for f, this is the same as _a_f_f_i_n_e_D_i_s_c_r_i_m_i_n_a_n_t(f).
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : (d,n) := (2,3);
│ │ │ │  i2 : time Disc = denseDiscriminant(d,n)
│ │ │ │ - -- used 0.408829s (cpu); 0.223024s (thread); 0s (gc)
│ │ │ │ + -- used 0.430955s (cpu); 0.241127s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = Disc
│ │ │ │  
│ │ │ │  o2 : SparseDiscriminant (sparse discriminant associated to | 0 0 0 0 0 0 1 1 1
│ │ │ │  2 |)
│ │ │ │                                                             | 0 0 0 1 1 2 0 0 1
│ │ │ │  0 |
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_dense__Resultant.html
│ │ │ @@ -90,27 +90,27 @@
│ │ │  
│ │ │  o1 : Sequence
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time denseResultant(f0,f1,f2); -- using Poisson formula
│ │ │ - -- used 0.163836s (cpu); 0.123645s (thread); 0s (gc)
    
│ │ │ + -- used 0.209212s (cpu); 0.151545s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay formula
│ │ │ - -- used 0.292245s (cpu); 0.237931s (thread); 0s (gc)
    
│ │ │ + -- used 0.344805s (cpu); 0.28602s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant
│ │ │ - -- used 0.365133s (cpu); 0.330814s (thread); 0s (gc)
    
│ │ │ + -- used 0.328512s (cpu); 0.29171s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : assert(o2 == o3 and o3 == o4)
    
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,20 +28,20 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                   2
│ │ │ │       c x x  + c x  + c x  + c x  + c )
│ │ │ │        4 1 2    2 2    3 1    1 2    0
│ │ │ │  
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : time denseResultant(f0,f1,f2); -- using Poisson formula
│ │ │ │ - -- used 0.163836s (cpu); 0.123645s (thread); 0s (gc)
│ │ │ │ + -- used 0.209212s (cpu); 0.151545s (thread); 0s (gc)
│ │ │ │  i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay
│ │ │ │  formula
│ │ │ │ - -- used 0.292245s (cpu); 0.237931s (thread); 0s (gc)
│ │ │ │ + -- used 0.344805s (cpu); 0.28602s (thread); 0s (gc)
│ │ │ │  i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant
│ │ │ │ - -- used 0.365133s (cpu); 0.330814s (thread); 0s (gc)
│ │ │ │ + -- used 0.328512s (cpu); 0.29171s (thread); 0s (gc)
│ │ │ │  i5 : assert(o2 == o3 and o3 == o4)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _s_p_a_r_s_e_R_e_s_u_l_t_a_n_t -- sparse resultant (A-resultant)
│ │ │ │      * _a_f_f_i_n_e_R_e_s_u_l_t_a_n_t -- affine resultant
│ │ │ │      * _d_e_n_s_e_D_i_s_c_r_i_m_i_n_a_n_t -- dense discriminant (classical discriminant)
│ │ │ │      * _e_x_p_o_n_e_n_t_s_M_a_t_r_i_x -- exponents in one or more polynomials
│ │ │ │      * _g_e_n_e_r_i_c_L_a_u_r_e_n_t_P_o_l_y_n_o_m_i_a_l_s -- generic (Laurent) polynomials
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_determinant_lp__Multidimensional__Matrix_rp.html
│ │ │ @@ -84,15 +84,15 @@
│ │ │  
│ │ │  o1 : 4-dimensional matrix of shape 2 x 2 x 2 x 2 over ZZ
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time det M
│ │ │ - -- used 0.0870547s (cpu); 0.0854223s (thread); 0s (gc)
│ │ │ + -- used 0.148646s (cpu); 0.10084s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 9698337990421512192
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : M = randomMultidimensionalMatrix(2,2,2,2,5)
│ │ │ @@ -109,15 +109,15 @@
│ │ │  
│ │ │  o3 : 5-dimensional matrix of shape 2 x 2 x 2 x 2 x 5 over ZZ
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time det M
│ │ │ - -- used 0.514606s (cpu); 0.440355s (thread); 0s (gc)
│ │ │ + -- used 0.46196s (cpu); 0.461782s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 912984499996938980479447727885644530753184525786986940737407301278806287
│ │ │       9257139493926586400187927813888
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,15 +25,15 @@
│ │ │ │  
│ │ │ │  o1 = {{{{8, 1}, {3, 7}}, {{8, 3}, {3, 7}}}, {{{8, 8}, {5, 7}}, {{8, 5}, {2,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       3}}}}
│ │ │ │  
│ │ │ │  o1 : 4-dimensional matrix of shape 2 x 2 x 2 x 2 over ZZ
│ │ │ │  i2 : time det M
│ │ │ │ - -- used 0.0870547s (cpu); 0.0854223s (thread); 0s (gc)
│ │ │ │ + -- used 0.148646s (cpu); 0.10084s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = 9698337990421512192
│ │ │ │  i3 : M = randomMultidimensionalMatrix(2,2,2,2,5)
│ │ │ │  
│ │ │ │  o3 = {{{{{6, 3, 6, 8, 6}, {9, 3, 7, 6, 9}}, {{6, 2, 6, 0, 2}, {6, 9, 3, 5,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       6}}}, {{{3, 5, 7, 7, 9}, {4, 5, 0, 4, 3}}, {{1, 8, 9, 1, 2}, {9, 6, 6,
│ │ │ │ @@ -42,15 +42,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       7, 4, 5}}}, {{{4, 0, 1, 4, 4}, {2, 6, 1, 1, 4}}, {{5, 4, 9, 7, 4}, {6,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       4, 8, 4, 2}}}}}
│ │ │ │  
│ │ │ │  o3 : 5-dimensional matrix of shape 2 x 2 x 2 x 2 x 5 over ZZ
│ │ │ │  i4 : time det M
│ │ │ │ - -- used 0.514606s (cpu); 0.440355s (thread); 0s (gc)
│ │ │ │ + -- used 0.46196s (cpu); 0.461782s (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 2.62265s (cpu); 2.22282s (thread); 0s (gc)
│ │ │ + -- used 2.76719s (cpu); 2.48392s (thread); 0s (gc)
│ │ │  
│ │ │                                                     2                        
│ │ │  o2 = a     a     a     a     a     a      - a     a     a     a     a      -
│ │ │        0,1,1 0,2,0 0,2,1 1,0,0 1,0,1 1,1,0    0,1,0 0,2,1 1,0,0 1,0,1 1,1,0  
│ │ │       ------------------------------------------------------------------------
│ │ │              2     2                                2            
│ │ │       a     a     a     a      + a     a     a     a     a      -
│ │ │ ├── html2text {}
│ │ │ │ @@ -37,15 +37,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       a     x y z  + a     x y z  + a     x y z
│ │ │ │        1,1,1 1 1 1    1,2,0 1 2 0    1,2,1 1 2 1
│ │ │ │  
│ │ │ │  o1 : ZZ[a     ..a     ][x ..x , y ..y , z ..z ]
│ │ │ │           0,0,0   1,2,1   0   1   0   2   0   1
│ │ │ │  i2 : time sparseDiscriminant f
│ │ │ │ - -- used 2.62265s (cpu); 2.22282s (thread); 0s (gc)
│ │ │ │ + -- used 2.76719s (cpu); 2.48392s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                                                     2
│ │ │ │  o2 = a     a     a     a     a     a      - a     a     a     a     a      -
│ │ │ │        0,1,1 0,2,0 0,2,1 1,0,0 1,0,1 1,1,0    0,1,0 0,2,1 1,0,0 1,0,1 1,1,0
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │              2     2                                2
│ │ │ │       a     a     a     a      + a     a     a     a     a      -
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Resultant.html
│ │ │ @@ -74,15 +74,15 @@
│ │ │          
    Description
    
│ │ │          
    Alternatively, one can apply the method directly to the list of Laurent polynomials $f_0,\ldots,f_n$. In this case, the matrices $A_0,\ldots,A_n$ are automatically determined by exponentsMatrix. If you want require that $A_0=\cdots=A_n$, then use the option Unmixed=>true (this could be faster). Below we consider some examples.
    
│ │ │          
    In the first example, we calculate the sparse (mixed) resultant associated to the three sets of monomials $(1,x y,x^2 y,x),(y,x^2 y^2,x^2 y,x),(1,y,x y,x)$. Then we evaluate it at the three polynomials $f = c_{(1,1)}+c_{(1,2)} x y+c_{(1,3)} x^2 y+c_{(1,4)} x, g = c_{(2,1)} y+c_{(2,2)} x^2 y^2+c_{(2,3)} x^2 y+c_{(2,4)} x, h = c_{(3,1)}+c_{(3,2)} y+c_{(3,3)} x y+c_{(3,4)} x$.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -104,15 +104,15 @@
│ │ │  
│ │ │  o3 : Sequence
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │          
    
│ │ │                
    i1 : time Res = sparseResultant(matrix{{0,1,1,2},{0,0,1,1}},matrix{{0,1,2,2},{1,0,1,2}},matrix{{0,0,1,1},{0,1,0,1}})
│ │ │ - -- used 0.507781s (cpu); 0.44829s (thread); 0s (gc)
│ │ │ + -- used 0.437655s (cpu); 0.417926s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = Res
│ │ │  
│ │ │  o1 : SparseResultant (sparse mixed resultant associated to {| 0 1 1 2 |, | 0 1 2 2 |, | 0 0 1 1 |})
│ │ │                                                              | 0 0 1 1 |  | 1 0 1 2 |  | 0 1 0 1 |
    
│ │ │              
    
│ │ │                
    i4 : time Res(f,g,h)
│ │ │ - -- used 0.00905261s (cpu); 0.00905263s (thread); 0s (gc)
│ │ │ + -- used 0.0105012s (cpu); 0.0105005s (thread); 0s (gc)
│ │ │  
│ │ │          2                       4      2   2               4    
│ │ │  o4 = - c   c   c   c   c   c   c    + c   c   c   c   c   c    +
│ │ │          1,2 1,3 1,4 2,1 2,2 2,3 3,1    1,2 1,3 2,1 2,2 2,4 3,1  
│ │ │       ------------------------------------------------------------------------
│ │ │        3       2       3               2                   3        
│ │ │       c   c   c   c   c   c    - 2c   c   c   c   c   c   c   c    +
│ │ │ @@ -825,15 +825,15 @@
│ │ │            
    
│ │ │          
    In the second example, we calculate the sparse unmixed resultant associated to the set of monomials $(1,x,y,xy)$. Then we evaluate it at the three polynomials $f = a_0 + a_1 x + a_2 y + a_3 x y, g = b_0 + b_1 x + b_2 y + b_3 x y, h = c_0 + c_1 x + c_2 y + c_3 x y$. Moreover, we perform all the computation over $\mathbb{Z}/3331$.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -68,15 +68,15 @@
│ │ │ │  
│ │ │ │  o4 = | 0 1 |
│ │ │ │       | 2 3 |
│ │ │ │       | 4 |
│ │ │ │  
│ │ │ │  o4 : YoungTableau
│ │ │ │  i5 : time higherSpechtPolynomial(S,T,R)
│ │ │ │ - -- used 0.0015725s (cpu); 0.00156896s (thread); 0s (gc)
│ │ │ │ + -- used 0.00166494s (cpu); 0.00166171s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        3 2          2 3      3     2        3 2    3   2      2   3
│ │ │ │  o5 = x x x x  - x x x x  - x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        3 2        3 2        2 3        2 3        3   2        3 2
│ │ │ │       x x x x  - x x x x  + x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ │ @@ -88,15 +88,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          2   3    2     3    2     3      2   3        2 3        2 3
│ │ │ │       x x x x  - x x x x  - x x x x  + x x x x  - x x x x  + x x x x
│ │ │ │        0 1 3 4    0 2 3 4    1 2 3 4    0 2 3 4    0 1 3 4    1 2 3 4
│ │ │ │  
│ │ │ │  o5 : R
│ │ │ │  i6 : time higherSpechtPolynomial(S,T,R, Robust => false)
│ │ │ │ - -- used 0.00127507s (cpu); 0.00127554s (thread); 0s (gc)
│ │ │ │ + -- used 0.00282548s (cpu); 0.00282517s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        3 2          2 3      3     2        3 2    3   2      2   3
│ │ │ │  o6 = x x x x  - x x x x  - x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        3 2        3 2        2 3        2 3        3   2        3 2
│ │ │ │       x x x x  - x x x x  + x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ │ @@ -108,15 +108,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          2   3    2     3    2     3      2   3        2 3        2 3
│ │ │ │       x x x x  - x x x x  - x x x x  + x x x x  - x x x x  + x x x x
│ │ │ │        0 1 3 4    0 2 3 4    1 2 3 4    0 2 3 4    0 1 3 4    1 2 3 4
│ │ │ │  
│ │ │ │  o6 : R
│ │ │ │  i7 : time higherSpechtPolynomial(S,T,R, Robust => false, AsExpression => true)
│ │ │ │ - -- used 0.00194803s (cpu); 0.00194842s (thread); 0s (gc)
│ │ │ │ + -- used 0.00215225s (cpu); 0.00215253s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = (- x  + x )(- x  + x )(- x  + x )(- x  + x )((x  + x  + x )(x )(x ) + (x )
│ │ │ │  (x )(x ))
│ │ │ │           0    2     0    4     2    4     1    3    0    2    4   3   1      4
│ │ │ │  2   0
│ │ │ │  
│ │ │ │  o7 : Expression of class Product
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/html/_representation__Multiplicity.html
│ │ │ @@ -126,15 +126,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -111,15 +111,15 @@
│ │ │  
│ │ │  o4 : ProjectiveVariety, a point in PP^5
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -121,15 +121,15 @@
│ │ │  
│ │ │  o5 : ProjectiveVariety, a point in PP^8
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i6 : time Res = sparseResultant(matrix{{0,0,1,1},{0,1,0,1}},CoefficientRing=>ZZ/3331);
│ │ │ - -- used 0.0304897s (cpu); 0.0292273s (thread); 0s (gc)
│ │ │ + -- used 0.174291s (cpu); 0.0527137s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : SparseResultant (sparse unmixed resultant associated to | 0 0 1 1 | over ZZ/3331)
│ │ │                                                               | 0 1 0 1 |
    
│ │ │              
    
│ │ │ @@ -849,15 +849,15 @@
│ │ │  
│ │ │  o8 : Sequence
│ │ │              
    
│ │ │                
    i9 : time Res(f,g,h)
│ │ │ - -- used 0.00316112s (cpu); 0.00316119s (thread); 0s (gc)
│ │ │ + -- used 0.00426482s (cpu); 0.00426298s (thread); 0s (gc)
│ │ │  
│ │ │        2     2            2            2        2 2    2          
│ │ │  o9 = a b b c  - a a b b c  - a a b b c  + a a b c  - a b b c c  -
│ │ │        3 1 2 0    2 3 1 3 0    1 3 2 3 0    1 2 3 0    3 0 2 0 1  
│ │ │       ------------------------------------------------------------------------
│ │ │                           2                       2                         
│ │ │       a a b b c c  + a a b c c  + a a b b c c  + a b b c c  - a a b b c c  +
│ │ │ @@ -938,15 +938,15 @@
│ │ │  
│ │ │  o11 : Sequence
    
│ │ │              
    
│ │ │                
    i12 : time (MixedRes,UnmixedRes) = (sparseResultant(f,g,h),sparseResultant(f,g,h,Unmixed=>true));
│ │ │ - -- used 0.260966s (cpu); 0.209622s (thread); 0s (gc)
    
│ │ │ + -- used 0.386656s (cpu); 0.211447s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i13 : quotientRemainder(UnmixedRes,MixedRes)
│ │ │  
│ │ │          2 2                   2    2                               2 2
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │  In the first example, we calculate the sparse (mixed) resultant associated to
│ │ │ │  the three sets of monomials $(1,x y,x^2 y,x),(y,x^2 y^2,x^2 y,x),(1,y,x y,x)$.
│ │ │ │  Then we evaluate it at the three polynomials $f = c_{(1,1)}+c_{(1,2)} x y+c_{
│ │ │ │  (1,3)} x^2 y+c_{(1,4)} x, g = c_{(2,1)} y+c_{(2,2)} x^2 y^2+c_{(2,3)} x^2 y+c_{
│ │ │ │  (2,4)} x, h = c_{(3,1)}+c_{(3,2)} y+c_{(3,3)} x y+c_{(3,4)} x$.
│ │ │ │  i1 : time Res = sparseResultant(matrix{{0,1,1,2},{0,0,1,1}},matrix{{0,1,2,2},
│ │ │ │  {1,0,1,2}},matrix{{0,0,1,1},{0,1,0,1}})
│ │ │ │ - -- used 0.507781s (cpu); 0.44829s (thread); 0s (gc)
│ │ │ │ + -- used 0.437655s (cpu); 0.417926s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = Res
│ │ │ │  
│ │ │ │  o1 : SparseResultant (sparse mixed resultant associated to {| 0 1 1 2 |, | 0 1
│ │ │ │  2 2 |, | 0 0 1 1 |})
│ │ │ │                                                              | 0 0 1 1 |  | 1 0
│ │ │ │  1 2 |  | 0 1 0 1 |
│ │ │ │ @@ -55,15 +55,15 @@
│ │ │ │         1,3       1,2       1,4     1,1   2,2        2,3       2,4     2,1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       c   x*y + c   x + c   y + c   )
│ │ │ │        3,3       3,4     3,2     3,1
│ │ │ │  
│ │ │ │  o3 : Sequence
│ │ │ │  i4 : time Res(f,g,h)
│ │ │ │ - -- used 0.00905261s (cpu); 0.00905263s (thread); 0s (gc)
│ │ │ │ + -- used 0.0105012s (cpu); 0.0105005s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          2                       4      2   2               4
│ │ │ │  o4 = - c   c   c   c   c   c   c    + c   c   c   c   c   c    +
│ │ │ │          1,2 1,3 1,4 2,1 2,2 2,3 3,1    1,2 1,3 2,1 2,2 2,4 3,1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        3       2       3               2                   3
│ │ │ │       c   c   c   c   c   c    - 2c   c   c   c   c   c   c   c    +
│ │ │ │ @@ -771,29 +771,29 @@
│ │ │ │  In the second example, we calculate the sparse unmixed resultant associated to
│ │ │ │  the set of monomials $(1,x,y,xy)$. Then we evaluate it at the three polynomials
│ │ │ │  $f = a_0 + a_1 x + a_2 y + a_3 x y, g = b_0 + b_1 x + b_2 y + b_3 x y, h = c_0
│ │ │ │  + c_1 x + c_2 y + c_3 x y$. Moreover, we perform all the computation over
│ │ │ │  $\mathbb{Z}/3331$.
│ │ │ │  i6 : time Res = sparseResultant(matrix{{0,0,1,1},
│ │ │ │  {0,1,0,1}},CoefficientRing=>ZZ/3331);
│ │ │ │ - -- used 0.0304897s (cpu); 0.0292273s (thread); 0s (gc)
│ │ │ │ + -- used 0.174291s (cpu); 0.0527137s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 : SparseResultant (sparse unmixed resultant associated to | 0 0 1 1 | over
│ │ │ │  ZZ/3331)
│ │ │ │                                                               | 0 1 0 1 |
│ │ │ │  i7 : ZZ/3331[a_0..a_3,b_0..b_3,c_0..c_3][x,y];
│ │ │ │  i8 : (f,g,h) = (a_0 + a_1*x + a_2*y + a_3*x*y, b_0 + b_1*x + b_2*y + b_3*x*y,
│ │ │ │  c_0 + c_1*x + c_2*y + c_3*x*y)
│ │ │ │  
│ │ │ │  o8 = (a x*y + a x + a y + a , b x*y + b x + b y + b , c x*y + c x + c y + c )
│ │ │ │         3       1     2     0   3       1     2     0   3       1     2     0
│ │ │ │  
│ │ │ │  o8 : Sequence
│ │ │ │  i9 : time Res(f,g,h)
│ │ │ │ - -- used 0.00316112s (cpu); 0.00316119s (thread); 0s (gc)
│ │ │ │ + -- used 0.00426482s (cpu); 0.00426298s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        2     2            2            2        2 2    2
│ │ │ │  o9 = a b b c  - a a b b c  - a a b b c  + a a b c  - a b b c c  -
│ │ │ │        3 1 2 0    2 3 1 3 0    1 3 2 3 0    1 2 3 0    3 0 2 0 1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                           2                       2
│ │ │ │       a a b b c c  + a a b c c  + a a b b c c  + a b b c c  - a a b b c c  +
│ │ │ │ @@ -863,15 +863,15 @@
│ │ │ │                    2
│ │ │ │        c x x  + c x  + c x  + c x  + c )
│ │ │ │         4 1 2    2 2    3 1    1 2    0
│ │ │ │  
│ │ │ │  o11 : Sequence
│ │ │ │  i12 : time (MixedRes,UnmixedRes) = (sparseResultant(f,g,h),sparseResultant
│ │ │ │  (f,g,h,Unmixed=>true));
│ │ │ │ - -- used 0.260966s (cpu); 0.209622s (thread); 0s (gc)
│ │ │ │ + -- used 0.386656s (cpu); 0.211447s (thread); 0s (gc)
│ │ │ │  i13 : quotientRemainder(UnmixedRes,MixedRes)
│ │ │ │  
│ │ │ │          2 2                   2    2                               2 2
│ │ │ │  o13 = (b c  - b b c c  + b b c  + b c c  - 2b b c c  - b b c c  + b c , 0)
│ │ │ │          5 2    4 5 2 4    2 5 4    4 2 5     2 5 2 5    2 4 4 5    2 5
│ │ │ │  
│ │ │ │  o13 : Sequence
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/example-output/_higher__Specht__Polynomial_lp__Young__Tableau_cm__Young__Tableau_cm__Polynomial__Ring_rp.out
│ │ │ @@ -25,15 +25,15 @@
│ │ │  o4 = | 0 1 |
│ │ │       | 2 3 |
│ │ │       | 4 |
│ │ │  
│ │ │  o4 : YoungTableau
│ │ │  
│ │ │  i5 : time higherSpechtPolynomial(S,T,R)
│ │ │ - -- used 0.0015725s (cpu); 0.00156896s (thread); 0s (gc)
│ │ │ + -- used 0.00166494s (cpu); 0.00166171s (thread); 0s (gc)
│ │ │  
│ │ │        3 2          2 3      3     2        3 2    3   2      2   3    
│ │ │  o5 = x x x x  - x x x x  - x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4  
│ │ │       ------------------------------------------------------------------------
│ │ │        3 2        3 2        2 3        2 3        3   2        3 2    
│ │ │       x x x x  - x x x x  + x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ @@ -46,15 +46,15 @@
│ │ │          2   3    2     3    2     3      2   3        2 3        2 3
│ │ │       x x x x  - x x x x  - x x x x  + x x x x  - x x x x  + x x x x
│ │ │        0 1 3 4    0 2 3 4    1 2 3 4    0 2 3 4    0 1 3 4    1 2 3 4
│ │ │  
│ │ │  o5 : R
│ │ │  
│ │ │  i6 : time higherSpechtPolynomial(S,T,R, Robust => false)
│ │ │ - -- used 0.00127507s (cpu); 0.00127554s (thread); 0s (gc)
│ │ │ + -- used 0.00282548s (cpu); 0.00282517s (thread); 0s (gc)
│ │ │  
│ │ │        3 2          2 3      3     2        3 2    3   2      2   3    
│ │ │  o6 = x x x x  - x x x x  - x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4  
│ │ │       ------------------------------------------------------------------------
│ │ │        3 2        3 2        2 3        2 3        3   2        3 2    
│ │ │       x x x x  - x x x x  + x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ @@ -67,15 +67,15 @@
│ │ │          2   3    2     3    2     3      2   3        2 3        2 3
│ │ │       x x x x  - x x x x  - x x x x  + x x x x  - x x x x  + x x x x
│ │ │        0 1 3 4    0 2 3 4    1 2 3 4    0 2 3 4    0 1 3 4    1 2 3 4
│ │ │  
│ │ │  o6 : R
│ │ │  
│ │ │  i7 : time higherSpechtPolynomial(S,T,R, Robust => false, AsExpression => true)
│ │ │ - -- used 0.00194803s (cpu); 0.00194842s (thread); 0s (gc)
│ │ │ + -- used 0.00215225s (cpu); 0.00215253s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = (- x  + x )(- x  + x )(- x  + x )(- x  + x )((x  + x  + x )(x )(x ) + (x )(x )(x ))
│ │ │           0    2     0    4     2    4     1    3    0    2    4   3   1      4   2   0
│ │ │  
│ │ │  o7 : Expression of class Product
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/example-output/_representation__Multiplicity.out
│ │ │ @@ -25,15 +25,15 @@
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : tal := tally apply (H,h->conjugacyClass h);
│ │ │  
│ │ │  i4 : partis = partitions 6;
│ │ │  
│ │ │  i5 : time multi = hashTable apply (partis, p-> p=> representationMultiplicity(tal,p))
│ │ │ - -- used 0.344281s (cpu); 0.287215s (thread); 0s (gc)
│ │ │ + -- used 0.373775s (cpu); 0.305037s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HashTable{Partition{1, 1, 1, 1, 1, 1} => 1}
│ │ │                 Partition{2, 1, 1, 1, 1} => 0
│ │ │                 Partition{2, 2, 1, 1} => 1
│ │ │                 Partition{2, 2, 2} => 1
│ │ │                 Partition{3, 1, 1, 1} => 0
│ │ │                 Partition{3, 2, 1} => 0
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/example-output/_secondary__Invariants_lp__List_cm__Polynomial__Ring_rp.out
│ │ │ @@ -20,15 +20,15 @@
│ │ │  (Partition{3, 3}, Ambient_Dimension, 5, Rank, 1)
│ │ │  (Partition{3, 2, 1}, Ambient_Dimension, 16, Rank, 0)
│ │ │  (Partition{3, 1, 1, 1}, Ambient_Dimension, 10, Rank, 0)
│ │ │  (Partition{2, 2, 2}, Ambient_Dimension, 5, Rank, 1)
│ │ │  (Partition{2, 2, 1, 1}, Ambient_Dimension, 9, Rank, 1)
│ │ │  (Partition{2, 1, 1, 1, 1}, Ambient_Dimension, 5, Rank, 0)
│ │ │  (Partition{1, 1, 1, 1, 1, 1}, Ambient_Dimension, 1, Rank, 1)
│ │ │ - -- used 0.68249s (cpu); 0.493249s (thread); 0s (gc)
│ │ │ + -- used 0.844622s (cpu); 0.633304s (thread); 0s (gc)
│ │ │  
│ │ │  i4 : seco#(new Partition from {2,2,2})
│ │ │  
│ │ │                                                        2 2 2       4 2   2     2   2 2     2 2     2   4   2   2   2     2 2   1 2 2       2 2   2     1   2 2     1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2     1   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2   2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1     2 2   1 2 2       2 2   2     1   2 2     1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2     1   2 2     1   2 2     2 2 2       4 2 2       2 2 2       2 2 2       4 2 2       2 2 2       1 2   2     1   2 2     2 2   2     1 2   2     2   2 2     1   2 2     1 2   2     2 2   2     1 2   2     1   2 2     2   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2   2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1     2 2   1 2     2   1   2   2   2 2     2   1 2     2   2   2   2   1   2   2   1 2     2   2 2     2   1 2     2   1   2   2   2   2   2   1   2   2   2     2 2   4     2 2   2     2 2   2     2 2   4     2 2   2     2 2
│ │ │  o4 = HashTable{{0, 1, 2, 3, 4, 5} => HashTable{0 => - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x }                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        }
│ │ │                                                        3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 5   3 1 2 3 5   3 1 2 3 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 4 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 3 5   3 1 2 3 5   3 1 2 3 5   3 1 2 4 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 3 6   3 1 2 3 6   3 1 2 3 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 4 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 2 5 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 2 3 6   3 1 2 3 6   3 1 2 3 6   3 1 2 4 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 5 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6
│ │ │                                                      2 3 2 2     4 2 3 2     2 2 2 3     4 3 2   2   2 2 3   2   2 3   2 2   2   3 2 2   2 2   3 2   4   2 3 2   2 2 2   3   4 2   2 3   2   2 2 3   1 3 2 2     2 2 3 2     1 2 2 3     2 3 2 2     1 2 3 2     1 3 2 2     1 3 2 2     1 2 3 2     2 2 3 2     1 2 2 3     2 2 2 3     1 2 2 3     2 3 2   2   1 2 3   2   1 3   2 2   1   3 2 2   1 2   3 2   2   2 3 2   1 3 2   2   2 2 3   2   1 3 2   2   2 3 2   2   1 2 3   2   1 2 3   2   1 3   2 2   1   3 2 2   2 3   2 2   1 3   2 2   2   3 2 2   1   3 2 2   2 2   3 2   1   2 3 2   1 2   3 2   1 2   3 2   1   2 3 2   2   2 3 2   1 2 2   3   2 2   2 3   1   2 2 3   1 2 2   3   2 2 2   3   1 2 2   3   1 2   2 3   2   2 2 3   1 2   2 3   2 2   2 3   1   2 2 3   1   2 2 3   1 3 2 2     2 2 3 2     1 2 2 3     2 3 2 2     1 2 3 2     1 3 2 2     1 3 2 2     1 2 3 2     2 2 3 2     1 2 2 3     2 2 2 3     1 2 2 3     1 3 2 2     1 2 3 2     2 3 2 2     1 3 2 2     2 2 3 2     1 2 3 2     1 3 2 2     2 3 2 2     1 3 2 2     1 2 3 2     2 2 3 2     1 2 3 2     2 2 2 3     4 2 2 3     2 2 2 3     2 2 2 3     4 2 2 3     2 2 2 3     2 3 2   2   1 2 3   2   1 3   2 2   1   3 2 2   1 2   3 2   2   2 3 2   1 3 2   2   2 2 3   2   1 3 2   2   2 3 2   2   1 2 3   2   1 2 3   2   1 3   2 2   1   3 2 2   2 3   2 2   1 3   2 2   2   3 2 2   1   3 2 2   2 2   3 2   1   2 3 2   1 2   3 2   1 2   3 2   1   2 3 2   2   2 3 2   1 3 2   2   1 2 3   2   2 3 2   2   1 3 2   2   2 2 3   2   1 2 3   2   1 3 2   2   2 3 2   2   1 3 2   2   1 2 3   2   2 2 3   2   1 2 3   2   2 3   2 2   2   3 2 2   4 3   2 2   2 3   2 2   4   3 2 2   2   3 2 2   2 3   2 2   4 3   2 2   2 3   2 2   2   3 2 2   4   3 2 2   2   3 2 2   1 2   3 2   1   2 3 2   2 2   3 2   1 2   3 2   2   2 3 2   1   2 3 2   1 2   3 2   2 2   3 2   1 2   3 2   1   2 3 2   2   2 3 2   1   2 3 2   1 2 2   3   2 2   2 3   1   2 2 3   1 2 2   3   2 2 2   3   1 2 2   3   1 2   2 3   2   2 2 3   1 2   2 3   2 2   2 3   1   2 2 3   1   2 2 3   2 2 2   3   4 2 2   3   2 2 2   3   2 2 2   3   4 2 2   3   2 2 2   3   1 2   2 3   1   2 2 3   2 2   2 3   1 2   2 3   2   2 2 3   1   2 2 3   1 2   2 3   2 2   2 3   1 2   2 3   1   2 2 3   2   2 2 3   1   2 2 3
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/html/_higher__Specht__Polynomial_lp__Young__Tableau_cm__Young__Tableau_cm__Polynomial__Ring_rp.html
│ │ │ @@ -125,15 +125,15 @@
│ │ │  
│ │ │  o4 : YoungTableau
    
│ │ │              
    
│ │ │                
    i5 : time higherSpechtPolynomial(S,T,R)
│ │ │ - -- used 0.0015725s (cpu); 0.00156896s (thread); 0s (gc)
│ │ │ + -- used 0.00166494s (cpu); 0.00166171s (thread); 0s (gc)
│ │ │  
│ │ │        3 2          2 3      3     2        3 2    3   2      2   3    
│ │ │  o5 = x x x x  - x x x x  - x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4  
│ │ │       ------------------------------------------------------------------------
│ │ │        3 2        3 2        2 3        2 3        3   2        3 2    
│ │ │       x x x x  - x x x x  + x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ @@ -149,15 +149,15 @@
│ │ │  
│ │ │  o5 : R
    
│ │ │              
    
│ │ │                
    i6 : time higherSpechtPolynomial(S,T,R, Robust => false)
│ │ │ - -- used 0.00127507s (cpu); 0.00127554s (thread); 0s (gc)
│ │ │ + -- used 0.00282548s (cpu); 0.00282517s (thread); 0s (gc)
│ │ │  
│ │ │        3 2          2 3      3     2        3 2    3   2      2   3    
│ │ │  o6 = x x x x  - x x x x  - x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4  
│ │ │       ------------------------------------------------------------------------
│ │ │        3 2        3 2        2 3        2 3        3   2        3 2    
│ │ │       x x x x  - x x x x  + x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ @@ -173,15 +173,15 @@
│ │ │  
│ │ │  o6 : R
    
│ │ │              
    
│ │ │                
    i7 : time higherSpechtPolynomial(S,T,R, Robust => false, AsExpression => true)
│ │ │ - -- used 0.00194803s (cpu); 0.00194842s (thread); 0s (gc)
│ │ │ + -- used 0.00215225s (cpu); 0.00215253s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = (- x  + x )(- x  + x )(- x  + x )(- x  + x )((x  + x  + x )(x )(x ) + (x )(x )(x ))
│ │ │           0    2     0    4     2    4     1    3    0    2    4   3   1      4   2   0
│ │ │  
│ │ │  o7 : Expression of class Product
    
│ │ │              
    
│ │ │                
    i4 : partis = partitions 6;
    
│ │ │              
    
│ │ │                
    i5 : time multi = hashTable apply (partis, p-> p=> representationMultiplicity(tal,p))
│ │ │ - -- used 0.344281s (cpu); 0.287215s (thread); 0s (gc)
│ │ │ + -- used 0.373775s (cpu); 0.305037s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HashTable{Partition{1, 1, 1, 1, 1, 1} => 1}
│ │ │                 Partition{2, 1, 1, 1, 1} => 0
│ │ │                 Partition{2, 2, 1, 1} => 1
│ │ │                 Partition{2, 2, 2} => 1
│ │ │                 Partition{3, 1, 1, 1} => 0
│ │ │                 Partition{3, 2, 1} => 0
│ │ │ ├── html2text {}
│ │ │ │ @@ -63,15 +63,15 @@
│ │ │ │  representations of $H$ in each irreducible representation of $S_6$. We take
│ │ │ │  into account that there are multiple copies of each representation by
│ │ │ │  multiplying the values with the number of copies which is given by the
│ │ │ │  hookLengthFormula.
│ │ │ │  i4 : partis = partitions 6;
│ │ │ │  i5 : time multi = hashTable apply (partis, p-> p=> representationMultiplicity
│ │ │ │  (tal,p))
│ │ │ │ - -- used 0.344281s (cpu); 0.287215s (thread); 0s (gc)
│ │ │ │ + -- used 0.373775s (cpu); 0.305037s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = HashTable{Partition{1, 1, 1, 1, 1, 1} => 1}
│ │ │ │                 Partition{2, 1, 1, 1, 1} => 0
│ │ │ │                 Partition{2, 2, 1, 1} => 1
│ │ │ │                 Partition{2, 2, 2} => 1
│ │ │ │                 Partition{3, 1, 1, 1} => 0
│ │ │ │                 Partition{3, 2, 1} => 0
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/html/_secondary__Invariants_lp__List_cm__Polynomial__Ring_rp.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │  (Partition{3, 3}, Ambient_Dimension, 5, Rank, 1)
│ │ │  (Partition{3, 2, 1}, Ambient_Dimension, 16, Rank, 0)
│ │ │  (Partition{3, 1, 1, 1}, Ambient_Dimension, 10, Rank, 0)
│ │ │  (Partition{2, 2, 2}, Ambient_Dimension, 5, Rank, 1)
│ │ │  (Partition{2, 2, 1, 1}, Ambient_Dimension, 9, Rank, 1)
│ │ │  (Partition{2, 1, 1, 1, 1}, Ambient_Dimension, 5, Rank, 0)
│ │ │  (Partition{1, 1, 1, 1, 1, 1}, Ambient_Dimension, 1, Rank, 1)
│ │ │ - -- used 0.68249s (cpu); 0.493249s (thread); 0s (gc)
    
│ │ │ + -- used 0.844622s (cpu); 0.633304s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i4 : seco#(new Partition from {2,2,2})
│ │ │  
│ │ │                                                        2 2 2       4 2   2     2   2 2     2 2     2   4   2   2   2     2 2   1 2 2       2 2   2     1   2 2     1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2     1   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2   2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1     2 2   1 2 2       2 2   2     1   2 2     1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2     1   2 2     1   2 2     2 2 2       4 2 2       2 2 2       2 2 2       4 2 2       2 2 2       1 2   2     1   2 2     2 2   2     1 2   2     2   2 2     1   2 2     1 2   2     2 2   2     1 2   2     1   2 2     2   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2   2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1     2 2   1 2     2   1   2   2   2 2     2   1 2     2   2   2   2   1   2   2   1 2     2   2 2     2   1 2     2   1   2   2   2   2   2   1   2   2   2     2 2   4     2 2   2     2 2   2     2 2   4     2 2   2     2 2
│ │ │ ├── html2text {}
│ │ │ │ @@ -56,15 +56,15 @@
│ │ │ │  (Partition{3, 3}, Ambient_Dimension, 5, Rank, 1)
│ │ │ │  (Partition{3, 2, 1}, Ambient_Dimension, 16, Rank, 0)
│ │ │ │  (Partition{3, 1, 1, 1}, Ambient_Dimension, 10, Rank, 0)
│ │ │ │  (Partition{2, 2, 2}, Ambient_Dimension, 5, Rank, 1)
│ │ │ │  (Partition{2, 2, 1, 1}, Ambient_Dimension, 9, Rank, 1)
│ │ │ │  (Partition{2, 1, 1, 1, 1}, Ambient_Dimension, 5, Rank, 0)
│ │ │ │  (Partition{1, 1, 1, 1, 1, 1}, Ambient_Dimension, 1, Rank, 1)
│ │ │ │ - -- used 0.68249s (cpu); 0.493249s (thread); 0s (gc)
│ │ │ │ + -- used 0.844622s (cpu); 0.633304s (thread); 0s (gc)
│ │ │ │  i4 : seco#(new Partition from {2,2,2})
│ │ │ │  
│ │ │ │                                                        2 2 2       4 2   2     2
│ │ │ │  2 2     2 2     2   4   2   2   2     2 2   1 2 2       2 2   2     1   2 2
│ │ │ │  1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2
│ │ │ │  1   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2
│ │ │ │  2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__Castelnuovo__Surface.out
│ │ │ @@ -10,15 +10,15 @@
│ │ │       of discriminant 31 = det| 8 1 |
│ │ │                               | 1 4 |
│ │ │       containing a surface of degree 1 and sectional genus 0
│ │ │       cut out by 5 hypersurfaces of degree 1
│ │ │       (This is a classical example of rational fourfold)
│ │ │  
│ │ │  i3 : time U' = associatedCastelnuovoSurface X;
│ │ │ - -- used 2.39568s (cpu); 1.11189s (thread); 0s (gc)
│ │ │ + -- used 2.7641s (cpu); 1.0622s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : ProjectiveVariety, Castelnuovo surface associated to X
│ │ │  
│ │ │  i4 : (mu,U,C,f) = building U';
│ │ │  
│ │ │  i5 : ? mu
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Special__Cubic__Fourfold_rp.out
│ │ │ @@ -7,15 +7,15 @@
│ │ │  i2 : describe X
│ │ │  
│ │ │  o2 = Special cubic fourfold of discriminant 14
│ │ │       containing a (smooth) surface of degree 4 and sectional genus 0
│ │ │       cut out by 6 hypersurfaces of degree 2
│ │ │  
│ │ │  i3 : time U' = associatedK3surface X;
│ │ │ - -- used 1.99111s (cpu); 1.02887s (thread); 0s (gc)
│ │ │ + -- used 2.70739s (cpu); 1.11711s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │  
│ │ │  i4 : (mu,U,C,f) = building U';
│ │ │  
│ │ │  i5 : ? mu
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Special__Gushel__Mukai__Fourfold_rp.out
│ │ │ @@ -10,15 +10,15 @@
│ │ │       containing a surface in PP^8 of degree 2 and sectional genus 0
│ │ │       cut out by 6 hypersurfaces of degrees (1,1,1,1,1,2)
│ │ │       and with class in G(1,4) given by s_(3,1)+s_(2,2)
│ │ │       Type: ordinary
│ │ │       (case 1 of Table 1 in arXiv:2002.07026)
│ │ │  
│ │ │  i3 : time U' = associatedK3surface X;
│ │ │ - -- used 7.14125s (cpu); 4.38468s (thread); 0s (gc)
│ │ │ + -- used 7.26543s (cpu); 4.3104s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │  
│ │ │  i4 : (mu,U,C,f) = building U';
│ │ │  
│ │ │  i5 : ? mu
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Special__Cubic__Fourfold_cm__Z__Z_rp.out
│ │ │ @@ -8,28 +8,28 @@
│ │ │  i2 : describe X
│ │ │  
│ │ │  o2 = Special cubic fourfold of discriminant 26
│ │ │       containing a 3-nodal surface of degree 7 and sectional genus 0
│ │ │       cut out by 13 hypersurfaces of degree 3
│ │ │  
│ │ │  i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 4.03482s (cpu); 2.10467s (thread); 0s (gc)
│ │ │ + -- used 3.13911s (cpu); 1.89488s (thread); 0s (gc)
│ │ │  number lines contained in the image of the cubic map and passing through a general point: 8
│ │ │  number 2-secant lines = 7
│ │ │  number 5-secant conics = 1
│ │ │  
│ │ │  o3 : Congruence of 5-secant conics to surface in PP^5
│ │ │  
│ │ │  i4 : p := point ambient X -- random point on P^5
│ │ │  
│ │ │  o4 = point of coordinates [15092, -9738, -3620, -15181, 12688, 1]
│ │ │  
│ │ │  o4 : ProjectiveVariety, a point in PP^5
│ │ │  
│ │ │  i5 : time C = f p; -- 5-secant conic to the surface
│ │ │ - -- used 0.429617s (cpu); 0.294228s (thread); 0s (gc)
│ │ │ + -- used 0.341292s (cpu); 0.28804s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, curve in PP^5
│ │ │  
│ │ │  i6 : assert(dim C == 1 and degree C == 2 and dim(C * surface X) == 0 and degree(C * surface X) == 5 and isSubset(p, C))
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Special__Gushel__Mukai__Fourfold_cm__Z__Z_rp.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │       containing a surface in PP^8 of degree 9 and sectional genus 2
│ │ │       cut out by 19 hypersurfaces of degree 2
│ │ │       and with class in G(1,4) given by 6*s_(3,1)+3*s_(2,2)
│ │ │       Type: ordinary
│ │ │       (case 17 of Table 1 in arXiv:2002.07026)
│ │ │  
│ │ │  i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 13.4563s (cpu); 7.009s (thread); 0s (gc)
│ │ │ + -- used 19.4691s (cpu); 7.34788s (thread); 0s (gc)
│ │ │  number lines contained in the image of the quadratic map and passing through a general point: 7
│ │ │  number 1-secant lines = 6
│ │ │  number 3-secant conics = 1
│ │ │  
│ │ │  o3 : Congruence of 3-secant conics to surface in a fivefold in PP^8
│ │ │  
│ │ │  i4 : Y = ambientFivefold X; -- del Pezzo fivefold containing X
│ │ │ @@ -29,15 +29,15 @@
│ │ │  i5 : p := point Y -- random point on Y
│ │ │  
│ │ │  o5 = point of coordinates [7214, -1460, 7057, -2440, 15907, -14345, -5937, 13402, 1]
│ │ │  
│ │ │  o5 : ProjectiveVariety, a point in PP^8
│ │ │  
│ │ │  i6 : time C = f p; -- 3-secant conic to the surface
│ │ │ - -- used 0.396s (cpu); 0.234093s (thread); 0s (gc)
│ │ │ + -- used 0.489967s (cpu); 0.302939s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : ProjectiveVariety, curve in PP^8 (subvariety of codimension 4 in Y)
│ │ │  
│ │ │  i7 : S = surface X;
│ │ │  
│ │ │  o7 : ProjectiveVariety, surface in PP^8 (subvariety of codimension 3 in Y)
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_discriminant_lp__Special__Cubic__Fourfold_rp.out
│ │ │ @@ -1,12 +1,12 @@
│ │ │  -- -*- M2-comint -*- hash: 1729890813579561111
│ │ │  
│ │ │  i1 : X = specialCubicFourfold "quintic del Pezzo surface";
│ │ │  
│ │ │  o1 : ProjectiveVariety, cubic fourfold containing a surface of degree 5 and sectional genus 1
│ │ │  
│ │ │  i2 : time discriminant X
│ │ │ - -- used 0.420819s (cpu); 0.147554s (thread); 0s (gc)
│ │ │ + -- used 0.36452s (cpu); 0.12517s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 14
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_discriminant_lp__Special__Gushel__Mukai__Fourfold_rp.out
│ │ │ @@ -1,12 +1,12 @@
│ │ │  -- -*- M2-comint -*- hash: 1730220932418738713
│ │ │  
│ │ │  i1 : X = specialGushelMukaiFourfold "tau-quadric";
│ │ │  
│ │ │  o1 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
│ │ │  
│ │ │  i2 : time discriminant X
│ │ │ - -- used 1.02747s (cpu); 0.458878s (thread); 0s (gc)
│ │ │ + -- used 1.07787s (cpu); 0.467581s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  o2 : ProjectiveVariety, curve in PP^5
│ │ │  
│ │ │  i3 : X = random({{2},{2},{2}},S);
│ │ │  
│ │ │  o3 : ProjectiveVariety, surface in PP^5
│ │ │  
│ │ │  i4 : time parameterCount(S,X,Verbose=>true)
│ │ │ - -- used 0.296112s (cpu); 0.200839s (thread); 0s (gc)
│ │ │ + -- used 0.465297s (cpu); 0.254661s (thread); 0s (gc)
│ │ │  S: rational normal curve of degree 5 in PP^5
│ │ │  X: smooth surface of degree 8 and sectional genus 5 in PP^5 cut out by 3 hypersurfaces of degree 2
│ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │  h^0(N_{S,P^5}) = 32
│ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,P^5}(2)) = 10 = h^0(O_(P^5)(2)) - \chi(O_S(2));
│ │ │  in particular, h^0(I_{S,P^5}(2)) is minimal
│ │ │  dim GG(2,9) = 21
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Special__Cubic__Fourfold_rp.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  o2 : ProjectiveVariety, surface in PP^5
│ │ │  
│ │ │  i3 : X = specialCubicFourfold V;
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
│ │ │  
│ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 0.641923s (cpu); 0.367693s (thread); 0s (gc)
│ │ │ + -- used 0.782566s (cpu); 0.507141s (thread); 0s (gc)
│ │ │  S: Veronese surface in PP^5
│ │ │  X: smooth cubic hypersurface in PP^5
│ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │  h^0(N_{S,P^5}) = 27
│ │ │  h^1(O_S(3)) = 0, and h^0(I_{S,P^5}(3)) = 28 = h^0(O_(P^5)(3)) - \chi(O_S(3));
│ │ │  in particular, h^0(I_{S,P^5}(3)) is minimal
│ │ │  h^0(N_{S,P^5}) + 27 = 54
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Special__Gushel__Mukai__Fourfold_rp.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │  o2 : ProjectiveVariety, surface in PP^9 (subvariety of codimension 4 in G)
│ │ │  
│ │ │  i3 : X = specialGushelMukaiFourfold S;
│ │ │  
│ │ │  o3 : ProjectiveVariety, GM fourfold containing a surface of degree 3 and sectional genus 0
│ │ │  
│ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 3.48579s (cpu); 2.16969s (thread); 0s (gc)
│ │ │ + -- used 3.80878s (cpu); 2.76661s (thread); 0s (gc)
│ │ │  S: cubic surface in PP^8 cut out by 7 hypersurfaces of degrees (1,1,1,1,2,2,2)
│ │ │  X: GM fourfold containing S
│ │ │  Y: del Pezzo fivefold containing X
│ │ │  h^1(N_{S,Y}) = 0
│ │ │  h^0(N_{S,Y}) = 11
│ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,Y}(2)) = 28 = h^0(O_Y(2)) - \chi(O_S(2));
│ │ │  in particular, h^0(I_{S,Y}(2)) is minimal
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parametrize__Fano__Fourfold.out
│ │ │ @@ -6,15 +6,15 @@
│ │ │  
│ │ │  i3 : ? X
│ │ │  
│ │ │  o3 = 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees
│ │ │       1^2 2^5
│ │ │  
│ │ │  i4 : time parametrizeFanoFourfold X
│ │ │ - -- used 1.79691s (cpu); 0.888229s (thread); 0s (gc)
│ │ │ + -- used 1.68186s (cpu); 0.773295s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = multi-rational map consisting of one single rational map
│ │ │       source variety: PP^4
│ │ │       target variety: 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees 1^2 2^5 
│ │ │       dominance: true
│ │ │       degree: 1
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Cubic__Fourfold.out
│ │ │ @@ -7,22 +7,22 @@
│ │ │  o3 : ProjectiveVariety, surface in PP^5
│ │ │  
│ │ │  i4 : X = projectiveVariety ideal(x_1^2*x_3+x_0*x_2*x_3-6*x_1*x_2*x_3-x_0*x_3^2-4*x_1*x_3^2-3*x_2*x_3^2+2*x_0^2*x_4-10*x_0*x_1*x_4+13*x_1^2*x_4-x_0*x_2*x_4-3*x_1*x_2*x_4+3*x_2^2*x_4+14*x_0*x_3*x_4-8*x_1*x_3*x_4-4*x_3^2*x_4+x_0*x_4^2-7*x_1*x_4^2+4*x_2*x_4^2-2*x_3*x_4^2-2*x_4^3-x_0*x_1*x_5+x_1^2*x_5+2*x_1*x_2*x_5+3*x_0*x_3*x_5+3*x_1*x_3*x_5-x_3^2*x_5-x_0*x_4*x_5-4*x_1*x_4*x_5+3*x_2*x_4*x_5+2*x_3*x_4*x_5-x_1*x_5^2);
│ │ │  
│ │ │  o4 : ProjectiveVariety, hypersurface in PP^5
│ │ │  
│ │ │  i5 : time F = specialCubicFourfold(S,X,NumNodes=>3);
│ │ │ - -- used 0.0119855s (cpu); 0.00881246s (thread); 0s (gc)
│ │ │ + -- used 0.00805083s (cpu); 0.00940807s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, cubic fourfold containing a surface of degree 7 and sectional genus 0
│ │ │  
│ │ │  i6 : time describe F
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 0.54403s (cpu); 0.233284s (thread); 0s (gc)
│ │ │ + -- used 0.930145s (cpu); 0.22887s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = Special cubic fourfold of discriminant 26
│ │ │       containing a 3-nodal surface of degree 7 and sectional genus 0
│ │ │       cut out by 13 hypersurfaces of degree 3
│ │ │  
│ │ │  i7 : assert(F == X)
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Gushel__Mukai__Fourfold.out
│ │ │ @@ -7,22 +7,22 @@
│ │ │  o3 : ProjectiveVariety, surface in PP^8
│ │ │  
│ │ │  i4 : X = projectiveVariety ideal(x_4*x_6-x_3*x_7+x_1*x_8, x_4*x_5-x_2*x_7+x_0*x_8, x_3*x_5-x_2*x_6+x_0*x_8+x_1*x_8-x_5*x_8, x_1*x_5-x_0*x_6+x_0*x_7+x_1*x_7-x_5*x_7, x_1*x_2-x_0*x_3+x_0*x_4+x_1*x_4-x_2*x_7+x_0*x_8, x_0^2+x_0*x_1+x_1^2+x_0*x_2+2*x_0*x_3+x_1*x_3+x_2*x_3+x_3^2-x_0*x_4-x_1*x_4-2*x_2*x_4-x_3*x_4-2*x_4^2+x_0*x_5+x_2*x_5+x_5^2+2*x_0*x_6+x_1*x_6+2*x_2*x_6+x_3*x_6+x_5*x_6+x_6^2-3*x_4*x_7+2*x_5*x_7-x_7^2+x_1*x_8+x_3*x_8-3*x_4*x_8+2*x_5*x_8+x_6*x_8-x_7*x_8);
│ │ │  
│ │ │  o4 : ProjectiveVariety, 4-dimensional subvariety of PP^8
│ │ │  
│ │ │  i5 : time F = specialGushelMukaiFourfold(S,X);
│ │ │ - -- used 2.29841s (cpu); 1.52131s (thread); 0s (gc)
│ │ │ + -- used 2.02782s (cpu); 1.66963s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
│ │ │  
│ │ │  i6 : time describe F
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 5.23566s (cpu); 3.15307s (thread); 0s (gc)
│ │ │ + -- used 6.22934s (cpu); 3.45221s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = Special Gushel-Mukai fourfold of discriminant 10(')
│ │ │       containing a surface in PP^8 of degree 2 and sectional genus 0
│ │ │       cut out by 6 hypersurfaces of degrees (1,1,1,1,1,2)
│ │ │       and with class in G(1,4) given by s_(3,1)+s_(2,2)
│ │ │       Type: ordinary
│ │ │       (case 1 of Table 1 in arXiv:2002.07026)
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__Grass.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  i2 : X = specialGushelMukaiFourfold(ideal(x_6-x_7, x_5, x_3-x_4, x_1, x_0-x_4, x_2*x_7-x_4*x_8), ideal(x_4*x_6-x_3*x_7+x_1*x_8, x_4*x_5-x_2*x_7+x_0*x_8, x_3*x_5-x_2*x_6+x_0*x_8+x_1*x_8-x_5*x_8, x_1*x_5-x_0*x_6+x_0*x_7+x_1*x_7-x_5*x_7, x_1*x_2-x_0*x_3+x_0*x_4+x_1*x_4-x_2*x_7+x_0*x_8, x_0^2+x_0*x_1+x_1^2+x_0*x_2+2*x_0*x_3+x_1*x_3+x_2*x_3+x_3^2-x_0*x_4-x_1*x_4-2*x_2*x_4-x_3*x_4-2*x_4^2+x_0*x_5+x_2*x_5+x_5^2+2*x_0*x_6+x_1*x_6+2*x_2*x_6+x_3*x_6+x_5*x_6+x_6^2-3*x_4*x_7+2*x_5*x_7-x_7^2+x_1*x_8+x_3*x_8-3*x_4*x_8+2*x_5*x_8+x_6*x_8-x_7*x_8));
│ │ │  
│ │ │  o2 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
│ │ │  
│ │ │  i3 : time toGrass X
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 4.03686s (cpu); 2.40993s (thread); 0s (gc)
│ │ │ + -- used 5.02065s (cpu); 2.80836s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │       source variety: 4-dimensional subvariety of PP^8 cut out by 6 hypersurfaces of degree 2
│ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__Grass_lp__Embedded__Projective__Variety_rp.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  i2 : X = projectiveVariety ideal(x_4*x_6-x_3*x_7+x_1*x_8, x_4*x_5-x_2*x_7+x_0*x_8, x_3*x_5-x_2*x_6+x_0*x_8+x_1*x_8-x_5*x_8, x_1*x_5-x_0*x_6+x_0*x_7+x_1*x_7-x_5*x_7, x_1*x_2-x_0*x_3+x_0*x_4+x_1*x_4-x_2*x_7+x_0*x_8);
│ │ │  
│ │ │  o2 : ProjectiveVariety, 5-dimensional subvariety of PP^8
│ │ │  
│ │ │  i3 : time toGrass X
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 4.71225s (cpu); 2.71013s (thread); 0s (gc)
│ │ │ + -- used 5.616s (cpu); 2.99969s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │       source variety: 5-dimensional subvariety of PP^8 cut out by 5 hypersurfaces of degree 2
│ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_unirational__Parametrization.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  o2 : ProjectiveVariety, surface in PP^5
│ │ │  
│ │ │  i3 : X = specialCubicFourfold S;
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
│ │ │  
│ │ │  i4 : time f = unirationalParametrization X;
│ │ │ - -- used 1.16218s (cpu); 0.624472s (thread); 0s (gc)
│ │ │ + -- used 1.14275s (cpu); 0.519467s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : MultirationalMap (rational map from PP^4 to X)
│ │ │  
│ │ │  i5 : degreeSequence f
│ │ │  
│ │ │  o5 = {[10]}
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__Castelnuovo__Surface.html
│ │ │ @@ -106,15 +106,15 @@
│ │ │       cut out by 5 hypersurfaces of degree 1
│ │ │       (This is a classical example of rational fourfold)
    
│ │ │              
    
│ │ │                
    i3 : time U' = associatedCastelnuovoSurface X;
│ │ │ - -- used 2.39568s (cpu); 1.11189s (thread); 0s (gc)
│ │ │ + -- used 2.7641s (cpu); 1.0622s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : ProjectiveVariety, Castelnuovo surface associated to X
    
│ │ │              
    
│ │ │                
    i4 : (mu,U,C,f) = building U';
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,15 +41,15 @@
│ │ │ │  o2 = Complete intersection of 3 quadrics in PP^7
│ │ │ │       of discriminant 31 = det| 8 1 |
│ │ │ │                               | 1 4 |
│ │ │ │       containing a surface of degree 1 and sectional genus 0
│ │ │ │       cut out by 5 hypersurfaces of degree 1
│ │ │ │       (This is a classical example of rational fourfold)
│ │ │ │  i3 : time U' = associatedCastelnuovoSurface X;
│ │ │ │ - -- used 2.39568s (cpu); 1.11189s (thread); 0s (gc)
│ │ │ │ + -- used 2.7641s (cpu); 1.0622s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, Castelnuovo surface associated to X
│ │ │ │  i4 : (mu,U,C,f) = building U';
│ │ │ │  i5 : ? mu
│ │ │ │  
│ │ │ │  o5 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: 5-dimensional subvariety of PP^7 cut out by 2
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__K3surface_lp__Special__Cubic__Fourfold_rp.html
│ │ │ @@ -104,15 +104,15 @@
│ │ │       containing a (smooth) surface of degree 4 and sectional genus 0
│ │ │       cut out by 6 hypersurfaces of degree 2
│ │ │              
    
│ │ │                
    i3 : time U' = associatedK3surface X;
│ │ │ - -- used 1.99111s (cpu); 1.02887s (thread); 0s (gc)
│ │ │ + -- used 2.70739s (cpu); 1.11711s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
    
│ │ │              
    
│ │ │                
    i4 : (mu,U,C,f) = building U';
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,15 +41,15 @@
│ │ │ │  sectional genus 0
│ │ │ │  i2 : describe X
│ │ │ │  
│ │ │ │  o2 = Special cubic fourfold of discriminant 14
│ │ │ │       containing a (smooth) surface of degree 4 and sectional genus 0
│ │ │ │       cut out by 6 hypersurfaces of degree 2
│ │ │ │  i3 : time U' = associatedK3surface X;
│ │ │ │ - -- used 1.99111s (cpu); 1.02887s (thread); 0s (gc)
│ │ │ │ + -- used 2.70739s (cpu); 1.11711s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │ │  i4 : (mu,U,C,f) = building U';
│ │ │ │  i5 : ? mu
│ │ │ │  
│ │ │ │  o5 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: PP^5
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__K3surface_lp__Special__Gushel__Mukai__Fourfold_rp.html
│ │ │ @@ -107,15 +107,15 @@
│ │ │       Type: ordinary
│ │ │       (case 1 of Table 1 in arXiv:2002.07026)
│ │ │              
    
│ │ │                
    i3 : time U' = associatedK3surface X;
│ │ │ - -- used 7.14125s (cpu); 4.38468s (thread); 0s (gc)
│ │ │ + -- used 7.26543s (cpu); 4.3104s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
    
│ │ │              
    
│ │ │                
    i4 : (mu,U,C,f) = building U';
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -43,15 +43,15 @@
│ │ │ │  o2 = Special Gushel-Mukai fourfold of discriminant 10(')
│ │ │ │       containing a surface in PP^8 of degree 2 and sectional genus 0
│ │ │ │       cut out by 6 hypersurfaces of degrees (1,1,1,1,1,2)
│ │ │ │       and with class in G(1,4) given by s_(3,1)+s_(2,2)
│ │ │ │       Type: ordinary
│ │ │ │       (case 1 of Table 1 in arXiv:2002.07026)
│ │ │ │  i3 : time U' = associatedK3surface X;
│ │ │ │ - -- used 7.14125s (cpu); 4.38468s (thread); 0s (gc)
│ │ │ │ + -- used 7.26543s (cpu); 4.3104s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │ │  i4 : (mu,U,C,f) = building U';
│ │ │ │  i5 : ? mu
│ │ │ │  
│ │ │ │  o5 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: 5-dimensional subvariety of PP^8 cut out by 5
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence_lp__Special__Cubic__Fourfold_cm__Z__Z_rp.html
│ │ │ @@ -91,15 +91,15 @@
│ │ │       containing a 3-nodal surface of degree 7 and sectional genus 0
│ │ │       cut out by 13 hypersurfaces of degree 3
│ │ │              
    
│ │ │                
    i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 4.03482s (cpu); 2.10467s (thread); 0s (gc)
│ │ │ + -- used 3.13911s (cpu); 1.89488s (thread); 0s (gc)
│ │ │  number lines contained in the image of the cubic map and passing through a general point: 8
│ │ │  number 2-secant lines = 7
│ │ │  number 5-secant conics = 1
│ │ │  
│ │ │  o3 : Congruence of 5-secant conics to surface in PP^5
    
│ │ │              
    
│ │ │                
    i5 : time C = f p; -- 5-secant conic to the surface
│ │ │ - -- used 0.429617s (cpu); 0.294228s (thread); 0s (gc)
│ │ │ + -- used 0.341292s (cpu); 0.28804s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, curve in PP^5
    
│ │ │              
    
│ │ │                
    i6 : assert(dim C == 1 and degree C == 2 and dim(C * surface X) == 0 and degree(C * surface X) == 5 and isSubset(p, C))
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,28 +30,28 @@
│ │ │ │  sectional genus 0
│ │ │ │  i2 : describe X
│ │ │ │  
│ │ │ │  o2 = Special cubic fourfold of discriminant 26
│ │ │ │       containing a 3-nodal surface of degree 7 and sectional genus 0
│ │ │ │       cut out by 13 hypersurfaces of degree 3
│ │ │ │  i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ │ - -- used 4.03482s (cpu); 2.10467s (thread); 0s (gc)
│ │ │ │ + -- used 3.13911s (cpu); 1.89488s (thread); 0s (gc)
│ │ │ │  number lines contained in the image of the cubic map and passing through a
│ │ │ │  general point: 8
│ │ │ │  number 2-secant lines = 7
│ │ │ │  number 5-secant conics = 1
│ │ │ │  
│ │ │ │  o3 : Congruence of 5-secant conics to surface in PP^5
│ │ │ │  i4 : p := point ambient X -- random point on P^5
│ │ │ │  
│ │ │ │  o4 = point of coordinates [15092, -9738, -3620, -15181, 12688, 1]
│ │ │ │  
│ │ │ │  o4 : ProjectiveVariety, a point in PP^5
│ │ │ │  i5 : time C = f p; -- 5-secant conic to the surface
│ │ │ │ - -- used 0.429617s (cpu); 0.294228s (thread); 0s (gc)
│ │ │ │ + -- used 0.341292s (cpu); 0.28804s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, curve in PP^5
│ │ │ │  i6 : assert(dim C == 1 and degree C == 2 and dim(C * surface X) == 0 and degree
│ │ │ │  (C * surface X) == 5 and isSubset(p, C))
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_e_t_e_c_t_C_o_n_g_r_u_e_n_c_e_(_S_p_e_c_i_a_l_G_u_s_h_e_l_M_u_k_a_i_F_o_u_r_f_o_l_d_,_Z_Z_) -- detect and return a
│ │ │ │        congruence of (2e-1)-secant curves of degree e inside a del Pezzo
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence_lp__Special__Gushel__Mukai__Fourfold_cm__Z__Z_rp.html
│ │ │ @@ -94,15 +94,15 @@
│ │ │       Type: ordinary
│ │ │       (case 17 of Table 1 in arXiv:2002.07026)
│ │ │              
    
│ │ │                
    i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 13.4563s (cpu); 7.009s (thread); 0s (gc)
│ │ │ + -- used 19.4691s (cpu); 7.34788s (thread); 0s (gc)
│ │ │  number lines contained in the image of the quadratic map and passing through a general point: 7
│ │ │  number 1-secant lines = 6
│ │ │  number 3-secant conics = 1
│ │ │  
│ │ │  o3 : Congruence of 3-secant conics to surface in a fivefold in PP^8
    
│ │ │              
    
│ │ │                
    i6 : time C = f p; -- 3-secant conic to the surface
│ │ │ - -- used 0.396s (cpu); 0.234093s (thread); 0s (gc)
│ │ │ + -- used 0.489967s (cpu); 0.302939s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : ProjectiveVariety, curve in PP^8 (subvariety of codimension 4 in Y)
    
│ │ │              
    
│ │ │                
    i7 : S = surface X;
│ │ │ ├── html2text {}
│ │ │ │ @@ -36,15 +36,15 @@
│ │ │ │  o2 = Special Gushel-Mukai fourfold of discriminant 20
│ │ │ │       containing a surface in PP^8 of degree 9 and sectional genus 2
│ │ │ │       cut out by 19 hypersurfaces of degree 2
│ │ │ │       and with class in G(1,4) given by 6*s_(3,1)+3*s_(2,2)
│ │ │ │       Type: ordinary
│ │ │ │       (case 17 of Table 1 in arXiv:2002.07026)
│ │ │ │  i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ │ - -- used 13.4563s (cpu); 7.009s (thread); 0s (gc)
│ │ │ │ + -- used 19.4691s (cpu); 7.34788s (thread); 0s (gc)
│ │ │ │  number lines contained in the image of the quadratic map and passing through a
│ │ │ │  general point: 7
│ │ │ │  number 1-secant lines = 6
│ │ │ │  number 3-secant conics = 1
│ │ │ │  
│ │ │ │  o3 : Congruence of 3-secant conics to surface in a fivefold in PP^8
│ │ │ │  i4 : Y = ambientFivefold X; -- del Pezzo fivefold containing X
│ │ │ │ @@ -53,15 +53,15 @@
│ │ │ │  i5 : p := point Y -- random point on Y
│ │ │ │  
│ │ │ │  o5 = point of coordinates [7214, -1460, 7057, -2440, 15907, -14345, -5937,
│ │ │ │  13402, 1]
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, a point in PP^8
│ │ │ │  i6 : time C = f p; -- 3-secant conic to the surface
│ │ │ │ - -- used 0.396s (cpu); 0.234093s (thread); 0s (gc)
│ │ │ │ + -- used 0.489967s (cpu); 0.302939s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 : ProjectiveVariety, curve in PP^8 (subvariety of codimension 4 in Y)
│ │ │ │  i7 : S = surface X;
│ │ │ │  
│ │ │ │  o7 : ProjectiveVariety, surface in PP^8 (subvariety of codimension 3 in Y)
│ │ │ │  i8 : assert(dim C == 1 and degree C == 2 and dim(C*S) == 0 and degree(C*S) == 3
│ │ │ │  and isSubset(p,C) and isSubset(C,Y))
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_discriminant_lp__Special__Cubic__Fourfold_rp.html
│ │ │ @@ -80,15 +80,15 @@
│ │ │  
│ │ │  o1 : ProjectiveVariety, cubic fourfold containing a surface of degree 5 and sectional genus 1
    
│ │ │              
    
│ │ │                
    i2 : time discriminant X
│ │ │ - -- used 0.420819s (cpu); 0.147554s (thread); 0s (gc)
│ │ │ + -- used 0.36452s (cpu); 0.12517s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 14
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │  thanks to the functions _E_u_l_e_r_C_h_a_r_a_c_t_e_r_i_s_t_i_c and _E_u_l_e_r (the option Algorithm
│ │ │ │  allows you to select the method).
│ │ │ │  i1 : X = specialCubicFourfold "quintic del Pezzo surface";
│ │ │ │  
│ │ │ │  o1 : ProjectiveVariety, cubic fourfold containing a surface of degree 5 and
│ │ │ │  sectional genus 1
│ │ │ │  i2 : time discriminant X
│ │ │ │ - -- used 0.420819s (cpu); 0.147554s (thread); 0s (gc)
│ │ │ │ + -- used 0.36452s (cpu); 0.12517s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = 14
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_i_s_c_r_i_m_i_n_a_n_t_(_S_p_e_c_i_a_l_G_u_s_h_e_l_M_u_k_a_i_F_o_u_r_f_o_l_d_) -- discriminant of a special
│ │ │ │        Gushel-Mukai fourfold
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * discriminant(HodgeSpecialFourfold)
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_discriminant_lp__Special__Gushel__Mukai__Fourfold_rp.html
│ │ │ @@ -80,15 +80,15 @@
│ │ │  
│ │ │  o1 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time discriminant X
│ │ │ - -- used 1.02747s (cpu); 0.458878s (thread); 0s (gc)
│ │ │ + -- used 1.07787s (cpu); 0.467581s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │  the functions _c_y_c_l_e_C_l_a_s_s, _E_u_l_e_r_C_h_a_r_a_c_t_e_r_i_s_t_i_c and _E_u_l_e_r (the option Algorithm
│ │ │ │  allows you to select the method).
│ │ │ │  i1 : X = specialGushelMukaiFourfold "tau-quadric";
│ │ │ │  
│ │ │ │  o1 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and
│ │ │ │  sectional genus 0
│ │ │ │  i2 : time discriminant X
│ │ │ │ - -- used 1.02747s (cpu); 0.458878s (thread); 0s (gc)
│ │ │ │ + -- used 1.07787s (cpu); 0.467581s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = 10
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_i_s_c_r_i_m_i_n_a_n_t_(_S_p_e_c_i_a_l_C_u_b_i_c_F_o_u_r_f_o_l_d_) -- discriminant of a special cubic
│ │ │ │        fourfold
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _d_i_s_c_r_i_m_i_n_a_n_t_(_S_p_e_c_i_a_l_G_u_s_h_e_l_M_u_k_a_i_F_o_u_r_f_o_l_d_) -- discriminant of a special
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count.html
│ │ │ @@ -88,15 +88,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, surface in PP^5
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time parameterCount(S,X,Verbose=>true)
│ │ │ - -- used 0.296112s (cpu); 0.200839s (thread); 0s (gc)
│ │ │ + -- used 0.465297s (cpu); 0.254661s (thread); 0s (gc)
│ │ │  S: rational normal curve of degree 5 in PP^5
│ │ │  X: smooth surface of degree 8 and sectional genus 5 in PP^5 cut out by 3 hypersurfaces of degree 2
│ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │  h^0(N_{S,P^5}) = 32
│ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,P^5}(2)) = 10 = h^0(O_(P^5)(2)) - \chi(O_S(2));
│ │ │  in particular, h^0(I_{S,P^5}(2)) is minimal
│ │ │  dim GG(2,9) = 21
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,15 +23,15 @@
│ │ │ │  i1 : K = ZZ/33331; S = PP_K^(1,5);
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, curve in PP^5
│ │ │ │  i3 : X = random({{2},{2},{2}},S);
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, surface in PP^5
│ │ │ │  i4 : time parameterCount(S,X,Verbose=>true)
│ │ │ │ - -- used 0.296112s (cpu); 0.200839s (thread); 0s (gc)
│ │ │ │ + -- used 0.465297s (cpu); 0.254661s (thread); 0s (gc)
│ │ │ │  S: rational normal curve of degree 5 in PP^5
│ │ │ │  X: smooth surface of degree 8 and sectional genus 5 in PP^5 cut out by 3
│ │ │ │  hypersurfaces of degree 2
│ │ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │ │  h^0(N_{S,P^5}) = 32
│ │ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,P^5}(2)) = 10 = h^0(O_(P^5)(2)) - \chi(O_S(2));
│ │ │ │  in particular, h^0(I_{S,P^5}(2)) is minimal
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Special__Cubic__Fourfold_rp.html
│ │ │ @@ -89,15 +89,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 0.641923s (cpu); 0.367693s (thread); 0s (gc)
│ │ │ + -- used 0.782566s (cpu); 0.507141s (thread); 0s (gc)
│ │ │  S: Veronese surface in PP^5
│ │ │  X: smooth cubic hypersurface in PP^5
│ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │  h^0(N_{S,P^5}) = 27
│ │ │  h^1(O_S(3)) = 0, and h^0(I_{S,P^5}(3)) = 28 = h^0(O_(P^5)(3)) - \chi(O_S(3));
│ │ │  in particular, h^0(I_{S,P^5}(3)) is minimal
│ │ │  h^0(N_{S,P^5}) + 27 = 54
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,15 +33,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, surface in PP^5
│ │ │ │  i3 : X = specialCubicFourfold V;
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and
│ │ │ │  sectional genus 0
│ │ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ │ - -- used 0.641923s (cpu); 0.367693s (thread); 0s (gc)
│ │ │ │ + -- used 0.782566s (cpu); 0.507141s (thread); 0s (gc)
│ │ │ │  S: Veronese surface in PP^5
│ │ │ │  X: smooth cubic hypersurface in PP^5
│ │ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │ │  h^0(N_{S,P^5}) = 27
│ │ │ │  h^1(O_S(3)) = 0, and h^0(I_{S,P^5}(3)) = 28 = h^0(O_(P^5)(3)) - \chi(O_S(3));
│ │ │ │  in particular, h^0(I_{S,P^5}(3)) is minimal
│ │ │ │  h^0(N_{S,P^5}) + 27 = 54
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Special__Gushel__Mukai__Fourfold_rp.html
│ │ │ @@ -98,15 +98,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, GM fourfold containing a surface of degree 3 and sectional genus 0
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 3.48579s (cpu); 2.16969s (thread); 0s (gc)
│ │ │ + -- used 3.80878s (cpu); 2.76661s (thread); 0s (gc)
│ │ │  S: cubic surface in PP^8 cut out by 7 hypersurfaces of degrees (1,1,1,1,2,2,2)
│ │ │  X: GM fourfold containing S
│ │ │  Y: del Pezzo fivefold containing X
│ │ │  h^1(N_{S,Y}) = 0
│ │ │  h^0(N_{S,Y}) = 11
│ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,Y}(2)) = 28 = h^0(O_Y(2)) - \chi(O_S(2));
│ │ │  in particular, h^0(I_{S,Y}(2)) is minimal
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,15 +35,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, surface in PP^9 (subvariety of codimension 4 in G)
│ │ │ │  i3 : X = specialGushelMukaiFourfold S;
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, GM fourfold containing a surface of degree 3 and
│ │ │ │  sectional genus 0
│ │ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ │ - -- used 3.48579s (cpu); 2.16969s (thread); 0s (gc)
│ │ │ │ + -- used 3.80878s (cpu); 2.76661s (thread); 0s (gc)
│ │ │ │  S: cubic surface in PP^8 cut out by 7 hypersurfaces of degrees (1,1,1,1,2,2,2)
│ │ │ │  X: GM fourfold containing S
│ │ │ │  Y: del Pezzo fivefold containing X
│ │ │ │  h^1(N_{S,Y}) = 0
│ │ │ │  h^0(N_{S,Y}) = 11
│ │ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,Y}(2)) = 28 = h^0(O_Y(2)) - \chi(O_S(2));
│ │ │ │  in particular, h^0(I_{S,Y}(2)) is minimal
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parametrize__Fano__Fourfold.html
│ │ │ @@ -88,15 +88,15 @@
│ │ │  o3 = 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees
│ │ │       1^2 2^5
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time parametrizeFanoFourfold X
│ │ │ - -- used 1.79691s (cpu); 0.888229s (thread); 0s (gc)
│ │ │ + -- used 1.68186s (cpu); 0.773295s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = multi-rational map consisting of one single rational map
│ │ │       source variety: PP^4
│ │ │       target variety: 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees 1^2 2^5 
│ │ │       dominance: true
│ │ │       degree: 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, 4-dimensional subvariety of PP^9
│ │ │ │  i3 : ? X
│ │ │ │  
│ │ │ │  o3 = 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees
│ │ │ │       1^2 2^5
│ │ │ │  i4 : time parametrizeFanoFourfold X
│ │ │ │ - -- used 1.79691s (cpu); 0.888229s (thread); 0s (gc)
│ │ │ │ + -- used 1.68186s (cpu); 0.773295s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: PP^4
│ │ │ │       target variety: 4-dimensional subvariety of PP^9 cut out by 7
│ │ │ │  hypersurfaces of degrees 1^2 2^5
│ │ │ │       dominance: true
│ │ │ │       degree: 1
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Cubic__Fourfold.html
│ │ │ @@ -95,25 +95,25 @@
│ │ │  
│ │ │  o4 : ProjectiveVariety, hypersurface in PP^5
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time F = specialCubicFourfold(S,X,NumNodes=>3);
│ │ │ - -- used 0.0119855s (cpu); 0.00881246s (thread); 0s (gc)
│ │ │ + -- used 0.00805083s (cpu); 0.00940807s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, cubic fourfold containing a surface of degree 7 and sectional genus 0
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time describe F
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 0.54403s (cpu); 0.233284s (thread); 0s (gc)
│ │ │ + -- used 0.930145s (cpu); 0.22887s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = Special cubic fourfold of discriminant 26
│ │ │       containing a 3-nodal surface of degree 7 and sectional genus 0
│ │ │       cut out by 13 hypersurfaces of degree 3
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -115,24 +115,24 @@
│ │ │ │  3*x_1*x_2*x_4+3*x_2^2*x_4+14*x_0*x_3*x_4-8*x_1*x_3*x_4-4*x_3^2*x_4+x_0*x_4^2-
│ │ │ │  7*x_1*x_4^2+4*x_2*x_4^2-2*x_3*x_4^2-2*x_4^3-
│ │ │ │  x_0*x_1*x_5+x_1^2*x_5+2*x_1*x_2*x_5+3*x_0*x_3*x_5+3*x_1*x_3*x_5-x_3^2*x_5-
│ │ │ │  x_0*x_4*x_5-4*x_1*x_4*x_5+3*x_2*x_4*x_5+2*x_3*x_4*x_5-x_1*x_5^2);
│ │ │ │  
│ │ │ │  o4 : ProjectiveVariety, hypersurface in PP^5
│ │ │ │  i5 : time F = specialCubicFourfold(S,X,NumNodes=>3);
│ │ │ │ - -- used 0.0119855s (cpu); 0.00881246s (thread); 0s (gc)
│ │ │ │ + -- used 0.00805083s (cpu); 0.00940807s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, cubic fourfold containing a surface of degree 7 and
│ │ │ │  sectional genus 0
│ │ │ │  i6 : time describe F
│ │ │ │  warning: clearing value of symbol x to allow access to subscripted variables
│ │ │ │  based on it
│ │ │ │         : debug with expression   debug 9868   or with command line option   --
│ │ │ │  debug 9868
│ │ │ │ - -- used 0.54403s (cpu); 0.233284s (thread); 0s (gc)
│ │ │ │ + -- used 0.930145s (cpu); 0.22887s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = Special cubic fourfold of discriminant 26
│ │ │ │       containing a 3-nodal surface of degree 7 and sectional genus 0
│ │ │ │       cut out by 13 hypersurfaces of degree 3
│ │ │ │  i7 : assert(F == X)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _s_p_e_c_i_a_l_C_u_b_i_c_F_o_u_r_f_o_l_d_(_E_m_b_e_d_d_e_d_P_r_o_j_e_c_t_i_v_e_V_a_r_i_e_t_y_) -- random special cubic
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Gushel__Mukai__Fourfold.html
│ │ │ @@ -93,25 +93,25 @@
│ │ │  
│ │ │  o4 : ProjectiveVariety, 4-dimensional subvariety of PP^8
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time F = specialGushelMukaiFourfold(S,X);
│ │ │ - -- used 2.29841s (cpu); 1.52131s (thread); 0s (gc)
│ │ │ + -- used 2.02782s (cpu); 1.66963s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time describe F
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 5.23566s (cpu); 3.15307s (thread); 0s (gc)
│ │ │ + -- used 6.22934s (cpu); 3.45221s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = Special Gushel-Mukai fourfold of discriminant 10(')
│ │ │       containing a surface in PP^8 of degree 2 and sectional genus 0
│ │ │       cut out by 6 hypersurfaces of degrees (1,1,1,1,1,2)
│ │ │       and with class in G(1,4) given by s_(3,1)+s_(2,2)
│ │ │       Type: ordinary
│ │ │       (case 1 of Table 1 in arXiv:2002.07026)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,24 +33,24 @@
│ │ │ │  x_2*x_7+x_0*x_8, x_0^2+x_0*x_1+x_1^2+x_0*x_2+2*x_0*x_3+x_1*x_3+x_2*x_3+x_3^2-
│ │ │ │  x_0*x_4-x_1*x_4-2*x_2*x_4-x_3*x_4-
│ │ │ │  2*x_4^2+x_0*x_5+x_2*x_5+x_5^2+2*x_0*x_6+x_1*x_6+2*x_2*x_6+x_3*x_6+x_5*x_6+x_6^2-
│ │ │ │  3*x_4*x_7+2*x_5*x_7-x_7^2+x_1*x_8+x_3*x_8-3*x_4*x_8+2*x_5*x_8+x_6*x_8-x_7*x_8);
│ │ │ │  
│ │ │ │  o4 : ProjectiveVariety, 4-dimensional subvariety of PP^8
│ │ │ │  i5 : time F = specialGushelMukaiFourfold(S,X);
│ │ │ │ - -- used 2.29841s (cpu); 1.52131s (thread); 0s (gc)
│ │ │ │ + -- used 2.02782s (cpu); 1.66963s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and
│ │ │ │  sectional genus 0
│ │ │ │  i6 : time describe F
│ │ │ │  warning: clearing value of symbol x to allow access to subscripted variables
│ │ │ │  based on it
│ │ │ │         : debug with expression   debug 9868   or with command line option   --
│ │ │ │  debug 9868
│ │ │ │ - -- used 5.23566s (cpu); 3.15307s (thread); 0s (gc)
│ │ │ │ + -- used 6.22934s (cpu); 3.45221s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = Special Gushel-Mukai fourfold of discriminant 10(')
│ │ │ │       containing a surface in PP^8 of degree 2 and sectional genus 0
│ │ │ │       cut out by 6 hypersurfaces of degrees (1,1,1,1,1,2)
│ │ │ │       and with class in G(1,4) given by s_(3,1)+s_(2,2)
│ │ │ │       Type: ordinary
│ │ │ │       (case 1 of Table 1 in arXiv:2002.07026)
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass.html
│ │ │ @@ -81,15 +81,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time toGrass X
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 4.03686s (cpu); 2.40993s (thread); 0s (gc)
│ │ │ + -- used 5.02065s (cpu); 2.80836s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │       source variety: 4-dimensional subvariety of PP^8 cut out by 6 hypersurfaces of degree 2
│ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
    
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │  o2 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and
│ │ │ │  sectional genus 0
│ │ │ │  i3 : time toGrass X
│ │ │ │  warning: clearing value of symbol x to allow access to subscripted variables
│ │ │ │  based on it
│ │ │ │         : debug with expression   debug 9868   or with command line option   --
│ │ │ │  debug 9868
│ │ │ │ - -- used 4.03686s (cpu); 2.40993s (thread); 0s (gc)
│ │ │ │ + -- used 5.02065s (cpu); 2.80836s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: 4-dimensional subvariety of PP^8 cut out by 6 hypersurfaces
│ │ │ │  of degree 2
│ │ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass_lp__Embedded__Projective__Variety_rp.html
│ │ │ @@ -82,15 +82,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time toGrass X
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 4.71225s (cpu); 2.71013s (thread); 0s (gc)
│ │ │ + -- used 5.616s (cpu); 2.99969s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │       source variety: 5-dimensional subvariety of PP^8 cut out by 5 hypersurfaces of degree 2
│ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
    
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,15 +25,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, 5-dimensional subvariety of PP^8
│ │ │ │  i3 : time toGrass X
│ │ │ │  warning: clearing value of symbol x to allow access to subscripted variables
│ │ │ │  based on it
│ │ │ │         : debug with expression   debug 9868   or with command line option   --
│ │ │ │  debug 9868
│ │ │ │ - -- used 4.71225s (cpu); 2.71013s (thread); 0s (gc)
│ │ │ │ + -- used 5.616s (cpu); 2.99969s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: 5-dimensional subvariety of PP^8 cut out by 5
│ │ │ │  hypersurfaces of degree 2
│ │ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_unirational__Parametrization.html
│ │ │ @@ -82,15 +82,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time f = unirationalParametrization X;
│ │ │ - -- used 1.16218s (cpu); 0.624472s (thread); 0s (gc)
│ │ │ + -- used 1.14275s (cpu); 0.519467s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : MultirationalMap (rational map from PP^4 to X)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : degreeSequence f
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,15 +18,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, surface in PP^5
│ │ │ │  i3 : X = specialCubicFourfold S;
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and
│ │ │ │  sectional genus 0
│ │ │ │  i4 : time f = unirationalParametrization X;
│ │ │ │ - -- used 1.16218s (cpu); 0.624472s (thread); 0s (gc)
│ │ │ │ + -- used 1.14275s (cpu); 0.519467s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : MultirationalMap (rational map from PP^4 to X)
│ │ │ │  i5 : degreeSequence f
│ │ │ │  
│ │ │ │  o5 = {[10]}
│ │ │ │  
│ │ │ │  o5 : List
│ │ ├── ./usr/share/doc/Macaulay2/StatGraphs/example-output/_graph_lp__Mixed__Graph_rp.out
│ │ │ @@ -30,15 +30,15 @@
│ │ │                                b => {a, c}
│ │ │                                c => {b}
│ │ │  
│ │ │  o2 : HashTable
│ │ │  
│ │ │  i3 : keys (graph G)
│ │ │  
│ │ │ -o3 = {Graph, Bigraph, Digraph}
│ │ │ +o3 = {Digraph, Graph, Bigraph}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : (graph G)#Bigraph === bigraph G
│ │ │  
│ │ │  o4 = true
│ │ ├── ./usr/share/doc/Macaulay2/StatGraphs/html/_graph_lp__Mixed__Graph_rp.html
│ │ │ @@ -116,15 +116,15 @@
│ │ │  o2 : HashTable
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : keys (graph G)
│ │ │  
│ │ │ -o3 = {Graph, Bigraph, Digraph}
│ │ │ +o3 = {Digraph, Graph, Bigraph}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : (graph G)#Bigraph === bigraph G
│ │ │ ├── html2text {}
│ │ │ │ @@ -46,15 +46,15 @@
│ │ │ │                 Graph => Graph{a => {b}   }
│ │ │ │                                b => {a, c}
│ │ │ │                                c => {b}
│ │ │ │  
│ │ │ │  o2 : HashTable
│ │ │ │  i3 : keys (graph G)
│ │ │ │  
│ │ │ │ -o3 = {Graph, Bigraph, Digraph}
│ │ │ │ +o3 = {Digraph, Graph, Bigraph}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : (graph G)#Bigraph === bigraph G
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _M_i_x_e_d_G_r_a_p_h -- a graph that has undirected, directed and bidirected edges
│ │ ├── ./usr/share/doc/Macaulay2/Style/example-output/_generate__Grammar.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 3455701143666534588
│ │ │  
│ │ │  i1 : outfile = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10069-0/0
│ │ │ +o1 = /tmp/M2-10109-0/0
│ │ │  
│ │ │  i2 : template = outfile | ".in"
│ │ │  
│ │ │ -o2 = /tmp/M2-10069-0/0.in
│ │ │ +o2 = /tmp/M2-10109-0/0.in
│ │ │  
│ │ │  i3 : template << "@M2BANNER@" << endl << endl;
│ │ │  
│ │ │  i4 : template << "This is an example file for the generateGrammar method!";
│ │ │  
│ │ │  i5 : template << endl;
│ │ │  
│ │ │ @@ -30,15 +30,15 @@
│ │ │        String regex: @M2STRINGS@
│ │ │        List of keywords: {
│ │ │            @M2KEYWORDS@
│ │ │        }
│ │ │  
│ │ │  
│ │ │  i11 : generateGrammar(template, outfile, x -> demark(",\n    ", x))
│ │ │ - -- generating /tmp/M2-10069-0/0
│ │ │ + -- generating /tmp/M2-10109-0/0
│ │ │  
│ │ │  i12 : get outfile
│ │ │  
│ │ │  o12 = Auto-generated for Macaulay2-1.25.11. Do not modify this file manually.
│ │ │  
│ │ │        This is an example file for the generateGrammar method!
│ │ │        String regex: "///\\(/?/?[^/]\\|\\(//\\)*////[^/]\\)*\\(//\\)*///"
│ │ ├── ./usr/share/doc/Macaulay2/Style/html/_generate__Grammar.html
│ │ │ @@ -82,22 +82,22 @@
│ │ │            
    The function demarkf indicates how the elements of each of the lists will be demarked in the resulting file.   The file outfile will then be generated, replacing each of these strings as indicated above.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -143,15 +143,15 @@
│ │ │            @M2KEYWORDS@
│ │ │        }
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : outfile = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10069-0/0
    
│ │ │ +o1 = /tmp/M2-10109-0/0
│ │ │              
    
│ │ │                
    i2 : template = outfile | ".in"
│ │ │  
│ │ │ -o2 = /tmp/M2-10069-0/0.in
    
│ │ │ +o2 = /tmp/M2-10109-0/0.in
│ │ │              
    
│ │ │                
    i3 : template << "@M2BANNER@" << endl << endl;
    
│ │ │              
    
│ │ │                
    i11 : generateGrammar(template, outfile, x -> demark(",\n    ", x))
│ │ │ - -- generating /tmp/M2-10069-0/0
    
│ │ │ + -- generating /tmp/M2-10109-0/0
│ │ │              
    
│ │ │                
    i12 : get outfile
│ │ │  
│ │ │  o12 = Auto-generated for Macaulay2-1.25.11. Do not modify this file manually.
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,18 +26,18 @@
│ │ │ │      * @M2CONSTANTS@, for a list of Macaulay2 symbols and packages.
│ │ │ │      * @M2STRINGS@, for a regular expression that matches Macaulay2 strings.
│ │ │ │  The function demarkf indicates how the elements of each of the lists will be
│ │ │ │  demarked in the resulting file. The file outfile will then be generated,
│ │ │ │  replacing each of these strings as indicated above.
│ │ │ │  i1 : outfile = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10069-0/0
│ │ │ │ +o1 = /tmp/M2-10109-0/0
│ │ │ │  i2 : template = outfile | ".in"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10069-0/0.in
│ │ │ │ +o2 = /tmp/M2-10109-0/0.in
│ │ │ │  i3 : template << "@M2BANNER@" << endl << endl;
│ │ │ │  i4 : template << "This is an example file for the generateGrammar method!";
│ │ │ │  i5 : template << endl;
│ │ │ │  i6 : template << "String regex: @M2STRINGS@" << endl;
│ │ │ │  i7 : template << "List of keywords: {" << endl;
│ │ │ │  i8 : template << "    @M2KEYWORDS@" << endl;
│ │ │ │  i9 : template << "}" << endl << close;
│ │ │ │ @@ -47,15 +47,15 @@
│ │ │ │  
│ │ │ │        This is an example file for the generateGrammar method!
│ │ │ │        String regex: @M2STRINGS@
│ │ │ │        List of keywords: {
│ │ │ │            @M2KEYWORDS@
│ │ │ │        }
│ │ │ │  i11 : generateGrammar(template, outfile, x -> demark(",\n    ", x))
│ │ │ │ - -- generating /tmp/M2-10069-0/0
│ │ │ │ + -- generating /tmp/M2-10109-0/0
│ │ │ │  i12 : get outfile
│ │ │ │  
│ │ │ │  o12 = Auto-generated for Macaulay2-1.25.11. Do not modify this file manually.
│ │ │ │  
│ │ │ │        This is an example file for the generateGrammar method!
│ │ │ │        String regex: "///\\(/?/?[^/]\\|\\(//\\)*////[^/]\\)*\\(//\\)*///"
│ │ │ │        List of keywords: {
│ │ ├── ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symbolic__Power.out
│ │ │ @@ -31,15 +31,15 @@
│ │ │  o5 : Ideal of QQ[x..z]
│ │ │  
│ │ │  i6 : isHomogeneous P
│ │ │  
│ │ │  o6 = false
│ │ │  
│ │ │  i7 : time symbolicPower(P,4);
│ │ │ - -- used 0.306855s (cpu); 0.192479s (thread); 0s (gc)
│ │ │ + -- used 0.275996s (cpu); 0.175133s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of QQ[x..z]
│ │ │  
│ │ │  i8 : Q = ker map(QQ[t],QQ[x,y,z, Degrees => {3,4,5}],{t^3,t^4,t^5})
│ │ │  
│ │ │               2         3         2     2
│ │ │  o8 = ideal (y  - x*z, x  - y*z, x y - z )
│ │ │ @@ -47,12 +47,12 @@
│ │ │  o8 : Ideal of QQ[x..z]
│ │ │  
│ │ │  i9 : isHomogeneous Q
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : time symbolicPower(Q,4);
│ │ │ - -- used 0.0330497s (cpu); 0.0330527s (thread); 0s (gc)
│ │ │ + -- used 0.0363619s (cpu); 0.036366s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of QQ[x..z]
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symbolic__Power.html
│ │ │ @@ -141,15 +141,15 @@
│ │ │  
│ │ │  o6 = false
    
│ │ │              
    
│ │ │                
    i7 : time symbolicPower(P,4);
│ │ │ - -- used 0.306855s (cpu); 0.192479s (thread); 0s (gc)
│ │ │ + -- used 0.275996s (cpu); 0.175133s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of QQ[x..z]
    
│ │ │              
    
│ │ │                
    i8 : Q = ker map(QQ[t],QQ[x,y,z, Degrees => {3,4,5}],{t^3,t^4,t^5})
│ │ │ @@ -166,15 +166,15 @@
│ │ │  
│ │ │  o9 = true
    
│ │ │              
    
│ │ │                
    i10 : time symbolicPower(Q,4);
│ │ │ - -- used 0.0330497s (cpu); 0.0330527s (thread); 0s (gc)
│ │ │ + -- used 0.0363619s (cpu); 0.036366s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of QQ[x..z]
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -59,28 +59,28 @@
│ │ │ │  o5 = ideal (y  - x*z, x y - z , x  - y*z)
│ │ │ │  
│ │ │ │  o5 : Ideal of QQ[x..z]
│ │ │ │  i6 : isHomogeneous P
│ │ │ │  
│ │ │ │  o6 = false
│ │ │ │  i7 : time symbolicPower(P,4);
│ │ │ │ - -- used 0.306855s (cpu); 0.192479s (thread); 0s (gc)
│ │ │ │ + -- used 0.275996s (cpu); 0.175133s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : Ideal of QQ[x..z]
│ │ │ │  i8 : Q = ker map(QQ[t],QQ[x,y,z, Degrees => {3,4,5}],{t^3,t^4,t^5})
│ │ │ │  
│ │ │ │               2         3         2     2
│ │ │ │  o8 = ideal (y  - x*z, x  - y*z, x y - z )
│ │ │ │  
│ │ │ │  o8 : Ideal of QQ[x..z]
│ │ │ │  i9 : isHomogeneous Q
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  i10 : time symbolicPower(Q,4);
│ │ │ │ - -- used 0.0330497s (cpu); 0.0330527s (thread); 0s (gc)
│ │ │ │ + -- used 0.0363619s (cpu); 0.036366s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 : Ideal of QQ[x..z]
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _s_y_m_b_P_o_w_e_r_P_r_i_m_e_P_o_s_C_h_a_r
│ │ │ │  ********** WWaayyss ttoo uussee ssyymmbboolliiccPPoowweerr:: **********
│ │ │ │      * symbolicPower(Ideal,ZZ)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_beilinson__Window.out
│ │ │ @@ -10,15 +10,15 @@
│ │ │  o3 = 0  <-- E  <-- 0
│ │ │                      
│ │ │       -1     0      1
│ │ │  
│ │ │  o3 : ChainComplex
│ │ │  
│ │ │  i4 : time T=tateExtension W;
│ │ │ - -- used 0.116725s (cpu); 0.116725s (thread); 0s (gc)
│ │ │ + -- used 0.130267s (cpu); 0.130268s (thread); 0s (gc)
│ │ │  
│ │ │  i5 : cohomologyMatrix(T,-{3,3},{3,3})
│ │ │  
│ │ │  o5 = | 8h  4h  0 4  8  12 16 |
│ │ │       | 6h  3h  0 3  6  9  12 |
│ │ │       | 4h  2h  0 2  4  6  8  |
│ │ │       | 2h  h   0 1  2  3  4  |
│ │ ├── ./usr/share/doc/Macaulay2/TateOnProducts/html/_beilinson__Window.html
│ │ │ @@ -92,15 +92,15 @@
│ │ │  
│ │ │  o3 : ChainComplex
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time T=tateExtension W;
│ │ │ - -- used 0.116725s (cpu); 0.116725s (thread); 0s (gc)
    
│ │ │ + -- used 0.130267s (cpu); 0.130268s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : cohomologyMatrix(T,-{3,3},{3,3})
│ │ │  
│ │ │  o5 = | 8h  4h  0 4  8  12 16 |
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,15 +23,15 @@
│ │ │ │               1
│ │ │ │  o3 = 0  <-- E  <-- 0
│ │ │ │  
│ │ │ │       -1     0      1
│ │ │ │  
│ │ │ │  o3 : ChainComplex
│ │ │ │  i4 : time T=tateExtension W;
│ │ │ │ - -- used 0.116725s (cpu); 0.116725s (thread); 0s (gc)
│ │ │ │ + -- used 0.130267s (cpu); 0.130268s (thread); 0s (gc)
│ │ │ │  i5 : cohomologyMatrix(T,-{3,3},{3,3})
│ │ │ │  
│ │ │ │  o5 = | 8h  4h  0 4  8  12 16 |
│ │ │ │       | 6h  3h  0 3  6  9  12 |
│ │ │ │       | 4h  2h  0 2  4  6  8  |
│ │ │ │       | 2h  h   0 1  2  3  4  |
│ │ │ │       | 0   0   0 0  0  0  0  |
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Root.out
│ │ │ @@ -63,20 +63,20 @@
│ │ │  o15 : Ideal of R
│ │ │  
│ │ │  i16 : I3 = ideal(x^50*y^50*z^50);
│ │ │  
│ │ │  o16 : Ideal of R
│ │ │  
│ │ │  i17 : time J1 = frobeniusRoot(1, {8, 10, 12}, {I1, I2, I3});
│ │ │ - -- used 0.854696s (cpu); 0.665482s (thread); 0s (gc)
│ │ │ + -- used 1.07015s (cpu); 0.783874s (thread); 0s (gc)
│ │ │  
│ │ │  o17 : Ideal of R
│ │ │  
│ │ │  i18 : time J2 = frobeniusRoot(1, I1^8*I2^10*I3^12);
│ │ │ - -- used 2.65944s (cpu); 2.30064s (thread); 0s (gc)
│ │ │ + -- used 2.81364s (cpu); 2.30175s (thread); 0s (gc)
│ │ │  
│ │ │  o18 : Ideal of R
│ │ │  
│ │ │  i19 : J1 == J2
│ │ │  
│ │ │  o19 = true
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__Cohen__Macaulay.out
│ │ │ @@ -7,20 +7,20 @@
│ │ │  i3 : g = map(T, S, {x^3, x^2*y, x*y^2, y^3});
│ │ │  
│ │ │  o3 : RingMap T <-- S
│ │ │  
│ │ │  i4 : R = S/(ker g);
│ │ │  
│ │ │  i5 : time isCohenMacaulay(R)
│ │ │ - -- used 0.00209879s (cpu); 0.00209348s (thread); 0s (gc)
│ │ │ + -- used 0.00221578s (cpu); 0.00220983s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
│ │ │  
│ │ │  i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ - -- used 0.00481737s (cpu); 0.0048183s (thread); 0s (gc)
│ │ │ + -- used 0.00582617s (cpu); 0.00583131s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : R = QQ[x,y,u,v]/(x*u, x*v, y*u, y*v);
│ │ │  
│ │ │  i8 : isCohenMacaulay(R)
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Injective.out
│ │ │ @@ -60,49 +60,49 @@
│ │ │  i19 : R = ZZ/5[x,y,z]/(y^2*z + x*y*z-x^3)
│ │ │  
│ │ │  o19 = R
│ │ │  
│ │ │  o19 : QuotientRing
│ │ │  
│ │ │  i20 : time isFInjective(R)
│ │ │ - -- used 0.0262795s (cpu); 0.0262787s (thread); 0s (gc)
│ │ │ + -- used 0.0329714s (cpu); 0.0329715s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = true
│ │ │  
│ │ │  i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ - -- used 2.16981s (cpu); 1.26905s (thread); 0s (gc)
│ │ │ + -- used 2.63713s (cpu); 1.43732s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = true
│ │ │  
│ │ │  i22 : R = ZZ/7[x,y,z]/((x-1)^5 + (y+1)^5 + z^5);
│ │ │  
│ │ │  i23 : time isFInjective(R)
│ │ │ - -- used 0.141185s (cpu); 0.0902546s (thread); 0s (gc)
│ │ │ + -- used 0.157892s (cpu); 0.0959184s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = false
│ │ │  
│ │ │  i24 : time isFInjective(R, AtOrigin => true)
│ │ │ - -- used 0.144332s (cpu); 0.0902735s (thread); 0s (gc)
│ │ │ + -- used 0.164486s (cpu); 0.104157s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
│ │ │  
│ │ │  i25 : S = ZZ/3[xs, ys, zs, xt, yt, zt];
│ │ │  
│ │ │  i26 : EP1 = ZZ/3[x,y,z,s,t]/(x^3 + y^2*z - x*z^2);
│ │ │  
│ │ │  i27 : f = map(EP1, S, {x*s, y*s, z*s, x*t, y*t, z*t});
│ │ │  
│ │ │  o27 : RingMap EP1 <-- S
│ │ │  
│ │ │  i28 : R = S/(ker f);
│ │ │  
│ │ │  i29 : time isFInjective(R)
│ │ │ - -- used 0.904683s (cpu); 0.715355s (thread); 0s (gc)
│ │ │ + -- used 1.0071s (cpu); 0.797444s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = false
│ │ │  
│ │ │  i30 : time isFInjective(R, AssumeCM => true)
│ │ │ - -- used 0.412784s (cpu); 0.287986s (thread); 0s (gc)
│ │ │ + -- used 0.400721s (cpu); 0.271248s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
│ │ │  
│ │ │  i31 :
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Regular.out
│ │ │ @@ -80,19 +80,19 @@
│ │ │  
│ │ │  o25 : Ideal of S
│ │ │  
│ │ │  i26 : debugLevel = 1;
│ │ │  
│ │ │  i27 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 1)
│ │ │  isFRegular: This ring does not appear to be F-regular.  Increasing DepthOfSearch will let the function search more deeply.
│ │ │ - -- used 0.120203s (cpu); 0.0782458s (thread); 0s (gc)
│ │ │ + -- used 0.151164s (cpu); 0.080519s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = false
│ │ │  
│ │ │  i28 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 2)
│ │ │ - -- used 0.228747s (cpu); 0.174944s (thread); 0s (gc)
│ │ │ + -- used 0.279722s (cpu); 0.211452s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = true
│ │ │  
│ │ │  i29 : debugLevel = 0;
│ │ │  
│ │ │  i30 :
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Ideal.out
│ │ │ @@ -81,21 +81,21 @@
│ │ │  i22 : testIdeal({3/4, 2/3, 3/5}, L)
│ │ │  
│ │ │  o22 = ideal (y, x)
│ │ │  
│ │ │  o22 : Ideal of R
│ │ │  
│ │ │  i23 : time testIdeal({3/4, 2/3, 3/5}, L)
│ │ │ - -- used 0.360889s (cpu); 0.177517s (thread); 0s (gc)
│ │ │ + -- used 0.403964s (cpu); 0.20971s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = ideal (y, x)
│ │ │  
│ │ │  o23 : Ideal of R
│ │ │  
│ │ │  i24 : time testIdeal(1/60, x^45*y^40*(x + y)^36)
│ │ │ - -- used 0.505166s (cpu); 0.327584s (thread); 0s (gc)
│ │ │ + -- used 0.502363s (cpu); 0.28764s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = ideal (y, x)
│ │ │  
│ │ │  o24 : Ideal of R
│ │ │  
│ │ │  i25 :
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Root.html
│ │ │ @@ -226,23 +226,23 @@
│ │ │  
│ │ │  o16 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i17 : time J1 = frobeniusRoot(1, {8, 10, 12}, {I1, I2, I3});
│ │ │ - -- used 0.854696s (cpu); 0.665482s (thread); 0s (gc)
│ │ │ + -- used 1.07015s (cpu); 0.783874s (thread); 0s (gc)
│ │ │  
│ │ │  o17 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i18 : time J2 = frobeniusRoot(1, I1^8*I2^10*I3^12);
│ │ │ - -- used 2.65944s (cpu); 2.30064s (thread); 0s (gc)
│ │ │ + -- used 2.81364s (cpu); 2.30175s (thread); 0s (gc)
│ │ │  
│ │ │  o18 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i19 : J1 == J2
│ │ │ ├── html2text {}
│ │ │ │ @@ -106,19 +106,19 @@
│ │ │ │  i15 : I2 = ideal(x^20*y^100, x + z^100);
│ │ │ │  
│ │ │ │  o15 : Ideal of R
│ │ │ │  i16 : I3 = ideal(x^50*y^50*z^50);
│ │ │ │  
│ │ │ │  o16 : Ideal of R
│ │ │ │  i17 : time J1 = frobeniusRoot(1, {8, 10, 12}, {I1, I2, I3});
│ │ │ │ - -- used 0.854696s (cpu); 0.665482s (thread); 0s (gc)
│ │ │ │ + -- used 1.07015s (cpu); 0.783874s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 : Ideal of R
│ │ │ │  i18 : time J2 = frobeniusRoot(1, I1^8*I2^10*I3^12);
│ │ │ │ - -- used 2.65944s (cpu); 2.30064s (thread); 0s (gc)
│ │ │ │ + -- used 2.81364s (cpu); 2.30175s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o18 : Ideal of R
│ │ │ │  i19 : J1 == J2
│ │ │ │  
│ │ │ │  o19 = true
│ │ │ │  For legacy reasons, the last ideal in the list can be specified separately,
│ │ │ │  using frobeniusRoot(e, \{a_1,\ldots,a_n\}, \{I_1,\ldots,I_n\}, I). The last
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_is__Cohen__Macaulay.html
│ │ │ @@ -96,23 +96,23 @@
│ │ │              
│ │ │                
    i4 : R = S/(ker g);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time isCohenMacaulay(R)
│ │ │ - -- used 0.00209879s (cpu); 0.00209348s (thread); 0s (gc)
│ │ │ + -- used 0.00221578s (cpu); 0.00220983s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ - -- used 0.00481737s (cpu); 0.0048183s (thread); 0s (gc)
│ │ │ + -- used 0.00582617s (cpu); 0.00583131s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,19 +23,19 @@
│ │ │ │  i1 : T = ZZ/5[x,y];
│ │ │ │  i2 : S = ZZ/5[a,b,c,d];
│ │ │ │  i3 : g = map(T, S, {x^3, x^2*y, x*y^2, y^3});
│ │ │ │  
│ │ │ │  o3 : RingMap T <-- S
│ │ │ │  i4 : R = S/(ker g);
│ │ │ │  i5 : time isCohenMacaulay(R)
│ │ │ │ - -- used 0.00209879s (cpu); 0.00209348s (thread); 0s (gc)
│ │ │ │ + -- used 0.00221578s (cpu); 0.00220983s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ │ - -- used 0.00481737s (cpu); 0.0048183s (thread); 0s (gc)
│ │ │ │ + -- used 0.00582617s (cpu); 0.00583131s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  i7 : R = QQ[x,y,u,v]/(x*u, x*v, y*u, y*v);
│ │ │ │  i8 : isCohenMacaulay(R)
│ │ │ │  
│ │ │ │  o8 = false
│ │ │ │  The function isCohenMacaulay considers $R$ as a quotient of a polynomial ring,
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Injective.html
│ │ │ @@ -214,23 +214,23 @@
│ │ │  
│ │ │  o19 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i20 : time isFInjective(R)
│ │ │ - -- used 0.0262795s (cpu); 0.0262787s (thread); 0s (gc)
│ │ │ + -- used 0.0329714s (cpu); 0.0329715s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = true
    
│ │ │              
    
│ │ │                
    i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ - -- used 2.16981s (cpu); 1.26905s (thread); 0s (gc)
│ │ │ + -- used 2.63713s (cpu); 1.43732s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = true
    
│ │ │              
    
│ │ │          
    
│ │ │            
    If the option AtOrigin (default value false) is set to true, isFInjective will only check $F$-injectivity at the origin. Otherwise, it will check $F$-injectivity globally. Note that checking $F$-injectivity at the origin can be slower than checking it globally. Consider the following example of a non-$F$-injective ring.
    
│ │ │ @@ -240,23 +240,23 @@
│ │ │              
│ │ │                
    i22 : R = ZZ/7[x,y,z]/((x-1)^5 + (y+1)^5 + z^5);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i23 : time isFInjective(R)
│ │ │ - -- used 0.141185s (cpu); 0.0902546s (thread); 0s (gc)
│ │ │ + -- used 0.157892s (cpu); 0.0959184s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = false
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i24 : time isFInjective(R, AtOrigin => true)
│ │ │ - -- used 0.144332s (cpu); 0.0902735s (thread); 0s (gc)
│ │ │ + -- used 0.164486s (cpu); 0.104157s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    If the option AssumeCM (default value false) is set to true, then isFInjective only checks the Frobenius action on top cohomology (which is typically much faster). Note that it can give an incorrect answer if the non-injective Frobenius occurs in a lower degree.  Consider the example of the cone over a supersingular elliptic curve times $\mathbb{P}^1$.
    
│ │ │ @@ -283,23 +283,23 @@
│ │ │              
│ │ │                
    i28 : R = S/(ker f);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i29 : time isFInjective(R)
│ │ │ - -- used 0.904683s (cpu); 0.715355s (thread); 0s (gc)
│ │ │ + -- used 1.0071s (cpu); 0.797444s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = false
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i30 : time isFInjective(R, AssumeCM => true)
│ │ │ - -- used 0.412784s (cpu); 0.287986s (thread); 0s (gc)
│ │ │ + -- used 0.400721s (cpu); 0.271248s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    If the option AssumedReduced is set to true (its default behavior), then the bottom local cohomology is avoided (this means the Frobenius action on the top potentially nonzero Ext is not computed).
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -81,52 +81,52 @@
│ │ │ │  much faster.
│ │ │ │  i19 : R = ZZ/5[x,y,z]/(y^2*z + x*y*z-x^3)
│ │ │ │  
│ │ │ │  o19 = R
│ │ │ │  
│ │ │ │  o19 : QuotientRing
│ │ │ │  i20 : time isFInjective(R)
│ │ │ │ - -- used 0.0262795s (cpu); 0.0262787s (thread); 0s (gc)
│ │ │ │ + -- used 0.0329714s (cpu); 0.0329715s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o20 = true
│ │ │ │  i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ │ - -- used 2.16981s (cpu); 1.26905s (thread); 0s (gc)
│ │ │ │ + -- used 2.63713s (cpu); 1.43732s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = true
│ │ │ │  If the option AtOrigin (default value false) is set to true, isFInjective will
│ │ │ │  only check $F$-injectivity at the origin. Otherwise, it will check $F$-
│ │ │ │  injectivity globally. Note that checking $F$-injectivity at the origin can be
│ │ │ │  slower than checking it globally. Consider the following example of a non-$F$-
│ │ │ │  injective ring.
│ │ │ │  i22 : R = ZZ/7[x,y,z]/((x-1)^5 + (y+1)^5 + z^5);
│ │ │ │  i23 : time isFInjective(R)
│ │ │ │ - -- used 0.141185s (cpu); 0.0902546s (thread); 0s (gc)
│ │ │ │ + -- used 0.157892s (cpu); 0.0959184s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = false
│ │ │ │  i24 : time isFInjective(R, AtOrigin => true)
│ │ │ │ - -- used 0.144332s (cpu); 0.0902735s (thread); 0s (gc)
│ │ │ │ + -- used 0.164486s (cpu); 0.104157s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o24 = true
│ │ │ │  If the option AssumeCM (default value false) is set to true, then isFInjective
│ │ │ │  only checks the Frobenius action on top cohomology (which is typically much
│ │ │ │  faster). Note that it can give an incorrect answer if the non-injective
│ │ │ │  Frobenius occurs in a lower degree. Consider the example of the cone over a
│ │ │ │  supersingular elliptic curve times $\mathbb{P}^1$.
│ │ │ │  i25 : S = ZZ/3[xs, ys, zs, xt, yt, zt];
│ │ │ │  i26 : EP1 = ZZ/3[x,y,z,s,t]/(x^3 + y^2*z - x*z^2);
│ │ │ │  i27 : f = map(EP1, S, {x*s, y*s, z*s, x*t, y*t, z*t});
│ │ │ │  
│ │ │ │  o27 : RingMap EP1 <-- S
│ │ │ │  i28 : R = S/(ker f);
│ │ │ │  i29 : time isFInjective(R)
│ │ │ │ - -- used 0.904683s (cpu); 0.715355s (thread); 0s (gc)
│ │ │ │ + -- used 1.0071s (cpu); 0.797444s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o29 = false
│ │ │ │  i30 : time isFInjective(R, AssumeCM => true)
│ │ │ │ - -- used 0.412784s (cpu); 0.287986s (thread); 0s (gc)
│ │ │ │ + -- used 0.400721s (cpu); 0.271248s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o30 = true
│ │ │ │  If the option AssumedReduced is set to true (its default behavior), then the
│ │ │ │  bottom local cohomology is avoided (this means the Frobenius action on the top
│ │ │ │  potentially nonzero Ext is not computed).
│ │ │ │  If the option AssumeNormal (default value false) is set to true, then the
│ │ │ │  bottom two local cohomology modules (or, rather, their duals) need not be
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Regular.html
│ │ │ @@ -273,23 +273,23 @@
│ │ │                
    i26 : debugLevel = 1;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i27 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 1)
│ │ │  isFRegular: This ring does not appear to be F-regular.  Increasing DepthOfSearch will let the function search more deeply.
│ │ │ - -- used 0.120203s (cpu); 0.0782458s (thread); 0s (gc)
│ │ │ + -- used 0.151164s (cpu); 0.080519s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = false
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i28 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 2)
│ │ │ - -- used 0.228747s (cpu); 0.174944s (thread); 0s (gc)
│ │ │ + -- used 0.279722s (cpu); 0.211452s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i29 : debugLevel = 0;
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -114,19 +114,19 @@
│ │ │ │  i25 : I = minors(2, matrix {{x, y, z}, {u, v, w}});
│ │ │ │  
│ │ │ │  o25 : Ideal of S
│ │ │ │  i26 : debugLevel = 1;
│ │ │ │  i27 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 1)
│ │ │ │  isFRegular: This ring does not appear to be F-regular.  Increasing
│ │ │ │  DepthOfSearch will let the function search more deeply.
│ │ │ │ - -- used 0.120203s (cpu); 0.0782458s (thread); 0s (gc)
│ │ │ │ + -- used 0.151164s (cpu); 0.080519s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = false
│ │ │ │  i28 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 2)
│ │ │ │ - -- used 0.228747s (cpu); 0.174944s (thread); 0s (gc)
│ │ │ │ + -- used 0.279722s (cpu); 0.211452s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o28 = true
│ │ │ │  i29 : debugLevel = 0;
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_e_s_t_I_d_e_a_l -- compute a test ideal in a Q-Gorenstein ring
│ │ │ │      * _i_s_F_R_a_t_i_o_n_a_l -- whether a ring is F-rational
│ │ │ │  ********** WWaayyss ttoo uussee iissFFRReegguullaarr:: **********
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Ideal.html
│ │ │ @@ -255,25 +255,25 @@
│ │ │          
    
│ │ │            
    It is often more efficient to pass a list, as opposed to finding a common denominator and passing a single element, since testIdeal can do things in a more intelligent way for such a list.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i23 : time testIdeal({3/4, 2/3, 3/5}, L)
│ │ │ - -- used 0.360889s (cpu); 0.177517s (thread); 0s (gc)
│ │ │ + -- used 0.403964s (cpu); 0.20971s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = ideal (y, x)
│ │ │  
│ │ │  o23 : Ideal of R
    
│ │ │              
    
│ │ │                
    i24 : time testIdeal(1/60, x^45*y^40*(x + y)^36)
│ │ │ - -- used 0.505166s (cpu); 0.327584s (thread); 0s (gc)
│ │ │ + -- used 0.502363s (cpu); 0.28764s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = ideal (y, x)
│ │ │  
│ │ │  o24 : Ideal of R
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -100,21 +100,21 @@
│ │ │ │  o22 = ideal (y, x)
│ │ │ │  
│ │ │ │  o22 : Ideal of R
│ │ │ │  It is often more efficient to pass a list, as opposed to finding a common
│ │ │ │  denominator and passing a single element, since testIdeal can do things in a
│ │ │ │  more intelligent way for such a list.
│ │ │ │  i23 : time testIdeal({3/4, 2/3, 3/5}, L)
│ │ │ │ - -- used 0.360889s (cpu); 0.177517s (thread); 0s (gc)
│ │ │ │ + -- used 0.403964s (cpu); 0.20971s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = ideal (y, x)
│ │ │ │  
│ │ │ │  o23 : Ideal of R
│ │ │ │  i24 : time testIdeal(1/60, x^45*y^40*(x + y)^36)
│ │ │ │ - -- used 0.505166s (cpu); 0.327584s (thread); 0s (gc)
│ │ │ │ + -- used 0.502363s (cpu); 0.28764s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o24 = ideal (y, x)
│ │ │ │  
│ │ │ │  o24 : Ideal of R
│ │ │ │  The option AssumeDomain (default value false) is used when finding a test
│ │ │ │  element. The option FrobeniusRootStrategy (default value Substitution) is
│ │ │ │  passed to internal _f_r_o_b_e_n_i_u_s_R_o_o_t calls.
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Minimal.out
│ │ │ @@ -2,16 +2,15 @@
│ │ │  
│ │ │  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
│ │ │ +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
│ │ │ @@ -2,18 +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), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │                                    2
│ │ │ -                  ((1, 2), 0) => c
│ │ │ +o3 = LineageTable{((1, 2), 0) => c }
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │                          2
│ │ │                    2 => b
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_minimize_lp__Lineage__Table_rp.out
│ │ │ @@ -4,16 +4,20 @@
│ │ │  
│ │ │  i2 : allowableThreads= 2;
│ │ │  
│ │ │  i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │                                     3
│ │ │  o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │ +                                    2
│ │ │ +                  ((0, 2), 1) => a*c
│ │ │                                     2
│ │ │                    ((1, 2), 0) => -c
│ │ │ +                             2
│ │ │ +                  (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │                                  2
│ │ │                    0 => a*b*c + c
│ │ │                            3       2
│ │ │                    1 => - b c + a*b  + a*c
│ │ │ @@ -21,16 +25,18 @@
│ │ │                    2 => b
│ │ │  
│ │ │  o3 : LineageTable
│ │ │  
│ │ │  i4 : minimize T
│ │ │  
│ │ │  o4 = LineageTable{((0, 2), 0) => null}
│ │ │ +                  ((0, 2), 1) => null
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │ +                  (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │                          2
│ │ │                    2 => b
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_reduce.out
│ │ │ @@ -2,44 +2,35 @@
│ │ │  
│ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │  
│ │ │  i2 : allowableThreads= 2;
│ │ │  
│ │ │  i3 : T = tgb ideal "abc+c2,ab2-b3c+ac,b2"
│ │ │  
│ │ │ -                                     2
│ │ │ -o3 = LineageTable{((0, 1), 0) => -a*c    }
│ │ │                                     3
│ │ │ -                  ((0, 2), 0) => -c
│ │ │ -                                    2
│ │ │ -                  ((0, 2), 1) => a*c
│ │ │ +o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │                                     2
│ │ │                    ((1, 2), 0) => -c
│ │ │ -                             2
│ │ │ -                  (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │                                  2
│ │ │                    0 => a*b*c + c
│ │ │                            3       2
│ │ │                    1 => - b c + a*b  + a*c
│ │ │                          2
│ │ │                    2 => b
│ │ │  
│ │ │  o3 : LineageTable
│ │ │  
│ │ │  i4 : reduce T
│ │ │  
│ │ │ -o4 = LineageTable{((0, 1), 0) => null}
│ │ │ -                  ((0, 2), 0) => null
│ │ │ -                  ((0, 2), 1) => null
│ │ │ +o4 = LineageTable{((0, 2), 0) => null}
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │ -                  (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │                          2
│ │ │                    2 => b
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb.out
│ │ │ @@ -6,62 +6,42 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : allowableThreads  = 4;
│ │ │  
│ │ │  i4 : H = tgb I
│ │ │  
│ │ │ -                                                                   2 9
│ │ │ -o4 = LineageTable{(((((0, 1), 2), 2), ((0, 1), 2)), (0, 1)) => -22y z                               }
│ │ │ -                                                             2 9
│ │ │ -                  (((((0, 1), 2), 2), ((0, 1), 2)), 2) => 16y z
│ │ │ -                                                         2 13
│ │ │ -                  (((((0, 1), 2), 2), 1), (0, 1)) => -22y z
│ │ │ -                                                   2 13
│ │ │ -                  (((((0, 1), 2), 2), 1), 2) => 16y z
│ │ │ -                                                   2 12
│ │ │ -                  (((((0, 1), 2), 2), 2), 2) => 16y z
│ │ │ -                                                                                                 2 4
│ │ │ -                  (((((0, 1), 2), 3), (((0, 1), 2), 2)), ((((0, 1), 2), 2), ((0, 1), 2))) => -43y z
│ │ │ -                                                                  2 7
│ │ │ -                  (((((0, 1), 2), 3), (((0, 1), 2), 2)), 2) => 16y z
│ │ │ -                                              4 13     4 9
│ │ │ -                  ((((0, 1), 2), 1), 2) => 23y z   + 6y z
│ │ │ -                                                        4 8      4 4
│ │ │ -                  ((((0, 1), 2), 2), ((0, 1), 2)) => 33y z  + 13y z
│ │ │ -                                              4 12      4 11
│ │ │ -                  ((((0, 1), 2), 2), 1) => 50y z   - 23y z
│ │ │ -                                                4 11     4 6
│ │ │ -                  ((((0, 1), 2), 2), 2) => - 26y z   + 9y z
│ │ │ -                                               4 6
│ │ │ -                  ((((0, 1), 2), 2), 3) => -13y z
│ │ │ -                                                             4 6      4 5
│ │ │ -                  ((((0, 1), 2), 3), (((0, 1), 2), 2)) => 10y z  + 31y z
│ │ │ -                                              3 17
│ │ │ -                  ((((0, 1), 2), 3), 1) => 11y z
│ │ │ -                                                4 5     3 16
│ │ │ -                  ((((0, 1), 2), 3), 2) => - 16y z  + 9y z
│ │ │ -                                              3 17
│ │ │ -                  ((((0, 1), 2), 3), 3) => 41y z
│ │ │ -                                         6 4      4 6
│ │ │ -                  (((0, 1), 2), 1) => 19y z  - 30y z
│ │ │ -                                         5 4     4 7
│ │ │ -                  (((0, 1), 2), 2) => 37y z  + 9y z
│ │ │ -                                         4 14      4 11
│ │ │ -                  (((0, 1), 2), 3) => 27y z   - 16y z
│ │ │ -                                      5 5     4 4
│ │ │ -                  ((0, 1), 2) => - 24y z  + 9y z
│ │ │ +                                           2 5      2 4
│ │ │ +o4 = LineageTable{((0, 1), (0, 2)) => - 40y z  - 22y z }
│ │ │ +                                           5       2 4
│ │ │ +                  ((0, 1), (0, 3)) => - 46y z - 40y z
│ │ │ +                                    2 7      2 4
│ │ │ +                  ((0, 1), 2) => 50y z  + 19y z
│ │ │ +                                      4 8      3 11
│ │ │ +                  ((0, 2), 1) => - 14y z  + 43y z
│ │ │ +                                    2 10      2 9
│ │ │ +                  ((0, 3), 1) => 12y z   + 47y z
│ │ │ +                                    2 6      2 4
│ │ │ +                  ((0, 3), 2) => 23y z  + 19y z
│ │ │ +                                        2 4
│ │ │ +                  ((1, 3), (0, 2)) => -y z
│ │ │ +                                    2 4
│ │ │ +                  ((2, 3), 1) => -7y z
│ │ │ +                                     2 4
│ │ │ +                  ((2, 3), 2) => -20y z
│ │ │                                   5 2      3 4
│ │ │                    (0, 1) => - 25y z  - 19y z
│ │ │ -                                 3 5     2 4
│ │ │ -                  (0, 2) => - 24y z  + 9y z
│ │ │ -                               5       3 4
│ │ │ -                  (0, 3) => 28y z - 24y z
│ │ │ -                                3 16
│ │ │ -                  (1, 2) => -19y z
│ │ │ +                              5 3     2 4
│ │ │ +                  (0, 2) => 5y z  + 9y z
│ │ │ +                              5 2      5
│ │ │ +                  (0, 3) => 5y z  + 28y z
│ │ │ +                                 4 5      2 7
│ │ │ +                  (1, 2) => - 45y z  - 14y z
│ │ │ +                                 7       2 6
│ │ │ +                  (1, 3) => - 24y z - 14y z
│ │ │                                3 4     2 4
│ │ │                    (2, 3) => 7y z  - 9y z
│ │ │                                 2
│ │ │                    0 => 2x + 10y z
│ │ │                           2           3
│ │ │                    1 => 8x y + 10x*y*z
│ │ │                             3 2       3
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/___Minimal.html
│ │ │ @@ -73,16 +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
│ │ │ +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 {}
│ │ │ │ @@ -12,16 +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
│ │ │ │ +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
│ │ │ @@ -86,18 +86,16 @@
│ │ │                
    i2 : allowableThreads= 2;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : T = reduce tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │                                    2
│ │ │ -                  ((1, 2), 0) => c
│ │ │ +o3 = LineageTable{((1, 2), 0) => c }
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │                          2
│ │ │                    2 => b
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,18 +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), 0) => null}
│ │ │ │ -                  ((0, 1), 0) => null
│ │ │ │                                    2
│ │ │ │ -                  ((1, 2), 0) => c
│ │ │ │ +o3 = LineageTable{((1, 2), 0) => c }
│ │ │ │                    (0, 1) => null
│ │ │ │                    (0, 2) => null
│ │ │ │                    (1, 2) => a*c
│ │ │ │                    0 => null
│ │ │ │                    1 => null
│ │ │ │                          2
│ │ │ │                    2 => b
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_minimize_lp__Lineage__Table_rp.html
│ │ │ @@ -84,16 +84,20 @@
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │                                     3
│ │ │  o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │ +                                    2
│ │ │ +                  ((0, 2), 1) => a*c
│ │ │                                     2
│ │ │                    ((1, 2), 0) => -c
│ │ │ +                             2
│ │ │ +                  (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │                                  2
│ │ │                    0 => a*b*c + c
│ │ │                            3       2
│ │ │                    1 => - b c + a*b  + a*c
│ │ │ @@ -104,16 +108,18 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : minimize T
│ │ │  
│ │ │  o4 = LineageTable{((0, 2), 0) => null}
│ │ │ +                  ((0, 2), 1) => null
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │ +                  (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │                          2
│ │ │                    2 => b
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,32 +21,38 @@
│ │ │ │  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
│ │ │ │ +                  ((0, 2), 1) => a*c
│ │ │ │                                     2
│ │ │ │                    ((1, 2), 0) => -c
│ │ │ │ +                             2
│ │ │ │ +                  (0, 1) => a c
│ │ │ │                                 2
│ │ │ │                    (0, 2) => b*c
│ │ │ │                    (1, 2) => -a*c
│ │ │ │                                  2
│ │ │ │                    0 => a*b*c + c
│ │ │ │                            3       2
│ │ │ │                    1 => - b c + a*b  + a*c
│ │ │ │                          2
│ │ │ │                    2 => b
│ │ │ │  
│ │ │ │  o3 : LineageTable
│ │ │ │  i4 : minimize T
│ │ │ │  
│ │ │ │  o4 = LineageTable{((0, 2), 0) => null}
│ │ │ │ +                  ((0, 2), 1) => null
│ │ │ │                                    2
│ │ │ │                    ((1, 2), 0) => c
│ │ │ │ +                  (0, 1) => null
│ │ │ │                    (0, 2) => null
│ │ │ │                    (1, 2) => a*c
│ │ │ │                    0 => null
│ │ │ │                    1 => null
│ │ │ │                          2
│ │ │ │                    2 => b
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_reduce.html
│ │ │ @@ -82,24 +82,18 @@
│ │ │                
    i2 : allowableThreads= 2;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : T = tgb ideal "abc+c2,ab2-b3c+ac,b2"
│ │ │  
│ │ │ -                                     2
│ │ │ -o3 = LineageTable{((0, 1), 0) => -a*c    }
│ │ │                                     3
│ │ │ -                  ((0, 2), 0) => -c
│ │ │ -                                    2
│ │ │ -                  ((0, 2), 1) => a*c
│ │ │ +o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │                                     2
│ │ │                    ((1, 2), 0) => -c
│ │ │ -                             2
│ │ │ -                  (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │                                  2
│ │ │                    0 => a*b*c + c
│ │ │                            3       2
│ │ │                    1 => - b c + a*b  + a*c
│ │ │ @@ -109,20 +103,17 @@
│ │ │  o3 : LineageTable
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : reduce T
│ │ │  
│ │ │ -o4 = LineageTable{((0, 1), 0) => null}
│ │ │ -                  ((0, 2), 0) => null
│ │ │ -                  ((0, 2), 1) => null
│ │ │ +o4 = LineageTable{((0, 2), 0) => null}
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │ -                  (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │                          2
│ │ │                    2 => b
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,43 +20,34 @@
│ │ │ │  remainder on the division by the remaining values H.
│ │ │ │  If values H constitute a Gr\"obner basis of the ideal they generate, this
│ │ │ │  method returns a reduced Gr\"obner basis.
│ │ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │ │  i2 : allowableThreads= 2;
│ │ │ │  i3 : T = tgb ideal "abc+c2,ab2-b3c+ac,b2"
│ │ │ │  
│ │ │ │ -                                     2
│ │ │ │ -o3 = LineageTable{((0, 1), 0) => -a*c    }
│ │ │ │                                     3
│ │ │ │ -                  ((0, 2), 0) => -c
│ │ │ │ -                                    2
│ │ │ │ -                  ((0, 2), 1) => a*c
│ │ │ │ +o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │ │                                     2
│ │ │ │                    ((1, 2), 0) => -c
│ │ │ │ -                             2
│ │ │ │ -                  (0, 1) => a c
│ │ │ │                                 2
│ │ │ │                    (0, 2) => b*c
│ │ │ │                    (1, 2) => -a*c
│ │ │ │                                  2
│ │ │ │                    0 => a*b*c + c
│ │ │ │                            3       2
│ │ │ │                    1 => - b c + a*b  + a*c
│ │ │ │                          2
│ │ │ │                    2 => b
│ │ │ │  
│ │ │ │  o3 : LineageTable
│ │ │ │  i4 : reduce T
│ │ │ │  
│ │ │ │ -o4 = LineageTable{((0, 1), 0) => null}
│ │ │ │ -                  ((0, 2), 0) => null
│ │ │ │ -                  ((0, 2), 1) => null
│ │ │ │ +o4 = LineageTable{((0, 2), 0) => null}
│ │ │ │                                    2
│ │ │ │                    ((1, 2), 0) => c
│ │ │ │ -                  (0, 1) => null
│ │ │ │                    (0, 2) => null
│ │ │ │                    (1, 2) => a*c
│ │ │ │                    0 => null
│ │ │ │                    1 => null
│ │ │ │                          2
│ │ │ │                    2 => b
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb.html
│ │ │ @@ -95,62 +95,42 @@
│ │ │                
    i3 : allowableThreads  = 4;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : H = tgb I
│ │ │  
│ │ │ -                                                                   2 9
│ │ │ -o4 = LineageTable{(((((0, 1), 2), 2), ((0, 1), 2)), (0, 1)) => -22y z                               }
│ │ │ -                                                             2 9
│ │ │ -                  (((((0, 1), 2), 2), ((0, 1), 2)), 2) => 16y z
│ │ │ -                                                         2 13
│ │ │ -                  (((((0, 1), 2), 2), 1), (0, 1)) => -22y z
│ │ │ -                                                   2 13
│ │ │ -                  (((((0, 1), 2), 2), 1), 2) => 16y z
│ │ │ -                                                   2 12
│ │ │ -                  (((((0, 1), 2), 2), 2), 2) => 16y z
│ │ │ -                                                                                                 2 4
│ │ │ -                  (((((0, 1), 2), 3), (((0, 1), 2), 2)), ((((0, 1), 2), 2), ((0, 1), 2))) => -43y z
│ │ │ -                                                                  2 7
│ │ │ -                  (((((0, 1), 2), 3), (((0, 1), 2), 2)), 2) => 16y z
│ │ │ -                                              4 13     4 9
│ │ │ -                  ((((0, 1), 2), 1), 2) => 23y z   + 6y z
│ │ │ -                                                        4 8      4 4
│ │ │ -                  ((((0, 1), 2), 2), ((0, 1), 2)) => 33y z  + 13y z
│ │ │ -                                              4 12      4 11
│ │ │ -                  ((((0, 1), 2), 2), 1) => 50y z   - 23y z
│ │ │ -                                                4 11     4 6
│ │ │ -                  ((((0, 1), 2), 2), 2) => - 26y z   + 9y z
│ │ │ -                                               4 6
│ │ │ -                  ((((0, 1), 2), 2), 3) => -13y z
│ │ │ -                                                             4 6      4 5
│ │ │ -                  ((((0, 1), 2), 3), (((0, 1), 2), 2)) => 10y z  + 31y z
│ │ │ -                                              3 17
│ │ │ -                  ((((0, 1), 2), 3), 1) => 11y z
│ │ │ -                                                4 5     3 16
│ │ │ -                  ((((0, 1), 2), 3), 2) => - 16y z  + 9y z
│ │ │ -                                              3 17
│ │ │ -                  ((((0, 1), 2), 3), 3) => 41y z
│ │ │ -                                         6 4      4 6
│ │ │ -                  (((0, 1), 2), 1) => 19y z  - 30y z
│ │ │ -                                         5 4     4 7
│ │ │ -                  (((0, 1), 2), 2) => 37y z  + 9y z
│ │ │ -                                         4 14      4 11
│ │ │ -                  (((0, 1), 2), 3) => 27y z   - 16y z
│ │ │ -                                      5 5     4 4
│ │ │ -                  ((0, 1), 2) => - 24y z  + 9y z
│ │ │ +                                           2 5      2 4
│ │ │ +o4 = LineageTable{((0, 1), (0, 2)) => - 40y z  - 22y z }
│ │ │ +                                           5       2 4
│ │ │ +                  ((0, 1), (0, 3)) => - 46y z - 40y z
│ │ │ +                                    2 7      2 4
│ │ │ +                  ((0, 1), 2) => 50y z  + 19y z
│ │ │ +                                      4 8      3 11
│ │ │ +                  ((0, 2), 1) => - 14y z  + 43y z
│ │ │ +                                    2 10      2 9
│ │ │ +                  ((0, 3), 1) => 12y z   + 47y z
│ │ │ +                                    2 6      2 4
│ │ │ +                  ((0, 3), 2) => 23y z  + 19y z
│ │ │ +                                        2 4
│ │ │ +                  ((1, 3), (0, 2)) => -y z
│ │ │ +                                    2 4
│ │ │ +                  ((2, 3), 1) => -7y z
│ │ │ +                                     2 4
│ │ │ +                  ((2, 3), 2) => -20y z
│ │ │                                   5 2      3 4
│ │ │                    (0, 1) => - 25y z  - 19y z
│ │ │ -                                 3 5     2 4
│ │ │ -                  (0, 2) => - 24y z  + 9y z
│ │ │ -                               5       3 4
│ │ │ -                  (0, 3) => 28y z - 24y z
│ │ │ -                                3 16
│ │ │ -                  (1, 2) => -19y z
│ │ │ +                              5 3     2 4
│ │ │ +                  (0, 2) => 5y z  + 9y z
│ │ │ +                              5 2      5
│ │ │ +                  (0, 3) => 5y z  + 28y z
│ │ │ +                                 4 5      2 7
│ │ │ +                  (1, 2) => - 45y z  - 14y z
│ │ │ +                                 7       2 6
│ │ │ +                  (1, 3) => - 24y z - 14y z
│ │ │                                3 4     2 4
│ │ │                    (2, 3) => 7y z  - 9y z
│ │ │                                 2
│ │ │                    0 => 2x + 10y z
│ │ │                           2           3
│ │ │                    1 => 8x y + 10x*y*z
│ │ │                             3 2       3
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,65 +26,42 @@
│ │ │ │  i2 : I = ideal {2*x + 10*y^2*z, 8*x^2*y + 10*x*y*z^3, 5*x*y^3*z^2 + 9*x*z^3,
│ │ │ │  9*x*y^3*z + 10*x*y^3};
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : allowableThreads  = 4;
│ │ │ │  i4 : H = tgb I
│ │ │ │  
│ │ │ │ -                                                                   2 9
│ │ │ │ -o4 = LineageTable{(((((0, 1), 2), 2), ((0, 1), 2)), (0, 1)) => -22y z
│ │ │ │ -}
│ │ │ │ -                                                             2 9
│ │ │ │ -                  (((((0, 1), 2), 2), ((0, 1), 2)), 2) => 16y z
│ │ │ │ -                                                         2 13
│ │ │ │ -                  (((((0, 1), 2), 2), 1), (0, 1)) => -22y z
│ │ │ │ -                                                   2 13
│ │ │ │ -                  (((((0, 1), 2), 2), 1), 2) => 16y z
│ │ │ │ -                                                   2 12
│ │ │ │ -                  (((((0, 1), 2), 2), 2), 2) => 16y z
│ │ │ │ -
│ │ │ │ -2 4
│ │ │ │ -                  (((((0, 1), 2), 3), (((0, 1), 2), 2)), ((((0, 1), 2), 2), (
│ │ │ │ -(0, 1), 2))) => -43y z
│ │ │ │ -                                                                  2 7
│ │ │ │ -                  (((((0, 1), 2), 3), (((0, 1), 2), 2)), 2) => 16y z
│ │ │ │ -                                              4 13     4 9
│ │ │ │ -                  ((((0, 1), 2), 1), 2) => 23y z   + 6y z
│ │ │ │ -                                                        4 8      4 4
│ │ │ │ -                  ((((0, 1), 2), 2), ((0, 1), 2)) => 33y z  + 13y z
│ │ │ │ -                                              4 12      4 11
│ │ │ │ -                  ((((0, 1), 2), 2), 1) => 50y z   - 23y z
│ │ │ │ -                                                4 11     4 6
│ │ │ │ -                  ((((0, 1), 2), 2), 2) => - 26y z   + 9y z
│ │ │ │ -                                               4 6
│ │ │ │ -                  ((((0, 1), 2), 2), 3) => -13y z
│ │ │ │ -                                                             4 6      4 5
│ │ │ │ -                  ((((0, 1), 2), 3), (((0, 1), 2), 2)) => 10y z  + 31y z
│ │ │ │ -                                              3 17
│ │ │ │ -                  ((((0, 1), 2), 3), 1) => 11y z
│ │ │ │ -                                                4 5     3 16
│ │ │ │ -                  ((((0, 1), 2), 3), 2) => - 16y z  + 9y z
│ │ │ │ -                                              3 17
│ │ │ │ -                  ((((0, 1), 2), 3), 3) => 41y z
│ │ │ │ -                                         6 4      4 6
│ │ │ │ -                  (((0, 1), 2), 1) => 19y z  - 30y z
│ │ │ │ -                                         5 4     4 7
│ │ │ │ -                  (((0, 1), 2), 2) => 37y z  + 9y z
│ │ │ │ -                                         4 14      4 11
│ │ │ │ -                  (((0, 1), 2), 3) => 27y z   - 16y z
│ │ │ │ -                                      5 5     4 4
│ │ │ │ -                  ((0, 1), 2) => - 24y z  + 9y z
│ │ │ │ +                                           2 5      2 4
│ │ │ │ +o4 = LineageTable{((0, 1), (0, 2)) => - 40y z  - 22y z }
│ │ │ │ +                                           5       2 4
│ │ │ │ +                  ((0, 1), (0, 3)) => - 46y z - 40y z
│ │ │ │ +                                    2 7      2 4
│ │ │ │ +                  ((0, 1), 2) => 50y z  + 19y z
│ │ │ │ +                                      4 8      3 11
│ │ │ │ +                  ((0, 2), 1) => - 14y z  + 43y z
│ │ │ │ +                                    2 10      2 9
│ │ │ │ +                  ((0, 3), 1) => 12y z   + 47y z
│ │ │ │ +                                    2 6      2 4
│ │ │ │ +                  ((0, 3), 2) => 23y z  + 19y z
│ │ │ │ +                                        2 4
│ │ │ │ +                  ((1, 3), (0, 2)) => -y z
│ │ │ │ +                                    2 4
│ │ │ │ +                  ((2, 3), 1) => -7y z
│ │ │ │ +                                     2 4
│ │ │ │ +                  ((2, 3), 2) => -20y z
│ │ │ │                                   5 2      3 4
│ │ │ │                    (0, 1) => - 25y z  - 19y z
│ │ │ │ -                                 3 5     2 4
│ │ │ │ -                  (0, 2) => - 24y z  + 9y z
│ │ │ │ -                               5       3 4
│ │ │ │ -                  (0, 3) => 28y z - 24y z
│ │ │ │ -                                3 16
│ │ │ │ -                  (1, 2) => -19y z
│ │ │ │ +                              5 3     2 4
│ │ │ │ +                  (0, 2) => 5y z  + 9y z
│ │ │ │ +                              5 2      5
│ │ │ │ +                  (0, 3) => 5y z  + 28y z
│ │ │ │ +                                 4 5      2 7
│ │ │ │ +                  (1, 2) => - 45y z  - 14y z
│ │ │ │ +                                 7       2 6
│ │ │ │ +                  (1, 3) => - 24y z - 14y z
│ │ │ │                                3 4     2 4
│ │ │ │                    (2, 3) => 7y z  - 9y z
│ │ │ │                                 2
│ │ │ │                    0 => 2x + 10y z
│ │ │ │                           2           3
│ │ │ │                    1 => 8x y + 10x*y*z
│ │ │ │                             3 2       3
│ │ ├── ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_ed__Deg.out
│ │ │ @@ -40,15 +40,15 @@
│ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │  ED Degree = 252
│ │ │  
│ │ │                         5      4      3      2
│ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │ - -- used 1.13366s (cpu); 0.779417s (thread); 0s (gc)
│ │ │ + -- used 1.21839s (cpu); 0.846414s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 252
│ │ │  
│ │ │  o4 : QQ
│ │ │  
│ │ │  i5 : time edDeg(A,ForceAmat=>true)
│ │ │  
│ │ │ @@ -56,14 +56,14 @@
│ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │  ED Degree = 252
│ │ │  
│ │ │                         5      4      3      2
│ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │ - -- used 4.61337s (cpu); 3.0038s (thread); 0s (gc)
│ │ │ + -- used 5.06829s (cpu); 3.28094s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 252
│ │ │  
│ │ │  o5 : QQ
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/ToricInvariants/html/_ed__Deg.html
│ │ │ @@ -131,15 +131,15 @@
│ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │  ED Degree = 252
│ │ │  
│ │ │                         5      4      3      2
│ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │ - -- used 1.13366s (cpu); 0.779417s (thread); 0s (gc)
│ │ │ + -- used 1.21839s (cpu); 0.846414s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 252
│ │ │  
│ │ │  o4 : QQ
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -150,15 +150,15 @@
│ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │  ED Degree = 252
│ │ │  
│ │ │                         5      4      3      2
│ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │ - -- used 4.61337s (cpu); 3.0038s (thread); 0s (gc)
│ │ │ + -- used 5.06829s (cpu); 3.28094s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 252
│ │ │  
│ │ │  o5 : QQ
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -66,30 +66,30 @@
│ │ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │ │  ED Degree = 252
│ │ │ │  
│ │ │ │                         5      4      3      2
│ │ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │ │ - -- used 1.13366s (cpu); 0.779417s (thread); 0s (gc)
│ │ │ │ + -- used 1.21839s (cpu); 0.846414s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 252
│ │ │ │  
│ │ │ │  o4 : QQ
│ │ │ │  i5 : time edDeg(A,ForceAmat=>true)
│ │ │ │  
│ │ │ │  The toric variety has degree = 28
│ │ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │ │  ED Degree = 252
│ │ │ │  
│ │ │ │                         5      4      3      2
│ │ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │ │ - -- used 4.61337s (cpu); 3.0038s (thread); 0s (gc)
│ │ │ │ + -- used 5.06829s (cpu); 3.28094s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 252
│ │ │ │  
│ │ │ │  o5 : QQ
│ │ │ │  ********** WWaayyss ttoo uussee eeddDDeegg:: **********
│ │ │ │      * edDeg(Matrix)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/Triangulations/example-output/___Triangulations.out
│ │ │ @@ -17,15 +17,15 @@
│ │ │       | -1 1  2  -1 -1 1 -1 1 0 0 |
│ │ │       | 1  0  -1 0  0  0 0  0 0 0 |
│ │ │  
│ │ │                4       10
│ │ │  o2 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i3 : elapsedTime Ts = allTriangulations(A, Fine => true);
│ │ │ - -- .0931162s elapsed
│ │ │ + -- .0990036s elapsed
│ │ │  
│ │ │  i4 : select(Ts, T -> isStar T)
│ │ │  
│ │ │  o4 = {triangulation {{0, 1, 2, 3, 9}, {0, 1, 2, 6, 9}, {0, 1, 3, 7, 9}, {0,
│ │ │       ------------------------------------------------------------------------
│ │ │       1, 6, 7, 9}, {0, 2, 3, 6, 9}, {0, 3, 4, 6, 9}, {0, 3, 4, 8, 9}, {0, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -50,14 +50,14 @@
│ │ │  i7 : T = regularFineTriangulation A
│ │ │  
│ │ │  o7 = triangulation {{0, 1, 2, 3, 9}, {0, 1, 2, 6, 9}, {0, 1, 3, 7, 9}, {0, 1, 6, 7, 9}, {0, 2, 3, 4, 6}, {0, 2, 3, 4, 9}, {0, 2, 4, 6, 9}, {0, 3, 4, 7, 8}, {0, 3, 4, 7, 9}, {0, 3, 5, 7, 8}, {0, 4, 6, 7, 8}, {0, 4, 6, 7, 9}, {0, 5, 6, 7, 8}, {1, 2, 3, 7, 9}, {1, 2, 6, 7, 9}, {2, 3, 4, 7, 8}, {2, 3, 4, 7, 9}, {2, 3, 5, 7, 8}, {2, 4, 6, 7, 8}, {2, 4, 6, 7, 9}, {2, 5, 6, 7, 8}}
│ │ │  
│ │ │  o7 : Triangulation
│ │ │  
│ │ │  i8 : elapsedTime Ts2 = generateTriangulations T;
│ │ │ - -- 1.28881s elapsed
│ │ │ + -- .933174s elapsed
│ │ │  
│ │ │  i9 : #Ts2 == #Ts
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 :
│ │ ├── ./usr/share/doc/Macaulay2/Triangulations/example-output/_generate__Triangulations.out
│ │ │ @@ -21,87 +21,15 @@
│ │ │  
│ │ │  o3 = triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}
│ │ │  
│ │ │  o3 : Triangulation
│ │ │  
│ │ │  i4 : Ts1 = generateTriangulations A -- list of Triangulation's.
│ │ │  
│ │ │ -o4 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ +o4 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -227,275 +155,275 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}}
│ │ │ -
│ │ │ -o4 : List
│ │ │ -
│ │ │ -i5 : Ts2 = generateTriangulations(A, T) -- list of list of subsets
│ │ │ -
│ │ │ -o5 = {{{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7},
│ │ │ +     {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │ +     {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 6, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 3, 5, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ +     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5},
│ │ │ +     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 5, 7},
│ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7},
│ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6},
│ │ │ +     {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 7},
│ │ │ +     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │ +     {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │ +     {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6},
│ │ │ +     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6},
│ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │ +     {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7}, {1, 4, 5, 7},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7},
│ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 4, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7},
│ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 7},
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}, {1, 2, 3, 7}},
│ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ +     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 4},
│ │ │ +     {2, 4, 6, 7}}}
│ │ │ +
│ │ │ +o4 : List
│ │ │ +
│ │ │ +i5 : Ts2 = generateTriangulations(A, T) -- list of list of subsets
│ │ │ +
│ │ │ +o5 = {{{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6},
│ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7},
│ │ │ +     {1, 2, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6},
│ │ │ +     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 6},
│ │ │ +     {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ +     {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5},
│ │ │ +     {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6},
│ │ │ +     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 2, 7},
│ │ │ +     {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}},
│ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ +     {2, 3, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7},
│ │ │ +     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6},
│ │ │ +     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6},
│ │ │ +     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 3, 5, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7},
│ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7},
│ │ │ +     {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6},
│ │ │ +     {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ +     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {2, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6},
│ │ │ +     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ +     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ +     {0, 2, 3, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6},
│ │ │ +     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}},
│ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 6},
│ │ │ +     {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7},
│ │ │ +     {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7},
│ │ │ +     {1, 2, 3, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ +     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}},
│ │ │ +     {1, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 6, 7},
│ │ │ +     {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7},
│ │ │ +     {1, 2, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ +     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 5}, {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ +     {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ +     {0, 5, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 6}, {0, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 6},
│ │ │ +     {0, 4, 6, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 6},
│ │ │ +     {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ +     {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │ +     {3, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6},
│ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7},
│ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {2, 5, 6, 7}}}
│ │ │ -
│ │ │ -o5 : List
│ │ │ -
│ │ │ -i6 : Ts3 = generateTriangulations triangulation(A, T) -- list of Triangulations
│ │ │ -
│ │ │ -o6 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +     {0, 1, 4, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ +     {2, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ +     {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     {0, 4, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ +     {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ +     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ +     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ +     {1, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ +     {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ +     {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ +     {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ +     {2, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ +     {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ +     {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ +     {0, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},
│ │ │ +     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ +     {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ +     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {1, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {0, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ +     {1, 4, 5, 7}, {2, 4, 6, 7}}}
│ │ │ +
│ │ │ +o5 : List
│ │ │ +
│ │ │ +i6 : Ts3 = generateTriangulations triangulation(A, T) -- list of Triangulations
│ │ │ +
│ │ │ +o6 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -621,93 +549,93 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}}
│ │ │ -
│ │ │ -o6 : List
│ │ │ -
│ │ │ -i7 : Ts4 = generateTriangulations tri -- list of Triangulations
│ │ │ -
│ │ │ -o7 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +     {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +     {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ +     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7},
│ │ │ +     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ +     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},
│ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +     {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ +     triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ +     {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ +     {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ +     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ +     {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},
│ │ │ +     {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ +     {2, 4, 6, 7}}}
│ │ │ +
│ │ │ +o6 : List
│ │ │ +
│ │ │ +i7 : Ts4 = generateTriangulations tri -- list of Triangulations
│ │ │ +
│ │ │ +o7 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -833,15 +761,87 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}}
│ │ │ +     {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 4, 6, 7}}}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : all(Ts4, isFine)
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │ @@ -858,191 +858,193 @@
│ │ │  o11 = Tally{false => 66}
│ │ │              true => 8
│ │ │  
│ │ │  o11 : Tally
│ │ │  
│ │ │  i12 : Ts4/gkzVector
│ │ │  
│ │ │ -        16     16  4     8  4       20     8  8     4  8          4  8     8 
│ │ │ -o12 = {{--, 4, --, -, 4, -, -, 8}, {--, 4, -, -, 4, -, -, 8}, {8, -, -, 4, -,
│ │ │ -         3      3  3     3  3        3     3  3     3  3          3  3     3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         8  20       16  16  4  16  4  4       8        4     16  4  16      
│ │ │ -      4, -, --}, {4, --, --, -, --, -, -, 8}, {-, 4, 8, -, 4, --, -, --}, {4,
│ │ │ -         3   3        3   3  3   3  3  3       3        3      3  3   3      
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         20  4  4  20          16        8  16  4  4          20  4        4 
│ │ │ -      4, --, -, -, --, 4, 4}, {--, 4, 4, -, --, -, -, 8}, {4, --, -, 4, 4, -,
│ │ │ -          3  3  3   3           3        3   3  3  3           3  3        3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      20       16  16  4     4        8       4  4  16  8        16    16  4 
│ │ │ -      --, 4}, {--, --, -, 4, -, 4, 8, -}, {8, -, -, --, -, 4, 4, --}, {--, -,
│ │ │ -       3        3   3  3     3        3       3  3   3  3         3     3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      16     4        8    4  16     4     16  8       8  8     8  8     8 
│ │ │ -      --, 4, -, 8, 4, -}, {-, --, 8, -, 4, --, -, 4}, {-, -, 8, -, -, 8, -,
│ │ │ -       3     3        3    3   3     3      3  3       3  3     3  3     3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8       8  4     20  8     8    16  4  4        16  16  4       4  16 
│ │ │ -      -}, {4, -, -, 8, --, -, 4, -}, {--, -, -, 8, 4, --, --, -}, {4, -, --,
│ │ │ -      3       3  3      3  3     3     3  3  3         3   3  3       3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      16  8        4    4  20              20  4    4        20  20       
│ │ │ -      --, -, 8, 4, -}, {-, --, 4, 4, 4, 4, --, -}, {-, 4, 4, --, --, 4, 4,
│ │ │ -       3  3        3    3   3               3  3    3         3   3       
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4    4        8  16  16  4          4     20  20     4             4 
│ │ │ -      -}, {-, 4, 8, -, --, --, -, 4}, {4, -, 4, --, --, 4, -, 4}, {4, 4, -,
│ │ │ -      3    3        3   3   3  3          3      3   3     3             3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      20  20  4          8        16     4  4  16    8        4  8  8  20 
│ │ │ -      --, --, -, 4, 4}, {-, 4, 4, --, 8, -, -, --}, {-, 8, 4, -, -, -, --,
│ │ │ -       3   3  3          3         3     3  3   3    3        3  3  3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -           8     8  8     4  20       20  4  4  20  4  20  20  4    4  20 
│ │ │ -      4}, {-, 8, -, -, 4, -, --, 4}, {--, -, -, --, -, --, --, -}, {-, --,
│ │ │ -           3     3  3     3   3        3  3  3   3  3   3   3  3    3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      20  4  20  4  4  20    20        4  4        20       8  4     4    
│ │ │ -      --, -, --, -, -, --}, {--, 4, 4, -, -, 4, 4, --}, {8, -, -, 4, -, 4,
│ │ │ -       3  3   3  3  3   3     3        3  3         3       3  3     3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      16  16    8        4     4  16  16    4  16     16     4  8       4    
│ │ │ -      --, --}, {-, 8, 4, -, 4, -, --, --}, {-, --, 4, --, 8, -, -, 4}, {-, 4,
│ │ │ -       3   3    3        3     3   3   3    3   3      3     3  3       3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         20     8  8  8    8  8  8     20        4    4        8     8  20 
│ │ │ -      4, --, 8, -, -, -}, {-, -, -, 8, --, 4, 4, -}, {-, 8, 4, -, 4, -, --,
│ │ │ -          3     3  3  3    3  3  3      3        3    3        3     3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8    4     16  4     8  16          4  20     8     8  8    16  4  4 
│ │ │ -      -}, {-, 8, --, -, 4, -, --, 4}, {4, -, --, 4, -, 8, -, -}, {--, -, -,
│ │ │ -      3    3      3  3     3   3          3   3     3     3  3     3  3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         16        8    16     4  16  4  16     4       8  20  8  4       
│ │ │ -      8, --, 4, 4, -}, {--, 4, -, --, -, --, 8, -}, {4, -, --, -, -, 8, 4,
│ │ │ -          3        3     3     3   3  3   3     3       3   3  3  3       
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8    8  8     8     20  4       4  16     4  16     4  16       20  8 
│ │ │ -      -}, {-, -, 8, -, 4, --, -, 4}, {-, --, 8, -, --, 4, -, --}, {4, --, -,
│ │ │ -      3    3  3     3      3  3       3   3     3   3     3   3        3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8  4        8    20        4  8  8  8          8  8  8  4        20  
│ │ │ -      -, -, 4, 8, -}, {--, 4, 4, -, -, -, -, 8}, {8, -, -, -, -, 4, 4, --},
│ │ │ -      3  3        3     3        3  3  3  3          3  3  3  3         3  
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -       16  16     4     4  8          8  4     8  8     20       20     4  4 
│ │ │ -      {--, --, 4, -, 4, -, -, 8}, {8, -, -, 4, -, -, 4, --}, {4, --, 4, -, -,
│ │ │ -        3   3     3     3  3          3  3     3  3      3        3     3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         20       20  4              4  20       4  20        20  4       8 
│ │ │ -      4, --, 4}, {--, -, 4, 4, 4, 4, -, --}, {4, -, --, 4, 4, --, -, 4}, {-,
│ │ │ -          3        3  3              3   3       3   3         3  3       3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         8  20     4  8          16  4  16  8        4    4     16  16     8 
│ │ │ -      4, -, --, 8, -, -, 4}, {4, --, -, --, -, 4, 8, -}, {-, 4, --, --, 8, -,
│ │ │ -         3   3     3  3           3  3   3  3        3    3      3   3     3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4          4  8     16  16     4    4     20        20     4    8  8 
│ │ │ -      -, 4}, {4, -, -, 8, --, --, 4, -}, {-, 4, --, 4, 4, --, 4, -}, {-, -,
│ │ │ -      3          3  3      3   3     3    3      3         3     3    3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8        8  8  8    8  20  8     8        4       16  8     4  16    
│ │ │ -      -, 8, 8, -, -, -}, {-, --, -, 4, -, 4, 8, -}, {4, --, -, 4, -, --, 8,
│ │ │ -      3        3  3  3    3   3  3     3        3        3  3     3   3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4    4        8  16  4  16       16  4     16  4     16  4    4     16 
│ │ │ -      -}, {-, 8, 4, -, --, -, --, 4}, {--, -, 4, --, -, 8, --, -}, {-, 8, --,
│ │ │ -      3    3        3   3  3   3        3  3      3  3      3  3    3      3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4  16  4     16       8  8  8  8  8  8       20  8     8     8  4     
│ │ │ -      -, --, -, 4, --}, {8, -, -, -, -, -, -, 8}, {--, -, 4, -, 4, -, -, 8},
│ │ │ -      3   3  3      3       3  3  3  3  3  3        3  3     3     3  3     
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -          4  8     4  16     16    20     4        4     20       4  4  16 
│ │ │ -      {8, -, -, 4, -, --, 4, --}, {--, 4, -, 4, 4, -, 4, --}, {8, -, -, --,
│ │ │ -          3  3     3   3      3     3     3        3      3       3  3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4  16  16          8  16     4     16  4    8  8     20     8  4     
│ │ │ -      -, --, --, 4}, {4, -, --, 4, -, 8, --, -}, {-, -, 4, --, 8, -, -, 4},
│ │ │ -      3   3   3          3   3     3      3  3    3  3      3     3  3     
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -          4  8     20     8  8       8  4     16     16  4    8  8  20     8 
│ │ │ -      {4, -, -, 8, --, 4, -, -}, {4, -, -, 8, --, 4, --, -}, {-, -, --, 4, -,
│ │ │ -          3  3      3     3  3       3  3      3      3  3    3  3   3     3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -            4    4        8     20  8  8    8     8  8  8  8     8    4  16 
│ │ │ -      8, 4, -}, {-, 4, 8, -, 4, --, -, -}, {-, 8, -, -, -, -, 8, -}, {-, --,
│ │ │ -            3    3        3      3  3  3    3     3  3  3  3     3    3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      16        4  4  16       20  4     8  8     8    8        4  8  20  8
│ │ │ -      --, 4, 8, -, -, --}, {4, --, -, 4, -, -, 8, -}, {-, 4, 8, -, -, --, -,
│ │ │ -       3        3  3   3        3  3     3  3     3    3        3  3   3  3
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4}}
│ │ │ +        20        4  4        20       8  4     4     16  16    8        4 
│ │ │ +o12 = {{--, 4, 4, -, -, 4, 4, --}, {8, -, -, 4, -, 4, --, --}, {-, 8, 4, -,
│ │ │ +         3        3  3         3       3  3     3      3   3    3        3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         4  16  16    4  16     16     4  8       4        20     8  8  8  
│ │ │ +      4, -, --, --}, {-, --, 4, --, 8, -, -, 4}, {-, 4, 4, --, 8, -, -, -},
│ │ │ +         3   3   3    3   3      3     3  3       3         3     3  3  3  
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +       8  8  8     20        4    4        8     8  20  8    4     16  4    
│ │ │ +      {-, -, -, 8, --, 4, 4, -}, {-, 8, 4, -, 4, -, --, -}, {-, 8, --, -, 4,
│ │ │ +       3  3  3      3        3    3        3     3   3  3    3      3  3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      8  16          4  20     8     8  8    16  4  4     16        8    16 
│ │ │ +      -, --, 4}, {4, -, --, 4, -, 8, -, -}, {--, -, -, 8, --, 4, 4, -}, {--,
│ │ │ +      3   3          3   3     3     3  3     3  3  3      3        3     3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         4  16  4  16     4       8  20  8  4        8    8  8     8     20 
│ │ │ +      4, -, --, -, --, 8, -}, {4, -, --, -, -, 8, 4, -}, {-, -, 8, -, 4, --,
│ │ │ +         3   3  3   3     3       3   3  3  3        3    3  3     3      3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4       4  16     4  16     4  16       20  8  8  4        8    20    
│ │ │ +      -, 4}, {-, --, 8, -, --, 4, -, --}, {4, --, -, -, -, 4, 8, -}, {--, 4,
│ │ │ +      3       3   3     3   3     3   3        3  3  3  3        3     3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         4  8  8  8          8  8  8  4        20    16  16     4     4  8 
│ │ │ +      4, -, -, -, -, 8}, {8, -, -, -, -, 4, 4, --}, {--, --, 4, -, 4, -, -,
│ │ │ +         3  3  3  3          3  3  3  3         3     3   3     3     3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +              8  4     8  8     20       20     4  4     20       20  4    
│ │ │ +      8}, {8, -, -, 4, -, -, 4, --}, {4, --, 4, -, -, 4, --, 4}, {--, -, 4,
│ │ │ +              3  3     3  3      3        3     3  3      3        3  3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +               4  20       4  20        20  4       8     8  20     4  8     
│ │ │ +      4, 4, 4, -, --}, {4, -, --, 4, 4, --, -, 4}, {-, 4, -, --, 8, -, -, 4},
│ │ │ +               3   3       3   3         3  3       3     3   3     3  3     
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +          16  4  16  8        4    4     16  16     8  4          4  8    
│ │ │ +      {4, --, -, --, -, 4, 8, -}, {-, 4, --, --, 8, -, -, 4}, {4, -, -, 8,
│ │ │ +           3  3   3  3        3    3      3   3     3  3          3  3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      16  16     4    4     20        20     4    8  8  8        8  8  8  
│ │ │ +      --, --, 4, -}, {-, 4, --, 4, 4, --, 4, -}, {-, -, -, 8, 8, -, -, -},
│ │ │ +       3   3     3    3      3         3     3    3  3  3        3  3  3  
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +       8  20  8     8        4       16  8     4  16     4    4        8  16 
│ │ │ +      {-, --, -, 4, -, 4, 8, -}, {4, --, -, 4, -, --, 8, -}, {-, 8, 4, -, --,
│ │ │ +       3   3  3     3        3        3  3     3   3     3    3        3   3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4  16       16  4     16  4     16  4    4     16  4  16  4     16  
│ │ │ +      -, --, 4}, {--, -, 4, --, -, 8, --, -}, {-, 8, --, -, --, -, 4, --},
│ │ │ +      3   3        3  3      3  3      3  3    3      3  3   3  3      3  
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +          8  8  8  8  8  8       20  8     8     8  4          4  8     4 
│ │ │ +      {8, -, -, -, -, -, -, 8}, {--, -, 4, -, 4, -, -, 8}, {8, -, -, 4, -,
│ │ │ +          3  3  3  3  3  3        3  3     3     3  3          3  3     3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      16     16    20     4        4     20       4  4  16  4  16  16     
│ │ │ +      --, 4, --}, {--, 4, -, 4, 4, -, 4, --}, {8, -, -, --, -, --, --, 4},
│ │ │ +       3      3     3     3        3      3       3  3   3  3   3   3     
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +          8  16     4     16  4    8  8     20     8  4          4  8     20 
│ │ │ +      {4, -, --, 4, -, 8, --, -}, {-, -, 4, --, 8, -, -, 4}, {4, -, -, 8, --,
│ │ │ +          3   3     3      3  3    3  3      3     3  3          3  3      3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         8  8       8  4     16     16  4    8  8  20     8        4    4    
│ │ │ +      4, -, -}, {4, -, -, 8, --, 4, --, -}, {-, -, --, 4, -, 8, 4, -}, {-, 4,
│ │ │ +         3  3       3  3      3      3  3    3  3   3     3        3    3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         8     20  8  8    8     8  8  8  8     8    4  16  16        4  4 
│ │ │ +      8, -, 4, --, -, -}, {-, 8, -, -, -, -, 8, -}, {-, --, --, 4, 8, -, -,
│ │ │ +         3      3  3  3    3     3  3  3  3     3    3   3   3        3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      16       20  4     8  8     8    8        4  8  20  8       16     16 
│ │ │ +      --}, {4, --, -, 4, -, -, 8, -}, {-, 4, 8, -, -, --, -, 4}, {--, 4, --,
│ │ │ +       3        3  3     3  3     3    3        3  3   3  3        3      3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4     8  4       20     8  8     4  8          4  8     8     8  20  
│ │ │ +      -, 4, -, -, 8}, {--, 4, -, -, 4, -, -, 8}, {8, -, -, 4, -, 4, -, --},
│ │ │ +      3     3  3        3     3  3     3  3          3  3     3     3   3  
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +          16  16  4  16  4  4       8        4     16  4  16          20  4 
│ │ │ +      {4, --, --, -, --, -, -, 8}, {-, 4, 8, -, 4, --, -, --}, {4, 4, --, -,
│ │ │ +           3   3  3   3  3  3       3        3      3  3   3           3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4  20          16        8  16  4  4          20  4        4  20     
│ │ │ +      -, --, 4, 4}, {--, 4, 4, -, --, -, -, 8}, {4, --, -, 4, 4, -, --, 4},
│ │ │ +      3   3           3        3   3  3  3           3  3        3   3     
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +       16  16  4     4        8       4  4  16  8        16    16  4  16    
│ │ │ +      {--, --, -, 4, -, 4, 8, -}, {8, -, -, --, -, 4, 4, --}, {--, -, --, 4,
│ │ │ +        3   3  3     3        3       3  3   3  3         3     3  3   3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4        8    4  16     4     16  8       8  8     8  8     8  8      
│ │ │ +      -, 8, 4, -}, {-, --, 8, -, 4, --, -, 4}, {-, -, 8, -, -, 8, -, -}, {4,
│ │ │ +      3        3    3   3     3      3  3       3  3     3  3     3  3      
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      8  4     20  8     8    16  4  4        16  16  4       4  16  16  8 
│ │ │ +      -, -, 8, --, -, 4, -}, {--, -, -, 8, 4, --, --, -}, {4, -, --, --, -,
│ │ │ +      3  3      3  3     3     3  3  3         3   3  3       3   3   3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +            4    4  20              20  4    4        20  20        4    4 
│ │ │ +      8, 4, -}, {-, --, 4, 4, 4, 4, --, -}, {-, 4, 4, --, --, 4, 4, -}, {-,
│ │ │ +            3    3   3               3  3    3         3   3        3    3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +            8  16  16  4          4     20  20     4             4  20  20 
│ │ │ +      4, 8, -, --, --, -, 4}, {4, -, 4, --, --, 4, -, 4}, {4, 4, -, --, --,
│ │ │ +            3   3   3  3          3      3   3     3             3   3   3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4          8        16     4  4  16    8        4  8  8  20       8    
│ │ │ +      -, 4, 4}, {-, 4, 4, --, 8, -, -, --}, {-, 8, 4, -, -, -, --, 4}, {-, 8,
│ │ │ +      3          3         3     3  3   3    3        3  3  3   3       3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      8  8     4  20       20  4  4  20  4  20  20  4    4  20  20  4  20  4 
│ │ │ +      -, -, 4, -, --, 4}, {--, -, -, --, -, --, --, -}, {-, --, --, -, --, -,
│ │ │ +      3  3     3   3        3  3  3   3  3   3   3  3    3   3   3  3   3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4  20
│ │ │ +      -, --}}
│ │ │ +      3   3
│ │ │  
│ │ │  o12 : List
│ │ │  
│ │ │  i13 : volume convexHull A -- 8
│ │ │  
│ │ │  o13 = 8
│ │ │  
│ │ │  o13 : QQ
│ │ │  
│ │ │  i14 : stars1 = select(Ts4, t -> (gkzVector t)#-1 == 8)
│ │ │  
│ │ │ -o14 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +o14 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +      {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ +      {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7},
│ │ │ +      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7},
│ │ │ +      {0, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │ +      {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}},
│ │ │ +      {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7},
│ │ │ +      triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ +      {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation
│ │ │ +      {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +      {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │        {2, 4, 6, 7}}}
│ │ │  
│ │ │  o14 : List
│ │ │  
│ │ │  i15 : stars2 = select(Ts4, isStar)
│ │ │  
│ │ │ -o15 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +o15 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +      {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ +      {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7},
│ │ │ +      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7},
│ │ │ +      {0, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │ +      {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}},
│ │ │ +      {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7},
│ │ │ +      triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ +      {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation
│ │ │ +      {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +      {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │        {2, 4, 6, 7}}}
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : stars1 == stars2
│ │ ├── ./usr/share/doc/Macaulay2/Triangulations/html/_generate__Triangulations.html
│ │ │ @@ -117,87 +117,15 @@
│ │ │  o3 : Triangulation
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : Ts1 = generateTriangulations A -- list of Triangulation's.
│ │ │  
│ │ │ -o4 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ +o4 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -323,281 +251,281 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}}
│ │ │ -
│ │ │ -o4 : List
    
│ │ │ -            
│ │ │ -          
│ │ │ -          
│ │ │ -            
│ │ │ -              
    i5 : Ts2 = generateTriangulations(A, T) -- list of list of subsets
│ │ │ -
│ │ │ -o5 = {{{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7},
│ │ │ +     {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │ +     {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 6, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 3, 5, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ +     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5},
│ │ │ +     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 5, 7},
│ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7},
│ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6},
│ │ │ +     {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 7},
│ │ │ +     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │ +     {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │ +     {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6},
│ │ │ +     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6},
│ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │ +     {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7}, {1, 4, 5, 7},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7},
│ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 4, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7},
│ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 7},
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}, {1, 2, 3, 7}},
│ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ +     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 4},
│ │ │ +     {2, 4, 6, 7}}}
│ │ │ +
│ │ │ +o4 : List
    
│ │ │ +            
│ │ │ +          
│ │ │ +          
│ │ │ +            
│ │ │ +              
    i5 : Ts2 = generateTriangulations(A, T) -- list of list of subsets
│ │ │ +
│ │ │ +o5 = {{{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6},
│ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7},
│ │ │ +     {1, 2, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6},
│ │ │ +     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 6},
│ │ │ +     {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ +     {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5},
│ │ │ +     {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6},
│ │ │ +     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 2, 7},
│ │ │ +     {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}},
│ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ +     {2, 3, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7},
│ │ │ +     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6},
│ │ │ +     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6},
│ │ │ +     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 3, 5, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7},
│ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7},
│ │ │ +     {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6},
│ │ │ +     {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ +     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {2, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6},
│ │ │ +     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ +     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ +     {0, 2, 3, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6},
│ │ │ +     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}},
│ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 6},
│ │ │ +     {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7},
│ │ │ +     {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7},
│ │ │ +     {1, 2, 3, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ +     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}},
│ │ │ +     {1, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 6, 7},
│ │ │ +     {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7},
│ │ │ +     {1, 2, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ +     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 5}, {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ +     {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ +     {0, 5, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 6}, {0, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 6},
│ │ │ +     {0, 4, 6, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 6},
│ │ │ +     {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ +     {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │ +     {3, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6},
│ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7},
│ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {2, 5, 6, 7}}}
│ │ │ -
│ │ │ -o5 : List
    
│ │ │ -            
│ │ │ -          
│ │ │ -          
│ │ │ -            
│ │ │ -              
    i6 : Ts3 = generateTriangulations triangulation(A, T) -- list of Triangulations
│ │ │ -
│ │ │ -o6 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +     {0, 1, 4, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ +     {2, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ +     {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     {0, 4, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ +     {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ +     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ +     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ +     {1, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ +     {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ +     {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ +     {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ +     {2, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ +     {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ +     {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ +     {0, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},
│ │ │ +     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ +     {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ +     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {1, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {0, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ +     {1, 4, 5, 7}, {2, 4, 6, 7}}}
│ │ │ +
│ │ │ +o5 : List
    
│ │ │ +            
│ │ │ +          
│ │ │ +          
│ │ │ +            
│ │ │ +              
    i6 : Ts3 = generateTriangulations triangulation(A, T) -- list of Triangulations
│ │ │ +
│ │ │ +o6 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -723,96 +651,96 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}}
│ │ │ -
│ │ │ -o6 : List
    
│ │ │ -            
│ │ │ -          
│ │ │ -          
│ │ │ -            
│ │ │ -              
    i7 : Ts4 = generateTriangulations tri -- list of Triangulations
│ │ │ -
│ │ │ -o7 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +     {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +     {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ +     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7},
│ │ │ +     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ +     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},
│ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +     {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ +     triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ +     {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ +     {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ +     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ +     {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},
│ │ │ +     {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ +     {2, 4, 6, 7}}}
│ │ │ +
│ │ │ +o6 : List
    
│ │ │ +            
│ │ │ +          
│ │ │ +          
│ │ │ +            
│ │ │ +              
    i7 : Ts4 = generateTriangulations tri -- list of Triangulations
│ │ │ +
│ │ │ +o7 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -938,15 +866,87 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}}
│ │ │ +     {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 4, 6, 7}}}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : all(Ts4, isFine)
│ │ │ @@ -978,131 +978,133 @@
│ │ │  o11 : Tally
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : Ts4/gkzVector
│ │ │  
│ │ │ -        16     16  4     8  4       20     8  8     4  8          4  8     8 
│ │ │ -o12 = {{--, 4, --, -, 4, -, -, 8}, {--, 4, -, -, 4, -, -, 8}, {8, -, -, 4, -,
│ │ │ -         3      3  3     3  3        3     3  3     3  3          3  3     3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         8  20       16  16  4  16  4  4       8        4     16  4  16      
│ │ │ -      4, -, --}, {4, --, --, -, --, -, -, 8}, {-, 4, 8, -, 4, --, -, --}, {4,
│ │ │ -         3   3        3   3  3   3  3  3       3        3      3  3   3      
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         20  4  4  20          16        8  16  4  4          20  4        4 
│ │ │ -      4, --, -, -, --, 4, 4}, {--, 4, 4, -, --, -, -, 8}, {4, --, -, 4, 4, -,
│ │ │ -          3  3  3   3           3        3   3  3  3           3  3        3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      20       16  16  4     4        8       4  4  16  8        16    16  4 
│ │ │ -      --, 4}, {--, --, -, 4, -, 4, 8, -}, {8, -, -, --, -, 4, 4, --}, {--, -,
│ │ │ -       3        3   3  3     3        3       3  3   3  3         3     3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      16     4        8    4  16     4     16  8       8  8     8  8     8 
│ │ │ -      --, 4, -, 8, 4, -}, {-, --, 8, -, 4, --, -, 4}, {-, -, 8, -, -, 8, -,
│ │ │ -       3     3        3    3   3     3      3  3       3  3     3  3     3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8       8  4     20  8     8    16  4  4        16  16  4       4  16 
│ │ │ -      -}, {4, -, -, 8, --, -, 4, -}, {--, -, -, 8, 4, --, --, -}, {4, -, --,
│ │ │ -      3       3  3      3  3     3     3  3  3         3   3  3       3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      16  8        4    4  20              20  4    4        20  20       
│ │ │ -      --, -, 8, 4, -}, {-, --, 4, 4, 4, 4, --, -}, {-, 4, 4, --, --, 4, 4,
│ │ │ -       3  3        3    3   3               3  3    3         3   3       
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4    4        8  16  16  4          4     20  20     4             4 
│ │ │ -      -}, {-, 4, 8, -, --, --, -, 4}, {4, -, 4, --, --, 4, -, 4}, {4, 4, -,
│ │ │ -      3    3        3   3   3  3          3      3   3     3             3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      20  20  4          8        16     4  4  16    8        4  8  8  20 
│ │ │ -      --, --, -, 4, 4}, {-, 4, 4, --, 8, -, -, --}, {-, 8, 4, -, -, -, --,
│ │ │ -       3   3  3          3         3     3  3   3    3        3  3  3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -           8     8  8     4  20       20  4  4  20  4  20  20  4    4  20 
│ │ │ -      4}, {-, 8, -, -, 4, -, --, 4}, {--, -, -, --, -, --, --, -}, {-, --,
│ │ │ -           3     3  3     3   3        3  3  3   3  3   3   3  3    3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      20  4  20  4  4  20    20        4  4        20       8  4     4    
│ │ │ -      --, -, --, -, -, --}, {--, 4, 4, -, -, 4, 4, --}, {8, -, -, 4, -, 4,
│ │ │ -       3  3   3  3  3   3     3        3  3         3       3  3     3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      16  16    8        4     4  16  16    4  16     16     4  8       4    
│ │ │ -      --, --}, {-, 8, 4, -, 4, -, --, --}, {-, --, 4, --, 8, -, -, 4}, {-, 4,
│ │ │ -       3   3    3        3     3   3   3    3   3      3     3  3       3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         20     8  8  8    8  8  8     20        4    4        8     8  20 
│ │ │ -      4, --, 8, -, -, -}, {-, -, -, 8, --, 4, 4, -}, {-, 8, 4, -, 4, -, --,
│ │ │ -          3     3  3  3    3  3  3      3        3    3        3     3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8    4     16  4     8  16          4  20     8     8  8    16  4  4 
│ │ │ -      -}, {-, 8, --, -, 4, -, --, 4}, {4, -, --, 4, -, 8, -, -}, {--, -, -,
│ │ │ -      3    3      3  3     3   3          3   3     3     3  3     3  3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         16        8    16     4  16  4  16     4       8  20  8  4       
│ │ │ -      8, --, 4, 4, -}, {--, 4, -, --, -, --, 8, -}, {4, -, --, -, -, 8, 4,
│ │ │ -          3        3     3     3   3  3   3     3       3   3  3  3       
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8    8  8     8     20  4       4  16     4  16     4  16       20  8 
│ │ │ -      -}, {-, -, 8, -, 4, --, -, 4}, {-, --, 8, -, --, 4, -, --}, {4, --, -,
│ │ │ -      3    3  3     3      3  3       3   3     3   3     3   3        3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8  4        8    20        4  8  8  8          8  8  8  4        20  
│ │ │ -      -, -, 4, 8, -}, {--, 4, 4, -, -, -, -, 8}, {8, -, -, -, -, 4, 4, --},
│ │ │ -      3  3        3     3        3  3  3  3          3  3  3  3         3  
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -       16  16     4     4  8          8  4     8  8     20       20     4  4 
│ │ │ -      {--, --, 4, -, 4, -, -, 8}, {8, -, -, 4, -, -, 4, --}, {4, --, 4, -, -,
│ │ │ -        3   3     3     3  3          3  3     3  3      3        3     3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         20       20  4              4  20       4  20        20  4       8 
│ │ │ -      4, --, 4}, {--, -, 4, 4, 4, 4, -, --}, {4, -, --, 4, 4, --, -, 4}, {-,
│ │ │ -          3        3  3              3   3       3   3         3  3       3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         8  20     4  8          16  4  16  8        4    4     16  16     8 
│ │ │ -      4, -, --, 8, -, -, 4}, {4, --, -, --, -, 4, 8, -}, {-, 4, --, --, 8, -,
│ │ │ -         3   3     3  3           3  3   3  3        3    3      3   3     3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4          4  8     16  16     4    4     20        20     4    8  8 
│ │ │ -      -, 4}, {4, -, -, 8, --, --, 4, -}, {-, 4, --, 4, 4, --, 4, -}, {-, -,
│ │ │ -      3          3  3      3   3     3    3      3         3     3    3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8        8  8  8    8  20  8     8        4       16  8     4  16    
│ │ │ -      -, 8, 8, -, -, -}, {-, --, -, 4, -, 4, 8, -}, {4, --, -, 4, -, --, 8,
│ │ │ -      3        3  3  3    3   3  3     3        3        3  3     3   3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4    4        8  16  4  16       16  4     16  4     16  4    4     16 
│ │ │ -      -}, {-, 8, 4, -, --, -, --, 4}, {--, -, 4, --, -, 8, --, -}, {-, 8, --,
│ │ │ -      3    3        3   3  3   3        3  3      3  3      3  3    3      3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4  16  4     16       8  8  8  8  8  8       20  8     8     8  4     
│ │ │ -      -, --, -, 4, --}, {8, -, -, -, -, -, -, 8}, {--, -, 4, -, 4, -, -, 8},
│ │ │ -      3   3  3      3       3  3  3  3  3  3        3  3     3     3  3     
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -          4  8     4  16     16    20     4        4     20       4  4  16 
│ │ │ -      {8, -, -, 4, -, --, 4, --}, {--, 4, -, 4, 4, -, 4, --}, {8, -, -, --,
│ │ │ -          3  3     3   3      3     3     3        3      3       3  3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4  16  16          8  16     4     16  4    8  8     20     8  4     
│ │ │ -      -, --, --, 4}, {4, -, --, 4, -, 8, --, -}, {-, -, 4, --, 8, -, -, 4},
│ │ │ -      3   3   3          3   3     3      3  3    3  3      3     3  3     
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -          4  8     20     8  8       8  4     16     16  4    8  8  20     8 
│ │ │ -      {4, -, -, 8, --, 4, -, -}, {4, -, -, 8, --, 4, --, -}, {-, -, --, 4, -,
│ │ │ -          3  3      3     3  3       3  3      3      3  3    3  3   3     3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -            4    4        8     20  8  8    8     8  8  8  8     8    4  16 
│ │ │ -      8, 4, -}, {-, 4, 8, -, 4, --, -, -}, {-, 8, -, -, -, -, 8, -}, {-, --,
│ │ │ -            3    3        3      3  3  3    3     3  3  3  3     3    3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      16        4  4  16       20  4     8  8     8    8        4  8  20  8
│ │ │ -      --, 4, 8, -, -, --}, {4, --, -, 4, -, -, 8, -}, {-, 4, 8, -, -, --, -,
│ │ │ -       3        3  3   3        3  3     3  3     3    3        3  3   3  3
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4}}
│ │ │ +        20        4  4        20       8  4     4     16  16    8        4 
│ │ │ +o12 = {{--, 4, 4, -, -, 4, 4, --}, {8, -, -, 4, -, 4, --, --}, {-, 8, 4, -,
│ │ │ +         3        3  3         3       3  3     3      3   3    3        3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         4  16  16    4  16     16     4  8       4        20     8  8  8  
│ │ │ +      4, -, --, --}, {-, --, 4, --, 8, -, -, 4}, {-, 4, 4, --, 8, -, -, -},
│ │ │ +         3   3   3    3   3      3     3  3       3         3     3  3  3  
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +       8  8  8     20        4    4        8     8  20  8    4     16  4    
│ │ │ +      {-, -, -, 8, --, 4, 4, -}, {-, 8, 4, -, 4, -, --, -}, {-, 8, --, -, 4,
│ │ │ +       3  3  3      3        3    3        3     3   3  3    3      3  3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      8  16          4  20     8     8  8    16  4  4     16        8    16 
│ │ │ +      -, --, 4}, {4, -, --, 4, -, 8, -, -}, {--, -, -, 8, --, 4, 4, -}, {--,
│ │ │ +      3   3          3   3     3     3  3     3  3  3      3        3     3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         4  16  4  16     4       8  20  8  4        8    8  8     8     20 
│ │ │ +      4, -, --, -, --, 8, -}, {4, -, --, -, -, 8, 4, -}, {-, -, 8, -, 4, --,
│ │ │ +         3   3  3   3     3       3   3  3  3        3    3  3     3      3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4       4  16     4  16     4  16       20  8  8  4        8    20    
│ │ │ +      -, 4}, {-, --, 8, -, --, 4, -, --}, {4, --, -, -, -, 4, 8, -}, {--, 4,
│ │ │ +      3       3   3     3   3     3   3        3  3  3  3        3     3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         4  8  8  8          8  8  8  4        20    16  16     4     4  8 
│ │ │ +      4, -, -, -, -, 8}, {8, -, -, -, -, 4, 4, --}, {--, --, 4, -, 4, -, -,
│ │ │ +         3  3  3  3          3  3  3  3         3     3   3     3     3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +              8  4     8  8     20       20     4  4     20       20  4    
│ │ │ +      8}, {8, -, -, 4, -, -, 4, --}, {4, --, 4, -, -, 4, --, 4}, {--, -, 4,
│ │ │ +              3  3     3  3      3        3     3  3      3        3  3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +               4  20       4  20        20  4       8     8  20     4  8     
│ │ │ +      4, 4, 4, -, --}, {4, -, --, 4, 4, --, -, 4}, {-, 4, -, --, 8, -, -, 4},
│ │ │ +               3   3       3   3         3  3       3     3   3     3  3     
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +          16  4  16  8        4    4     16  16     8  4          4  8    
│ │ │ +      {4, --, -, --, -, 4, 8, -}, {-, 4, --, --, 8, -, -, 4}, {4, -, -, 8,
│ │ │ +           3  3   3  3        3    3      3   3     3  3          3  3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      16  16     4    4     20        20     4    8  8  8        8  8  8  
│ │ │ +      --, --, 4, -}, {-, 4, --, 4, 4, --, 4, -}, {-, -, -, 8, 8, -, -, -},
│ │ │ +       3   3     3    3      3         3     3    3  3  3        3  3  3  
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +       8  20  8     8        4       16  8     4  16     4    4        8  16 
│ │ │ +      {-, --, -, 4, -, 4, 8, -}, {4, --, -, 4, -, --, 8, -}, {-, 8, 4, -, --,
│ │ │ +       3   3  3     3        3        3  3     3   3     3    3        3   3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4  16       16  4     16  4     16  4    4     16  4  16  4     16  
│ │ │ +      -, --, 4}, {--, -, 4, --, -, 8, --, -}, {-, 8, --, -, --, -, 4, --},
│ │ │ +      3   3        3  3      3  3      3  3    3      3  3   3  3      3  
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +          8  8  8  8  8  8       20  8     8     8  4          4  8     4 
│ │ │ +      {8, -, -, -, -, -, -, 8}, {--, -, 4, -, 4, -, -, 8}, {8, -, -, 4, -,
│ │ │ +          3  3  3  3  3  3        3  3     3     3  3          3  3     3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      16     16    20     4        4     20       4  4  16  4  16  16     
│ │ │ +      --, 4, --}, {--, 4, -, 4, 4, -, 4, --}, {8, -, -, --, -, --, --, 4},
│ │ │ +       3      3     3     3        3      3       3  3   3  3   3   3     
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +          8  16     4     16  4    8  8     20     8  4          4  8     20 
│ │ │ +      {4, -, --, 4, -, 8, --, -}, {-, -, 4, --, 8, -, -, 4}, {4, -, -, 8, --,
│ │ │ +          3   3     3      3  3    3  3      3     3  3          3  3      3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         8  8       8  4     16     16  4    8  8  20     8        4    4    
│ │ │ +      4, -, -}, {4, -, -, 8, --, 4, --, -}, {-, -, --, 4, -, 8, 4, -}, {-, 4,
│ │ │ +         3  3       3  3      3      3  3    3  3   3     3        3    3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         8     20  8  8    8     8  8  8  8     8    4  16  16        4  4 
│ │ │ +      8, -, 4, --, -, -}, {-, 8, -, -, -, -, 8, -}, {-, --, --, 4, 8, -, -,
│ │ │ +         3      3  3  3    3     3  3  3  3     3    3   3   3        3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      16       20  4     8  8     8    8        4  8  20  8       16     16 
│ │ │ +      --}, {4, --, -, 4, -, -, 8, -}, {-, 4, 8, -, -, --, -, 4}, {--, 4, --,
│ │ │ +       3        3  3     3  3     3    3        3  3   3  3        3      3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4     8  4       20     8  8     4  8          4  8     8     8  20  
│ │ │ +      -, 4, -, -, 8}, {--, 4, -, -, 4, -, -, 8}, {8, -, -, 4, -, 4, -, --},
│ │ │ +      3     3  3        3     3  3     3  3          3  3     3     3   3  
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +          16  16  4  16  4  4       8        4     16  4  16          20  4 
│ │ │ +      {4, --, --, -, --, -, -, 8}, {-, 4, 8, -, 4, --, -, --}, {4, 4, --, -,
│ │ │ +           3   3  3   3  3  3       3        3      3  3   3           3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4  20          16        8  16  4  4          20  4        4  20     
│ │ │ +      -, --, 4, 4}, {--, 4, 4, -, --, -, -, 8}, {4, --, -, 4, 4, -, --, 4},
│ │ │ +      3   3           3        3   3  3  3           3  3        3   3     
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +       16  16  4     4        8       4  4  16  8        16    16  4  16    
│ │ │ +      {--, --, -, 4, -, 4, 8, -}, {8, -, -, --, -, 4, 4, --}, {--, -, --, 4,
│ │ │ +        3   3  3     3        3       3  3   3  3         3     3  3   3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4        8    4  16     4     16  8       8  8     8  8     8  8      
│ │ │ +      -, 8, 4, -}, {-, --, 8, -, 4, --, -, 4}, {-, -, 8, -, -, 8, -, -}, {4,
│ │ │ +      3        3    3   3     3      3  3       3  3     3  3     3  3      
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      8  4     20  8     8    16  4  4        16  16  4       4  16  16  8 
│ │ │ +      -, -, 8, --, -, 4, -}, {--, -, -, 8, 4, --, --, -}, {4, -, --, --, -,
│ │ │ +      3  3      3  3     3     3  3  3         3   3  3       3   3   3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +            4    4  20              20  4    4        20  20        4    4 
│ │ │ +      8, 4, -}, {-, --, 4, 4, 4, 4, --, -}, {-, 4, 4, --, --, 4, 4, -}, {-,
│ │ │ +            3    3   3               3  3    3         3   3        3    3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +            8  16  16  4          4     20  20     4             4  20  20 
│ │ │ +      4, 8, -, --, --, -, 4}, {4, -, 4, --, --, 4, -, 4}, {4, 4, -, --, --,
│ │ │ +            3   3   3  3          3      3   3     3             3   3   3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4          8        16     4  4  16    8        4  8  8  20       8    
│ │ │ +      -, 4, 4}, {-, 4, 4, --, 8, -, -, --}, {-, 8, 4, -, -, -, --, 4}, {-, 8,
│ │ │ +      3          3         3     3  3   3    3        3  3  3   3       3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      8  8     4  20       20  4  4  20  4  20  20  4    4  20  20  4  20  4 
│ │ │ +      -, -, 4, -, --, 4}, {--, -, -, --, -, --, --, -}, {-, --, --, -, --, -,
│ │ │ +      3  3     3   3        3  3  3   3  3   3   3  3    3   3   3  3   3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4  20
│ │ │ +      -, --}}
│ │ │ +      3   3
│ │ │  
│ │ │  o12 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : volume convexHull A -- 8
│ │ │ @@ -1112,66 +1114,66 @@
│ │ │  o13 : QQ
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : stars1 = select(Ts4, t -> (gkzVector t)#-1 == 8)
│ │ │  
│ │ │ -o14 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +o14 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +      {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ +      {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7},
│ │ │ +      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7},
│ │ │ +      {0, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │ +      {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}},
│ │ │ +      {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7},
│ │ │ +      triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ +      {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation
│ │ │ +      {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +      {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │        {2, 4, 6, 7}}}
│ │ │  
│ │ │  o14 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : stars2 = select(Ts4, isStar)
│ │ │  
│ │ │ -o15 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +o15 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +      {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ +      {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7},
│ │ │ +      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7},
│ │ │ +      {0, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │ +      {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}},
│ │ │ +      {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7},
│ │ │ +      triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ +      {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation
│ │ │ +      {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +      {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │        {2, 4, 6, 7}}}
│ │ │  
│ │ │  o15 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -57,87 +57,15 @@
│ │ │ │  
│ │ │ │  o3 = triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3,
│ │ │ │  4, 5, 6}, {3, 5, 6, 7}}
│ │ │ │  
│ │ │ │  o3 : Triangulation
│ │ │ │  i4 : Ts1 = generateTriangulations A -- list of Triangulation's.
│ │ │ │  
│ │ │ │ -o4 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ │ +o4 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -263,273 +191,273 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 5, 6, 7}}}
│ │ │ │ -
│ │ │ │ -o4 : List
│ │ │ │ -i5 : Ts2 = generateTriangulations(A, T) -- list of list of subsets
│ │ │ │ -
│ │ │ │ -o5 = {{{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7},
│ │ │ │ +     {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │ │ +     {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 6, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 3, 5, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ │ +     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5},
│ │ │ │ +     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │ +     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 5, 7},
│ │ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 5, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7},
│ │ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6},
│ │ │ │ +     {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │ │ +     triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 7},
│ │ │ │ +     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │ │ +     {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │ │ +     {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6},
│ │ │ │ +     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6},
│ │ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │ │ +     {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7}, {1, 4, 5, 7},
│ │ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {3, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7},
│ │ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 4, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7},
│ │ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ │ +     {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 7},
│ │ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}, {1, 2, 3, 7}},
│ │ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6},
│ │ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ │ +     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 4},
│ │ │ │ +     {2, 4, 6, 7}}}
│ │ │ │ +
│ │ │ │ +o4 : List
│ │ │ │ +i5 : Ts2 = generateTriangulations(A, T) -- list of list of subsets
│ │ │ │ +
│ │ │ │ +o5 = {{{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │ +     {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6},
│ │ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7},
│ │ │ │ +     {1, 2, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 2, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6},
│ │ │ │ +     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 6},
│ │ │ │ +     {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}},
│ │ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ │ +     {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5},
│ │ │ │ +     {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6},
│ │ │ │ +     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 2, 7},
│ │ │ │ +     {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}},
│ │ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ │ +     {2, 3, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7},
│ │ │ │ +     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6},
│ │ │ │ +     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6},
│ │ │ │ +     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │ │ +     {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 3, 5, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │ +     {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7},
│ │ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7},
│ │ │ │ +     {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6},
│ │ │ │ +     {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ │ +     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 5},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │ +     {2, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6},
│ │ │ │ +     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ │ +     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ │ +     {0, 2, 3, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6},
│ │ │ │ +     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}},
│ │ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 6},
│ │ │ │ +     {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7},
│ │ │ │ +     {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7},
│ │ │ │ +     {1, 2, 3, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ │ +     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}},
│ │ │ │ +     {1, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 6, 7},
│ │ │ │ +     {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7},
│ │ │ │ +     {1, 2, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ │ +     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 5}, {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ │ +     {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 5},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ │ +     {0, 5, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 6}, {0, 3, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 6},
│ │ │ │ +     {0, 4, 6, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 6},
│ │ │ │ +     {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ │ +     {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │ │ +     {3, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6},
│ │ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7},
│ │ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 5, 6}, {2, 5, 6, 7}}}
│ │ │ │ -
│ │ │ │ -o5 : List
│ │ │ │ -i6 : Ts3 = generateTriangulations triangulation(A, T) -- list of Triangulations
│ │ │ │ -
│ │ │ │ -o6 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ │ +     {0, 1, 4, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ │ +     {2, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7},
│ │ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ │ +     {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │ +     {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},
│ │ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │ +     {0, 4, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ │ +     {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ │ +     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ +     {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ │ +     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ │ +     {1, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ │ +     {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ │ +     {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ │ +     {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ │ +     {2, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ │ +     {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ │ +     {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ │ +     {0, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},
│ │ │ │ +     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ │ +     {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ │ +     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ +     {1, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, {1, 3, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │ +     {0, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ │ +     {1, 4, 5, 7}, {2, 4, 6, 7}}}
│ │ │ │ +
│ │ │ │ +o5 : List
│ │ │ │ +i6 : Ts3 = generateTriangulations triangulation(A, T) -- list of Triangulations
│ │ │ │ +
│ │ │ │ +o6 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -655,92 +583,92 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 5, 6, 7}}}
│ │ │ │ -
│ │ │ │ -o6 : List
│ │ │ │ -i7 : Ts4 = generateTriangulations tri -- list of Triangulations
│ │ │ │ -
│ │ │ │ -o7 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ │ +     {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ │ +     {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ │ +     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7},
│ │ │ │ +     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ │ +     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},
│ │ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ │ +     {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ │ +     triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ +     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ │ +     {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ │ +     {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ │ +     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ │ +     {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},
│ │ │ │ +     {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ │ +     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ │ +     {2, 4, 6, 7}}}
│ │ │ │ +
│ │ │ │ +o6 : List
│ │ │ │ +i7 : Ts4 = generateTriangulations tri -- list of Triangulations
│ │ │ │ +
│ │ │ │ +o7 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -866,15 +794,87 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, 5, 6, 7}}}
│ │ │ │ +     {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │ │ +     ------------------------------------------------------------------------
│ │ │ │ +     {2, 4, 6, 7}}}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  i8 : all(Ts4, isFine)
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  i9 : all(Ts4, isStar)
│ │ │ │  
│ │ │ │ @@ -886,188 +886,190 @@
│ │ │ │  
│ │ │ │  o11 = Tally{false => 66}
│ │ │ │              true => 8
│ │ │ │  
│ │ │ │  o11 : Tally
│ │ │ │  i12 : Ts4/gkzVector
│ │ │ │  
│ │ │ │ -        16     16  4     8  4       20     8  8     4  8          4  8     8
│ │ │ │ -o12 = {{--, 4, --, -, 4, -, -, 8}, {--, 4, -, -, 4, -, -, 8}, {8, -, -, 4, -,
│ │ │ │ -         3      3  3     3  3        3     3  3     3  3          3  3     3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -         8  20       16  16  4  16  4  4       8        4     16  4  16
│ │ │ │ -      4, -, --}, {4, --, --, -, --, -, -, 8}, {-, 4, 8, -, 4, --, -, --}, {4,
│ │ │ │ -         3   3        3   3  3   3  3  3       3        3      3  3   3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -         20  4  4  20          16        8  16  4  4          20  4        4
│ │ │ │ -      4, --, -, -, --, 4, 4}, {--, 4, 4, -, --, -, -, 8}, {4, --, -, 4, 4, -,
│ │ │ │ -          3  3  3   3           3        3   3  3  3           3  3        3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      20       16  16  4     4        8       4  4  16  8        16    16  4
│ │ │ │ -      --, 4}, {--, --, -, 4, -, 4, 8, -}, {8, -, -, --, -, 4, 4, --}, {--, -,
│ │ │ │ -       3        3   3  3     3        3       3  3   3  3         3     3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      16     4        8    4  16     4     16  8       8  8     8  8     8
│ │ │ │ -      --, 4, -, 8, 4, -}, {-, --, 8, -, 4, --, -, 4}, {-, -, 8, -, -, 8, -,
│ │ │ │ -       3     3        3    3   3     3      3  3       3  3     3  3     3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      8       8  4     20  8     8    16  4  4        16  16  4       4  16
│ │ │ │ -      -}, {4, -, -, 8, --, -, 4, -}, {--, -, -, 8, 4, --, --, -}, {4, -, --,
│ │ │ │ -      3       3  3      3  3     3     3  3  3         3   3  3       3   3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      16  8        4    4  20              20  4    4        20  20
│ │ │ │ -      --, -, 8, 4, -}, {-, --, 4, 4, 4, 4, --, -}, {-, 4, 4, --, --, 4, 4,
│ │ │ │ -       3  3        3    3   3               3  3    3         3   3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      4    4        8  16  16  4          4     20  20     4             4
│ │ │ │ -      -}, {-, 4, 8, -, --, --, -, 4}, {4, -, 4, --, --, 4, -, 4}, {4, 4, -,
│ │ │ │ -      3    3        3   3   3  3          3      3   3     3             3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      20  20  4          8        16     4  4  16    8        4  8  8  20
│ │ │ │ -      --, --, -, 4, 4}, {-, 4, 4, --, 8, -, -, --}, {-, 8, 4, -, -, -, --,
│ │ │ │ -       3   3  3          3         3     3  3   3    3        3  3  3   3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -           8     8  8     4  20       20  4  4  20  4  20  20  4    4  20
│ │ │ │ -      4}, {-, 8, -, -, 4, -, --, 4}, {--, -, -, --, -, --, --, -}, {-, --,
│ │ │ │ -           3     3  3     3   3        3  3  3   3  3   3   3  3    3   3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      20  4  20  4  4  20    20        4  4        20       8  4     4
│ │ │ │ -      --, -, --, -, -, --}, {--, 4, 4, -, -, 4, 4, --}, {8, -, -, 4, -, 4,
│ │ │ │ -       3  3   3  3  3   3     3        3  3         3       3  3     3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      16  16    8        4     4  16  16    4  16     16     4  8       4
│ │ │ │ -      --, --}, {-, 8, 4, -, 4, -, --, --}, {-, --, 4, --, 8, -, -, 4}, {-, 4,
│ │ │ │ -       3   3    3        3     3   3   3    3   3      3     3  3       3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -         20     8  8  8    8  8  8     20        4    4        8     8  20
│ │ │ │ -      4, --, 8, -, -, -}, {-, -, -, 8, --, 4, 4, -}, {-, 8, 4, -, 4, -, --,
│ │ │ │ -          3     3  3  3    3  3  3      3        3    3        3     3   3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      8    4     16  4     8  16          4  20     8     8  8    16  4  4
│ │ │ │ -      -}, {-, 8, --, -, 4, -, --, 4}, {4, -, --, 4, -, 8, -, -}, {--, -, -,
│ │ │ │ -      3    3      3  3     3   3          3   3     3     3  3     3  3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -         16        8    16     4  16  4  16     4       8  20  8  4
│ │ │ │ -      8, --, 4, 4, -}, {--, 4, -, --, -, --, 8, -}, {4, -, --, -, -, 8, 4,
│ │ │ │ -          3        3     3     3   3  3   3     3       3   3  3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      8    8  8     8     20  4       4  16     4  16     4  16       20  8
│ │ │ │ -      -}, {-, -, 8, -, 4, --, -, 4}, {-, --, 8, -, --, 4, -, --}, {4, --, -,
│ │ │ │ -      3    3  3     3      3  3       3   3     3   3     3   3        3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      8  4        8    20        4  8  8  8          8  8  8  4        20
│ │ │ │ -      -, -, 4, 8, -}, {--, 4, 4, -, -, -, -, 8}, {8, -, -, -, -, 4, 4, --},
│ │ │ │ -      3  3        3     3        3  3  3  3          3  3  3  3         3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -       16  16     4     4  8          8  4     8  8     20       20     4  4
│ │ │ │ -      {--, --, 4, -, 4, -, -, 8}, {8, -, -, 4, -, -, 4, --}, {4, --, 4, -, -,
│ │ │ │ -        3   3     3     3  3          3  3     3  3      3        3     3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -         20       20  4              4  20       4  20        20  4       8
│ │ │ │ -      4, --, 4}, {--, -, 4, 4, 4, 4, -, --}, {4, -, --, 4, 4, --, -, 4}, {-,
│ │ │ │ -          3        3  3              3   3       3   3         3  3       3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -         8  20     4  8          16  4  16  8        4    4     16  16     8
│ │ │ │ -      4, -, --, 8, -, -, 4}, {4, --, -, --, -, 4, 8, -}, {-, 4, --, --, 8, -,
│ │ │ │ -         3   3     3  3           3  3   3  3        3    3      3   3     3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      4          4  8     16  16     4    4     20        20     4    8  8
│ │ │ │ -      -, 4}, {4, -, -, 8, --, --, 4, -}, {-, 4, --, 4, 4, --, 4, -}, {-, -,
│ │ │ │ -      3          3  3      3   3     3    3      3         3     3    3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      8        8  8  8    8  20  8     8        4       16  8     4  16
│ │ │ │ -      -, 8, 8, -, -, -}, {-, --, -, 4, -, 4, 8, -}, {4, --, -, 4, -, --, 8,
│ │ │ │ -      3        3  3  3    3   3  3     3        3        3  3     3   3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      4    4        8  16  4  16       16  4     16  4     16  4    4     16
│ │ │ │ -      -}, {-, 8, 4, -, --, -, --, 4}, {--, -, 4, --, -, 8, --, -}, {-, 8, --,
│ │ │ │ -      3    3        3   3  3   3        3  3      3  3      3  3    3      3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      4  16  4     16       8  8  8  8  8  8       20  8     8     8  4
│ │ │ │ -      -, --, -, 4, --}, {8, -, -, -, -, -, -, 8}, {--, -, 4, -, 4, -, -, 8},
│ │ │ │ -      3   3  3      3       3  3  3  3  3  3        3  3     3     3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -          4  8     4  16     16    20     4        4     20       4  4  16
│ │ │ │ -      {8, -, -, 4, -, --, 4, --}, {--, 4, -, 4, 4, -, 4, --}, {8, -, -, --,
│ │ │ │ -          3  3     3   3      3     3     3        3      3       3  3   3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      4  16  16          8  16     4     16  4    8  8     20     8  4
│ │ │ │ -      -, --, --, 4}, {4, -, --, 4, -, 8, --, -}, {-, -, 4, --, 8, -, -, 4},
│ │ │ │ -      3   3   3          3   3     3      3  3    3  3      3     3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -          4  8     20     8  8       8  4     16     16  4    8  8  20     8
│ │ │ │ -      {4, -, -, 8, --, 4, -, -}, {4, -, -, 8, --, 4, --, -}, {-, -, --, 4, -,
│ │ │ │ -          3  3      3     3  3       3  3      3      3  3    3  3   3     3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -            4    4        8     20  8  8    8     8  8  8  8     8    4  16
│ │ │ │ -      8, 4, -}, {-, 4, 8, -, 4, --, -, -}, {-, 8, -, -, -, -, 8, -}, {-, --,
│ │ │ │ -            3    3        3      3  3  3    3     3  3  3  3     3    3   3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      16        4  4  16       20  4     8  8     8    8        4  8  20  8
│ │ │ │ -      --, 4, 8, -, -, --}, {4, --, -, 4, -, -, 8, -}, {-, 4, 8, -, -, --, -,
│ │ │ │ -       3        3  3   3        3  3     3  3     3    3        3  3   3  3
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      4}}
│ │ │ │ +        20        4  4        20       8  4     4     16  16    8        4
│ │ │ │ +o12 = {{--, 4, 4, -, -, 4, 4, --}, {8, -, -, 4, -, 4, --, --}, {-, 8, 4, -,
│ │ │ │ +         3        3  3         3       3  3     3      3   3    3        3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +         4  16  16    4  16     16     4  8       4        20     8  8  8
│ │ │ │ +      4, -, --, --}, {-, --, 4, --, 8, -, -, 4}, {-, 4, 4, --, 8, -, -, -},
│ │ │ │ +         3   3   3    3   3      3     3  3       3         3     3  3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +       8  8  8     20        4    4        8     8  20  8    4     16  4
│ │ │ │ +      {-, -, -, 8, --, 4, 4, -}, {-, 8, 4, -, 4, -, --, -}, {-, 8, --, -, 4,
│ │ │ │ +       3  3  3      3        3    3        3     3   3  3    3      3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      8  16          4  20     8     8  8    16  4  4     16        8    16
│ │ │ │ +      -, --, 4}, {4, -, --, 4, -, 8, -, -}, {--, -, -, 8, --, 4, 4, -}, {--,
│ │ │ │ +      3   3          3   3     3     3  3     3  3  3      3        3     3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +         4  16  4  16     4       8  20  8  4        8    8  8     8     20
│ │ │ │ +      4, -, --, -, --, 8, -}, {4, -, --, -, -, 8, 4, -}, {-, -, 8, -, 4, --,
│ │ │ │ +         3   3  3   3     3       3   3  3  3        3    3  3     3      3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      4       4  16     4  16     4  16       20  8  8  4        8    20
│ │ │ │ +      -, 4}, {-, --, 8, -, --, 4, -, --}, {4, --, -, -, -, 4, 8, -}, {--, 4,
│ │ │ │ +      3       3   3     3   3     3   3        3  3  3  3        3     3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +         4  8  8  8          8  8  8  4        20    16  16     4     4  8
│ │ │ │ +      4, -, -, -, -, 8}, {8, -, -, -, -, 4, 4, --}, {--, --, 4, -, 4, -, -,
│ │ │ │ +         3  3  3  3          3  3  3  3         3     3   3     3     3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +              8  4     8  8     20       20     4  4     20       20  4
│ │ │ │ +      8}, {8, -, -, 4, -, -, 4, --}, {4, --, 4, -, -, 4, --, 4}, {--, -, 4,
│ │ │ │ +              3  3     3  3      3        3     3  3      3        3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +               4  20       4  20        20  4       8     8  20     4  8
│ │ │ │ +      4, 4, 4, -, --}, {4, -, --, 4, 4, --, -, 4}, {-, 4, -, --, 8, -, -, 4},
│ │ │ │ +               3   3       3   3         3  3       3     3   3     3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +          16  4  16  8        4    4     16  16     8  4          4  8
│ │ │ │ +      {4, --, -, --, -, 4, 8, -}, {-, 4, --, --, 8, -, -, 4}, {4, -, -, 8,
│ │ │ │ +           3  3   3  3        3    3      3   3     3  3          3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      16  16     4    4     20        20     4    8  8  8        8  8  8
│ │ │ │ +      --, --, 4, -}, {-, 4, --, 4, 4, --, 4, -}, {-, -, -, 8, 8, -, -, -},
│ │ │ │ +       3   3     3    3      3         3     3    3  3  3        3  3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +       8  20  8     8        4       16  8     4  16     4    4        8  16
│ │ │ │ +      {-, --, -, 4, -, 4, 8, -}, {4, --, -, 4, -, --, 8, -}, {-, 8, 4, -, --,
│ │ │ │ +       3   3  3     3        3        3  3     3   3     3    3        3   3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      4  16       16  4     16  4     16  4    4     16  4  16  4     16
│ │ │ │ +      -, --, 4}, {--, -, 4, --, -, 8, --, -}, {-, 8, --, -, --, -, 4, --},
│ │ │ │ +      3   3        3  3      3  3      3  3    3      3  3   3  3      3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +          8  8  8  8  8  8       20  8     8     8  4          4  8     4
│ │ │ │ +      {8, -, -, -, -, -, -, 8}, {--, -, 4, -, 4, -, -, 8}, {8, -, -, 4, -,
│ │ │ │ +          3  3  3  3  3  3        3  3     3     3  3          3  3     3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      16     16    20     4        4     20       4  4  16  4  16  16
│ │ │ │ +      --, 4, --}, {--, 4, -, 4, 4, -, 4, --}, {8, -, -, --, -, --, --, 4},
│ │ │ │ +       3      3     3     3        3      3       3  3   3  3   3   3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +          8  16     4     16  4    8  8     20     8  4          4  8     20
│ │ │ │ +      {4, -, --, 4, -, 8, --, -}, {-, -, 4, --, 8, -, -, 4}, {4, -, -, 8, --,
│ │ │ │ +          3   3     3      3  3    3  3      3     3  3          3  3      3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +         8  8       8  4     16     16  4    8  8  20     8        4    4
│ │ │ │ +      4, -, -}, {4, -, -, 8, --, 4, --, -}, {-, -, --, 4, -, 8, 4, -}, {-, 4,
│ │ │ │ +         3  3       3  3      3      3  3    3  3   3     3        3    3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +         8     20  8  8    8     8  8  8  8     8    4  16  16        4  4
│ │ │ │ +      8, -, 4, --, -, -}, {-, 8, -, -, -, -, 8, -}, {-, --, --, 4, 8, -, -,
│ │ │ │ +         3      3  3  3    3     3  3  3  3     3    3   3   3        3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      16       20  4     8  8     8    8        4  8  20  8       16     16
│ │ │ │ +      --}, {4, --, -, 4, -, -, 8, -}, {-, 4, 8, -, -, --, -, 4}, {--, 4, --,
│ │ │ │ +       3        3  3     3  3     3    3        3  3   3  3        3      3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      4     8  4       20     8  8     4  8          4  8     8     8  20
│ │ │ │ +      -, 4, -, -, 8}, {--, 4, -, -, 4, -, -, 8}, {8, -, -, 4, -, 4, -, --},
│ │ │ │ +      3     3  3        3     3  3     3  3          3  3     3     3   3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +          16  16  4  16  4  4       8        4     16  4  16          20  4
│ │ │ │ +      {4, --, --, -, --, -, -, 8}, {-, 4, 8, -, 4, --, -, --}, {4, 4, --, -,
│ │ │ │ +           3   3  3   3  3  3       3        3      3  3   3           3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      4  20          16        8  16  4  4          20  4        4  20
│ │ │ │ +      -, --, 4, 4}, {--, 4, 4, -, --, -, -, 8}, {4, --, -, 4, 4, -, --, 4},
│ │ │ │ +      3   3           3        3   3  3  3           3  3        3   3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +       16  16  4     4        8       4  4  16  8        16    16  4  16
│ │ │ │ +      {--, --, -, 4, -, 4, 8, -}, {8, -, -, --, -, 4, 4, --}, {--, -, --, 4,
│ │ │ │ +        3   3  3     3        3       3  3   3  3         3     3  3   3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      4        8    4  16     4     16  8       8  8     8  8     8  8
│ │ │ │ +      -, 8, 4, -}, {-, --, 8, -, 4, --, -, 4}, {-, -, 8, -, -, 8, -, -}, {4,
│ │ │ │ +      3        3    3   3     3      3  3       3  3     3  3     3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      8  4     20  8     8    16  4  4        16  16  4       4  16  16  8
│ │ │ │ +      -, -, 8, --, -, 4, -}, {--, -, -, 8, 4, --, --, -}, {4, -, --, --, -,
│ │ │ │ +      3  3      3  3     3     3  3  3         3   3  3       3   3   3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +            4    4  20              20  4    4        20  20        4    4
│ │ │ │ +      8, 4, -}, {-, --, 4, 4, 4, 4, --, -}, {-, 4, 4, --, --, 4, 4, -}, {-,
│ │ │ │ +            3    3   3               3  3    3         3   3        3    3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +            8  16  16  4          4     20  20     4             4  20  20
│ │ │ │ +      4, 8, -, --, --, -, 4}, {4, -, 4, --, --, 4, -, 4}, {4, 4, -, --, --,
│ │ │ │ +            3   3   3  3          3      3   3     3             3   3   3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      4          8        16     4  4  16    8        4  8  8  20       8
│ │ │ │ +      -, 4, 4}, {-, 4, 4, --, 8, -, -, --}, {-, 8, 4, -, -, -, --, 4}, {-, 8,
│ │ │ │ +      3          3         3     3  3   3    3        3  3  3   3       3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      8  8     4  20       20  4  4  20  4  20  20  4    4  20  20  4  20  4
│ │ │ │ +      -, -, 4, -, --, 4}, {--, -, -, --, -, --, --, -}, {-, --, --, -, --, -,
│ │ │ │ +      3  3     3   3        3  3  3   3  3   3   3  3    3   3   3  3   3  3
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      4  20
│ │ │ │ +      -, --}}
│ │ │ │ +      3   3
│ │ │ │  
│ │ │ │  o12 : List
│ │ │ │  i13 : volume convexHull A -- 8
│ │ │ │  
│ │ │ │  o13 = 8
│ │ │ │  
│ │ │ │  o13 : QQ
│ │ │ │  i14 : stars1 = select(Ts4, t -> (gkzVector t)#-1 == 8)
│ │ │ │  
│ │ │ │ -o14 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ │ +o14 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ │ +      {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ │ +      {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7},
│ │ │ │ +      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7},
│ │ │ │ +      {0, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │ │ +      {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}},
│ │ │ │ +      {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7},
│ │ │ │ +      triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ │ +      {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation
│ │ │ │ +      {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ │ +      {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        {2, 4, 6, 7}}}
│ │ │ │  
│ │ │ │  o14 : List
│ │ │ │  i15 : stars2 = select(Ts4, isStar)
│ │ │ │  
│ │ │ │ -o15 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ │ +o15 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ │ +      {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ │ +      {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7},
│ │ │ │ +      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7},
│ │ │ │ +      {0, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │ │ +      {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}},
│ │ │ │ +      {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7},
│ │ │ │ +      triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ │ +      {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation
│ │ │ │ +      {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ │ +      {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        {2, 4, 6, 7}}}
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  i16 : stars1 == stars2
│ │ │ │  
│ │ │ │  o16 = true
│ │ ├── ./usr/share/doc/Macaulay2/Triangulations/html/index.html
│ │ │ @@ -150,15 +150,15 @@
│ │ │                4       10
│ │ │  o2 : Matrix ZZ  <-- ZZ
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime Ts = allTriangulations(A, Fine => true);
│ │ │ - -- .0931162s elapsed
    
│ │ │ + -- .0990036s elapsed
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : select(Ts, T -> isStar T)
│ │ │  
│ │ │  o4 = {triangulation {{0, 1, 2, 3, 9}, {0, 1, 2, 6, 9}, {0, 1, 3, 7, 9}, {0,
│ │ │ @@ -198,15 +198,15 @@
│ │ │  
│ │ │  o7 : Triangulation
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : elapsedTime Ts2 = generateTriangulations T;
│ │ │ - -- 1.28881s elapsed
    
│ │ │ + -- .933174s elapsed
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : #Ts2 == #Ts
│ │ │  
│ │ │  o9 = true
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -54,15 +54,15 @@
│ │ │ │       | 0  0  0  1  0  0 -1 0 0 0 |
│ │ │ │       | -1 1  2  -1 -1 1 -1 1 0 0 |
│ │ │ │       | 1  0  -1 0  0  0 0  0 0 0 |
│ │ │ │  
│ │ │ │                4       10
│ │ │ │  o2 : Matrix ZZ  <-- ZZ
│ │ │ │  i3 : elapsedTime Ts = allTriangulations(A, Fine => true);
│ │ │ │ - -- .0931162s elapsed
│ │ │ │ + -- .0990036s elapsed
│ │ │ │  i4 : select(Ts, T -> isStar T)
│ │ │ │  
│ │ │ │  o4 = {triangulation {{0, 1, 2, 3, 9}, {0, 1, 2, 6, 9}, {0, 1, 3, 7, 9}, {0,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       1, 6, 7, 9}, {0, 2, 3, 6, 9}, {0, 3, 4, 6, 9}, {0, 3, 4, 8, 9}, {0, 3,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       5, 7, 9}, {0, 3, 5, 8, 9}, {0, 4, 6, 8, 9}, {0, 5, 6, 7, 9}, {0, 5, 6,
│ │ │ │ @@ -86,15 +86,15 @@
│ │ │ │  6, 7, 9}, {0, 2, 3, 4, 6}, {0, 2, 3, 4, 9}, {0, 2, 4, 6, 9}, {0, 3, 4, 7, 8},
│ │ │ │  {0, 3, 4, 7, 9}, {0, 3, 5, 7, 8}, {0, 4, 6, 7, 8}, {0, 4, 6, 7, 9}, {0, 5, 6,
│ │ │ │  7, 8}, {1, 2, 3, 7, 9}, {1, 2, 6, 7, 9}, {2, 3, 4, 7, 8}, {2, 3, 4, 7, 9}, {2,
│ │ │ │  3, 5, 7, 8}, {2, 4, 6, 7, 8}, {2, 4, 6, 7, 9}, {2, 5, 6, 7, 8}}
│ │ │ │  
│ │ │ │  o7 : Triangulation
│ │ │ │  i8 : elapsedTime Ts2 = generateTriangulations T;
│ │ │ │ - -- 1.28881s elapsed
│ │ │ │ + -- .933174s elapsed
│ │ │ │  i9 : #Ts2 == #Ts
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _P_o_l_y_h_e_d_r_a -- for computations with convex polyhedra, cones, and fans
│ │ │ │      * _T_o_p_c_o_m -- interface to selected functions from topcom package
│ │ │ │      * _R_e_f_l_e_x_i_v_e_P_o_l_y_t_o_p_e_s_D_B -- simple access to Kreuzer-Skarke database of
│ │ ├── ./usr/share/doc/Macaulay2/VersalDeformations/dump/rawdocumentation.dump
│ │ │ @@ -1,8 +1,8 @@
│ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sun Dec 14 14:09:53 2025
│ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sun Dec 14 14:09:54 2025
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │  #:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=33
│ │ │  bGlmdERlZm9ybWF0aW9uKC4uLixWZXJib3NlPT4uLi4p
│ │ ├── ./usr/share/doc/Macaulay2/VersalDeformations/example-output/___Smart__Lift.out
│ │ │ @@ -6,30 +6,30 @@
│ │ │  
│ │ │  o2 = | xz yz z2 x3 |
│ │ │  
│ │ │               1      4
│ │ │  o2 : Matrix S  <-- S
│ │ │  
│ │ │  i3 : time (F,R,G,C)=localHilbertScheme(F0);
│ │ │ - -- used 0.91597s (cpu); 0.578592s (thread); 0s (gc)
│ │ │ + -- used 1.06518s (cpu); 0.680372s (thread); 0s (gc)
│ │ │  
│ │ │  i4 : T=ring first G;
│ │ │  
│ │ │  i5 : sum G
│ │ │  
│ │ │  o5 = | t_1t_16             |
│ │ │       | t_9t_16             |
│ │ │       | -t_4t_16            |
│ │ │       | -2t_14t_16+t_15t_16 |
│ │ │  
│ │ │               4      1
│ │ │  o5 : Matrix T  <-- T
│ │ │  
│ │ │  i6 : time (F,R,G,C)=localHilbertScheme(F0,SmartLift=>false);
│ │ │ - -- used 0.654992s (cpu); 0.406513s (thread); 0s (gc)
│ │ │ + -- used 0.716249s (cpu); 0.449181s (thread); 0s (gc)
│ │ │  
│ │ │  i7 : sum G
│ │ │  
│ │ │  o7 = | t_1t_16                                                             
│ │ │       | 2t_5t_10t_11t_16+t_7t_11^2t_16-2t_6t_10t_16+3t_10^2t_16-t_8t_11t_16+
│ │ │       | -t_5t_10^2t_16-2t_7t_10t_11t_16-3t_2t_11^2t_16+t_8t_10t_16+2t_3t_11t
│ │ │       | 2t_5t_10t_16^2+2t_7t_11t_16^2+4t_10t_12t_16+2t_11t_13t_16-t_8t_16^2-
│ │ ├── ./usr/share/doc/Macaulay2/VersalDeformations/html/___Smart__Lift.html
│ │ │ @@ -71,15 +71,15 @@
│ │ │            
│ │ │          
│ │ │          
    With the default setting SmartLift=>true we get very nice equations for the base space:
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -98,15 +98,15 @@
│ │ │            
│ │ │          
    
│ │ │                
    i3 : time (F,R,G,C)=localHilbertScheme(F0);
│ │ │ - -- used 0.91597s (cpu); 0.578592s (thread); 0s (gc)
    
│ │ │ + -- used 1.06518s (cpu); 0.680372s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i4 : T=ring first G;
    
│ │ │              
    
│ │ │          
    With the setting SmartLift=>false the calculation is faster, but the equations are no longer homogeneous:
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i6 : time (F,R,G,C)=localHilbertScheme(F0,SmartLift=>false);
│ │ │ - -- used 0.654992s (cpu); 0.406513s (thread); 0s (gc)
    
│ │ │ + -- used 0.716249s (cpu); 0.449181s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i7 : sum G
│ │ │  
│ │ │  o7 = | t_1t_16
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,29 +18,29 @@
│ │ │ │  o2 = | xz yz z2 x3 |
│ │ │ │  
│ │ │ │               1      4
│ │ │ │  o2 : Matrix S  <-- S
│ │ │ │  With the default setting SmartLift=>true we get very nice equations for the
│ │ │ │  base space:
│ │ │ │  i3 : time (F,R,G,C)=localHilbertScheme(F0);
│ │ │ │ - -- used 0.91597s (cpu); 0.578592s (thread); 0s (gc)
│ │ │ │ + -- used 1.06518s (cpu); 0.680372s (thread); 0s (gc)
│ │ │ │  i4 : T=ring first G;
│ │ │ │  i5 : sum G
│ │ │ │  
│ │ │ │  o5 = | t_1t_16             |
│ │ │ │       | t_9t_16             |
│ │ │ │       | -t_4t_16            |
│ │ │ │       | -2t_14t_16+t_15t_16 |
│ │ │ │  
│ │ │ │               4      1
│ │ │ │  o5 : Matrix T  <-- T
│ │ │ │  With the setting SmartLift=>false the calculation is faster, but the equations
│ │ │ │  are no longer homogeneous:
│ │ │ │  i6 : time (F,R,G,C)=localHilbertScheme(F0,SmartLift=>false);
│ │ │ │ - -- used 0.654992s (cpu); 0.406513s (thread); 0s (gc)
│ │ │ │ + -- used 0.716249s (cpu); 0.449181s (thread); 0s (gc)
│ │ │ │  i7 : sum G
│ │ │ │  
│ │ │ │  o7 = | t_1t_16
│ │ │ │       | 2t_5t_10t_11t_16+t_7t_11^2t_16-2t_6t_10t_16+3t_10^2t_16-t_8t_11t_16+
│ │ │ │       | -t_5t_10^2t_16-2t_7t_10t_11t_16-3t_2t_11^2t_16+t_8t_10t_16+2t_3t_11t
│ │ │ │       | 2t_5t_10t_16^2+2t_7t_11t_16^2+4t_10t_12t_16+2t_11t_13t_16-t_8t_16^2-
│ │ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/___Basic__Divisor.out
│ │ │ @@ -1,14 +1,14 @@
│ │ │  -- -*- M2-comint -*- hash: 18380296066161043289
│ │ │  
│ │ │  i1 : R = QQ[x,y,z];
│ │ │  
│ │ │  i2 : D = divisor(x*y^2*z^3)
│ │ │  
│ │ │ -o2 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │ +o2 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : H = new HashTable from D
│ │ │  
│ │ │  o3 = HashTable{{x} => {1}                  }
│ │ │                 {y} => {2}
│ │ │ @@ -16,18 +16,18 @@
│ │ │                 cache => CacheTable{...1...}
│ │ │                 ring => R
│ │ │  
│ │ │  o3 : HashTable
│ │ │  
│ │ │  i4 : (2/3)*D
│ │ │  
│ │ │ -o4 = 2/3*Div(x) + 4/3*Div(y) + 2*Div(z)
│ │ │ +o4 = 2*Div(z) + 4/3*Div(y) + 2/3*Div(x)
│ │ │  
│ │ │  o4 : QWeilDivisor on R
│ │ │  
│ │ │  i5 : 0.6*D
│ │ │  
│ │ │ -o5 = .6*Div(x) + 1.2*Div(y) + 1.8*Div(z)
│ │ │ +o5 = 1.8*Div(z) + 1.2*Div(y) + .6*Div(x)
│ │ │  
│ │ │  o5 : RWeilDivisor on R
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/___Number_sp_st_sp__Basic__Divisor.out
│ │ │ @@ -16,21 +16,21 @@
│ │ │  
│ │ │  o4 = -3.2*Div(-y^3+x^2) + 1.5*Div(x) + 0*Div(y)
│ │ │  
│ │ │  o4 : RWeilDivisor on R
│ │ │  
│ │ │  i5 : 8*D
│ │ │  
│ │ │ -o5 = -8*Div(x+y) + 16*Div(x) + 8*Div(y)
│ │ │ +o5 = -8*Div(x+y) + 8*Div(y) + 16*Div(x)
│ │ │  
│ │ │  o5 : WeilDivisor on R
│ │ │  
│ │ │  i6 : (-2/3)*D
│ │ │  
│ │ │ -o6 = 2/3*Div(x+y) + -4/3*Div(x) + -2/3*Div(y)
│ │ │ +o6 = 2/3*Div(x+y) + -2/3*Div(y) + -4/3*Div(x)
│ │ │  
│ │ │  o6 : QWeilDivisor on R
│ │ │  
│ │ │  i7 : 0.0*D
│ │ │  
│ │ │  o7 = 0, the zero divisor
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_ceiling_lp__R__Weil__Divisor_rp.out
│ │ │ @@ -1,32 +1,32 @@
│ │ │  -- -*- M2-comint -*- hash: 992133077988949640
│ │ │  
│ │ │  i1 : R = QQ[x, y, z] / ideal(x *y - z^2);
│ │ │  
│ │ │  i2 : D = divisor({1/2, 4/3}, {ideal(x, z), ideal(y, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o2 = 4/3*Div(y, z) + 1/2*Div(x, z)
│ │ │ +o2 = 1/2*Div(x, z) + 4/3*Div(y, z)
│ │ │  
│ │ │  o2 : QWeilDivisor on R
│ │ │  
│ │ │  i3 : ceiling( D )
│ │ │  
│ │ │ -o3 = 2*Div(y, z) + Div(x, z)
│ │ │ +o3 = Div(x, z) + 2*Div(y, z)
│ │ │  
│ │ │  o3 : WeilDivisor on R
│ │ │  
│ │ │  i4 : floor( D )
│ │ │  
│ │ │  o4 = Div(y, z)
│ │ │  
│ │ │  o4 : WeilDivisor on R
│ │ │  
│ │ │  i5 : E = divisor({0.3, -0.7}, {ideal(x, z), ideal(y,z)}, CoefficientType => RR)
│ │ │  
│ │ │ -o5 = -.7*Div(y, z) + .3*Div(x, z)
│ │ │ +o5 = .3*Div(x, z) + -.7*Div(y, z)
│ │ │  
│ │ │  o5 : RWeilDivisor on R
│ │ │  
│ │ │  i6 : ceiling( E )
│ │ │  
│ │ │  o6 = Div(x, z)
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_divisor.out
│ │ │ @@ -1,38 +1,38 @@
│ │ │  -- -*- M2-comint -*- hash: 16935688116980988371
│ │ │  
│ │ │  i1 : R = QQ[x,y,z];
│ │ │  
│ │ │  i2 : D = divisor({1,2,3}, {ideal(x), ideal(y), ideal(z)})
│ │ │  
│ │ │ -o2 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ +o2 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : E = divisor(x*y^2*z^3)
│ │ │  
│ │ │ -o3 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ +o3 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │  
│ │ │  o3 : WeilDivisor on R
│ │ │  
│ │ │  i4 : F = divisor(ideal(x*y^2*z^3))
│ │ │  
│ │ │ -o4 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ +o4 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │  
│ │ │  o4 : WeilDivisor on R
│ │ │  
│ │ │  i5 : G = divisor({{1, ideal(x)}, {2, ideal(y)}, {3, ideal(z)}})
│ │ │  
│ │ │ -o5 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ +o5 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │  
│ │ │  o5 : WeilDivisor on R
│ │ │  
│ │ │  i6 : H = divisor(x) + 2*divisor(y) + 3*divisor(z)
│ │ │  
│ │ │ -o6 = 3*Div(z) + 2*Div(y) + Div(x)
│ │ │ +o6 = Div(x) + 3*Div(z) + 2*Div(y)
│ │ │  
│ │ │  o6 : WeilDivisor on R
│ │ │  
│ │ │  i7 : R = QQ[x,y,z]/ideal(x^2-y*z);
│ │ │  
│ │ │  i8 : D = divisor({2}, {ideal(x,y)})
│ │ │  
│ │ │ @@ -60,29 +60,29 @@
│ │ │  
│ │ │  o14 = 3*Div(xz2, xyz, xy2, x2z, x2y, x3)
│ │ │  
│ │ │  o14 : WeilDivisor on A
│ │ │  
│ │ │  i15 : E = divisor(y2z)
│ │ │  
│ │ │ -o15 = 2*Div(yz2, y2z, y3, xyz, xy2, x2y) + Div(z3, yz2, y2z, xz2, xyz, x2z)
│ │ │ +o15 = Div(z3, yz2, y2z, xz2, xyz, x2z) + 2*Div(yz2, y2z, y3, xyz, xy2, x2y)
│ │ │  
│ │ │  o15 : WeilDivisor on A
│ │ │  
│ │ │  i16 : R = ZZ/7[x,y];
│ │ │  
│ │ │  i17 : D = divisor({-1/2, 2/1}, {ideal(y^2-x^3), ideal(x)}, CoefficientType=>QQ)
│ │ │  
│ │ │ -o17 = -1/2*Div(-x^3+y^2) + 2*Div(x)
│ │ │ +o17 = 2*Div(x) + -1/2*Div(-x^3+y^2)
│ │ │  
│ │ │  o17 : QWeilDivisor on R
│ │ │  
│ │ │  i18 : D = (-1/2)*divisor(y^2-x^3) + (2/1)*divisor(x)
│ │ │  
│ │ │ -o18 = -1/2*Div(-x^3+y^2) + 2*Div(x)
│ │ │ +o18 = 2*Div(x) + -1/2*Div(-x^3+y^2)
│ │ │  
│ │ │  o18 : QWeilDivisor on R
│ │ │  
│ │ │  i19 : R = ZZ/11[x,y,u,v]/ideal(x*y-u*v);
│ │ │  
│ │ │  i20 : D = divisor({1.1, -3.14159}, {ideal(x,u), ideal(x, v)}, CoefficientType=>RR)
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_dualize.out
│ │ │ @@ -44,51 +44,51 @@
│ │ │  i10 : J = m^9;
│ │ │  
│ │ │  o10 : Ideal of R
│ │ │  
│ │ │  i11 : M = J*R^1;
│ │ │  
│ │ │  i12 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ - -- used 0.135013s (cpu); 0.0782845s (thread); 0s (gc)
│ │ │ + -- used 0.127632s (cpu); 0.0718664s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of R
│ │ │  
│ │ │  i13 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.455344s (cpu); 0.455348s (thread); 0s (gc)
│ │ │ + -- used 0.495472s (cpu); 0.495481s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
│ │ │  
│ │ │  i14 : time dualize(M, Strategy=>IdealStrategy);
│ │ │ - -- used 0.57369s (cpu); 0.498594s (thread); 0s (gc)
│ │ │ + -- used 0.627359s (cpu); 0.55998s (thread); 0s (gc)
│ │ │  
│ │ │  i15 : time dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00285362s (cpu); 0.00285446s (thread); 0s (gc)
│ │ │ + -- used 0.00316931s (cpu); 0.00317578s (thread); 0s (gc)
│ │ │  
│ │ │  i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00222665s (cpu); 0.0022275s (thread); 0s (gc)
│ │ │ + -- used 0.00282471s (cpu); 0.00283097s (thread); 0s (gc)
│ │ │  
│ │ │  o16 : Ideal of R
│ │ │  
│ │ │  i17 : R = ZZ/7[x,y,u,v]/ideal(x*y-u*v);
│ │ │  
│ │ │  i18 : I = ideal(x,u);
│ │ │  
│ │ │  o18 : Ideal of R
│ │ │  
│ │ │  i19 : J = I^15;
│ │ │  
│ │ │  o19 : Ideal of R
│ │ │  
│ │ │  i20 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ - -- used 0.279007s (cpu); 0.150681s (thread); 0s (gc)
│ │ │ + -- used 0.290507s (cpu); 0.138215s (thread); 0s (gc)
│ │ │  
│ │ │  o20 : Ideal of R
│ │ │  
│ │ │  i21 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00604243s (cpu); 0.00604328s (thread); 0s (gc)
│ │ │ + -- used 0.00726241s (cpu); 0.00726781s (thread); 0s (gc)
│ │ │  
│ │ │  o21 : Ideal of R
│ │ │  
│ │ │  i22 : R = QQ[x,y]/ideal(x*y);
│ │ │  
│ │ │  i23 : J = ideal(x,y);
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_is__Cartier.out
│ │ │ @@ -12,15 +12,15 @@
│ │ │  
│ │ │  o3 = false
│ │ │  
│ │ │  i4 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │  
│ │ │  i5 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o5 = Div(x, z) + 2*Div(y, z)
│ │ │ +o5 = 2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o5 : WeilDivisor on R
│ │ │  
│ │ │  i6 : isCartier( D )
│ │ │  
│ │ │  o6 = false
│ │ │  
│ │ │ @@ -48,15 +48,15 @@
│ │ │  
│ │ │  o12 = true
│ │ │  
│ │ │  i13 : R = QQ[x, y, z] / ideal(x * y - z^2);
│ │ │  
│ │ │  i14 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o14 = Div(x, z) + 2*Div(y, z)
│ │ │ +o14 = 2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o14 : WeilDivisor on R
│ │ │  
│ │ │  i15 : isCartier(D, IsGraded => true)
│ │ │  
│ │ │  o15 = true
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_is__Homogeneous_lp__Basic__Divisor_rp.out
│ │ │ @@ -1,14 +1,14 @@
│ │ │  -- -*- M2-comint -*- hash: 18048197335381839324
│ │ │  
│ │ │  i1 : R = QQ[x, y, z];
│ │ │  
│ │ │  i2 : D = divisor({1, 2, 3}, {ideal(x * y - z^2), ideal(y * z - x^2), ideal(x * z - y^2)})
│ │ │  
│ │ │ -o2 = 3*Div(-y^2+x*z) + Div(x*y-z^2) + 2*Div(-x^2+y*z)
│ │ │ +o2 = Div(x*y-z^2) + 2*Div(-x^2+y*z) + 3*Div(-y^2+x*z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : isHomogeneous( D )
│ │ │  
│ │ │  o3 = true
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_is__Linear__Equivalent.out
│ │ │ @@ -1,38 +1,38 @@
│ │ │  -- -*- M2-comint -*- hash: 6019119347082811396
│ │ │  
│ │ │  i1 : R = QQ[x, y, z]/ ideal(x * y - z^2);
│ │ │  
│ │ │  i2 : D1 = divisor({3, 8}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = 8*Div(y, z) + 3*Div(x, z)
│ │ │ +o2 = 3*Div(x, z) + 8*Div(y, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : D2 = divisor({8, 1}, {ideal(y, z), ideal(x, z)})
│ │ │  
│ │ │ -o3 = 8*Div(y, z) + Div(x, z)
│ │ │ +o3 = Div(x, z) + 8*Div(y, z)
│ │ │  
│ │ │  o3 : WeilDivisor on R
│ │ │  
│ │ │  i4 : isLinearEquivalent(D1, D2)
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 : R = QQ[x, y, z]/ ideal(x * y - z^2);
│ │ │  
│ │ │  i6 : D1 = divisor({3, 8}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o6 = 3*Div(x, z) + 8*Div(y, z)
│ │ │ +o6 = 8*Div(y, z) + 3*Div(x, z)
│ │ │  
│ │ │  o6 : WeilDivisor on R
│ │ │  
│ │ │  i7 : D2 = divisor({8, 1}, {ideal(y, z), ideal(x, z)})
│ │ │  
│ │ │ -o7 = Div(x, z) + 8*Div(y, z)
│ │ │ +o7 = 8*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o7 : WeilDivisor on R
│ │ │  
│ │ │  i8 : isLinearEquivalent(D1, D2, IsGraded => true)
│ │ │  
│ │ │  o8 = false
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_is__Q__Cartier.out
│ │ │ @@ -1,20 +1,20 @@
│ │ │  -- -*- M2-comint -*- hash: 13719144060491348416
│ │ │  
│ │ │  i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │  
│ │ │  i2 : D1 = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = 2*Div(y, z) + Div(x, z)
│ │ │ +o2 = Div(x, z) + 2*Div(y, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : D2 = divisor({1/2, 3/4}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o3 = 1/2*Div(y, z) + 3/4*Div(x, z)
│ │ │ +o3 = 3/4*Div(x, z) + 1/2*Div(y, z)
│ │ │  
│ │ │  o3 : QWeilDivisor on R
│ │ │  
│ │ │  i4 : isQCartier(10, D1)
│ │ │  
│ │ │  o4 = 2
│ │ │  
│ │ │ @@ -44,21 +44,21 @@
│ │ │  
│ │ │  o10 = 0
│ │ │  
│ │ │  i11 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │  
│ │ │  i12 : D1 = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o12 = 2*Div(y, z) + Div(x, z)
│ │ │ +o12 = Div(x, z) + 2*Div(y, z)
│ │ │  
│ │ │  o12 : WeilDivisor on R
│ │ │  
│ │ │  i13 : D2 = divisor({1/2, 3/4}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o13 = 1/2*Div(y, z) + 3/4*Div(x, z)
│ │ │ +o13 = 3/4*Div(x, z) + 1/2*Div(y, z)
│ │ │  
│ │ │  o13 : QWeilDivisor on R
│ │ │  
│ │ │  i14 : isQCartier(10, D1, IsGraded => true)
│ │ │  
│ │ │  o14 = 1
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_is__Q__Linear__Equivalent.out
│ │ │ @@ -36,21 +36,21 @@
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : R = QQ[x, y, z] / ideal(x * y - z^2);
│ │ │  
│ │ │  i11 : D = divisor({1/2, 3/4}, {ideal(x, z), ideal(y, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o11 = 3/4*Div(y, z) + 1/2*Div(x, z)
│ │ │ +o11 = 1/2*Div(x, z) + 3/4*Div(y, z)
│ │ │  
│ │ │  o11 : QWeilDivisor on R
│ │ │  
│ │ │  i12 : E = divisor({3/2, -1/4}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o12 = 3/2*Div(y, z) + -1/4*Div(x, z)
│ │ │ +o12 = -1/4*Div(x, z) + 3/2*Div(y, z)
│ │ │  
│ │ │  o12 : QWeilDivisor on R
│ │ │  
│ │ │  i13 : isQLinearEquivalent(10, D, E, IsGraded => true)
│ │ │  
│ │ │  o13 = true
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_is__S__N__C.out
│ │ │ @@ -1,14 +1,14 @@
│ │ │  -- -*- M2-comint -*- hash: 2360371518304120718
│ │ │  
│ │ │  i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │  
│ │ │  i2 : D = divisor({1, -2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = Div(x, z) + -2*Div(y, z)
│ │ │ +o2 = -2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : isSNC( D )
│ │ │  
│ │ │  o3 = false
│ │ │  
│ │ │ @@ -36,15 +36,15 @@
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │  
│ │ │  i11 : D = divisor({1, -2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o11 = -2*Div(y, z) + Div(x, z)
│ │ │ +o11 = Div(x, z) + -2*Div(y, z)
│ │ │  
│ │ │  o11 : WeilDivisor on R
│ │ │  
│ │ │  i12 : isSNC( D, IsGraded => true )
│ │ │  
│ │ │  o12 = true
│ │ │  
│ │ │ @@ -60,15 +60,15 @@
│ │ │  
│ │ │  o15 = true
│ │ │  
│ │ │  i16 : R = QQ[x,y,z];
│ │ │  
│ │ │  i17 : D = divisor(x*y*(x+y))
│ │ │  
│ │ │ -o17 = Div(x) + Div(x+y) + Div(y)
│ │ │ +o17 = Div(x+y) + Div(y) + Div(x)
│ │ │  
│ │ │  o17 : WeilDivisor on R
│ │ │  
│ │ │  i18 : isSNC( D, IsGraded => true)
│ │ │  
│ │ │  o18 = false
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_map__To__Projective__Space.out
│ │ │ @@ -16,15 +16,15 @@
│ │ │  o3 : RingMap R <-- QQ[YY ..YY ]
│ │ │                          1    2
│ │ │  
│ │ │  i4 : R = ZZ/7[x,y,z];
│ │ │  
│ │ │  i5 : D = divisor(x*y)
│ │ │  
│ │ │ -o5 = Div(y) + Div(x)
│ │ │ +o5 = Div(x) + Div(y)
│ │ │  
│ │ │  o5 : WeilDivisor on R
│ │ │  
│ │ │  i6 : mapToProjectiveSpace(D, Variable=>"Z")
│ │ │  
│ │ │               ZZ            2             2        2
│ │ │  o6 = map (R, --[Z ..Z ], {x , x*y, x*z, y , y*z, z })
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_pullback_lp__Ring__Map_cm__R__Weil__Divisor_rp.out
│ │ │ @@ -6,21 +6,21 @@
│ │ │  
│ │ │  i3 : f = map(T, R, {a^3, a^2*b, a*b^2, b^3});
│ │ │  
│ │ │  o3 : RingMap T <-- R
│ │ │  
│ │ │  i4 : D = divisor(y*z)
│ │ │  
│ │ │ -o4 = 3*Div(w, z, y) + 3*Div(z, y, x)
│ │ │ +o4 = 3*Div(z, y, x) + 3*Div(w, z, y)
│ │ │  
│ │ │  o4 : WeilDivisor on R
│ │ │  
│ │ │  i5 : pullback(f, D, Strategy=>Primes)
│ │ │  
│ │ │ -o5 = 3*Div(b) + 3*Div(a)
│ │ │ +o5 = 3*Div(a) + 3*Div(b)
│ │ │  
│ │ │  o5 : WeilDivisor on T
│ │ │  
│ │ │  i6 : pullback(f, D, Strategy=>Sheaves)
│ │ │  
│ │ │  o6 = 3*Div(b) + 3*Div(a)
│ │ │  
│ │ │ @@ -36,18 +36,18 @@
│ │ │  
│ │ │  i10 : D = divisor(x*y*(x+y));
│ │ │  
│ │ │  o10 : WeilDivisor on R
│ │ │  
│ │ │  i11 : D1 = pullback(f, D)
│ │ │  
│ │ │ -o11 = Div(a+1) + 3*Div(b) + Div(a)
│ │ │ +o11 = Div(a) + 3*Div(b) + Div(a+1)
│ │ │  
│ │ │  o11 : WeilDivisor on S
│ │ │  
│ │ │  i12 : f^* D
│ │ │  
│ │ │ -o12 = Div(a+1) + 3*Div(b) + Div(a)
│ │ │ +o12 = Div(a) + 3*Div(b) + Div(a+1)
│ │ │  
│ │ │  o12 : WeilDivisor on S
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_reflexify.out
│ │ │ @@ -103,104 +103,104 @@
│ │ │  o21 : Ideal of R
│ │ │  
│ │ │  i22 : J = I^21;
│ │ │  
│ │ │  o22 : Ideal of R
│ │ │  
│ │ │  i23 : time reflexify(J);
│ │ │ - -- used 0.260716s (cpu); 0.201186s (thread); 0s (gc)
│ │ │ + -- used 0.271189s (cpu); 0.210021s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of R
│ │ │  
│ │ │  i24 : time reflexify(J*R^1);
│ │ │ - -- used 0.527272s (cpu); 0.394308s (thread); 0s (gc)
│ │ │ + -- used 0.44102s (cpu); 0.361892s (thread); 0s (gc)
│ │ │  
│ │ │  i25 : R = ZZ/13[x,y,z]/ideal(x^3 + y^3-z^11*x*y);
│ │ │  
│ │ │  i26 : I = ideal(x-4*y, z);
│ │ │  
│ │ │  o26 : Ideal of R
│ │ │  
│ │ │  i27 : J = I^20;
│ │ │  
│ │ │  o27 : Ideal of R
│ │ │  
│ │ │  i28 : M = J*R^1;
│ │ │  
│ │ │  i29 : J1 = time reflexify( J, Strategy=>IdealStrategy )
│ │ │ - -- used 0.273432s (cpu); 0.158699s (thread); 0s (gc)
│ │ │ + -- used 0.284278s (cpu); 0.146299s (thread); 0s (gc)
│ │ │  
│ │ │                2            2     9       9   11
│ │ │  o29 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │  
│ │ │  o29 : Ideal of R
│ │ │  
│ │ │  i30 : J2 = time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ - -- used 6.95165s (cpu); 4.6997s (thread); 0s (gc)
│ │ │ + -- used 5.49105s (cpu); 4.34638s (thread); 0s (gc)
│ │ │  
│ │ │                2            2     9       9   11
│ │ │  o30 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │  
│ │ │  o30 : Ideal of R
│ │ │  
│ │ │  i31 : J1 == J2
│ │ │  
│ │ │  o31 = true
│ │ │  
│ │ │  i32 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ - -- used 5.67659s (cpu); 4.46311s (thread); 0s (gc)
│ │ │ + -- used 5.68386s (cpu); 4.59503s (thread); 0s (gc)
│ │ │  
│ │ │  i33 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.775911s (cpu); 0.448237s (thread); 0s (gc)
│ │ │ + -- used 0.539944s (cpu); 0.372248s (thread); 0s (gc)
│ │ │  
│ │ │  i34 : R = QQ[x,y,u,v]/ideal(x*y-u*v);
│ │ │  
│ │ │  i35 : I = ideal(x,u);
│ │ │  
│ │ │  o35 : Ideal of R
│ │ │  
│ │ │  i36 : J = I^20;
│ │ │  
│ │ │  o36 : Ideal of R
│ │ │  
│ │ │  i37 : M = I^20*R^1;
│ │ │  
│ │ │  i38 : time reflexify( J, Strategy=>IdealStrategy )
│ │ │ - -- used 1.05331s (cpu); 0.384322s (thread); 0s (gc)
│ │ │ + -- used 1.08794s (cpu); 0.386032s (thread); 0s (gc)
│ │ │  
│ │ │                20     19   2 18   3 17   4 16   5 15   6 14   7 13   8 12 
│ │ │  o38 = ideal (u  , x*u  , x u  , x u  , x u  , x u  , x u  , x u  , x u  ,
│ │ │        -----------------------------------------------------------------------
│ │ │         9 11   10 10   11 9   12 8   13 7   14 6   15 5   16 4   17 3   18 2 
│ │ │        x u  , x  u  , x  u , x  u , x  u , x  u , x  u , x  u , x  u , x  u ,
│ │ │        -----------------------------------------------------------------------
│ │ │         19    20
│ │ │        x  u, x  )
│ │ │  
│ │ │  o38 : Ideal of R
│ │ │  
│ │ │  i39 : time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ - -- used 0.238734s (cpu); 0.0653637s (thread); 0s (gc)
│ │ │ + -- used 0.226139s (cpu); 0.0553398s (thread); 0s (gc)
│ │ │  
│ │ │                20     19   2 18   3 17   4 16   5 15   6 14   7 13   8 12 
│ │ │  o39 = ideal (u  , x*u  , x u  , x u  , x u  , x u  , x u  , x u  , x u  ,
│ │ │        -----------------------------------------------------------------------
│ │ │         9 11   10 10   11 9   12 8   13 7   14 6   15 5   16 4   17 3   18 2 
│ │ │        x u  , x  u  , x  u , x  u , x  u , x  u , x  u , x  u , x  u , x  u ,
│ │ │        -----------------------------------------------------------------------
│ │ │         19    20
│ │ │        x  u, x  )
│ │ │  
│ │ │  o39 : Ideal of R
│ │ │  
│ │ │  i40 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ - -- used 0.276556s (cpu); 0.103744s (thread); 0s (gc)
│ │ │ + -- used 0.254852s (cpu); 0.0903637s (thread); 0s (gc)
│ │ │  
│ │ │  i41 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.00801402s (cpu); 0.00801143s (thread); 0s (gc)
│ │ │ + -- used 0.00672375s (cpu); 0.00672976s (thread); 0s (gc)
│ │ │  
│ │ │  i42 : R = QQ[x,y]/ideal(x*y);
│ │ │  
│ │ │  i43 : I = ideal(x,y);
│ │ │  
│ │ │  o43 : Ideal of R
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_reflexive__Power.out
│ │ │ @@ -23,44 +23,44 @@
│ │ │  i5 : R = QQ[x,y,z]/ideal(-y^2*z +x^3 + x^2*z + x*z^2+z^3);
│ │ │  
│ │ │  i6 : I = ideal(x-z,y-2*z);
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : time J20a = reflexivePower(20, I);
│ │ │ - -- used 0.0323817s (cpu); 0.0323825s (thread); 0s (gc)
│ │ │ + -- used 0.0325992s (cpu); 0.0325983s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 : I20 = I^20;
│ │ │  
│ │ │  o8 : Ideal of R
│ │ │  
│ │ │  i9 : time J20b = reflexify(I20);
│ │ │ - -- used 0.204644s (cpu); 0.153031s (thread); 0s (gc)
│ │ │ + -- used 0.219399s (cpu); 0.145855s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of R
│ │ │  
│ │ │  i10 : J20a == J20b
│ │ │  
│ │ │  o10 = true
│ │ │  
│ │ │  i11 : R = QQ[x,y,z]/ideal(-y^2*z +x^3 + x^2*z + x*z^2+z^3);
│ │ │  
│ │ │  i12 : I = ideal(x-z,y-2*z);
│ │ │  
│ │ │  o12 : Ideal of R
│ │ │  
│ │ │  i13 : time J1 = reflexivePower(20, I, Strategy=>IdealStrategy);
│ │ │ - -- used 0.029315s (cpu); 0.0293196s (thread); 0s (gc)
│ │ │ + -- used 0.0363386s (cpu); 0.0363264s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
│ │ │  
│ │ │  i14 : time J2 = reflexivePower(20, I, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.143657s (cpu); 0.0818326s (thread); 0s (gc)
│ │ │ + -- used 0.162794s (cpu); 0.0890308s (thread); 0s (gc)
│ │ │  
│ │ │  o14 : Ideal of R
│ │ │  
│ │ │  i15 : J1 == J2
│ │ │  
│ │ │  o15 = true
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_ring_lp__Basic__Divisor_rp.out
│ │ │ @@ -1,14 +1,14 @@
│ │ │  -- -*- M2-comint -*- hash: 5006859181202351713
│ │ │  
│ │ │  i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │  
│ │ │  i2 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = 2*Div(y, z) + Div(x, z)
│ │ │ +o2 = Div(x, z) + 2*Div(y, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : ring( D )
│ │ │  
│ │ │  o3 = R
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_to__R__Weil__Divisor.out
│ │ │ @@ -1,32 +1,32 @@
│ │ │  -- -*- M2-comint -*- hash: 12819564349892123361
│ │ │  
│ │ │  i1 : R = ZZ/5[x,y];
│ │ │  
│ │ │  i2 : D = divisor({2, 0, -4}, {ideal(x), ideal(y), ideal(x-y)})
│ │ │  
│ │ │ -o2 = -4*Div(x-y) + 2*Div(x) + 0*Div(y)
│ │ │ +o2 = 2*Div(x) + 0*Div(y) + -4*Div(x-y)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : E = (1/2)*D
│ │ │  
│ │ │ -o3 = -2*Div(x-y) + Div(x)
│ │ │ +o3 = Div(x) + -2*Div(x-y)
│ │ │  
│ │ │  o3 : QWeilDivisor on R
│ │ │  
│ │ │  i4 : F = toRWeilDivisor(D)
│ │ │  
│ │ │ -o4 = -4*Div(x-y) + 2*Div(x)
│ │ │ +o4 = 2*Div(x) + -4*Div(x-y)
│ │ │  
│ │ │  o4 : RWeilDivisor on R
│ │ │  
│ │ │  i5 : G = toRWeilDivisor(E)
│ │ │  
│ │ │ -o5 = -2*Div(x-y) + Div(x)
│ │ │ +o5 = Div(x) + -2*Div(x-y)
│ │ │  
│ │ │  o5 : RWeilDivisor on R
│ │ │  
│ │ │  i6 : F == 2*G
│ │ │  
│ │ │  o6 = true
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/___Basic__Divisor.html
│ │ │ @@ -72,15 +72,15 @@
│ │ │                
    i1 : R = QQ[x,y,z];
    
│ │ │              
    
│ │ │                
    i2 : D = divisor(x*y^2*z^3)
│ │ │  
│ │ │ -o2 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │ +o2 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    
│ │ │              
    
│ │ │                
    i3 : H = new HashTable from D
│ │ │ @@ -94,24 +94,24 @@
│ │ │  o3 : HashTable
    
│ │ │              
    
│ │ │                
    i4 : (2/3)*D
│ │ │  
│ │ │ -o4 = 2/3*Div(x) + 4/3*Div(y) + 2*Div(z)
│ │ │ +o4 = 2*Div(z) + 4/3*Div(y) + 2/3*Div(x)
│ │ │  
│ │ │  o4 : QWeilDivisor on R
    
│ │ │              
    
│ │ │                
    i5 : 0.6*D
│ │ │  
│ │ │ -o5 = .6*Div(x) + 1.2*Div(y) + 1.8*Div(z)
│ │ │ +o5 = 1.8*Div(z) + 1.2*Div(y) + .6*Div(x)
│ │ │  
│ │ │  o5 : RWeilDivisor on R
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,34 +15,34 @@
│ │ │ │  specifies the ambient ring. Another key is cache which points to a CacheTable.
│ │ │ │  The remaining keys are a Groebner basis $L$ for each prime ideal $P$ in the
│ │ │ │  support with corresponding value a list with one entry {$n$} where $n$ is the
│ │ │ │  coefficient of the height one prime.
│ │ │ │  i1 : R = QQ[x,y,z];
│ │ │ │  i2 : D = divisor(x*y^2*z^3)
│ │ │ │  
│ │ │ │ -o2 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │ │ +o2 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : H = new HashTable from D
│ │ │ │  
│ │ │ │  o3 = HashTable{{x} => {1}                  }
│ │ │ │                 {y} => {2}
│ │ │ │                 {z} => {3}
│ │ │ │                 cache => CacheTable{...1...}
│ │ │ │                 ring => R
│ │ │ │  
│ │ │ │  o3 : HashTable
│ │ │ │  i4 : (2/3)*D
│ │ │ │  
│ │ │ │ -o4 = 2/3*Div(x) + 4/3*Div(y) + 2*Div(z)
│ │ │ │ +o4 = 2*Div(z) + 4/3*Div(y) + 2/3*Div(x)
│ │ │ │  
│ │ │ │  o4 : QWeilDivisor on R
│ │ │ │  i5 : 0.6*D
│ │ │ │  
│ │ │ │ -o5 = .6*Div(x) + 1.2*Div(y) + 1.8*Div(z)
│ │ │ │ +o5 = 1.8*Div(z) + 1.2*Div(y) + .6*Div(x)
│ │ │ │  
│ │ │ │  o5 : RWeilDivisor on R
│ │ │ │  ********** TTyyppeess ooff BBaassiiccDDiivviissoorr:: **********
│ │ │ │      * RWeilDivisor
│ │ │ │  ********** FFuunnccttiioonnss aanndd mmeetthhooddss rreettuurrnniinngg aann oobbjjeecctt ooff ccllaassss BBaassiiccDDiivviissoorr:: **********
│ │ │ │      * applyToCoefficients(BasicDivisor,Function) -- see _a_p_p_l_y_T_o_C_o_e_f_f_i_c_i_e_n_t_s -
│ │ │ │        - apply a function to the coefficients of a divisor
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/___Number_sp_st_sp__Basic__Divisor.html
│ │ │ @@ -103,24 +103,24 @@
│ │ │  o4 : RWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : 8*D
│ │ │  
│ │ │ -o5 = -8*Div(x+y) + 16*Div(x) + 8*Div(y)
│ │ │ +o5 = -8*Div(x+y) + 8*Div(y) + 16*Div(x)
│ │ │  
│ │ │  o5 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : (-2/3)*D
│ │ │  
│ │ │ -o6 = 2/3*Div(x+y) + -4/3*Div(x) + -2/3*Div(y)
│ │ │ +o6 = 2/3*Div(x+y) + -2/3*Div(y) + -4/3*Div(x)
│ │ │  
│ │ │  o6 : QWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : 0.0*D
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,20 +27,20 @@
│ │ │ │  CoefficientType=>RR)
│ │ │ │  
│ │ │ │  o4 = -3.2*Div(-y^3+x^2) + 1.5*Div(x) + 0*Div(y)
│ │ │ │  
│ │ │ │  o4 : RWeilDivisor on R
│ │ │ │  i5 : 8*D
│ │ │ │  
│ │ │ │ -o5 = -8*Div(x+y) + 16*Div(x) + 8*Div(y)
│ │ │ │ +o5 = -8*Div(x+y) + 8*Div(y) + 16*Div(x)
│ │ │ │  
│ │ │ │  o5 : WeilDivisor on R
│ │ │ │  i6 : (-2/3)*D
│ │ │ │  
│ │ │ │ -o6 = 2/3*Div(x+y) + -4/3*Div(x) + -2/3*Div(y)
│ │ │ │ +o6 = 2/3*Div(x+y) + -2/3*Div(y) + -4/3*Div(x)
│ │ │ │  
│ │ │ │  o6 : QWeilDivisor on R
│ │ │ │  i7 : 0.0*D
│ │ │ │  
│ │ │ │  o7 = 0, the zero divisor
│ │ │ │  
│ │ │ │  o7 : RWeilDivisor on R
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_ceiling_lp__R__Weil__Divisor_rp.html
│ │ │ @@ -78,24 +78,24 @@
│ │ │                
    i1 : R = QQ[x, y, z] / ideal(x *y - z^2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : D = divisor({1/2, 4/3}, {ideal(x, z), ideal(y, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o2 = 4/3*Div(y, z) + 1/2*Div(x, z)
│ │ │ +o2 = 1/2*Div(x, z) + 4/3*Div(y, z)
│ │ │  
│ │ │  o2 : QWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : ceiling( D )
│ │ │  
│ │ │ -o3 = 2*Div(y, z) + Div(x, z)
│ │ │ +o3 = Div(x, z) + 2*Div(y, z)
│ │ │  
│ │ │  o3 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : floor( D )
│ │ │ @@ -105,15 +105,15 @@
│ │ │  o4 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : E = divisor({0.3, -0.7}, {ideal(x, z), ideal(y,z)}, CoefficientType => RR)
│ │ │  
│ │ │ -o5 = -.7*Div(y, z) + .3*Div(x, z)
│ │ │ +o5 = .3*Div(x, z) + -.7*Div(y, z)
│ │ │  
│ │ │  o5 : RWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : ceiling( E )
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,30 +15,30 @@
│ │ │ │            o an instance of the type _W_e_i_l_D_i_v_i_s_o_r,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Start with a rational or real Weil divisor. We form a new divisor whose
│ │ │ │  coefficients are obtained by applying the ceiling or floor function to them.
│ │ │ │  i1 : R = QQ[x, y, z] / ideal(x *y - z^2);
│ │ │ │  i2 : D = divisor({1/2, 4/3}, {ideal(x, z), ideal(y, z)}, CoefficientType => QQ)
│ │ │ │  
│ │ │ │ -o2 = 4/3*Div(y, z) + 1/2*Div(x, z)
│ │ │ │ +o2 = 1/2*Div(x, z) + 4/3*Div(y, z)
│ │ │ │  
│ │ │ │  o2 : QWeilDivisor on R
│ │ │ │  i3 : ceiling( D )
│ │ │ │  
│ │ │ │ -o3 = 2*Div(y, z) + Div(x, z)
│ │ │ │ +o3 = Div(x, z) + 2*Div(y, z)
│ │ │ │  
│ │ │ │  o3 : WeilDivisor on R
│ │ │ │  i4 : floor( D )
│ │ │ │  
│ │ │ │  o4 = Div(y, z)
│ │ │ │  
│ │ │ │  o4 : WeilDivisor on R
│ │ │ │  i5 : E = divisor({0.3, -0.7}, {ideal(x, z), ideal(y,z)}, CoefficientType => RR)
│ │ │ │  
│ │ │ │ -o5 = -.7*Div(y, z) + .3*Div(x, z)
│ │ │ │ +o5 = .3*Div(x, z) + -.7*Div(y, z)
│ │ │ │  
│ │ │ │  o5 : RWeilDivisor on R
│ │ │ │  i6 : ceiling( E )
│ │ │ │  
│ │ │ │  o6 = Div(x, z)
│ │ │ │  
│ │ │ │  o6 : WeilDivisor on R
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_divisor.html
│ │ │ @@ -95,51 +95,51 @@
│ │ │                
    i1 : R = QQ[x,y,z];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : D = divisor({1,2,3}, {ideal(x), ideal(y), ideal(z)})
│ │ │  
│ │ │ -o2 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ +o2 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : E = divisor(x*y^2*z^3)
│ │ │  
│ │ │ -o3 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ +o3 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │  
│ │ │  o3 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : F = divisor(ideal(x*y^2*z^3))
│ │ │  
│ │ │ -o4 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ +o4 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │  
│ │ │  o4 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : G = divisor({{1, ideal(x)}, {2, ideal(y)}, {3, ideal(z)}})
│ │ │  
│ │ │ -o5 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ +o5 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │  
│ │ │  o5 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : H = divisor(x) + 2*divisor(y) + 3*divisor(z)
│ │ │  
│ │ │ -o6 = 3*Div(z) + 2*Div(y) + Div(x)
│ │ │ +o6 = Div(x) + 3*Div(z) + 2*Div(y)
│ │ │  
│ │ │  o6 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    Next we construct the same divisor in two different ways.  We are working on the quadric cone, and we are working with a divisor of a ruling of the cone.  This divisor is not Cartier, but 2 times it is.
    
│ │ │ @@ -204,15 +204,15 @@
│ │ │  o14 : WeilDivisor on A
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : E = divisor(y2z)
│ │ │  
│ │ │ -o15 = 2*Div(yz2, y2z, y3, xyz, xy2, x2y) + Div(z3, yz2, y2z, xz2, xyz, x2z)
│ │ │ +o15 = Div(z3, yz2, y2z, xz2, xyz, x2z) + 2*Div(yz2, y2z, y3, xyz, xy2, x2y)
│ │ │  
│ │ │  o15 : WeilDivisor on A
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    We can construct a Q-divisor as well.  Here are two ways to do it (we work in $A^2$ this time).
    
│ │ │ @@ -223,24 +223,24 @@
│ │ │                
    i16 : R = ZZ/7[x,y];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i17 : D = divisor({-1/2, 2/1}, {ideal(y^2-x^3), ideal(x)}, CoefficientType=>QQ)
│ │ │  
│ │ │ -o17 = -1/2*Div(-x^3+y^2) + 2*Div(x)
│ │ │ +o17 = 2*Div(x) + -1/2*Div(-x^3+y^2)
│ │ │  
│ │ │  o17 : QWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i18 : D = (-1/2)*divisor(y^2-x^3) + (2/1)*divisor(x)
│ │ │  
│ │ │ -o18 = -1/2*Div(-x^3+y^2) + 2*Div(x)
│ │ │ +o18 = 2*Div(x) + -1/2*Div(-x^3+y^2)
│ │ │  
│ │ │  o18 : QWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    Or an R-divisor.  This time we work in the cone over $P^1 \times P^1$.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -43,35 +43,35 @@
│ │ │ │  call it. In our first example, we construct divisors on $A^3$ (which can also
│ │ │ │  be viewed as divisors on $P^2$ since the ideals are homogeneous). The following
│ │ │ │  creates the same Weil divisor with coefficients 1, 2 and 3 in five different
│ │ │ │  ways.
│ │ │ │  i1 : R = QQ[x,y,z];
│ │ │ │  i2 : D = divisor({1,2,3}, {ideal(x), ideal(y), ideal(z)})
│ │ │ │  
│ │ │ │ -o2 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ │ +o2 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : E = divisor(x*y^2*z^3)
│ │ │ │  
│ │ │ │ -o3 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ │ +o3 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │ │  
│ │ │ │  o3 : WeilDivisor on R
│ │ │ │  i4 : F = divisor(ideal(x*y^2*z^3))
│ │ │ │  
│ │ │ │ -o4 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ │ +o4 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │ │  
│ │ │ │  o4 : WeilDivisor on R
│ │ │ │  i5 : G = divisor({{1, ideal(x)}, {2, ideal(y)}, {3, ideal(z)}})
│ │ │ │  
│ │ │ │ -o5 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ │ +o5 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │ │  
│ │ │ │  o5 : WeilDivisor on R
│ │ │ │  i6 : H = divisor(x) + 2*divisor(y) + 3*divisor(z)
│ │ │ │  
│ │ │ │ -o6 = 3*Div(z) + 2*Div(y) + Div(x)
│ │ │ │ +o6 = Div(x) + 3*Div(z) + 2*Div(y)
│ │ │ │  
│ │ │ │  o6 : WeilDivisor on R
│ │ │ │  Next we construct the same divisor in two different ways. We are working on the
│ │ │ │  quadric cone, and we are working with a divisor of a ruling of the cone. This
│ │ │ │  divisor is not Cartier, but 2 times it is.
│ │ │ │  i7 : R = QQ[x,y,z]/ideal(x^2-y*z);
│ │ │ │  i8 : D = divisor({2}, {ideal(x,y)})
│ │ │ │ @@ -95,28 +95,28 @@
│ │ │ │  i14 : D = divisor(x3)
│ │ │ │  
│ │ │ │  o14 = 3*Div(xz2, xyz, xy2, x2z, x2y, x3)
│ │ │ │  
│ │ │ │  o14 : WeilDivisor on A
│ │ │ │  i15 : E = divisor(y2z)
│ │ │ │  
│ │ │ │ -o15 = 2*Div(yz2, y2z, y3, xyz, xy2, x2y) + Div(z3, yz2, y2z, xz2, xyz, x2z)
│ │ │ │ +o15 = Div(z3, yz2, y2z, xz2, xyz, x2z) + 2*Div(yz2, y2z, y3, xyz, xy2, x2y)
│ │ │ │  
│ │ │ │  o15 : WeilDivisor on A
│ │ │ │  We can construct a Q-divisor as well. Here are two ways to do it (we work in
│ │ │ │  $A^2$ this time).
│ │ │ │  i16 : R = ZZ/7[x,y];
│ │ │ │  i17 : D = divisor({-1/2, 2/1}, {ideal(y^2-x^3), ideal(x)}, CoefficientType=>QQ)
│ │ │ │  
│ │ │ │ -o17 = -1/2*Div(-x^3+y^2) + 2*Div(x)
│ │ │ │ +o17 = 2*Div(x) + -1/2*Div(-x^3+y^2)
│ │ │ │  
│ │ │ │  o17 : QWeilDivisor on R
│ │ │ │  i18 : D = (-1/2)*divisor(y^2-x^3) + (2/1)*divisor(x)
│ │ │ │  
│ │ │ │ -o18 = -1/2*Div(-x^3+y^2) + 2*Div(x)
│ │ │ │ +o18 = 2*Div(x) + -1/2*Div(-x^3+y^2)
│ │ │ │  
│ │ │ │  o18 : QWeilDivisor on R
│ │ │ │  Or an R-divisor. This time we work in the cone over $P^1 \times P^1$.
│ │ │ │  i19 : R = ZZ/11[x,y,u,v]/ideal(x*y-u*v);
│ │ │ │  i20 : D = divisor({1.1, -3.14159}, {ideal(x,u), ideal(x, v)},
│ │ │ │  CoefficientType=>RR)
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_dualize.html
│ │ │ @@ -163,43 +163,43 @@
│ │ │              
│ │ │                
    i11 : M = J*R^1;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ - -- used 0.135013s (cpu); 0.0782845s (thread); 0s (gc)
│ │ │ + -- used 0.127632s (cpu); 0.0718664s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.455344s (cpu); 0.455348s (thread); 0s (gc)
│ │ │ + -- used 0.495472s (cpu); 0.495481s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : time dualize(M, Strategy=>IdealStrategy);
│ │ │ - -- used 0.57369s (cpu); 0.498594s (thread); 0s (gc)
    
│ │ │ + -- used 0.627359s (cpu); 0.55998s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : time dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00285362s (cpu); 0.00285446s (thread); 0s (gc)
    
│ │ │ + -- used 0.00316931s (cpu); 0.00317578s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00222665s (cpu); 0.0022275s (thread); 0s (gc)
│ │ │ + -- used 0.00282471s (cpu); 0.00283097s (thread); 0s (gc)
│ │ │  
│ │ │  o16 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    For monomial ideals in toric rings, frequently ModuleStrategy appears faster.
    
│ │ │ @@ -223,23 +223,23 @@
│ │ │  
│ │ │  o19 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i20 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ - -- used 0.279007s (cpu); 0.150681s (thread); 0s (gc)
│ │ │ + -- used 0.290507s (cpu); 0.138215s (thread); 0s (gc)
│ │ │  
│ │ │  o20 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i21 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00604243s (cpu); 0.00604328s (thread); 0s (gc)
│ │ │ + -- used 0.00726241s (cpu); 0.00726781s (thread); 0s (gc)
│ │ │  
│ │ │  o21 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    KnownDomain is an option for dualize.  If it is false (default is true), then the computer will first check whether the ring is a domain, if it is not then it will revert to ModuleStrategy.  If KnownDomain is set to true for a non-domain, then the function can return an incorrect answer.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -60,43 +60,43 @@
│ │ │ │  
│ │ │ │  o9 : Ideal of R
│ │ │ │  i10 : J = m^9;
│ │ │ │  
│ │ │ │  o10 : Ideal of R
│ │ │ │  i11 : M = J*R^1;
│ │ │ │  i12 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ │ - -- used 0.135013s (cpu); 0.0782845s (thread); 0s (gc)
│ │ │ │ + -- used 0.127632s (cpu); 0.0718664s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o12 : Ideal of R
│ │ │ │  i13 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ │ - -- used 0.455344s (cpu); 0.455348s (thread); 0s (gc)
│ │ │ │ + -- used 0.495472s (cpu); 0.495481s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 : Ideal of R
│ │ │ │  i14 : time dualize(M, Strategy=>IdealStrategy);
│ │ │ │ - -- used 0.57369s (cpu); 0.498594s (thread); 0s (gc)
│ │ │ │ + -- used 0.627359s (cpu); 0.55998s (thread); 0s (gc)
│ │ │ │  i15 : time dualize(M, Strategy=>ModuleStrategy);
│ │ │ │ - -- used 0.00285362s (cpu); 0.00285446s (thread); 0s (gc)
│ │ │ │ + -- used 0.00316931s (cpu); 0.00317578s (thread); 0s (gc)
│ │ │ │  i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy);
│ │ │ │ - -- used 0.00222665s (cpu); 0.0022275s (thread); 0s (gc)
│ │ │ │ + -- used 0.00282471s (cpu); 0.00283097s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 : Ideal of R
│ │ │ │  For monomial ideals in toric rings, frequently ModuleStrategy appears faster.
│ │ │ │  i17 : R = ZZ/7[x,y,u,v]/ideal(x*y-u*v);
│ │ │ │  i18 : I = ideal(x,u);
│ │ │ │  
│ │ │ │  o18 : Ideal of R
│ │ │ │  i19 : J = I^15;
│ │ │ │  
│ │ │ │  o19 : Ideal of R
│ │ │ │  i20 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ │ - -- used 0.279007s (cpu); 0.150681s (thread); 0s (gc)
│ │ │ │ + -- used 0.290507s (cpu); 0.138215s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o20 : Ideal of R
│ │ │ │  i21 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ │ - -- used 0.00604243s (cpu); 0.00604328s (thread); 0s (gc)
│ │ │ │ + -- used 0.00726241s (cpu); 0.00726781s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 : Ideal of R
│ │ │ │  KnownDomain is an option for dualize. If it is false (default is true), then
│ │ │ │  the computer will first check whether the ring is a domain, if it is not then
│ │ │ │  it will revert to ModuleStrategy. If KnownDomain is set to true for a non-
│ │ │ │  domain, then the function can return an incorrect answer.
│ │ │ │  i22 : R = QQ[x,y]/ideal(x*y);
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_is__Cartier.html
│ │ │ @@ -106,15 +106,15 @@
│ │ │                
    i4 : R = QQ[x, y, z] / ideal(x * y - z^2 );
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o5 = Div(x, z) + 2*Div(y, z)
│ │ │ +o5 = 2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o5 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : isCartier( D )
│ │ │ @@ -181,15 +181,15 @@
│ │ │                
    i13 : R = QQ[x, y, z] / ideal(x * y - z^2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o14 = Div(x, z) + 2*Div(y, z)
│ │ │ +o14 = 2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o14 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : isCartier(D, IsGraded => true)
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,15 +25,15 @@
│ │ │ │  i3 : isCartier( D )
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  Neither is this divisor.
│ │ │ │  i4 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │ │  i5 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │ │  
│ │ │ │ -o5 = Div(x, z) + 2*Div(y, z)
│ │ │ │ +o5 = 2*Div(y, z) + Div(x, z)
│ │ │ │  
│ │ │ │  o5 : WeilDivisor on R
│ │ │ │  i6 : isCartier( D )
│ │ │ │  
│ │ │ │  o6 = false
│ │ │ │  Of course the next divisor is Cartier.
│ │ │ │  i7 : R = QQ[x, y, z];
│ │ │ │ @@ -55,15 +55,15 @@
│ │ │ │  o11 : WeilDivisor on R
│ │ │ │  i12 : isCartier(D, IsGraded => true)
│ │ │ │  
│ │ │ │  o12 = true
│ │ │ │  i13 : R = QQ[x, y, z] / ideal(x * y - z^2);
│ │ │ │  i14 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │ │  
│ │ │ │ -o14 = Div(x, z) + 2*Div(y, z)
│ │ │ │ +o14 = 2*Div(y, z) + Div(x, z)
│ │ │ │  
│ │ │ │  o14 : WeilDivisor on R
│ │ │ │  i15 : isCartier(D, IsGraded => true)
│ │ │ │  
│ │ │ │  o15 = true
│ │ │ │  The output value of this function is stored in the divisor's cache with the
│ │ │ │  value of the last IsGraded option. If you change the IsGraded option, the value
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_is__Homogeneous_lp__Basic__Divisor_rp.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │                
    i1 : R = QQ[x, y, z];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : D = divisor({1, 2, 3}, {ideal(x * y - z^2), ideal(y * z - x^2), ideal(x * z - y^2)})
│ │ │  
│ │ │ -o2 = 3*Div(-y^2+x*z) + Div(x*y-z^2) + 2*Div(-x^2+y*z)
│ │ │ +o2 = Div(x*y-z^2) + 2*Div(-x^2+y*z) + 3*Div(-y^2+x*z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : isHomogeneous( D )
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This function returns true if the divisor is graded (homogeneous), otherwise it
│ │ │ │  returns false.
│ │ │ │  i1 : R = QQ[x, y, z];
│ │ │ │  i2 : D = divisor({1, 2, 3}, {ideal(x * y - z^2), ideal(y * z - x^2), ideal(x *
│ │ │ │  z - y^2)})
│ │ │ │  
│ │ │ │ -o2 = 3*Div(-y^2+x*z) + Div(x*y-z^2) + 2*Div(-x^2+y*z)
│ │ │ │ +o2 = Div(x*y-z^2) + 2*Div(-x^2+y*z) + 3*Div(-y^2+x*z)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : isHomogeneous( D )
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : R = QQ[x, y, z];
│ │ │ │  i5 : D = divisor({1, 2}, {ideal(x * y - z^2), ideal(y^2 - z^3)})
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_is__Linear__Equivalent.html
│ │ │ @@ -81,24 +81,24 @@
│ │ │                
    i1 : R = QQ[x, y, z]/ ideal(x * y - z^2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : D1 = divisor({3, 8}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = 8*Div(y, z) + 3*Div(x, z)
│ │ │ +o2 = 3*Div(x, z) + 8*Div(y, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : D2 = divisor({8, 1}, {ideal(y, z), ideal(x, z)})
│ │ │  
│ │ │ -o3 = 8*Div(y, z) + Div(x, z)
│ │ │ +o3 = Div(x, z) + 8*Div(y, z)
│ │ │  
│ │ │  o3 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : isLinearEquivalent(D1, D2)
│ │ │ @@ -116,24 +116,24 @@
│ │ │                
    i5 : R = QQ[x, y, z]/ ideal(x * y - z^2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : D1 = divisor({3, 8}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o6 = 3*Div(x, z) + 8*Div(y, z)
│ │ │ +o6 = 8*Div(y, z) + 3*Div(x, z)
│ │ │  
│ │ │  o6 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : D2 = divisor({8, 1}, {ideal(y, z), ideal(x, z)})
│ │ │  
│ │ │ -o7 = Div(x, z) + 8*Div(y, z)
│ │ │ +o7 = 8*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o7 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : isLinearEquivalent(D1, D2, IsGraded => true)
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,36 +17,36 @@
│ │ │ │            o flag, a _B_o_o_l_e_a_n_ _v_a_l_u_e,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Given two Weil divisors, this method checks whether they are linearly
│ │ │ │  equivalent.
│ │ │ │  i1 : R = QQ[x, y, z]/ ideal(x * y - z^2);
│ │ │ │  i2 : D1 = divisor({3, 8}, {ideal(x, z), ideal(y, z)})
│ │ │ │  
│ │ │ │ -o2 = 8*Div(y, z) + 3*Div(x, z)
│ │ │ │ +o2 = 3*Div(x, z) + 8*Div(y, z)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : D2 = divisor({8, 1}, {ideal(y, z), ideal(x, z)})
│ │ │ │  
│ │ │ │ -o3 = 8*Div(y, z) + Div(x, z)
│ │ │ │ +o3 = Div(x, z) + 8*Div(y, z)
│ │ │ │  
│ │ │ │  o3 : WeilDivisor on R
│ │ │ │  i4 : isLinearEquivalent(D1, D2)
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  If IsGraded is set to true (by default it is false), then it treats the
│ │ │ │  divisors as divisors on the $Proj$ of their ambient ring.
│ │ │ │  i5 : R = QQ[x, y, z]/ ideal(x * y - z^2);
│ │ │ │  i6 : D1 = divisor({3, 8}, {ideal(x, z), ideal(y, z)})
│ │ │ │  
│ │ │ │ -o6 = 3*Div(x, z) + 8*Div(y, z)
│ │ │ │ +o6 = 8*Div(y, z) + 3*Div(x, z)
│ │ │ │  
│ │ │ │  o6 : WeilDivisor on R
│ │ │ │  i7 : D2 = divisor({8, 1}, {ideal(y, z), ideal(x, z)})
│ │ │ │  
│ │ │ │ -o7 = Div(x, z) + 8*Div(y, z)
│ │ │ │ +o7 = 8*Div(y, z) + Div(x, z)
│ │ │ │  
│ │ │ │  o7 : WeilDivisor on R
│ │ │ │  i8 : isLinearEquivalent(D1, D2, IsGraded => true)
│ │ │ │  
│ │ │ │  o8 = false
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _O_O_ _R_W_e_i_l_D_i_v_i_s_o_r
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_is__Q__Cartier.html
│ │ │ @@ -83,24 +83,24 @@
│ │ │                
    i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : D1 = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = 2*Div(y, z) + Div(x, z)
│ │ │ +o2 = Div(x, z) + 2*Div(y, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : D2 = divisor({1/2, 3/4}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o3 = 1/2*Div(y, z) + 3/4*Div(x, z)
│ │ │ +o3 = 3/4*Div(x, z) + 1/2*Div(y, z)
│ │ │  
│ │ │  o3 : QWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : isQCartier(10, D1)
│ │ │ @@ -164,24 +164,24 @@
│ │ │                
    i11 : R = QQ[x, y, z] / ideal(x * y - z^2 );
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : D1 = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o12 = 2*Div(y, z) + Div(x, z)
│ │ │ +o12 = Div(x, z) + 2*Div(y, z)
│ │ │  
│ │ │  o12 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : D2 = divisor({1/2, 3/4}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o13 = 1/2*Div(y, z) + 3/4*Div(x, z)
│ │ │ +o13 = 3/4*Div(x, z) + 1/2*Div(y, z)
│ │ │  
│ │ │  o13 : QWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : isQCartier(10, D1, IsGraded => true)
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,21 +21,21 @@
│ │ │ │  Check whether $m$ times a Weil or Q-divisor $D$ is Cartier for each $m$ from 1
│ │ │ │  to a fixed positive integer n1 (if the divisor is a QWeilDivisor, it can search
│ │ │ │  slightly higher than n1). If m * D1 is Cartier, it returns m. If it fails to
│ │ │ │  find an m, it returns 0.
│ │ │ │  i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │ │  i2 : D1 = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │ │  
│ │ │ │ -o2 = 2*Div(y, z) + Div(x, z)
│ │ │ │ +o2 = Div(x, z) + 2*Div(y, z)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : D2 = divisor({1/2, 3/4}, {ideal(y, z), ideal(x, z)}, CoefficientType =>
│ │ │ │  QQ)
│ │ │ │  
│ │ │ │ -o3 = 1/2*Div(y, z) + 3/4*Div(x, z)
│ │ │ │ +o3 = 3/4*Div(x, z) + 1/2*Div(y, z)
│ │ │ │  
│ │ │ │  o3 : QWeilDivisor on R
│ │ │ │  i4 : isQCartier(10, D1)
│ │ │ │  
│ │ │ │  o4 = 2
│ │ │ │  i5 : isQCartier(10, D2)
│ │ │ │  
│ │ │ │ @@ -59,21 +59,21 @@
│ │ │ │  
│ │ │ │  o10 = 0
│ │ │ │  If the option IsGraded is set to true (by default it is false), then it treats
│ │ │ │  the divisor as a divisor on the $Proj$ of their ambient ring.
│ │ │ │  i11 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │ │  i12 : D1 = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │ │  
│ │ │ │ -o12 = 2*Div(y, z) + Div(x, z)
│ │ │ │ +o12 = Div(x, z) + 2*Div(y, z)
│ │ │ │  
│ │ │ │  o12 : WeilDivisor on R
│ │ │ │  i13 : D2 = divisor({1/2, 3/4}, {ideal(y, z), ideal(x, z)}, CoefficientType =>
│ │ │ │  QQ)
│ │ │ │  
│ │ │ │ -o13 = 1/2*Div(y, z) + 3/4*Div(x, z)
│ │ │ │ +o13 = 3/4*Div(x, z) + 1/2*Div(y, z)
│ │ │ │  
│ │ │ │  o13 : QWeilDivisor on R
│ │ │ │  i14 : isQCartier(10, D1, IsGraded => true)
│ │ │ │  
│ │ │ │  o14 = 1
│ │ │ │  i15 : isQCartier(10, D2, IsGraded => true)
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_is__Q__Linear__Equivalent.html
│ │ │ @@ -155,24 +155,24 @@
│ │ │                
    i10 : R = QQ[x, y, z] / ideal(x * y - z^2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : D = divisor({1/2, 3/4}, {ideal(x, z), ideal(y, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o11 = 3/4*Div(y, z) + 1/2*Div(x, z)
│ │ │ +o11 = 1/2*Div(x, z) + 3/4*Div(y, z)
│ │ │  
│ │ │  o11 : QWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : E = divisor({3/2, -1/4}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o12 = 3/2*Div(y, z) + -1/4*Div(x, z)
│ │ │ +o12 = -1/4*Div(x, z) + 3/2*Div(y, z)
│ │ │  
│ │ │  o12 : QWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : isQLinearEquivalent(10, D, E, IsGraded => true)
│ │ │ ├── html2text {}
│ │ │ │ @@ -52,21 +52,21 @@
│ │ │ │  o9 = true
│ │ │ │  If IsGraded=>true (the default is false), then it treats the divisors as if
│ │ │ │  they are divisors on the $Proj$ of their ambient ring.
│ │ │ │  i10 : R = QQ[x, y, z] / ideal(x * y - z^2);
│ │ │ │  i11 : D = divisor({1/2, 3/4}, {ideal(x, z), ideal(y, z)}, CoefficientType =>
│ │ │ │  QQ)
│ │ │ │  
│ │ │ │ -o11 = 3/4*Div(y, z) + 1/2*Div(x, z)
│ │ │ │ +o11 = 1/2*Div(x, z) + 3/4*Div(y, z)
│ │ │ │  
│ │ │ │  o11 : QWeilDivisor on R
│ │ │ │  i12 : E = divisor({3/2, -1/4}, {ideal(y, z), ideal(x, z)}, CoefficientType =>
│ │ │ │  QQ)
│ │ │ │  
│ │ │ │ -o12 = 3/2*Div(y, z) + -1/4*Div(x, z)
│ │ │ │ +o12 = -1/4*Div(x, z) + 3/2*Div(y, z)
│ │ │ │  
│ │ │ │  o12 : QWeilDivisor on R
│ │ │ │  i13 : isQLinearEquivalent(10, D, E, IsGraded => true)
│ │ │ │  
│ │ │ │  o13 = true
│ │ │ │  i14 : isQLinearEquivalent(10, 3*D, E, IsGraded => true)
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_is__S__N__C.html
│ │ │ @@ -80,15 +80,15 @@
│ │ │                
    i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : D = divisor({1, -2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = Div(x, z) + -2*Div(y, z)
│ │ │ +o2 = -2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : isSNC( D )
│ │ │ @@ -152,15 +152,15 @@
│ │ │                
    i10 : R = QQ[x, y, z] / ideal(x * y - z^2 );
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : D = divisor({1, -2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o11 = -2*Div(y, z) + Div(x, z)
│ │ │ +o11 = Div(x, z) + -2*Div(y, z)
│ │ │  
│ │ │  o11 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : isSNC( D, IsGraded => true )
│ │ │ @@ -198,15 +198,15 @@
│ │ │                
    i16 : R = QQ[x,y,z];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i17 : D = divisor(x*y*(x+y))
│ │ │  
│ │ │ -o17 = Div(x) + Div(x+y) + Div(y)
│ │ │ +o17 = Div(x+y) + Div(y) + Div(x)
│ │ │  
│ │ │  o17 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i18 : isSNC( D, IsGraded => true)
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │            o a _B_o_o_l_e_a_n_ _v_a_l_u_e,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This function returns true if the divisor is simple normal crossings, this
│ │ │ │  includes checking that the ambient ring is regular.
│ │ │ │  i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │ │  i2 : D = divisor({1, -2}, {ideal(x, z), ideal(y, z)})
│ │ │ │  
│ │ │ │ -o2 = Div(x, z) + -2*Div(y, z)
│ │ │ │ +o2 = -2*Div(y, z) + Div(x, z)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : isSNC( D )
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : R = QQ[x, y];
│ │ │ │  i5 : D = divisor(x*y*(x+y))
│ │ │ │ @@ -45,15 +45,15 @@
│ │ │ │  o9 = true
│ │ │ │  If IsGraded is set to true (default false), then the divisor is treated as if
│ │ │ │  it is on the $Proj$ of the ambient ring. In particular, non-SNC behavior at the
│ │ │ │  origin is ignored.
│ │ │ │  i10 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │ │  i11 : D = divisor({1, -2}, {ideal(x, z), ideal(y, z)})
│ │ │ │  
│ │ │ │ -o11 = -2*Div(y, z) + Div(x, z)
│ │ │ │ +o11 = Div(x, z) + -2*Div(y, z)
│ │ │ │  
│ │ │ │  o11 : WeilDivisor on R
│ │ │ │  i12 : isSNC( D, IsGraded => true )
│ │ │ │  
│ │ │ │  o12 = true
│ │ │ │  i13 : R = QQ[x, y];
│ │ │ │  i14 : D = divisor(x*y*(x+y))
│ │ │ │ @@ -63,15 +63,15 @@
│ │ │ │  o14 : WeilDivisor on R
│ │ │ │  i15 : isSNC( D, IsGraded => true )
│ │ │ │  
│ │ │ │  o15 = true
│ │ │ │  i16 : R = QQ[x,y,z];
│ │ │ │  i17 : D = divisor(x*y*(x+y))
│ │ │ │  
│ │ │ │ -o17 = Div(x) + Div(x+y) + Div(y)
│ │ │ │ +o17 = Div(x+y) + Div(y) + Div(x)
│ │ │ │  
│ │ │ │  o17 : WeilDivisor on R
│ │ │ │  i18 : isSNC( D, IsGraded => true)
│ │ │ │  
│ │ │ │  o18 = false
│ │ │ │  The output value of this function is stored in the divisor's cache with the
│ │ │ │  value of the last IsGraded option. If you change the IsGraded option, the value
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_map__To__Projective__Space.html
│ │ │ @@ -112,15 +112,15 @@
│ │ │                
    i4 : R = ZZ/7[x,y,z];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : D = divisor(x*y)
│ │ │  
│ │ │ -o5 = Div(y) + Div(x)
│ │ │ +o5 = Div(x) + Div(y)
│ │ │  
│ │ │  o5 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : mapToProjectiveSpace(D, Variable=>"Z")
│ │ │ ├── html2text {}
│ │ │ │ @@ -36,15 +36,15 @@
│ │ │ │  
│ │ │ │  o3 : RingMap R <-- QQ[YY ..YY ]
│ │ │ │                          1    2
│ │ │ │  The user may also specify the variable name of the new projective space.
│ │ │ │  i4 : R = ZZ/7[x,y,z];
│ │ │ │  i5 : D = divisor(x*y)
│ │ │ │  
│ │ │ │ -o5 = Div(y) + Div(x)
│ │ │ │ +o5 = Div(x) + Div(y)
│ │ │ │  
│ │ │ │  o5 : WeilDivisor on R
│ │ │ │  i6 : mapToProjectiveSpace(D, Variable=>"Z")
│ │ │ │  
│ │ │ │               ZZ            2             2        2
│ │ │ │  o6 = map (R, --[Z ..Z ], {x , x*y, x*z, y , y*z, z })
│ │ │ │                7  1   6
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_pullback_lp__Ring__Map_cm__R__Weil__Divisor_rp.html
│ │ │ @@ -94,24 +94,24 @@
│ │ │  o3 : RingMap T <-- R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : D = divisor(y*z)
│ │ │  
│ │ │ -o4 = 3*Div(w, z, y) + 3*Div(z, y, x)
│ │ │ +o4 = 3*Div(z, y, x) + 3*Div(w, z, y)
│ │ │  
│ │ │  o4 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : pullback(f, D, Strategy=>Primes)
│ │ │  
│ │ │ -o5 = 3*Div(b) + 3*Div(a)
│ │ │ +o5 = 3*Div(a) + 3*Div(b)
│ │ │  
│ │ │  o5 : WeilDivisor on T
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : pullback(f, D, Strategy=>Sheaves)
│ │ │ @@ -150,24 +150,24 @@
│ │ │  o10 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : D1 = pullback(f, D)
│ │ │  
│ │ │ -o11 = Div(a+1) + 3*Div(b) + Div(a)
│ │ │ +o11 = Div(a) + 3*Div(b) + Div(a+1)
│ │ │  
│ │ │  o11 : WeilDivisor on S
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : f^* D
│ │ │  
│ │ │ -o12 = Div(a+1) + 3*Div(b) + Div(a)
│ │ │ +o12 = Div(a) + 3*Div(b) + Div(a+1)
│ │ │  
│ │ │  o12 : WeilDivisor on S
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    As illustrated by the previous example, the same functionality can also be accomplished by f^* (which creates a function which sends a divisor $D$ to $f^* D$).
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,20 +29,20 @@
│ │ │ │  i1 : R = QQ[x,y,z,w]/ideal(z^2-y*w,y*z-x*w,y^2-x*z);
│ │ │ │  i2 : T = QQ[a,b];
│ │ │ │  i3 : f = map(T, R, {a^3, a^2*b, a*b^2, b^3});
│ │ │ │  
│ │ │ │  o3 : RingMap T <-- R
│ │ │ │  i4 : D = divisor(y*z)
│ │ │ │  
│ │ │ │ -o4 = 3*Div(w, z, y) + 3*Div(z, y, x)
│ │ │ │ +o4 = 3*Div(z, y, x) + 3*Div(w, z, y)
│ │ │ │  
│ │ │ │  o4 : WeilDivisor on R
│ │ │ │  i5 : pullback(f, D, Strategy=>Primes)
│ │ │ │  
│ │ │ │ -o5 = 3*Div(b) + 3*Div(a)
│ │ │ │ +o5 = 3*Div(a) + 3*Div(b)
│ │ │ │  
│ │ │ │  o5 : WeilDivisor on T
│ │ │ │  i6 : pullback(f, D, Strategy=>Sheaves)
│ │ │ │  
│ │ │ │  o6 = 3*Div(b) + 3*Div(a)
│ │ │ │  
│ │ │ │  o6 : WeilDivisor on T
│ │ │ │ @@ -53,20 +53,20 @@
│ │ │ │  
│ │ │ │  o9 : RingMap S <-- R
│ │ │ │  i10 : D = divisor(x*y*(x+y));
│ │ │ │  
│ │ │ │  o10 : WeilDivisor on R
│ │ │ │  i11 : D1 = pullback(f, D)
│ │ │ │  
│ │ │ │ -o11 = Div(a+1) + 3*Div(b) + Div(a)
│ │ │ │ +o11 = Div(a) + 3*Div(b) + Div(a+1)
│ │ │ │  
│ │ │ │  o11 : WeilDivisor on S
│ │ │ │  i12 : f^* D
│ │ │ │  
│ │ │ │ -o12 = Div(a+1) + 3*Div(b) + Div(a)
│ │ │ │ +o12 = Div(a) + 3*Div(b) + Div(a+1)
│ │ │ │  
│ │ │ │  o12 : WeilDivisor on S
│ │ │ │  As illustrated by the previous example, the same functionality can also be
│ │ │ │  accomplished by f^* (which creates a function which sends a divisor $D$ to $f^*
│ │ │ │  D$).
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _P_r_i_m_e_s -- a value for the option Strategy for the pullback method
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_reflexify.html
│ │ │ @@ -267,23 +267,23 @@
│ │ │  
│ │ │  o22 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i23 : time reflexify(J);
│ │ │ - -- used 0.260716s (cpu); 0.201186s (thread); 0s (gc)
│ │ │ + -- used 0.271189s (cpu); 0.210021s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i24 : time reflexify(J*R^1);
│ │ │ - -- used 0.527272s (cpu); 0.394308s (thread); 0s (gc)
    
│ │ │ + -- used 0.44102s (cpu); 0.361892s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    Because of this, there are two strategies for computing a reflexification (at least if the module embeds as an ideal).
    
│ │ │          
    
│ │ │          
    
│ │ │ @@ -319,26 +319,26 @@
│ │ │              
│ │ │                
    i28 : M = J*R^1;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i29 : J1 = time reflexify( J, Strategy=>IdealStrategy )
│ │ │ - -- used 0.273432s (cpu); 0.158699s (thread); 0s (gc)
│ │ │ + -- used 0.284278s (cpu); 0.146299s (thread); 0s (gc)
│ │ │  
│ │ │                2            2     9       9   11
│ │ │  o29 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │  
│ │ │  o29 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i30 : J2 = time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ - -- used 6.95165s (cpu); 4.6997s (thread); 0s (gc)
│ │ │ + -- used 5.49105s (cpu); 4.34638s (thread); 0s (gc)
│ │ │  
│ │ │                2            2     9       9   11
│ │ │  o30 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │  
│ │ │  o30 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │ @@ -348,21 +348,21 @@
│ │ │  
│ │ │  o31 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i32 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ - -- used 5.67659s (cpu); 4.46311s (thread); 0s (gc)
    
│ │ │ + -- used 5.68386s (cpu); 4.59503s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i33 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.775911s (cpu); 0.448237s (thread); 0s (gc)
    
│ │ │ + -- used 0.539944s (cpu); 0.372248s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    However, sometimes ModuleStrategy is faster, especially for Monomial ideals.
    
│ │ │          
    
│ │ │          
│ │ │ @@ -389,15 +389,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i37 : M = I^20*R^1;
    
│ │ │              
    
│ │ │                
    i38 : time reflexify( J, Strategy=>IdealStrategy )
│ │ │ - -- used 1.05331s (cpu); 0.384322s (thread); 0s (gc)
│ │ │ + -- used 1.08794s (cpu); 0.386032s (thread); 0s (gc)
│ │ │  
│ │ │                20     19   2 18   3 17   4 16   5 15   6 14   7 13   8 12 
│ │ │  o38 = ideal (u  , x*u  , x u  , x u  , x u  , x u  , x u  , x u  , x u  ,
│ │ │        -----------------------------------------------------------------------
│ │ │         9 11   10 10   11 9   12 8   13 7   14 6   15 5   16 4   17 3   18 2 
│ │ │        x u  , x  u  , x  u , x  u , x  u , x  u , x  u , x  u , x  u , x  u ,
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -406,15 +406,15 @@
│ │ │  
│ │ │  o38 : Ideal of R
    
│ │ │              
    
│ │ │                
    i39 : time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ - -- used 0.238734s (cpu); 0.0653637s (thread); 0s (gc)
│ │ │ + -- used 0.226139s (cpu); 0.0553398s (thread); 0s (gc)
│ │ │  
│ │ │                20     19   2 18   3 17   4 16   5 15   6 14   7 13   8 12 
│ │ │  o39 = ideal (u  , x*u  , x u  , x u  , x u  , x u  , x u  , x u  , x u  ,
│ │ │        -----------------------------------------------------------------------
│ │ │         9 11   10 10   11 9   12 8   13 7   14 6   15 5   16 4   17 3   18 2 
│ │ │        x u  , x  u  , x  u , x  u , x  u , x  u , x  u , x  u , x  u , x  u ,
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -423,21 +423,21 @@
│ │ │  
│ │ │  o39 : Ideal of R
    
│ │ │              
    
│ │ │                
    i40 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ - -- used 0.276556s (cpu); 0.103744s (thread); 0s (gc)
    
│ │ │ + -- used 0.254852s (cpu); 0.0903637s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i41 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.00801402s (cpu); 0.00801143s (thread); 0s (gc)
    
│ │ │ + -- used 0.00672375s (cpu); 0.00672976s (thread); 0s (gc)
│ │ │              
    
│ │ │          
    
│ │ │            
    For ideals, if KnownDomain is false (default value is true), then the function will check whether it is a domain.  If it is a domain (or assumed to be a domain), it will reflexify using a strategy which can speed up computation, if not it will compute using a sometimes slower method which is essentially reflexifying it as a module.
    
│ │ │          
    
│ │ │          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -114,19 +114,19 @@
│ │ │ │  i21 : I = ideal(x-z,y-2*z);
│ │ │ │  
│ │ │ │  o21 : Ideal of R
│ │ │ │  i22 : J = I^21;
│ │ │ │  
│ │ │ │  o22 : Ideal of R
│ │ │ │  i23 : time reflexify(J);
│ │ │ │ - -- used 0.260716s (cpu); 0.201186s (thread); 0s (gc)
│ │ │ │ + -- used 0.271189s (cpu); 0.210021s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 : Ideal of R
│ │ │ │  i24 : time reflexify(J*R^1);
│ │ │ │ - -- used 0.527272s (cpu); 0.394308s (thread); 0s (gc)
│ │ │ │ + -- used 0.44102s (cpu); 0.361892s (thread); 0s (gc)
│ │ │ │  Because of this, there are two strategies for computing a reflexification (at
│ │ │ │  least if the module embeds as an ideal).
│ │ │ │  IdealStrategy. In the case that $R$ is a domain, and our module is isomorphic
│ │ │ │  to an ideal $I$, then one can compute the reflexification by computing colons.
│ │ │ │  ModuleStrategy. This computes the reflexification simply by computing $Hom$
│ │ │ │  twice.
│ │ │ │  ModuleStrategy is the default strategy for modules, IdealStrategy is the
│ │ │ │ @@ -139,73 +139,73 @@
│ │ │ │  
│ │ │ │  o26 : Ideal of R
│ │ │ │  i27 : J = I^20;
│ │ │ │  
│ │ │ │  o27 : Ideal of R
│ │ │ │  i28 : M = J*R^1;
│ │ │ │  i29 : J1 = time reflexify( J, Strategy=>IdealStrategy )
│ │ │ │ - -- used 0.273432s (cpu); 0.158699s (thread); 0s (gc)
│ │ │ │ + -- used 0.284278s (cpu); 0.146299s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                2            2     9       9   11
│ │ │ │  o29 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │ │  
│ │ │ │  o29 : Ideal of R
│ │ │ │  i30 : J2 = time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ │ - -- used 6.95165s (cpu); 4.6997s (thread); 0s (gc)
│ │ │ │ + -- used 5.49105s (cpu); 4.34638s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                2            2     9       9   11
│ │ │ │  o30 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │ │  
│ │ │ │  o30 : Ideal of R
│ │ │ │  i31 : J1 == J2
│ │ │ │  
│ │ │ │  o31 = true
│ │ │ │  i32 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ │ - -- used 5.67659s (cpu); 4.46311s (thread); 0s (gc)
│ │ │ │ + -- used 5.68386s (cpu); 4.59503s (thread); 0s (gc)
│ │ │ │  i33 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ │ - -- used 0.775911s (cpu); 0.448237s (thread); 0s (gc)
│ │ │ │ + -- used 0.539944s (cpu); 0.372248s (thread); 0s (gc)
│ │ │ │  However, sometimes ModuleStrategy is faster, especially for Monomial ideals.
│ │ │ │  i34 : R = QQ[x,y,u,v]/ideal(x*y-u*v);
│ │ │ │  i35 : I = ideal(x,u);
│ │ │ │  
│ │ │ │  o35 : Ideal of R
│ │ │ │  i36 : J = I^20;
│ │ │ │  
│ │ │ │  o36 : Ideal of R
│ │ │ │  i37 : M = I^20*R^1;
│ │ │ │  i38 : time reflexify( J, Strategy=>IdealStrategy )
│ │ │ │ - -- used 1.05331s (cpu); 0.384322s (thread); 0s (gc)
│ │ │ │ + -- used 1.08794s (cpu); 0.386032s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                20     19   2 18   3 17   4 16   5 15   6 14   7 13   8 12
│ │ │ │  o38 = ideal (u  , x*u  , x u  , x u  , x u  , x u  , x u  , x u  , x u  ,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │         9 11   10 10   11 9   12 8   13 7   14 6   15 5   16 4   17 3   18 2
│ │ │ │        x u  , x  u  , x  u , x  u , x  u , x  u , x  u , x  u , x  u , x  u ,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │         19    20
│ │ │ │        x  u, x  )
│ │ │ │  
│ │ │ │  o38 : Ideal of R
│ │ │ │  i39 : time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ │ - -- used 0.238734s (cpu); 0.0653637s (thread); 0s (gc)
│ │ │ │ + -- used 0.226139s (cpu); 0.0553398s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                20     19   2 18   3 17   4 16   5 15   6 14   7 13   8 12
│ │ │ │  o39 = ideal (u  , x*u  , x u  , x u  , x u  , x u  , x u  , x u  , x u  ,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │         9 11   10 10   11 9   12 8   13 7   14 6   15 5   16 4   17 3   18 2
│ │ │ │        x u  , x  u  , x  u , x  u , x  u , x  u , x  u , x  u , x  u , x  u ,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │         19    20
│ │ │ │        x  u, x  )
│ │ │ │  
│ │ │ │  o39 : Ideal of R
│ │ │ │  i40 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ │ - -- used 0.276556s (cpu); 0.103744s (thread); 0s (gc)
│ │ │ │ + -- used 0.254852s (cpu); 0.0903637s (thread); 0s (gc)
│ │ │ │  i41 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ │ - -- used 0.00801402s (cpu); 0.00801143s (thread); 0s (gc)
│ │ │ │ + -- used 0.00672375s (cpu); 0.00672976s (thread); 0s (gc)
│ │ │ │  For ideals, if KnownDomain is false (default value is true), then the function
│ │ │ │  will check whether it is a domain. If it is a domain (or assumed to be a
│ │ │ │  domain), it will reflexify using a strategy which can speed up computation, if
│ │ │ │  not it will compute using a sometimes slower method which is essentially
│ │ │ │  reflexifying it as a module.
│ │ │ │  Consider the following example showing the importance of making the correct
│ │ │ │  assumption about the ring being a domain.
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_reflexive__Power.html
│ │ │ @@ -124,30 +124,30 @@
│ │ │  
│ │ │  o6 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time J20a = reflexivePower(20, I);
│ │ │ - -- used 0.0323817s (cpu); 0.0323825s (thread); 0s (gc)
│ │ │ + -- used 0.0325992s (cpu); 0.0325983s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : I20 = I^20;
│ │ │  
│ │ │  o8 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : time J20b = reflexify(I20);
│ │ │ - -- used 0.204644s (cpu); 0.153031s (thread); 0s (gc)
│ │ │ + -- used 0.219399s (cpu); 0.145855s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : J20a == J20b
│ │ │ @@ -171,23 +171,23 @@
│ │ │  
│ │ │  o12 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : time J1 = reflexivePower(20, I, Strategy=>IdealStrategy);
│ │ │ - -- used 0.029315s (cpu); 0.0293196s (thread); 0s (gc)
│ │ │ + -- used 0.0363386s (cpu); 0.0363264s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : time J2 = reflexivePower(20, I, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.143657s (cpu); 0.0818326s (thread); 0s (gc)
│ │ │ + -- used 0.162794s (cpu); 0.0890308s (thread); 0s (gc)
│ │ │  
│ │ │  o14 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : J1 == J2
│ │ │ ├── html2text {}
│ │ │ │ @@ -40,39 +40,39 @@
│ │ │ │  of the generators of $I$. Consider the example of a cone over a point on an
│ │ │ │  elliptic curve.
│ │ │ │  i5 : R = QQ[x,y,z]/ideal(-y^2*z +x^3 + x^2*z + x*z^2+z^3);
│ │ │ │  i6 : I = ideal(x-z,y-2*z);
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : time J20a = reflexivePower(20, I);
│ │ │ │ - -- used 0.0323817s (cpu); 0.0323825s (thread); 0s (gc)
│ │ │ │ + -- used 0.0325992s (cpu); 0.0325983s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  i8 : I20 = I^20;
│ │ │ │  
│ │ │ │  o8 : Ideal of R
│ │ │ │  i9 : time J20b = reflexify(I20);
│ │ │ │ - -- used 0.204644s (cpu); 0.153031s (thread); 0s (gc)
│ │ │ │ + -- used 0.219399s (cpu); 0.145855s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 : Ideal of R
│ │ │ │  i10 : J20a == J20b
│ │ │ │  
│ │ │ │  o10 = true
│ │ │ │  This passes the Strategy option to a reflexify call. Valid options are
│ │ │ │  IdealStrategy and ModuleStrategy.
│ │ │ │  i11 : R = QQ[x,y,z]/ideal(-y^2*z +x^3 + x^2*z + x*z^2+z^3);
│ │ │ │  i12 : I = ideal(x-z,y-2*z);
│ │ │ │  
│ │ │ │  o12 : Ideal of R
│ │ │ │  i13 : time J1 = reflexivePower(20, I, Strategy=>IdealStrategy);
│ │ │ │ - -- used 0.029315s (cpu); 0.0293196s (thread); 0s (gc)
│ │ │ │ + -- used 0.0363386s (cpu); 0.0363264s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 : Ideal of R
│ │ │ │  i14 : time J2 = reflexivePower(20, I, Strategy=>ModuleStrategy);
│ │ │ │ - -- used 0.143657s (cpu); 0.0818326s (thread); 0s (gc)
│ │ │ │ + -- used 0.162794s (cpu); 0.0890308s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 : Ideal of R
│ │ │ │  i15 : J1 == J2
│ │ │ │  
│ │ │ │  o15 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_f_l_e_x_i_f_y -- calculate the double dual of an ideal or module Hom(Hom(M,
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_ring_lp__Basic__Divisor_rp.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │                
    i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = 2*Div(y, z) + Div(x, z)
│ │ │ +o2 = Div(x, z) + 2*Div(y, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : ring( D )
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,15 +12,15 @@
│ │ │ │      * Outputs:
│ │ │ │            o a _r_i_n_g,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This function returns the ambient ring of a divisor.
│ │ │ │  i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │ │  i2 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │ │  
│ │ │ │ -o2 = 2*Div(y, z) + Div(x, z)
│ │ │ │ +o2 = Div(x, z) + 2*Div(y, z)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : ring( D )
│ │ │ │  
│ │ │ │  o3 = R
│ │ │ │  
│ │ │ │  o3 : QuotientRing
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_to__R__Weil__Divisor.html
│ │ │ @@ -79,42 +79,42 @@
│ │ │                
    i1 : R = ZZ/5[x,y];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : D = divisor({2, 0, -4}, {ideal(x), ideal(y), ideal(x-y)})
│ │ │  
│ │ │ -o2 = -4*Div(x-y) + 2*Div(x) + 0*Div(y)
│ │ │ +o2 = 2*Div(x) + 0*Div(y) + -4*Div(x-y)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : E = (1/2)*D
│ │ │  
│ │ │ -o3 = -2*Div(x-y) + Div(x)
│ │ │ +o3 = Div(x) + -2*Div(x-y)
│ │ │  
│ │ │  o3 : QWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : F = toRWeilDivisor(D)
│ │ │  
│ │ │ -o4 = -4*Div(x-y) + 2*Div(x)
│ │ │ +o4 = 2*Div(x) + -4*Div(x-y)
│ │ │  
│ │ │  o4 : RWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : G = toRWeilDivisor(E)
│ │ │  
│ │ │ -o5 = -2*Div(x-y) + Div(x)
│ │ │ +o5 = Div(x) + -2*Div(x-y)
│ │ │  
│ │ │  o5 : RWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : F == 2*G
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,30 +15,30 @@
│ │ │ │            o an instance of the type _R_W_e_i_l_D_i_v_i_s_o_r,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Turn a Weil divisor or a Q-divisor into a R-divisor (or do nothing to a R-
│ │ │ │  divisor).
│ │ │ │  i1 : R = ZZ/5[x,y];
│ │ │ │  i2 : D = divisor({2, 0, -4}, {ideal(x), ideal(y), ideal(x-y)})
│ │ │ │  
│ │ │ │ -o2 = -4*Div(x-y) + 2*Div(x) + 0*Div(y)
│ │ │ │ +o2 = 2*Div(x) + 0*Div(y) + -4*Div(x-y)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : E = (1/2)*D
│ │ │ │  
│ │ │ │ -o3 = -2*Div(x-y) + Div(x)
│ │ │ │ +o3 = Div(x) + -2*Div(x-y)
│ │ │ │  
│ │ │ │  o3 : QWeilDivisor on R
│ │ │ │  i4 : F = toRWeilDivisor(D)
│ │ │ │  
│ │ │ │ -o4 = -4*Div(x-y) + 2*Div(x)
│ │ │ │ +o4 = 2*Div(x) + -4*Div(x-y)
│ │ │ │  
│ │ │ │  o4 : RWeilDivisor on R
│ │ │ │  i5 : G = toRWeilDivisor(E)
│ │ │ │  
│ │ │ │ -o5 = -2*Div(x-y) + Div(x)
│ │ │ │ +o5 = Div(x) + -2*Div(x-y)
│ │ │ │  
│ │ │ │  o5 : RWeilDivisor on R
│ │ │ │  i6 : F == 2*G
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_o_W_e_i_l_D_i_v_i_s_o_r -- create a Weil divisor from a Q or R-divisor
│ │ ├── ./usr/share/doc/Macaulay2/WeylAlgebras/example-output/_factor__Weyl__Algebra.out
│ │ │ @@ -4,19 +4,19 @@
│ │ │  
│ │ │  o1 = R
│ │ │  
│ │ │  o1 : PolynomialRing, 1 differential variable(s)
│ │ │  
│ │ │  i2 : factorWA(x^5*dx^2+7*x^4*dx+8*x^3-x*dx^2+dx)
│ │ │  
│ │ │ -                                    2         3       2                  2  
│ │ │ -o2 = {(x*dx - 1)(dx)(x - 1)(x + 1)(x  + 1), (x dx + 3x  - x*dx + 1)(dx)(x  +
│ │ │ +        3       2                                      3                  
│ │ │ +o2 = {(x dx + 3x  + x*dx - 1)(dx)(x - 1)(x + 1), (dx)(x dx + x*dx - 2)(x -
│ │ │       ------------------------------------------------------------------------
│ │ │ -               3                2         3       2                         
│ │ │ -     1), (dx)(x dx - x*dx + 2)(x  + 1), (x dx + 3x  + x*dx - 1)(dx)(x - 1)(x
│ │ │ +                  3       2                  2             3                2
│ │ │ +     1)(x + 1), (x dx + 3x  - x*dx + 1)(dx)(x  + 1), (dx)(x dx - x*dx + 2)(x 
│ │ │       ------------------------------------------------------------------------
│ │ │ -                 3
│ │ │ -     + 1), (dx)(x dx + x*dx - 2)(x - 1)(x + 1)}
│ │ │ +                                         2
│ │ │ +     + 1), (x*dx - 1)(dx)(x - 1)(x + 1)(x  + 1)}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylAlgebras/html/_factor__Weyl__Algebra.html
│ │ │ @@ -84,22 +84,22 @@
│ │ │  o1 : PolynomialRing, 1 differential variable(s)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : factorWA(x^5*dx^2+7*x^4*dx+8*x^3-x*dx^2+dx)
│ │ │  
│ │ │ -                                    2         3       2                  2  
│ │ │ -o2 = {(x*dx - 1)(dx)(x - 1)(x + 1)(x  + 1), (x dx + 3x  - x*dx + 1)(dx)(x  +
│ │ │ +        3       2                                      3                  
│ │ │ +o2 = {(x dx + 3x  + x*dx - 1)(dx)(x - 1)(x + 1), (dx)(x dx + x*dx - 2)(x -
│ │ │       ------------------------------------------------------------------------
│ │ │ -               3                2         3       2                         
│ │ │ -     1), (dx)(x dx - x*dx + 2)(x  + 1), (x dx + 3x  + x*dx - 1)(dx)(x - 1)(x
│ │ │ +                  3       2                  2             3                2
│ │ │ +     1)(x + 1), (x dx + 3x  - x*dx + 1)(dx)(x  + 1), (dx)(x dx - x*dx + 2)(x 
│ │ │       ------------------------------------------------------------------------
│ │ │ -                 3
│ │ │ -     + 1), (dx)(x dx + x*dx - 2)(x - 1)(x + 1)}
│ │ │ +                                         2
│ │ │ +     + 1), (x*dx - 1)(dx)(x - 1)(x + 1)(x  + 1)}
│ │ │  
│ │ │  o2 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    To reduce their number, two factorisations are considered equivalent if they can be related by (1)	switching commuting irreducible factors or (2) switching monomials and degree 0 factors; a normal order is chosen where commuting factors are sorted, and monomials are pushed to the right/left if they're differential/not.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,22 +20,22 @@
│ │ │ │  i1 : R = makeWA(QQ[x])
│ │ │ │  
│ │ │ │  o1 = R
│ │ │ │  
│ │ │ │  o1 : PolynomialRing, 1 differential variable(s)
│ │ │ │  i2 : factorWA(x^5*dx^2+7*x^4*dx+8*x^3-x*dx^2+dx)
│ │ │ │  
│ │ │ │ -                                    2         3       2                  2
│ │ │ │ -o2 = {(x*dx - 1)(dx)(x - 1)(x + 1)(x  + 1), (x dx + 3x  - x*dx + 1)(dx)(x  +
│ │ │ │ +        3       2                                      3
│ │ │ │ +o2 = {(x dx + 3x  + x*dx - 1)(dx)(x - 1)(x + 1), (dx)(x dx + x*dx - 2)(x -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -               3                2         3       2
│ │ │ │ -     1), (dx)(x dx - x*dx + 2)(x  + 1), (x dx + 3x  + x*dx - 1)(dx)(x - 1)(x
│ │ │ │ +                  3       2                  2             3                2
│ │ │ │ +     1)(x + 1), (x dx + 3x  - x*dx + 1)(dx)(x  + 1), (dx)(x dx - x*dx + 2)(x
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -                 3
│ │ │ │ -     + 1), (dx)(x dx + x*dx - 2)(x - 1)(x + 1)}
│ │ │ │ +                                         2
│ │ │ │ +     + 1), (x*dx - 1)(dx)(x - 1)(x + 1)(x  + 1)}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  To reduce their number, two factorisations are considered equivalent if they
│ │ │ │  can be related by (1) switching commuting irreducible factors or (2) switching
│ │ │ │  monomials and degree 0 factors; a normal order is chosen where commuting
│ │ │ │  factors are sorted, and monomials are pushed to the right/left if they're
│ │ │ │  differential/not.
│ │ ├── ./usr/share/doc/Macaulay2/WhitneyStratifications/example-output/_map__Stratify.out
│ │ │ @@ -122,15 +122,15 @@
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  loop over components of JY=ideal 1
│ │ │ - -- used 1.9378s (cpu); 1.16676s (thread); 0s (gc)
│ │ │ + -- used 2.14729s (cpu); 1.1965s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o23 : List
│ │ │  
│ │ │  i24 : peek last ms
│ │ │  
│ │ │ @@ -142,15 +142,15 @@
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  loop over components of JY=ideal 1
│ │ │ - -- used 4.77653s (cpu); 2.98611s (thread); 0s (gc)
│ │ │ + -- used 6.60694s (cpu); 3.03429s (thread); 0s (gc)
│ │ │  
│ │ │  o25 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o25 : List
│ │ │  
│ │ │  i26 : peek last ms
│ │ │  
│ │ │ @@ -162,15 +162,15 @@
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  loop over components of JY=ideal 1
│ │ │ - -- used 5.07329s (cpu); 2.97775s (thread); 0s (gc)
│ │ │ + -- used 7.85829s (cpu); 3.26666s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o27 : List
│ │ │  
│ │ │  i28 : peek last ms
│ │ ├── ./usr/share/doc/Macaulay2/WhitneyStratifications/html/_map__Stratify.html
│ │ │ @@ -292,15 +292,15 @@
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  loop over components of JY=ideal 1
│ │ │ - -- used 1.9378s (cpu); 1.16676s (thread); 0s (gc)
│ │ │ + -- used 2.14729s (cpu); 1.1965s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o23 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -318,15 +318,15 @@
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  loop over components of JY=ideal 1
│ │ │ - -- used 4.77653s (cpu); 2.98611s (thread); 0s (gc)
│ │ │ + -- used 6.60694s (cpu); 3.03429s (thread); 0s (gc)
│ │ │  
│ │ │  o25 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o25 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -344,15 +344,15 @@
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  loop over components of JY=ideal 1
│ │ │ - -- used 5.07329s (cpu); 2.97775s (thread); 0s (gc)
│ │ │ + -- used 7.85829s (cpu); 3.26666s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o27 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -184,15 +184,15 @@
│ │ │ │  for j2....
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │  for j2....
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │  for j2....
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │ - -- used 1.9378s (cpu); 1.16676s (thread); 0s (gc)
│ │ │ │ + -- used 2.14729s (cpu); 1.1965s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │ │  
│ │ │ │  o23 : List
│ │ │ │  i24 : peek last ms
│ │ │ │  
│ │ │ │  o24 = MutableHashTable{0 => {ideal (P, M1)}                    }
│ │ │ │ @@ -202,15 +202,15 @@
│ │ │ │  for j2....
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │  for j2....
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │  for j2....
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │ - -- used 4.77653s (cpu); 2.98611s (thread); 0s (gc)
│ │ │ │ + -- used 6.60694s (cpu); 3.03429s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o25 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │ │  
│ │ │ │  o25 : List
│ │ │ │  i26 : peek last ms
│ │ │ │  
│ │ │ │  o26 = MutableHashTable{0 => {ideal (P, M1)}                    }
│ │ │ │ @@ -220,15 +220,15 @@
│ │ │ │  for j2....
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │  for j2....
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │  for j2....
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │ - -- used 5.07329s (cpu); 2.97775s (thread); 0s (gc)
│ │ │ │ + -- used 7.85829s (cpu); 3.26666s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │ │  
│ │ │ │  o27 : List
│ │ │ │  i28 : peek last ms
│ │ │ │  
│ │ │ │  o28 = MutableHashTable{0 => {ideal (P, M1)}                    }
│ │ ├── ./usr/share/doc/Macaulay2/gfanInterface/example-output/___Installation_spand_sp__Configuration_spof_spgfan__Interface.out
│ │ │ @@ -19,15 +19,15 @@
│ │ │  i4 : prefixDirectory | currentLayout#"programs"
│ │ │  
│ │ │  o4 = /usr/x86_64-Linux-
│ │ │       Debian-forky/libexec/Macaulay2/bin/
│ │ │  
│ │ │  i5 : loadPackage("gfanInterface", Configuration => { "keepfiles" => true, "verbose" => true}, Reload => true);
│ │ │   -- warning: reloading gfanInterface; recreate instances of types from this package
│ │ │ - -- running: /usr/bin/gfan gfan --help < /tmp/M2-16949-0/172
│ │ │ + -- running: /usr/bin/gfan gfan --help < /tmp/M2-22465-0/172
│ │ │  This is a program for computing all reduced Groebner bases of a polynomial ideal. It takes the ring and a generating set for the ideal as input. By default the enumeration is done by an almost memoryless reverse search. If the ideal is symmetric the symmetry option is useful and enumeration will be done up to symmetry using a breadth first search. The program needs a starting Groebner basis to do its computations. If the -g option is not specified it will compute one using Buchberger's algorithm.
│ │ │  Options:
│ │ │  -g:
│ │ │   Tells the program that the input is already a Groebner basis (with the initial term of each polynomial being the first ones listed). Use this option if it takes too much time to compute the starting (standard degree lexicographic) Groebner basis and the input is already a Groebner basis.
│ │ │  
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ideal. The program checks that the ideal stays fixed when permuting the variables with respect to elements in the group. The program uses breadth first search to compute the set of reduced Groebner bases up to symmetry with respect to the specified subgroup.
│ │ │ @@ -38,16 +38,16 @@
│ │ │  --disableSymmetryTest:
│ │ │   When using --symmetry this option will disable the check that the group read off from the input actually is a symmetry group with respect to the input ideal.
│ │ │  
│ │ │  --parameters value:
│ │ │   With this option you can specify how many variables to treat as parameters instead of variables. This makes it possible to do computations where the coefficient field is the field of rational functions in the parameters.
│ │ │  --interrupt value:
│ │ │   Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans).
│ │ │ -using temporary file /tmp/M2-16949-0/172
│ │ │ - -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-16949-0/174
│ │ │ +using temporary file /tmp/M2-22465-0/172
│ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-22465-0/174
│ │ │  This program computes a reduced lexicographic Groebner basis of the polynomial ideal given as input. The default behavior is to use Buchberger's algorithm. The ordering of the variables is $a>b>c...$ (assuming that the ring is Q[a,b,c,...]).
│ │ │  Options:
│ │ │  -w:
│ │ │   Compute a Groebner basis with respect to a degree lexicographic order with $a>b>c...$ instead. The degrees are given by a weight vector which is read from the input after the generating set has been read.
│ │ │  
│ │ │  -r:
│ │ │   Use the reverse lexicographic order (or the reverse lexicographic order as a tie breaker if -w is used). The input must be homogeneous if the pure reverse lexicographic order is chosen. Ignored if -W is used.
│ │ │ @@ -56,69 +56,69 @@
│ │ │   Do a Groebner walk. The input must be a minimal Groebner basis. If -W is used -w is ignored.
│ │ │  
│ │ │  -g:
│ │ │   Do a generic Groebner walk. The input must be homogeneous and must be a minimal Groebner basis with respect to the reverse lexicographic term order. The target term order is always lexicographic. The -W option must be used.
│ │ │  
│ │ │  --parameters value:
│ │ │   With this option you can specify how many variables to treat as parameters instead of variables. This makes it possible to do computations where the coefficient field is the field of rational functions in the parameters.
│ │ │ -using temporary file /tmp/M2-16949-0/174
│ │ │ - -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-16949-0/176
│ │ │ +using temporary file /tmp/M2-22465-0/174
│ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-22465-0/176
│ │ │  This program takes a marked Groebner basis of an ideal I and a set of polynomials on its input and tests if the polynomial set is contained in I by applying the division algorithm for each element. The output is 1 for true and 0 for false.
│ │ │  Options:
│ │ │  --remainder:
│ │ │   Tell the program to output the remainders of the divisions rather than outputting 0 or 1.
│ │ │  --multiplier:
│ │ │   Reads in a polynomial that will be multiplied to the polynomial to be divided before doing the division.
│ │ │ -using temporary file /tmp/M2-16949-0/176
│ │ │ - -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-16949-0/178
│ │ │ +using temporary file /tmp/M2-22465-0/176
│ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-22465-0/178
│ │ │  This program takes two polyhedral fans and computes their common refinement.
│ │ │  Options:
│ │ │  -i1 value:
│ │ │   Specify the name of the first input file.
│ │ │  -i2 value:
│ │ │   Specify the name of the second input file.
│ │ │  --stable:
│ │ │   Compute the stable intersection.
│ │ │ -using temporary file /tmp/M2-16949-0/178
│ │ │ - -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-16949-0/180
│ │ │ +using temporary file /tmp/M2-22465-0/178
│ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-22465-0/180
│ │ │  This program takes a polyhedral fan and a vector and computes the link of the polyhedral fan around that vertex. The link will have lineality space dimension equal to the dimension of the relative open polyhedral cone of the original fan containing the vector.
│ │ │  Options:
│ │ │  -i value:
│ │ │   Specify the name of the input file.
│ │ │  --symmetry:
│ │ │   Reads in a fan stored with symmetry. The generators of the symmetry group must be given on the standard input.
│ │ │  
│ │ │  --star:
│ │ │   Computes the star instead. The star is defined as the smallest polyhedral fan containing all cones of the original fan containing the vector.
│ │ │ -using temporary file /tmp/M2-16949-0/180
│ │ │ - -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-16949-0/182
│ │ │ +using temporary file /tmp/M2-22465-0/180
│ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-22465-0/182
│ │ │  This program takes two polyhedral fans and computes their product.
│ │ │  Options:
│ │ │  -i1 value:
│ │ │   Specify the name of the first input file.
│ │ │  -i2 value:
│ │ │   Specify the name of the second input file.
│ │ │ -using temporary file /tmp/M2-16949-0/182
│ │ │ - -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-16949-0/184
│ │ │ +using temporary file /tmp/M2-22465-0/182
│ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-22465-0/184
│ │ │  This program computes a Groebner cone. Three different cases are handled. The input may be a marked reduced Groebner basis in which case its Groebner cone is computed. The input may be just a marked minimal basis in which case the cone computed is not a Groebner cone in the usual sense but smaller. (These cones are described in [Fukuda, Jensen, Lauritzen, Thomas]). The third possible case is that the Groebner cone is possibly lower dimensional and given by a pair of Groebner bases as it is useful to do for tropical varieties, see option --pair. The facets of the cone can be read off in section FACETS and the equations in section IMPLIED_EQUATIONS.
│ │ │  Options:
│ │ │  --restrict:
│ │ │   Add an inequality for each coordinate, so that the the cone is restricted to the non-negative orthant.
│ │ │  --pair:
│ │ │   The Groebner cone is given by a pair of compatible Groebner bases. The first basis is for the initial ideal and the second for the ideal itself. See the tropical section of the manual.
│ │ │  --asfan:
│ │ │   Writes the cone as a polyhedral fan with all its faces instead. In this way the extreme rays of the cone are also computed.
│ │ │  --vectorinput:
│ │ │   Compute a cone given list of inequalities rather than a Groebner cone. The input is an integer which specifies the dimension of the ambient space, a list of inequalities given as vectors and a list of equations.
│ │ │ -using temporary file /tmp/M2-16949-0/184
│ │ │ - -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-16949-0/186
│ │ │ +using temporary file /tmp/M2-22465-0/184
│ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-22465-0/186
│ │ │  This program computes the homogeneity space of a list of polynomials - as a cone. Thus generators for the homogeneity space are found in the section LINEALITY_SPACE. If you wish the homogeneity space of an ideal you should first compute a set of homogeneous generators and call the program on these. A reduced Groebner basis will always suffice for this purpose.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16949-0/186
│ │ │ - -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-16949-0/188
│ │ │ +using temporary file /tmp/M2-22465-0/186
│ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-22465-0/188
│ │ │  This program homogenises a list of polynomials by introducing an extra variable. The name of the variable to be introduced is read from the input after the list of polynomials. Without the -w option the homogenisation is done with respect to total degree.
│ │ │  Example:
│ │ │  Input:
│ │ │  Q[x,y]{y-1}
│ │ │  z
│ │ │  Output:
│ │ │  Q[x,y,z]{y-z}
│ │ │ @@ -126,30 +126,30 @@
│ │ │  -i:
│ │ │   Treat input as an ideal. This will make the program compute the homogenisation of the input ideal. This is done by computing a degree Groebner basis and homogenising it.
│ │ │  -w:
│ │ │   Specify a homogenisation vector. The length of the vector must be the same as the number of variables in the ring. The vector is read from the input after the list of polynomials.
│ │ │  
│ │ │  -H:
│ │ │   Let the name of the new variable be H rather than reading in a name from the input.
│ │ │ -using temporary file /tmp/M2-16949-0/188
│ │ │ - -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-16949-0/190
│ │ │ +using temporary file /tmp/M2-22465-0/188
│ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-22465-0/190
│ │ │  This program converts a list of polynomials to a list of their initial forms with respect to the vector given after the list.
│ │ │  Options:
│ │ │  --ideal:
│ │ │   Treat input as an ideal. This will make the program compute the initial ideal of the ideal generated by the input polynomials. The computation is done by computing a Groebner basis with respect to the given vector. The vector must be positive or the input polynomials must be homogeneous in a positive grading. None of these conditions are checked by the program.
│ │ │  
│ │ │  --pair:
│ │ │   Produce a pair of polynomial lists. Used together with --ideal this option will also write a compatible reduced Groebner basis for the input ideal to the output. This is useful for finding the Groebner cone of a non-monomial initial ideal.
│ │ │  
│ │ │  --mark:
│ │ │   If the --pair option is and the --ideal option is not used this option will still make sure that the second output basis is marked consistently with the vector.
│ │ │  --list:
│ │ │   Read in a list of vectors instead of a single vector and produce a list of polynomial sets as output.
│ │ │ -using temporary file /tmp/M2-16949-0/190
│ │ │ - -- running: /usr/bin/gfan _interactive --help < /tmp/M2-16949-0/192
│ │ │ +using temporary file /tmp/M2-22465-0/190
│ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-22465-0/192
│ │ │  This is a program for doing interactive walks in the Groebner fan of an ideal. The input is a Groebner basis defining the starting Groebner cone of the walk. The program will list all flippable facets of the Groebner cone and ask the user to choose one. The user types in the index (number) of the facet in the list. The program will walk through the selected facet and display the new Groebner basis and a list of new facet normals for the user to choose from. Since the program reads the user's choices through the the standard input it is recommended not to redirect the standard input for this program.
│ │ │  Options:
│ │ │  -L:
│ │ │   Latex mode. The program will try to show the current Groebner basis in a readable form by invoking LaTeX and xdvi.
│ │ │  
│ │ │  -x:
│ │ │   Exit immediately.
│ │ │ @@ -164,57 +164,57 @@
│ │ │   Tell the program to list the defining set of inequalities of the non-restricted Groebner cone as a set of vectors after having listed the current Groebner basis.
│ │ │  
│ │ │  -W:
│ │ │   Print weight vector. This will make the program print an interior vector of the current Groebner cone and a relative interior point for each flippable facet of the current Groebner cone.
│ │ │  
│ │ │  --tropical:
│ │ │   Traverse a tropical variety interactively.
│ │ │ -using temporary file /tmp/M2-16949-0/192
│ │ │ - -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-16949-0/194
│ │ │ +using temporary file /tmp/M2-22465-0/192
│ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-22465-0/194
│ │ │  This program checks if a set of marked polynomials is a Groebner basis with respect to its marking. First it is checked if the markings are consistent with respect to a positive vector. Then Buchberger's S-criterion is checked. The output is boolean value.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16949-0/194
│ │ │ - -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-16949-0/196
│ │ │ +using temporary file /tmp/M2-22465-0/194
│ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-22465-0/196
│ │ │  Takes an ideal $I$ and computes the Krull dimension of R/I where R is the polynomial ring. This is done by first computing a Groebner basis.
│ │ │  Options:
│ │ │  -g:
│ │ │   Tell the program that the input is already a reduced Groebner basis.
│ │ │ -using temporary file /tmp/M2-16949-0/196
│ │ │ - -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-16949-0/198
│ │ │ +using temporary file /tmp/M2-22465-0/196
│ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-22465-0/198
│ │ │  This program computes the lattice ideal of a lattice. The input is a list of generators for the lattice.
│ │ │  Options:
│ │ │  -t:
│ │ │   Compute the toric ideal of the matrix whose rows are given on the input instead.
│ │ │  --convert:
│ │ │   Does not do any computation, but just converts the vectors to binomials.
│ │ │ -using temporary file /tmp/M2-16949-0/198
│ │ │ - -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-16949-0/200
│ │ │ +using temporary file /tmp/M2-22465-0/198
│ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-22465-0/200
│ │ │  This program converts a list of polynomials to a list of their leading terms.
│ │ │  Options:
│ │ │  -m:
│ │ │   Do the same thing for a list of polynomial sets. That is, output the set of sets of leading terms.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16949-0/200
│ │ │ - -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-16949-0/202
│ │ │ +using temporary file /tmp/M2-22465-0/200
│ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-22465-0/202
│ │ │  This program marks a set of polynomials with respect to the vector given at the end of the input, meaning that the largest terms are moved to the front. In case of a tie the lexicographic term order with $a>b>c...$ is used to break it.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16949-0/202
│ │ │ - -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-16949-0/204
│ │ │ +using temporary file /tmp/M2-22465-0/202
│ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-22465-0/204
│ │ │  This is a program for computing the normal fan of the Minkowski sum of the Newton polytopes of a list of polynomials.
│ │ │  Options:
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ideal. The program checks that the ideal stays fixed when permuting the variables with respect to elements in the group. The program uses breadth first search to compute the set of reduced Groebner bases up to symmetry with respect to the specified subgroup.
│ │ │  
│ │ │  --disableSymmetryTest:
│ │ │   When using --symmetry this option will disable the check that the group read off from the input actually is a symmetry group with respect to the input ideal.
│ │ │  
│ │ │  --nocones:
│ │ │   Tell the program to not list cones in the output.
│ │ │ -using temporary file /tmp/M2-16949-0/204
│ │ │ - -- running: /usr/bin/gfan _minors --help < /tmp/M2-16949-0/206
│ │ │ +using temporary file /tmp/M2-22465-0/204
│ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-22465-0/206
│ │ │  This program will generate the r*r minors of a d*n matrix of indeterminates.
│ │ │  Options:
│ │ │  -r value:
│ │ │   Specify r.
│ │ │  -d value:
│ │ │   Specify d.
│ │ │  -n value:
│ │ │ @@ -229,16 +229,16 @@
│ │ │   Do nothing but produce symmetry generators for the Pluecker ideal.
│ │ │  --symmetry:
│ │ │   Produces a list of generators for the group of symmetries keeping the set of minors fixed. (Only without --names).
│ │ │  --parametrize:
│ │ │   Parametrize the set of d times n matrices of Barvinok rank less than or equal to r-1 by a list of tropical polynomials.
│ │ │  --ultrametric:
│ │ │   Produce tropical equations cutting out the ultrametrics.
│ │ │ -using temporary file /tmp/M2-16949-0/206
│ │ │ - -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-16949-0/208
│ │ │ +using temporary file /tmp/M2-22465-0/206
│ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-22465-0/208
│ │ │  This program computes the mixed volume of the Newton polytopes of a list of polynomials. The ring is specified on the input. After this follows the list of polynomials.
│ │ │  Options:
│ │ │  --vectorinput:
│ │ │   Read in a list of point configurations instead of a polynomial ring and a list of polynomials.
│ │ │  --cyclic value:
│ │ │   Use cyclic-n example instead of reading input.
│ │ │  --noon value:
│ │ │ @@ -249,44 +249,44 @@
│ │ │   Use Katsura-n example instead of reading input.
│ │ │  --gaukwa value:
│ │ │   Use Gaukwa-n example instead of reading input.
│ │ │  --eco value:
│ │ │   Use Eco-n example instead of reading input.
│ │ │  -j value:
│ │ │   Number of threads
│ │ │ -using temporary file /tmp/M2-16949-0/208
│ │ │ - -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-16949-0/210
│ │ │ +using temporary file /tmp/M2-22465-0/208
│ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-22465-0/210
│ │ │  This program computes the union of a list of polynomial sets given as input. The polynomials must all belong to the same ring. The ring is specified on the input. After this follows the list of polynomial sets.
│ │ │  Options:
│ │ │  -s:
│ │ │   Sort output by degree.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16949-0/210
│ │ │ - -- running: /usr/bin/gfan _render --help < /tmp/M2-16949-0/212
│ │ │ +using temporary file /tmp/M2-22465-0/210
│ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-22465-0/212
│ │ │  This program renders a Groebner fan as an xfig file. To be more precise, the input is the list of all reduced Groebner bases of an ideal. The output is a drawing of the Groebner fan intersected with a triangle. The corners of the triangle are (1,0,0) to the right, (0,1,0) to the left and (0,0,1) at the top. If there are more than three variables in the ring these coordinates are extended with zeros. It is possible to shift the 1 entry cyclic with the option --shiftVariables.
│ │ │  Options:
│ │ │  -L:
│ │ │   Make the triangle larger so that the shape of the Groebner region appears.
│ │ │  --shiftVariables value:
│ │ │   Shift the positions of the variables in the drawing. For example with the value equal to 1 the corners will be right: (0,1,0,0,...), left: (0,0,1,0,...) and top: (0,0,0,1,...). The shifting is done modulo the number of variables in the polynomial ring. The default value is 0.
│ │ │ -using temporary file /tmp/M2-16949-0/212
│ │ │ - -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-16949-0/214
│ │ │ +using temporary file /tmp/M2-22465-0/212
│ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-22465-0/214
│ │ │  This program renders a staircase diagram of a monomial initial ideal to an xfig file. The input is a Groebner basis of a (not necessarily monomial) polynomial ideal. The initial ideal is given by the leading terms in the Groebner basis. Using the -m option it is possible to render more than one staircase diagram. The program only works for ideals in a polynomial ring with three variables.
│ │ │  Options:
│ │ │  -m:
│ │ │   Read multiple ideals from the input. The ideals are given as a list of lists of polynomials. For each polynomial list in the list a staircase diagram is drawn.
│ │ │  
│ │ │  -d value:
│ │ │   Specifies the number of boxes being shown along each axis. Be sure that this number is large enough to give a correct picture of the standard monomials. The default value is 8.
│ │ │  
│ │ │  -w value:
│ │ │   Width. Specifies the number of staircase diagrams per row in the xfig file. The default value is 5.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16949-0/214
│ │ │ - -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-16949-0/216
│ │ │ +using temporary file /tmp/M2-22465-0/214
│ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-22465-0/216
│ │ │  This program computes the resultant fan as defined in "Computing Tropical Resultants" by Jensen and Yu. The input is a polynomial ring followed by polynomials, whose coefficients are ignored. The output is the fan of coefficients such that the input system has a tropical solution.
│ │ │  Options:
│ │ │  --codimension:
│ │ │   Compute only the codimension of the resultant fan and return.
│ │ │  
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the vector configuration. The program DOES NOT checks that the configuration stays fixed when permuting the variables with respect to elements in the group. The output is grouped according to the symmetry.
│ │ │ @@ -299,25 +299,25 @@
│ │ │  
│ │ │  --vectorinput:
│ │ │   Read in a list of point configurations instead of a polynomial ring and a list of polynomials.
│ │ │  
│ │ │  --projection:
│ │ │   Use the projection method to compute the resultant fan. This works only if the resultant fan is a hypersurface. If this option is combined with --special, then the output fan lives in the subspace of the non-specialized coordinates.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16949-0/216
│ │ │ - -- running: /usr/bin/gfan _saturation --help < /tmp/M2-16949-0/218
│ │ │ +using temporary file /tmp/M2-22465-0/216
│ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-22465-0/218
│ │ │  This program computes the saturation of the input ideal with the product of the variables x_1,...,x_n. The ideal does not have to be homogeneous.
│ │ │  Options:
│ │ │  -h:
│ │ │   Tell the program that the input is a homogeneous ideal (with homogeneous generators).
│ │ │  
│ │ │  --noideal:
│ │ │   Do not treat input as an ideal but just factor out common monomial factors of the input polynomials.
│ │ │ -using temporary file /tmp/M2-16949-0/218
│ │ │ - -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-16949-0/220
│ │ │ +using temporary file /tmp/M2-22465-0/218
│ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-22465-0/220
│ │ │  This program computes the secondary fan of a vector configuration. The configuration is given as an ordered list of vectors. In order to compute the secondary fan of a point configuration an additional coordinate of ones must be added. For example {(1,0),(1,1),(1,2),(1,3)}.
│ │ │  Options:
│ │ │  --unimodular:
│ │ │   Use heuristics to search for unimodular triangulation rather than computing the complete secondary fan
│ │ │  --scale value:
│ │ │   Assuming that the first coordinate of each vector is 1, this option will take the polytope in the 1 plane and scale it. The point configuration will be all lattice points in that scaled polytope. The polytope must have maximal dimension. When this option is used the vector configuration must have full rank. This option may be removed in the future.
│ │ │  --restrictingfan value:
│ │ │ @@ -326,70 +326,70 @@
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the vector configuration. The program checks that the configuration stays fixed when permuting the variables with respect to elements in the group. The output is grouped according to the symmetry.
│ │ │  
│ │ │  --nocones:
│ │ │   Tells the program not to output the CONES and MAXIMAL_CONES sections, but still output CONES_COMPRESSED and MAXIMAL_CONES_COMPRESSED if --symmetry is used.
│ │ │  --interrupt value:
│ │ │   Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans).
│ │ │ -using temporary file /tmp/M2-16949-0/220
│ │ │ - -- running: /usr/bin/gfan _stats --help < /tmp/M2-16949-0/222
│ │ │ +using temporary file /tmp/M2-22465-0/220
│ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-22465-0/222
│ │ │  This program takes a list of reduced Groebner bases for the same ideal and computes various statistics. The following information is listed: the number of bases in the input, the number of variables, the dimension of the homogeneity space, the maximal total degree of any polynomial in the input and the minimal total degree of any basis in the input, the maximal number of polynomials and terms in a basis in the input.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16949-0/222
│ │ │ - -- running: /usr/bin/gfan _substitute --help < /tmp/M2-16949-0/224
│ │ │ +using temporary file /tmp/M2-22465-0/222
│ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-22465-0/224
│ │ │  This program changes the variable names of a polynomial ring. The input is a polynomial ring, a polynomial set in the ring and a new polynomial ring with the same coefficient field but different variable names. The output is the polynomial set written with the variable names of the second polynomial ring.
│ │ │  Example:
│ │ │  Input:
│ │ │  Q[a,b,c,d]{2a-3b,c+d}Q[b,a,c,x]
│ │ │  Output:
│ │ │  Q[b,a,c,x]{2*b-3*a,c+x}
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16949-0/224
│ │ │ - -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-16949-0/226
│ │ │ +using temporary file /tmp/M2-22465-0/224
│ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-22465-0/226
│ │ │  This program converts ASCII math to TeX math. The data-type is specified by the options.
│ │ │  Options:
│ │ │  -h:
│ │ │   Add a header to the output. Using this option the output will be LaTeXable right away.
│ │ │  --polynomialset_:
│ │ │   The data to be converted is a list of polynomials.
│ │ │  --polynomialsetlist_:
│ │ │   The data to be converted is a list of lists of polynomials.
│ │ │ -using temporary file /tmp/M2-16949-0/226
│ │ │ - -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-16949-0/228
│ │ │ +using temporary file /tmp/M2-22465-0/226
│ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-22465-0/228
│ │ │  This program takes a list of reduced Groebner bases and produces the fan of all faces of these. In this way by giving the complete list of reduced Groebner bases, the Groebner fan can be computed as a polyhedral complex. The option --restrict lets the user choose between computing the Groebner fan or the restricted Groebner fan.
│ │ │  Options:
│ │ │  --restrict:
│ │ │   Add an inequality for each coordinate, so that the the cones are restricted to the non-negative orthant.
│ │ │  --symmetry:
│ │ │   Tell the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ring. The output is grouped according to these symmetries. Only one representative for each orbit is needed on the input.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16949-0/228
│ │ │ - -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-16949-0/230
│ │ │ +using temporary file /tmp/M2-22465-0/228
│ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-22465-0/230
│ │ │  This program computes a tropical basis for an ideal defining a tropical curve. Defining a tropical curve means that the Krull dimension of R/I is at most 1 + the dimension of the homogeneity space of I where R is the polynomial ring. The input is a generating set for the ideal. If the input is not homogeneous option -h must be used.
│ │ │  Options:
│ │ │  -h:
│ │ │   Homogenise the input before computing a tropical basis and dehomogenise the output. This is needed if the input generators are not already homogeneous.
│ │ │ -using temporary file /tmp/M2-16949-0/230
│ │ │ - -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-16949-0/232
│ │ │ +using temporary file /tmp/M2-22465-0/230
│ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-22465-0/232
│ │ │  This program takes a marked reduced Groebner basis for a homogeneous ideal and computes the tropical variety of the ideal as a subfan of the Groebner fan. The program is slow but works for any homogeneous ideal. If you know that your ideal is prime over the complex numbers or you simply know that its tropical variety is pure and connected in codimension one then use gfan_tropicalstartingcone and gfan_tropicaltraverse instead.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16949-0/232
│ │ │ - -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-16949-0/234
│ │ │ +using temporary file /tmp/M2-22465-0/232
│ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-22465-0/234
│ │ │  This program evaluates a tropical polynomial function in a given set of points.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16949-0/234
│ │ │ - -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-16949-0/236
│ │ │ +using temporary file /tmp/M2-22465-0/234
│ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-22465-0/236
│ │ │  This program takes a polynomial and tropicalizes it. The output is piecewise linear function represented by a fan whose cones are the linear regions. Each ray of the fan gets the value of the tropical function assigned to it. In other words this program computes the normal fan of the Newton polytope of the input polynomial with additional information.Options:
│ │ │  --exponents:
│ │ │   Tell program to read a list of exponent vectors instead.
│ │ │ -using temporary file /tmp/M2-16949-0/236
│ │ │ - -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-16949-0/238
│ │ │ +using temporary file /tmp/M2-22465-0/236
│ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-22465-0/238
│ │ │  This program computes the tropical hypersurface defined by a principal ideal. The input is the polynomial ring followed by a set containing just a generator of the ideal.Options:
│ │ │ -using temporary file /tmp/M2-16949-0/238
│ │ │ - -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-16949-0/240
│ │ │ +using temporary file /tmp/M2-22465-0/238
│ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-22465-0/240
│ │ │  This program computes the set theoretical intersection of a set of tropical hypersurfaces (or to be precise, their common refinement as a fan). The input is a list of polynomials with each polynomial defining a hypersurface. Considering tropical hypersurfaces as fans, the intersection can be computed as the common refinement of these. Thus the output is a fan whose support is the intersection of the tropical hypersurfaces.
│ │ │  Options:
│ │ │  --tropicalbasistest:
│ │ │   This option will test that the input polynomials for a tropical basis of the ideal they generate by computing the tropical prevariety of the input polynomials and then refine each cone with the Groebner fan and testing whether each cone in the refinement has an associated monomial free initial ideal. If so, then we have a tropical basis and 1 is written as output. If not, then a zero is written to the output together with a vector in the tropical prevariety but not in the variety. The actual check is done on a homogenization of the input ideal, but this does not affect the result. (This option replaces the -t option from earlier gfan versions.)
│ │ │  
│ │ │  --tplane:
│ │ │   This option intersects the resulting fan with the plane x_0=-1, where x_0 is the first variable. To simplify the implementation the output is actually the common refinement with the non-negative half space. This means that "stuff at infinity" (where x_0=0) is not removed.
│ │ │ @@ -401,16 +401,16 @@
│ │ │   Tells the program not to output the CONES and MAXIMAL_CONES sections, but still output CONES_COMPRESSED and MAXIMAL_CONES_COMPRESSED if --symmetry is used.
│ │ │  --restrict:
│ │ │   Restrict the computation to a full-dimensional cone given by a list of marked polynomials. The cone is the closure of all weight vectors choosing these marked terms.
│ │ │  --stable:
│ │ │   Find the stable intersection of the input polynomials using tropical intersection theory. This can be slow. Most other options are ignored.
│ │ │  --parameters value:
│ │ │   With this option you can specify how many variables to treat as parameters instead of variables. This makes it possible to do computations where the coefficient field is the field of rational functions in the parameters.
│ │ │ -using temporary file /tmp/M2-16949-0/240
│ │ │ - -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-16949-0/242
│ │ │ +using temporary file /tmp/M2-22465-0/240
│ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-22465-0/242
│ │ │  This program is part of the Puiseux lifting algorithm implemented in Gfan and Singular. The Singular part of the implementation can be found in:
│ │ │  
│ │ │  Anders Nedergaard Jensen, Hannah Markwig, Thomas Markwig:
│ │ │   tropical.lib. A SINGULAR 3.0 library for computations in tropical geometry, 2007 
│ │ │  
│ │ │  See also
│ │ │  
│ │ │ @@ -435,48 +435,48 @@
│ │ │  Options:
│ │ │  --noMult:
│ │ │   Disable the multiplicity computation.
│ │ │  -n value:
│ │ │   Number of variables that should have negative weight.
│ │ │  -c:
│ │ │   Only output a list of vectors being the possible choices.
│ │ │ -using temporary file /tmp/M2-16949-0/242
│ │ │ - -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-16949-0/244
│ │ │ +using temporary file /tmp/M2-22465-0/242
│ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-22465-0/244
│ │ │  This program generates tropical equations for a tropical linear space in the Speyer sense given the tropical Pluecker coordinates as input.
│ │ │  Options:
│ │ │  -d value:
│ │ │   Specify d.
│ │ │  -n value:
│ │ │   Specify n.
│ │ │  --trees:
│ │ │   list the boundary trees (assumes d=3)
│ │ │ -using temporary file /tmp/M2-16949-0/244
│ │ │ - -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-16949-0/246
│ │ │ +using temporary file /tmp/M2-22465-0/244
│ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-22465-0/246
│ │ │  This program computes the multiplicity of a tropical cone given a marked reduced Groebner basis for its initial ideal.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16949-0/246
│ │ │ - -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-16949-0/248
│ │ │ +using temporary file /tmp/M2-22465-0/246
│ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-22465-0/248
│ │ │  This program will compute the tropical rank of matrix given as input. Tropical addition is MAXIMUM.
│ │ │  Options:
│ │ │  --kapranov:
│ │ │   Compute Kapranov rank instead of tropical rank.
│ │ │  --determinant:
│ │ │   Compute the tropical determinant instead.
│ │ │ -using temporary file /tmp/M2-16949-0/248
│ │ │ - -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-16949-0/250
│ │ │ +using temporary file /tmp/M2-22465-0/248
│ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-22465-0/250
│ │ │  This program computes a starting pair of marked reduced Groebner bases to be used as input for gfan_tropicaltraverse. The input is a homogeneous ideal whose tropical variety is a pure d-dimensional polyhedral complex.
│ │ │  Options:
│ │ │  -g:
│ │ │   Tell the program that the input is already a reduced Groebner basis.
│ │ │  -d:
│ │ │   Output dimension information to standard error.
│ │ │  --stable:
│ │ │   Find starting cone in the stable intersection or, equivalently, pretend that the coefficients are genereric.
│ │ │ -using temporary file /tmp/M2-16949-0/250
│ │ │ - -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-16949-0/252
│ │ │ +using temporary file /tmp/M2-22465-0/250
│ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-22465-0/252
│ │ │  This program computes a polyhedral fan representation of the tropical variety of a homogeneous prime ideal $I$. Let $d$ be the Krull dimension of $I$ and let $\omega$ be a relative interior point of $d$-dimensional Groebner cone contained in the tropical variety. The input for this program is a pair of marked reduced Groebner bases with respect to the term order represented by $\omega$, tie-broken in some way. The first one is for the initial ideal $in_\omega(I)$ the second one for $I$ itself. The pair is the starting point for a traversal of the $d$-dimensional Groebner cones contained in the tropical variety. If the ideal is not prime but with the tropical variety still being pure $d$-dimensional the program will only compute a codimension $1$ connected component of the tropical variety.
│ │ │  Options:
│ │ │  --symmetry:
│ │ │   Do computations up to symmetry and group the output accordingly. If this option is used the program will read in a list of generators for a symmetry group after the pair of Groebner bases have been read. Two advantages of using this option is that the output is nicely grouped and that the computation can be done faster.
│ │ │  --symsigns:
│ │ │   Specify for each generator of the symmetry group an element of ${-1,+1}^n$ which by its multiplication on the variables together with the permutation will keep the ideal fixed. The vectors are given as the rows of a matrix.
│ │ │  --nocones:
│ │ │ @@ -484,24 +484,24 @@
│ │ │  --disableSymmetryTest:
│ │ │   When using --symmetry this option will disable the check that the group read off from the input actually is a symmetry group with respect to the input ideal.
│ │ │  
│ │ │  --stable:
│ │ │   Traverse the stable intersection or, equivalently, pretend that the coefficients are genereric.
│ │ │  --interrupt value:
│ │ │   Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans).
│ │ │ -using temporary file /tmp/M2-16949-0/252
│ │ │ - -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-16949-0/254
│ │ │ +using temporary file /tmp/M2-22465-0/252
│ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-22465-0/254
│ │ │  This program computes the tropical Weil divisor of piecewise linear (or tropical rational) function on a tropical k-cycle. See the Gfan manual for more information.
│ │ │  Options:
│ │ │  -i1 value:
│ │ │   Specify the name of the Polymake input file containing the k-cycle.
│ │ │  -i2 value:
│ │ │   Specify the name of the Polymake input file containing the piecewise linear function.
│ │ │ -using temporary file /tmp/M2-16949-0/254
│ │ │ - -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-16949-0/256
│ │ │ +using temporary file /tmp/M2-22465-0/254
│ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-22465-0/256
│ │ │  This program is an experimental implementation of Groebner bases for ideals in Z[x_1,...,x_n].
│ │ │  Several operations are supported by specifying the appropriate option:
│ │ │   (1) computation of the reduced Groebner basis with respect to a given vector (tiebroken lexicographically),
│ │ │   (2) computation of an initial ideal,
│ │ │   (3) computation of the Groebner fan,
│ │ │   (4) computation of a single Groebner cone.
│ │ │  Since Gfan only knows polynomial rings with coefficients being elements of a field, the ideal is specified by giving a set of polynomials in the polynomial ring Q[x_1,...,x_n]. That is, by using Q instead of Z when specifying the ring. The ideal MUST BE HOMOGENEOUS (in a positive grading) for computation of the Groebner fan. Non-homogeneous ideals are allowed for the other computations if the specified weight vectors are positive.
│ │ │ @@ -521,21 +521,21 @@
│ │ │  --groebnerCone:
│ │ │   Asks the program to compute a single Groebner cone containing the specified vector in its relative interior. The output is stored as a fan. The input order is: Ring ideal vector.
│ │ │  -m:
│ │ │   For the operations taking a vector as input, read in a list of vectors instead, and perform the operation for each vector in the list.
│ │ │  -g:
│ │ │   Tells the program that the input is already a Groebner basis (with the initial term of each polynomial being the first ones listed). Use this option if the usual --groebnerFan is too slow.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16949-0/256
│ │ │ +using temporary file /tmp/M2-22465-0/256
│ │ │  
│ │ │  i6 : QQ[x,y];
│ │ │  
│ │ │  i7 : gfan {x,y};
│ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-16949-0/258
│ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-22465-0/258
│ │ │  Q[x1,x2]
│ │ │  {{
│ │ │  x2,
│ │ │  x1}
│ │ │  }
│ │ │ -using temporary file /tmp/M2-16949-0/258
│ │ │ +using temporary file /tmp/M2-22465-0/258
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/gfanInterface/html/___Installation_spand_sp__Configuration_spof_spgfan__Interface.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │            
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : loadPackage("gfanInterface", Configuration => { "keepfiles" => true, "verbose" => true}, Reload => true);
│ │ │   -- warning: reloading gfanInterface; recreate instances of types from this package
│ │ │ - -- running: /usr/bin/gfan gfan --help < /tmp/M2-16949-0/172
│ │ │ + -- running: /usr/bin/gfan gfan --help < /tmp/M2-22465-0/172
│ │ │  This is a program for computing all reduced Groebner bases of a polynomial ideal. It takes the ring and a generating set for the ideal as input. By default the enumeration is done by an almost memoryless reverse search. If the ideal is symmetric the symmetry option is useful and enumeration will be done up to symmetry using a breadth first search. The program needs a starting Groebner basis to do its computations. If the -g option is not specified it will compute one using Buchberger's algorithm.
│ │ │  Options:
│ │ │  -g:
│ │ │   Tells the program that the input is already a Groebner basis (with the initial term of each polynomial being the first ones listed). Use this option if it takes too much time to compute the starting (standard degree lexicographic) Groebner basis and the input is already a Groebner basis.
│ │ │  
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ideal. The program checks that the ideal stays fixed when permuting the variables with respect to elements in the group. The program uses breadth first search to compute the set of reduced Groebner bases up to symmetry with respect to the specified subgroup.
│ │ │ @@ -128,16 +128,16 @@
│ │ │  --disableSymmetryTest:
│ │ │   When using --symmetry this option will disable the check that the group read off from the input actually is a symmetry group with respect to the input ideal.
│ │ │  
│ │ │  --parameters value:
│ │ │   With this option you can specify how many variables to treat as parameters instead of variables. This makes it possible to do computations where the coefficient field is the field of rational functions in the parameters.
│ │ │  --interrupt value:
│ │ │   Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans).
│ │ │ -using temporary file /tmp/M2-16949-0/172
│ │ │ - -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-16949-0/174
│ │ │ +using temporary file /tmp/M2-22465-0/172
│ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-22465-0/174
│ │ │  This program computes a reduced lexicographic Groebner basis of the polynomial ideal given as input. The default behavior is to use Buchberger's algorithm. The ordering of the variables is $a>b>c...$ (assuming that the ring is Q[a,b,c,...]).
│ │ │  Options:
│ │ │  -w:
│ │ │   Compute a Groebner basis with respect to a degree lexicographic order with $a>b>c...$ instead. The degrees are given by a weight vector which is read from the input after the generating set has been read.
│ │ │  
│ │ │  -r:
│ │ │   Use the reverse lexicographic order (or the reverse lexicographic order as a tie breaker if -w is used). The input must be homogeneous if the pure reverse lexicographic order is chosen. Ignored if -W is used.
│ │ │ @@ -146,69 +146,69 @@
│ │ │   Do a Groebner walk. The input must be a minimal Groebner basis. If -W is used -w is ignored.
│ │ │  
│ │ │  -g:
│ │ │   Do a generic Groebner walk. The input must be homogeneous and must be a minimal Groebner basis with respect to the reverse lexicographic term order. The target term order is always lexicographic. The -W option must be used.
│ │ │  
│ │ │  --parameters value:
│ │ │   With this option you can specify how many variables to treat as parameters instead of variables. This makes it possible to do computations where the coefficient field is the field of rational functions in the parameters.
│ │ │ -using temporary file /tmp/M2-16949-0/174
│ │ │ - -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-16949-0/176
│ │ │ +using temporary file /tmp/M2-22465-0/174
│ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-22465-0/176
│ │ │  This program takes a marked Groebner basis of an ideal I and a set of polynomials on its input and tests if the polynomial set is contained in I by applying the division algorithm for each element. The output is 1 for true and 0 for false.
│ │ │  Options:
│ │ │  --remainder:
│ │ │   Tell the program to output the remainders of the divisions rather than outputting 0 or 1.
│ │ │  --multiplier:
│ │ │   Reads in a polynomial that will be multiplied to the polynomial to be divided before doing the division.
│ │ │ -using temporary file /tmp/M2-16949-0/176
│ │ │ - -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-16949-0/178
│ │ │ +using temporary file /tmp/M2-22465-0/176
│ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-22465-0/178
│ │ │  This program takes two polyhedral fans and computes their common refinement.
│ │ │  Options:
│ │ │  -i1 value:
│ │ │   Specify the name of the first input file.
│ │ │  -i2 value:
│ │ │   Specify the name of the second input file.
│ │ │  --stable:
│ │ │   Compute the stable intersection.
│ │ │ -using temporary file /tmp/M2-16949-0/178
│ │ │ - -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-16949-0/180
│ │ │ +using temporary file /tmp/M2-22465-0/178
│ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-22465-0/180
│ │ │  This program takes a polyhedral fan and a vector and computes the link of the polyhedral fan around that vertex. The link will have lineality space dimension equal to the dimension of the relative open polyhedral cone of the original fan containing the vector.
│ │ │  Options:
│ │ │  -i value:
│ │ │   Specify the name of the input file.
│ │ │  --symmetry:
│ │ │   Reads in a fan stored with symmetry. The generators of the symmetry group must be given on the standard input.
│ │ │  
│ │ │  --star:
│ │ │   Computes the star instead. The star is defined as the smallest polyhedral fan containing all cones of the original fan containing the vector.
│ │ │ -using temporary file /tmp/M2-16949-0/180
│ │ │ - -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-16949-0/182
│ │ │ +using temporary file /tmp/M2-22465-0/180
│ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-22465-0/182
│ │ │  This program takes two polyhedral fans and computes their product.
│ │ │  Options:
│ │ │  -i1 value:
│ │ │   Specify the name of the first input file.
│ │ │  -i2 value:
│ │ │   Specify the name of the second input file.
│ │ │ -using temporary file /tmp/M2-16949-0/182
│ │ │ - -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-16949-0/184
│ │ │ +using temporary file /tmp/M2-22465-0/182
│ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-22465-0/184
│ │ │  This program computes a Groebner cone. Three different cases are handled. The input may be a marked reduced Groebner basis in which case its Groebner cone is computed. The input may be just a marked minimal basis in which case the cone computed is not a Groebner cone in the usual sense but smaller. (These cones are described in [Fukuda, Jensen, Lauritzen, Thomas]). The third possible case is that the Groebner cone is possibly lower dimensional and given by a pair of Groebner bases as it is useful to do for tropical varieties, see option --pair. The facets of the cone can be read off in section FACETS and the equations in section IMPLIED_EQUATIONS.
│ │ │  Options:
│ │ │  --restrict:
│ │ │   Add an inequality for each coordinate, so that the the cone is restricted to the non-negative orthant.
│ │ │  --pair:
│ │ │   The Groebner cone is given by a pair of compatible Groebner bases. The first basis is for the initial ideal and the second for the ideal itself. See the tropical section of the manual.
│ │ │  --asfan:
│ │ │   Writes the cone as a polyhedral fan with all its faces instead. In this way the extreme rays of the cone are also computed.
│ │ │  --vectorinput:
│ │ │   Compute a cone given list of inequalities rather than a Groebner cone. The input is an integer which specifies the dimension of the ambient space, a list of inequalities given as vectors and a list of equations.
│ │ │ -using temporary file /tmp/M2-16949-0/184
│ │ │ - -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-16949-0/186
│ │ │ +using temporary file /tmp/M2-22465-0/184
│ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-22465-0/186
│ │ │  This program computes the homogeneity space of a list of polynomials - as a cone. Thus generators for the homogeneity space are found in the section LINEALITY_SPACE. If you wish the homogeneity space of an ideal you should first compute a set of homogeneous generators and call the program on these. A reduced Groebner basis will always suffice for this purpose.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16949-0/186
│ │ │ - -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-16949-0/188
│ │ │ +using temporary file /tmp/M2-22465-0/186
│ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-22465-0/188
│ │ │  This program homogenises a list of polynomials by introducing an extra variable. The name of the variable to be introduced is read from the input after the list of polynomials. Without the -w option the homogenisation is done with respect to total degree.
│ │ │  Example:
│ │ │  Input:
│ │ │  Q[x,y]{y-1}
│ │ │  z
│ │ │  Output:
│ │ │  Q[x,y,z]{y-z}
│ │ │ @@ -216,30 +216,30 @@
│ │ │  -i:
│ │ │   Treat input as an ideal. This will make the program compute the homogenisation of the input ideal. This is done by computing a degree Groebner basis and homogenising it.
│ │ │  -w:
│ │ │   Specify a homogenisation vector. The length of the vector must be the same as the number of variables in the ring. The vector is read from the input after the list of polynomials.
│ │ │  
│ │ │  -H:
│ │ │   Let the name of the new variable be H rather than reading in a name from the input.
│ │ │ -using temporary file /tmp/M2-16949-0/188
│ │ │ - -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-16949-0/190
│ │ │ +using temporary file /tmp/M2-22465-0/188
│ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-22465-0/190
│ │ │  This program converts a list of polynomials to a list of their initial forms with respect to the vector given after the list.
│ │ │  Options:
│ │ │  --ideal:
│ │ │   Treat input as an ideal. This will make the program compute the initial ideal of the ideal generated by the input polynomials. The computation is done by computing a Groebner basis with respect to the given vector. The vector must be positive or the input polynomials must be homogeneous in a positive grading. None of these conditions are checked by the program.
│ │ │  
│ │ │  --pair:
│ │ │   Produce a pair of polynomial lists. Used together with --ideal this option will also write a compatible reduced Groebner basis for the input ideal to the output. This is useful for finding the Groebner cone of a non-monomial initial ideal.
│ │ │  
│ │ │  --mark:
│ │ │   If the --pair option is and the --ideal option is not used this option will still make sure that the second output basis is marked consistently with the vector.
│ │ │  --list:
│ │ │   Read in a list of vectors instead of a single vector and produce a list of polynomial sets as output.
│ │ │ -using temporary file /tmp/M2-16949-0/190
│ │ │ - -- running: /usr/bin/gfan _interactive --help < /tmp/M2-16949-0/192
│ │ │ +using temporary file /tmp/M2-22465-0/190
│ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-22465-0/192
│ │ │  This is a program for doing interactive walks in the Groebner fan of an ideal. The input is a Groebner basis defining the starting Groebner cone of the walk. The program will list all flippable facets of the Groebner cone and ask the user to choose one. The user types in the index (number) of the facet in the list. The program will walk through the selected facet and display the new Groebner basis and a list of new facet normals for the user to choose from. Since the program reads the user's choices through the the standard input it is recommended not to redirect the standard input for this program.
│ │ │  Options:
│ │ │  -L:
│ │ │   Latex mode. The program will try to show the current Groebner basis in a readable form by invoking LaTeX and xdvi.
│ │ │  
│ │ │  -x:
│ │ │   Exit immediately.
│ │ │ @@ -254,57 +254,57 @@
│ │ │   Tell the program to list the defining set of inequalities of the non-restricted Groebner cone as a set of vectors after having listed the current Groebner basis.
│ │ │  
│ │ │  -W:
│ │ │   Print weight vector. This will make the program print an interior vector of the current Groebner cone and a relative interior point for each flippable facet of the current Groebner cone.
│ │ │  
│ │ │  --tropical:
│ │ │   Traverse a tropical variety interactively.
│ │ │ -using temporary file /tmp/M2-16949-0/192
│ │ │ - -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-16949-0/194
│ │ │ +using temporary file /tmp/M2-22465-0/192
│ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-22465-0/194
│ │ │  This program checks if a set of marked polynomials is a Groebner basis with respect to its marking. First it is checked if the markings are consistent with respect to a positive vector. Then Buchberger's S-criterion is checked. The output is boolean value.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16949-0/194
│ │ │ - -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-16949-0/196
│ │ │ +using temporary file /tmp/M2-22465-0/194
│ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-22465-0/196
│ │ │  Takes an ideal $I$ and computes the Krull dimension of R/I where R is the polynomial ring. This is done by first computing a Groebner basis.
│ │ │  Options:
│ │ │  -g:
│ │ │   Tell the program that the input is already a reduced Groebner basis.
│ │ │ -using temporary file /tmp/M2-16949-0/196
│ │ │ - -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-16949-0/198
│ │ │ +using temporary file /tmp/M2-22465-0/196
│ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-22465-0/198
│ │ │  This program computes the lattice ideal of a lattice. The input is a list of generators for the lattice.
│ │ │  Options:
│ │ │  -t:
│ │ │   Compute the toric ideal of the matrix whose rows are given on the input instead.
│ │ │  --convert:
│ │ │   Does not do any computation, but just converts the vectors to binomials.
│ │ │ -using temporary file /tmp/M2-16949-0/198
│ │ │ - -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-16949-0/200
│ │ │ +using temporary file /tmp/M2-22465-0/198
│ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-22465-0/200
│ │ │  This program converts a list of polynomials to a list of their leading terms.
│ │ │  Options:
│ │ │  -m:
│ │ │   Do the same thing for a list of polynomial sets. That is, output the set of sets of leading terms.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16949-0/200
│ │ │ - -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-16949-0/202
│ │ │ +using temporary file /tmp/M2-22465-0/200
│ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-22465-0/202
│ │ │  This program marks a set of polynomials with respect to the vector given at the end of the input, meaning that the largest terms are moved to the front. In case of a tie the lexicographic term order with $a>b>c...$ is used to break it.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16949-0/202
│ │ │ - -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-16949-0/204
│ │ │ +using temporary file /tmp/M2-22465-0/202
│ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-22465-0/204
│ │ │  This is a program for computing the normal fan of the Minkowski sum of the Newton polytopes of a list of polynomials.
│ │ │  Options:
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ideal. The program checks that the ideal stays fixed when permuting the variables with respect to elements in the group. The program uses breadth first search to compute the set of reduced Groebner bases up to symmetry with respect to the specified subgroup.
│ │ │  
│ │ │  --disableSymmetryTest:
│ │ │   When using --symmetry this option will disable the check that the group read off from the input actually is a symmetry group with respect to the input ideal.
│ │ │  
│ │ │  --nocones:
│ │ │   Tell the program to not list cones in the output.
│ │ │ -using temporary file /tmp/M2-16949-0/204
│ │ │ - -- running: /usr/bin/gfan _minors --help < /tmp/M2-16949-0/206
│ │ │ +using temporary file /tmp/M2-22465-0/204
│ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-22465-0/206
│ │ │  This program will generate the r*r minors of a d*n matrix of indeterminates.
│ │ │  Options:
│ │ │  -r value:
│ │ │   Specify r.
│ │ │  -d value:
│ │ │   Specify d.
│ │ │  -n value:
│ │ │ @@ -319,16 +319,16 @@
│ │ │   Do nothing but produce symmetry generators for the Pluecker ideal.
│ │ │  --symmetry:
│ │ │   Produces a list of generators for the group of symmetries keeping the set of minors fixed. (Only without --names).
│ │ │  --parametrize:
│ │ │   Parametrize the set of d times n matrices of Barvinok rank less than or equal to r-1 by a list of tropical polynomials.
│ │ │  --ultrametric:
│ │ │   Produce tropical equations cutting out the ultrametrics.
│ │ │ -using temporary file /tmp/M2-16949-0/206
│ │ │ - -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-16949-0/208
│ │ │ +using temporary file /tmp/M2-22465-0/206
│ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-22465-0/208
│ │ │  This program computes the mixed volume of the Newton polytopes of a list of polynomials. The ring is specified on the input. After this follows the list of polynomials.
│ │ │  Options:
│ │ │  --vectorinput:
│ │ │   Read in a list of point configurations instead of a polynomial ring and a list of polynomials.
│ │ │  --cyclic value:
│ │ │   Use cyclic-n example instead of reading input.
│ │ │  --noon value:
│ │ │ @@ -339,44 +339,44 @@
│ │ │   Use Katsura-n example instead of reading input.
│ │ │  --gaukwa value:
│ │ │   Use Gaukwa-n example instead of reading input.
│ │ │  --eco value:
│ │ │   Use Eco-n example instead of reading input.
│ │ │  -j value:
│ │ │   Number of threads
│ │ │ -using temporary file /tmp/M2-16949-0/208
│ │ │ - -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-16949-0/210
│ │ │ +using temporary file /tmp/M2-22465-0/208
│ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-22465-0/210
│ │ │  This program computes the union of a list of polynomial sets given as input. The polynomials must all belong to the same ring. The ring is specified on the input. After this follows the list of polynomial sets.
│ │ │  Options:
│ │ │  -s:
│ │ │   Sort output by degree.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16949-0/210
│ │ │ - -- running: /usr/bin/gfan _render --help < /tmp/M2-16949-0/212
│ │ │ +using temporary file /tmp/M2-22465-0/210
│ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-22465-0/212
│ │ │  This program renders a Groebner fan as an xfig file. To be more precise, the input is the list of all reduced Groebner bases of an ideal. The output is a drawing of the Groebner fan intersected with a triangle. The corners of the triangle are (1,0,0) to the right, (0,1,0) to the left and (0,0,1) at the top. If there are more than three variables in the ring these coordinates are extended with zeros. It is possible to shift the 1 entry cyclic with the option --shiftVariables.
│ │ │  Options:
│ │ │  -L:
│ │ │   Make the triangle larger so that the shape of the Groebner region appears.
│ │ │  --shiftVariables value:
│ │ │   Shift the positions of the variables in the drawing. For example with the value equal to 1 the corners will be right: (0,1,0,0,...), left: (0,0,1,0,...) and top: (0,0,0,1,...). The shifting is done modulo the number of variables in the polynomial ring. The default value is 0.
│ │ │ -using temporary file /tmp/M2-16949-0/212
│ │ │ - -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-16949-0/214
│ │ │ +using temporary file /tmp/M2-22465-0/212
│ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-22465-0/214
│ │ │  This program renders a staircase diagram of a monomial initial ideal to an xfig file. The input is a Groebner basis of a (not necessarily monomial) polynomial ideal. The initial ideal is given by the leading terms in the Groebner basis. Using the -m option it is possible to render more than one staircase diagram. The program only works for ideals in a polynomial ring with three variables.
│ │ │  Options:
│ │ │  -m:
│ │ │   Read multiple ideals from the input. The ideals are given as a list of lists of polynomials. For each polynomial list in the list a staircase diagram is drawn.
│ │ │  
│ │ │  -d value:
│ │ │   Specifies the number of boxes being shown along each axis. Be sure that this number is large enough to give a correct picture of the standard monomials. The default value is 8.
│ │ │  
│ │ │  -w value:
│ │ │   Width. Specifies the number of staircase diagrams per row in the xfig file. The default value is 5.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16949-0/214
│ │ │ - -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-16949-0/216
│ │ │ +using temporary file /tmp/M2-22465-0/214
│ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-22465-0/216
│ │ │  This program computes the resultant fan as defined in "Computing Tropical Resultants" by Jensen and Yu. The input is a polynomial ring followed by polynomials, whose coefficients are ignored. The output is the fan of coefficients such that the input system has a tropical solution.
│ │ │  Options:
│ │ │  --codimension:
│ │ │   Compute only the codimension of the resultant fan and return.
│ │ │  
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the vector configuration. The program DOES NOT checks that the configuration stays fixed when permuting the variables with respect to elements in the group. The output is grouped according to the symmetry.
│ │ │ @@ -389,25 +389,25 @@
│ │ │  
│ │ │  --vectorinput:
│ │ │   Read in a list of point configurations instead of a polynomial ring and a list of polynomials.
│ │ │  
│ │ │  --projection:
│ │ │   Use the projection method to compute the resultant fan. This works only if the resultant fan is a hypersurface. If this option is combined with --special, then the output fan lives in the subspace of the non-specialized coordinates.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16949-0/216
│ │ │ - -- running: /usr/bin/gfan _saturation --help < /tmp/M2-16949-0/218
│ │ │ +using temporary file /tmp/M2-22465-0/216
│ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-22465-0/218
│ │ │  This program computes the saturation of the input ideal with the product of the variables x_1,...,x_n. The ideal does not have to be homogeneous.
│ │ │  Options:
│ │ │  -h:
│ │ │   Tell the program that the input is a homogeneous ideal (with homogeneous generators).
│ │ │  
│ │ │  --noideal:
│ │ │   Do not treat input as an ideal but just factor out common monomial factors of the input polynomials.
│ │ │ -using temporary file /tmp/M2-16949-0/218
│ │ │ - -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-16949-0/220
│ │ │ +using temporary file /tmp/M2-22465-0/218
│ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-22465-0/220
│ │ │  This program computes the secondary fan of a vector configuration. The configuration is given as an ordered list of vectors. In order to compute the secondary fan of a point configuration an additional coordinate of ones must be added. For example {(1,0),(1,1),(1,2),(1,3)}.
│ │ │  Options:
│ │ │  --unimodular:
│ │ │   Use heuristics to search for unimodular triangulation rather than computing the complete secondary fan
│ │ │  --scale value:
│ │ │   Assuming that the first coordinate of each vector is 1, this option will take the polytope in the 1 plane and scale it. The point configuration will be all lattice points in that scaled polytope. The polytope must have maximal dimension. When this option is used the vector configuration must have full rank. This option may be removed in the future.
│ │ │  --restrictingfan value:
│ │ │ @@ -416,70 +416,70 @@
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the vector configuration. The program checks that the configuration stays fixed when permuting the variables with respect to elements in the group. The output is grouped according to the symmetry.
│ │ │  
│ │ │  --nocones:
│ │ │   Tells the program not to output the CONES and MAXIMAL_CONES sections, but still output CONES_COMPRESSED and MAXIMAL_CONES_COMPRESSED if --symmetry is used.
│ │ │  --interrupt value:
│ │ │   Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans).
│ │ │ -using temporary file /tmp/M2-16949-0/220
│ │ │ - -- running: /usr/bin/gfan _stats --help < /tmp/M2-16949-0/222
│ │ │ +using temporary file /tmp/M2-22465-0/220
│ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-22465-0/222
│ │ │  This program takes a list of reduced Groebner bases for the same ideal and computes various statistics. The following information is listed: the number of bases in the input, the number of variables, the dimension of the homogeneity space, the maximal total degree of any polynomial in the input and the minimal total degree of any basis in the input, the maximal number of polynomials and terms in a basis in the input.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16949-0/222
│ │ │ - -- running: /usr/bin/gfan _substitute --help < /tmp/M2-16949-0/224
│ │ │ +using temporary file /tmp/M2-22465-0/222
│ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-22465-0/224
│ │ │  This program changes the variable names of a polynomial ring. The input is a polynomial ring, a polynomial set in the ring and a new polynomial ring with the same coefficient field but different variable names. The output is the polynomial set written with the variable names of the second polynomial ring.
│ │ │  Example:
│ │ │  Input:
│ │ │  Q[a,b,c,d]{2a-3b,c+d}Q[b,a,c,x]
│ │ │  Output:
│ │ │  Q[b,a,c,x]{2*b-3*a,c+x}
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16949-0/224
│ │ │ - -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-16949-0/226
│ │ │ +using temporary file /tmp/M2-22465-0/224
│ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-22465-0/226
│ │ │  This program converts ASCII math to TeX math. The data-type is specified by the options.
│ │ │  Options:
│ │ │  -h:
│ │ │   Add a header to the output. Using this option the output will be LaTeXable right away.
│ │ │  --polynomialset_:
│ │ │   The data to be converted is a list of polynomials.
│ │ │  --polynomialsetlist_:
│ │ │   The data to be converted is a list of lists of polynomials.
│ │ │ -using temporary file /tmp/M2-16949-0/226
│ │ │ - -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-16949-0/228
│ │ │ +using temporary file /tmp/M2-22465-0/226
│ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-22465-0/228
│ │ │  This program takes a list of reduced Groebner bases and produces the fan of all faces of these. In this way by giving the complete list of reduced Groebner bases, the Groebner fan can be computed as a polyhedral complex. The option --restrict lets the user choose between computing the Groebner fan or the restricted Groebner fan.
│ │ │  Options:
│ │ │  --restrict:
│ │ │   Add an inequality for each coordinate, so that the the cones are restricted to the non-negative orthant.
│ │ │  --symmetry:
│ │ │   Tell the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ring. The output is grouped according to these symmetries. Only one representative for each orbit is needed on the input.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16949-0/228
│ │ │ - -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-16949-0/230
│ │ │ +using temporary file /tmp/M2-22465-0/228
│ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-22465-0/230
│ │ │  This program computes a tropical basis for an ideal defining a tropical curve. Defining a tropical curve means that the Krull dimension of R/I is at most 1 + the dimension of the homogeneity space of I where R is the polynomial ring. The input is a generating set for the ideal. If the input is not homogeneous option -h must be used.
│ │ │  Options:
│ │ │  -h:
│ │ │   Homogenise the input before computing a tropical basis and dehomogenise the output. This is needed if the input generators are not already homogeneous.
│ │ │ -using temporary file /tmp/M2-16949-0/230
│ │ │ - -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-16949-0/232
│ │ │ +using temporary file /tmp/M2-22465-0/230
│ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-22465-0/232
│ │ │  This program takes a marked reduced Groebner basis for a homogeneous ideal and computes the tropical variety of the ideal as a subfan of the Groebner fan. The program is slow but works for any homogeneous ideal. If you know that your ideal is prime over the complex numbers or you simply know that its tropical variety is pure and connected in codimension one then use gfan_tropicalstartingcone and gfan_tropicaltraverse instead.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16949-0/232
│ │ │ - -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-16949-0/234
│ │ │ +using temporary file /tmp/M2-22465-0/232
│ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-22465-0/234
│ │ │  This program evaluates a tropical polynomial function in a given set of points.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16949-0/234
│ │ │ - -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-16949-0/236
│ │ │ +using temporary file /tmp/M2-22465-0/234
│ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-22465-0/236
│ │ │  This program takes a polynomial and tropicalizes it. The output is piecewise linear function represented by a fan whose cones are the linear regions. Each ray of the fan gets the value of the tropical function assigned to it. In other words this program computes the normal fan of the Newton polytope of the input polynomial with additional information.Options:
│ │ │  --exponents:
│ │ │   Tell program to read a list of exponent vectors instead.
│ │ │ -using temporary file /tmp/M2-16949-0/236
│ │ │ - -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-16949-0/238
│ │ │ +using temporary file /tmp/M2-22465-0/236
│ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-22465-0/238
│ │ │  This program computes the tropical hypersurface defined by a principal ideal. The input is the polynomial ring followed by a set containing just a generator of the ideal.Options:
│ │ │ -using temporary file /tmp/M2-16949-0/238
│ │ │ - -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-16949-0/240
│ │ │ +using temporary file /tmp/M2-22465-0/238
│ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-22465-0/240
│ │ │  This program computes the set theoretical intersection of a set of tropical hypersurfaces (or to be precise, their common refinement as a fan). The input is a list of polynomials with each polynomial defining a hypersurface. Considering tropical hypersurfaces as fans, the intersection can be computed as the common refinement of these. Thus the output is a fan whose support is the intersection of the tropical hypersurfaces.
│ │ │  Options:
│ │ │  --tropicalbasistest:
│ │ │   This option will test that the input polynomials for a tropical basis of the ideal they generate by computing the tropical prevariety of the input polynomials and then refine each cone with the Groebner fan and testing whether each cone in the refinement has an associated monomial free initial ideal. If so, then we have a tropical basis and 1 is written as output. If not, then a zero is written to the output together with a vector in the tropical prevariety but not in the variety. The actual check is done on a homogenization of the input ideal, but this does not affect the result. (This option replaces the -t option from earlier gfan versions.)
│ │ │  
│ │ │  --tplane:
│ │ │   This option intersects the resulting fan with the plane x_0=-1, where x_0 is the first variable. To simplify the implementation the output is actually the common refinement with the non-negative half space. This means that "stuff at infinity" (where x_0=0) is not removed.
│ │ │ @@ -491,16 +491,16 @@
│ │ │   Tells the program not to output the CONES and MAXIMAL_CONES sections, but still output CONES_COMPRESSED and MAXIMAL_CONES_COMPRESSED if --symmetry is used.
│ │ │  --restrict:
│ │ │   Restrict the computation to a full-dimensional cone given by a list of marked polynomials. The cone is the closure of all weight vectors choosing these marked terms.
│ │ │  --stable:
│ │ │   Find the stable intersection of the input polynomials using tropical intersection theory. This can be slow. Most other options are ignored.
│ │ │  --parameters value:
│ │ │   With this option you can specify how many variables to treat as parameters instead of variables. This makes it possible to do computations where the coefficient field is the field of rational functions in the parameters.
│ │ │ -using temporary file /tmp/M2-16949-0/240
│ │ │ - -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-16949-0/242
│ │ │ +using temporary file /tmp/M2-22465-0/240
│ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-22465-0/242
│ │ │  This program is part of the Puiseux lifting algorithm implemented in Gfan and Singular. The Singular part of the implementation can be found in:
│ │ │  
│ │ │  Anders Nedergaard Jensen, Hannah Markwig, Thomas Markwig:
│ │ │   tropical.lib. A SINGULAR 3.0 library for computations in tropical geometry, 2007 
│ │ │  
│ │ │  See also
│ │ │  
│ │ │ @@ -525,48 +525,48 @@
│ │ │  Options:
│ │ │  --noMult:
│ │ │   Disable the multiplicity computation.
│ │ │  -n value:
│ │ │   Number of variables that should have negative weight.
│ │ │  -c:
│ │ │   Only output a list of vectors being the possible choices.
│ │ │ -using temporary file /tmp/M2-16949-0/242
│ │ │ - -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-16949-0/244
│ │ │ +using temporary file /tmp/M2-22465-0/242
│ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-22465-0/244
│ │ │  This program generates tropical equations for a tropical linear space in the Speyer sense given the tropical Pluecker coordinates as input.
│ │ │  Options:
│ │ │  -d value:
│ │ │   Specify d.
│ │ │  -n value:
│ │ │   Specify n.
│ │ │  --trees:
│ │ │   list the boundary trees (assumes d=3)
│ │ │ -using temporary file /tmp/M2-16949-0/244
│ │ │ - -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-16949-0/246
│ │ │ +using temporary file /tmp/M2-22465-0/244
│ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-22465-0/246
│ │ │  This program computes the multiplicity of a tropical cone given a marked reduced Groebner basis for its initial ideal.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16949-0/246
│ │ │ - -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-16949-0/248
│ │ │ +using temporary file /tmp/M2-22465-0/246
│ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-22465-0/248
│ │ │  This program will compute the tropical rank of matrix given as input. Tropical addition is MAXIMUM.
│ │ │  Options:
│ │ │  --kapranov:
│ │ │   Compute Kapranov rank instead of tropical rank.
│ │ │  --determinant:
│ │ │   Compute the tropical determinant instead.
│ │ │ -using temporary file /tmp/M2-16949-0/248
│ │ │ - -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-16949-0/250
│ │ │ +using temporary file /tmp/M2-22465-0/248
│ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-22465-0/250
│ │ │  This program computes a starting pair of marked reduced Groebner bases to be used as input for gfan_tropicaltraverse. The input is a homogeneous ideal whose tropical variety is a pure d-dimensional polyhedral complex.
│ │ │  Options:
│ │ │  -g:
│ │ │   Tell the program that the input is already a reduced Groebner basis.
│ │ │  -d:
│ │ │   Output dimension information to standard error.
│ │ │  --stable:
│ │ │   Find starting cone in the stable intersection or, equivalently, pretend that the coefficients are genereric.
│ │ │ -using temporary file /tmp/M2-16949-0/250
│ │ │ - -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-16949-0/252
│ │ │ +using temporary file /tmp/M2-22465-0/250
│ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-22465-0/252
│ │ │  This program computes a polyhedral fan representation of the tropical variety of a homogeneous prime ideal $I$. Let $d$ be the Krull dimension of $I$ and let $\omega$ be a relative interior point of $d$-dimensional Groebner cone contained in the tropical variety. The input for this program is a pair of marked reduced Groebner bases with respect to the term order represented by $\omega$, tie-broken in some way. The first one is for the initial ideal $in_\omega(I)$ the second one for $I$ itself. The pair is the starting point for a traversal of the $d$-dimensional Groebner cones contained in the tropical variety. If the ideal is not prime but with the tropical variety still being pure $d$-dimensional the program will only compute a codimension $1$ connected component of the tropical variety.
│ │ │  Options:
│ │ │  --symmetry:
│ │ │   Do computations up to symmetry and group the output accordingly. If this option is used the program will read in a list of generators for a symmetry group after the pair of Groebner bases have been read. Two advantages of using this option is that the output is nicely grouped and that the computation can be done faster.
│ │ │  --symsigns:
│ │ │   Specify for each generator of the symmetry group an element of ${-1,+1}^n$ which by its multiplication on the variables together with the permutation will keep the ideal fixed. The vectors are given as the rows of a matrix.
│ │ │  --nocones:
│ │ │ @@ -574,24 +574,24 @@
│ │ │  --disableSymmetryTest:
│ │ │   When using --symmetry this option will disable the check that the group read off from the input actually is a symmetry group with respect to the input ideal.
│ │ │  
│ │ │  --stable:
│ │ │   Traverse the stable intersection or, equivalently, pretend that the coefficients are genereric.
│ │ │  --interrupt value:
│ │ │   Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans).
│ │ │ -using temporary file /tmp/M2-16949-0/252
│ │ │ - -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-16949-0/254
│ │ │ +using temporary file /tmp/M2-22465-0/252
│ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-22465-0/254
│ │ │  This program computes the tropical Weil divisor of piecewise linear (or tropical rational) function on a tropical k-cycle. See the Gfan manual for more information.
│ │ │  Options:
│ │ │  -i1 value:
│ │ │   Specify the name of the Polymake input file containing the k-cycle.
│ │ │  -i2 value:
│ │ │   Specify the name of the Polymake input file containing the piecewise linear function.
│ │ │ -using temporary file /tmp/M2-16949-0/254
│ │ │ - -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-16949-0/256
│ │ │ +using temporary file /tmp/M2-22465-0/254
│ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-22465-0/256
│ │ │  This program is an experimental implementation of Groebner bases for ideals in Z[x_1,...,x_n].
│ │ │  Several operations are supported by specifying the appropriate option:
│ │ │   (1) computation of the reduced Groebner basis with respect to a given vector (tiebroken lexicographically),
│ │ │   (2) computation of an initial ideal,
│ │ │   (3) computation of the Groebner fan,
│ │ │   (4) computation of a single Groebner cone.
│ │ │  Since Gfan only knows polynomial rings with coefficients being elements of a field, the ideal is specified by giving a set of polynomials in the polynomial ring Q[x_1,...,x_n]. That is, by using Q instead of Z when specifying the ring. The ideal MUST BE HOMOGENEOUS (in a positive grading) for computation of the Groebner fan. Non-homogeneous ideals are allowed for the other computations if the specified weight vectors are positive.
│ │ │ @@ -611,32 +611,32 @@
│ │ │  --groebnerCone:
│ │ │   Asks the program to compute a single Groebner cone containing the specified vector in its relative interior. The output is stored as a fan. The input order is: Ring ideal vector.
│ │ │  -m:
│ │ │   For the operations taking a vector as input, read in a list of vectors instead, and perform the operation for each vector in the list.
│ │ │  -g:
│ │ │   Tells the program that the input is already a Groebner basis (with the initial term of each polynomial being the first ones listed). Use this option if the usual --groebnerFan is too slow.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16949-0/256
    
│ │ │ +using temporary file /tmp/M2-22465-0/256
│ │ │              
    
│ │ │                
    i6 : QQ[x,y];
    
│ │ │              
    
│ │ │                
    i7 : gfan {x,y};
│ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-16949-0/258
│ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-22465-0/258
│ │ │  Q[x1,x2]
│ │ │  {{
│ │ │  x2,
│ │ │  x1}
│ │ │  }
│ │ │ -using temporary file /tmp/M2-16949-0/258
    
│ │ │ +using temporary file /tmp/M2-22465-0/258
│ │ │              
    
│ │ │          
    
│ │ │            
    Finally, if you want to be able to render Groebner fans and monomial staircases to .png files, you should install fig2dev.  If it is installed in a non-standard location, then you may specify its path using programPaths.
    
│ │ │          
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -43,15 +43,15 @@
│ │ │ │  If you would like to see the input and output files used to communicate with
│ │ │ │  gfan you can set the "keepfiles" configuration option to true. If "verbose" is
│ │ │ │  set to true, gfanInterface will output the names of the temporary files used.
│ │ │ │  i5 : loadPackage("gfanInterface", Configuration => { "keepfiles" => true,
│ │ │ │  "verbose" => true}, Reload => true);
│ │ │ │   -- warning: reloading gfanInterface; recreate instances of types from this
│ │ │ │  package
│ │ │ │ - -- running: /usr/bin/gfan gfan --help < /tmp/M2-16949-0/172
│ │ │ │ + -- running: /usr/bin/gfan gfan --help < /tmp/M2-22465-0/172
│ │ │ │  This is a program for computing all reduced Groebner bases of a polynomial
│ │ │ │  ideal. It takes the ring and a generating set for the ideal as input. By
│ │ │ │  default the enumeration is done by an almost memoryless reverse search. If the
│ │ │ │  ideal is symmetric the symmetry option is useful and enumeration will be done
│ │ │ │  up to symmetry using a breadth first search. The program needs a starting
│ │ │ │  Groebner basis to do its computations. If the -g option is not specified it
│ │ │ │  will compute one using Buchberger's algorithm.
│ │ │ │ @@ -81,16 +81,16 @@
│ │ │ │   With this option you can specify how many variables to treat as parameters
│ │ │ │  instead of variables. This makes it possible to do computations where the
│ │ │ │  coefficient field is the field of rational functions in the parameters.
│ │ │ │  --interrupt value:
│ │ │ │   Interrupt the enumeration after a specified number of facets have been
│ │ │ │  computed (works for usual symmetric traversals, but may not work in general for
│ │ │ │  non-symmetric traversals or for traversals restricted to fans).
│ │ │ │ -using temporary file /tmp/M2-16949-0/172
│ │ │ │ - -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-16949-0/174
│ │ │ │ +using temporary file /tmp/M2-22465-0/172
│ │ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-22465-0/174
│ │ │ │  This program computes a reduced lexicographic Groebner basis of the polynomial
│ │ │ │  ideal given as input. The default behavior is to use Buchberger's algorithm.
│ │ │ │  The ordering of the variables is $a>b>c...$ (assuming that the ring is Q
│ │ │ │  [a,b,c,...]).
│ │ │ │  Options:
│ │ │ │  -w:
│ │ │ │   Compute a Groebner basis with respect to a degree lexicographic order with
│ │ │ │ @@ -111,63 +111,63 @@
│ │ │ │  minimal Groebner basis with respect to the reverse lexicographic term order.
│ │ │ │  The target term order is always lexicographic. The -W option must be used.
│ │ │ │  
│ │ │ │  --parameters value:
│ │ │ │   With this option you can specify how many variables to treat as parameters
│ │ │ │  instead of variables. This makes it possible to do computations where the
│ │ │ │  coefficient field is the field of rational functions in the parameters.
│ │ │ │ -using temporary file /tmp/M2-16949-0/174
│ │ │ │ - -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-16949-0/176
│ │ │ │ +using temporary file /tmp/M2-22465-0/174
│ │ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-22465-0/176
│ │ │ │  This program takes a marked Groebner basis of an ideal I and a set of
│ │ │ │  polynomials on its input and tests if the polynomial set is contained in I by
│ │ │ │  applying the division algorithm for each element. The output is 1 for true and
│ │ │ │  0 for false.
│ │ │ │  Options:
│ │ │ │  --remainder:
│ │ │ │   Tell the program to output the remainders of the divisions rather than
│ │ │ │  outputting 0 or 1.
│ │ │ │  --multiplier:
│ │ │ │   Reads in a polynomial that will be multiplied to the polynomial to be divided
│ │ │ │  before doing the division.
│ │ │ │ -using temporary file /tmp/M2-16949-0/176
│ │ │ │ - -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-16949-0/178
│ │ │ │ +using temporary file /tmp/M2-22465-0/176
│ │ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-22465-0/178
│ │ │ │  This program takes two polyhedral fans and computes their common refinement.
│ │ │ │  Options:
│ │ │ │  -i1 value:
│ │ │ │   Specify the name of the first input file.
│ │ │ │  -i2 value:
│ │ │ │   Specify the name of the second input file.
│ │ │ │  --stable:
│ │ │ │   Compute the stable intersection.
│ │ │ │ -using temporary file /tmp/M2-16949-0/178
│ │ │ │ - -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-16949-0/180
│ │ │ │ +using temporary file /tmp/M2-22465-0/178
│ │ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-22465-0/180
│ │ │ │  This program takes a polyhedral fan and a vector and computes the link of the
│ │ │ │  polyhedral fan around that vertex. The link will have lineality space dimension
│ │ │ │  equal to the dimension of the relative open polyhedral cone of the original fan
│ │ │ │  containing the vector.
│ │ │ │  Options:
│ │ │ │  -i value:
│ │ │ │   Specify the name of the input file.
│ │ │ │  --symmetry:
│ │ │ │   Reads in a fan stored with symmetry. The generators of the symmetry group must
│ │ │ │  be given on the standard input.
│ │ │ │  
│ │ │ │  --star:
│ │ │ │   Computes the star instead. The star is defined as the smallest polyhedral fan
│ │ │ │  containing all cones of the original fan containing the vector.
│ │ │ │ -using temporary file /tmp/M2-16949-0/180
│ │ │ │ - -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-16949-0/182
│ │ │ │ +using temporary file /tmp/M2-22465-0/180
│ │ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-22465-0/182
│ │ │ │  This program takes two polyhedral fans and computes their product.
│ │ │ │  Options:
│ │ │ │  -i1 value:
│ │ │ │   Specify the name of the first input file.
│ │ │ │  -i2 value:
│ │ │ │   Specify the name of the second input file.
│ │ │ │ -using temporary file /tmp/M2-16949-0/182
│ │ │ │ - -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-16949-0/184
│ │ │ │ +using temporary file /tmp/M2-22465-0/182
│ │ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-22465-0/184
│ │ │ │  This program computes a Groebner cone. Three different cases are handled. The
│ │ │ │  input may be a marked reduced Groebner basis in which case its Groebner cone is
│ │ │ │  computed. The input may be just a marked minimal basis in which case the cone
│ │ │ │  computed is not a Groebner cone in the usual sense but smaller. (These cones
│ │ │ │  are described in [Fukuda, Jensen, Lauritzen, Thomas]). The third possible case
│ │ │ │  is that the Groebner cone is possibly lower dimensional and given by a pair of
│ │ │ │  Groebner bases as it is useful to do for tropical varieties, see option --pair.
│ │ │ │ @@ -184,24 +184,24 @@
│ │ │ │  --asfan:
│ │ │ │   Writes the cone as a polyhedral fan with all its faces instead. In this way
│ │ │ │  the extreme rays of the cone are also computed.
│ │ │ │  --vectorinput:
│ │ │ │   Compute a cone given list of inequalities rather than a Groebner cone. The
│ │ │ │  input is an integer which specifies the dimension of the ambient space, a list
│ │ │ │  of inequalities given as vectors and a list of equations.
│ │ │ │ -using temporary file /tmp/M2-16949-0/184
│ │ │ │ - -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-16949-0/186
│ │ │ │ +using temporary file /tmp/M2-22465-0/184
│ │ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-22465-0/186
│ │ │ │  This program computes the homogeneity space of a list of polynomials - as a
│ │ │ │  cone. Thus generators for the homogeneity space are found in the section
│ │ │ │  LINEALITY_SPACE. If you wish the homogeneity space of an ideal you should first
│ │ │ │  compute a set of homogeneous generators and call the program on these. A
│ │ │ │  reduced Groebner basis will always suffice for this purpose.
│ │ │ │  Options:
│ │ │ │ -using temporary file /tmp/M2-16949-0/186
│ │ │ │ - -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-16949-0/188
│ │ │ │ +using temporary file /tmp/M2-22465-0/186
│ │ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-22465-0/188
│ │ │ │  This program homogenises a list of polynomials by introducing an extra
│ │ │ │  variable. The name of the variable to be introduced is read from the input
│ │ │ │  after the list of polynomials. Without the -w option the homogenisation is done
│ │ │ │  with respect to total degree.
│ │ │ │  Example:
│ │ │ │  Input:
│ │ │ │  Q[x,y]{y-1}
│ │ │ │ @@ -217,16 +217,16 @@
│ │ │ │   Specify a homogenisation vector. The length of the vector must be the same as
│ │ │ │  the number of variables in the ring. The vector is read from the input after
│ │ │ │  the list of polynomials.
│ │ │ │  
│ │ │ │  -H:
│ │ │ │   Let the name of the new variable be H rather than reading in a name from the
│ │ │ │  input.
│ │ │ │ -using temporary file /tmp/M2-16949-0/188
│ │ │ │ - -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-16949-0/190
│ │ │ │ +using temporary file /tmp/M2-22465-0/188
│ │ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-22465-0/190
│ │ │ │  This program converts a list of polynomials to a list of their initial forms
│ │ │ │  with respect to the vector given after the list.
│ │ │ │  Options:
│ │ │ │  --ideal:
│ │ │ │   Treat input as an ideal. This will make the program compute the initial ideal
│ │ │ │  of the ideal generated by the input polynomials. The computation is done by
│ │ │ │  computing a Groebner basis with respect to the given vector. The vector must be
│ │ │ │ @@ -242,16 +242,16 @@
│ │ │ │  --mark:
│ │ │ │   If the --pair option is and the --ideal option is not used this option will
│ │ │ │  still make sure that the second output basis is marked consistently with the
│ │ │ │  vector.
│ │ │ │  --list:
│ │ │ │   Read in a list of vectors instead of a single vector and produce a list of
│ │ │ │  polynomial sets as output.
│ │ │ │ -using temporary file /tmp/M2-16949-0/190
│ │ │ │ - -- running: /usr/bin/gfan _interactive --help < /tmp/M2-16949-0/192
│ │ │ │ +using temporary file /tmp/M2-22465-0/190
│ │ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-22465-0/192
│ │ │ │  This is a program for doing interactive walks in the Groebner fan of an ideal.
│ │ │ │  The input is a Groebner basis defining the starting Groebner cone of the walk.
│ │ │ │  The program will list all flippable facets of the Groebner cone and ask the
│ │ │ │  user to choose one. The user types in the index (number) of the facet in the
│ │ │ │  list. The program will walk through the selected facet and display the new
│ │ │ │  Groebner basis and a list of new facet normals for the user to choose from.
│ │ │ │  Since the program reads the user's choices through the the standard input it is
│ │ │ │ @@ -281,54 +281,54 @@
│ │ │ │  -W:
│ │ │ │   Print weight vector. This will make the program print an interior vector of
│ │ │ │  the current Groebner cone and a relative interior point for each flippable
│ │ │ │  facet of the current Groebner cone.
│ │ │ │  
│ │ │ │  --tropical:
│ │ │ │   Traverse a tropical variety interactively.
│ │ │ │ -using temporary file /tmp/M2-16949-0/192
│ │ │ │ - -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-16949-0/194
│ │ │ │ +using temporary file /tmp/M2-22465-0/192
│ │ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-22465-0/194
│ │ │ │  This program checks if a set of marked polynomials is a Groebner basis with
│ │ │ │  respect to its marking. First it is checked if the markings are consistent with
│ │ │ │  respect to a positive vector. Then Buchberger's S-criterion is checked. The
│ │ │ │  output is boolean value.
│ │ │ │  Options:
│ │ │ │ -using temporary file /tmp/M2-16949-0/194
│ │ │ │ - -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-16949-0/196
│ │ │ │ +using temporary file /tmp/M2-22465-0/194
│ │ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-22465-0/196
│ │ │ │  Takes an ideal $I$ and computes the Krull dimension of R/I where R is the
│ │ │ │  polynomial ring. This is done by first computing a Groebner basis.
│ │ │ │  Options:
│ │ │ │  -g:
│ │ │ │   Tell the program that the input is already a reduced Groebner basis.
│ │ │ │ -using temporary file /tmp/M2-16949-0/196
│ │ │ │ - -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-16949-0/198
│ │ │ │ +using temporary file /tmp/M2-22465-0/196
│ │ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-22465-0/198
│ │ │ │  This program computes the lattice ideal of a lattice. The input is a list of
│ │ │ │  generators for the lattice.
│ │ │ │  Options:
│ │ │ │  -t:
│ │ │ │   Compute the toric ideal of the matrix whose rows are given on the input
│ │ │ │  instead.
│ │ │ │  --convert:
│ │ │ │   Does not do any computation, but just converts the vectors to binomials.
│ │ │ │ -using temporary file /tmp/M2-16949-0/198
│ │ │ │ - -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-16949-0/200
│ │ │ │ +using temporary file /tmp/M2-22465-0/198
│ │ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-22465-0/200
│ │ │ │  This program converts a list of polynomials to a list of their leading terms.
│ │ │ │  Options:
│ │ │ │  -m:
│ │ │ │   Do the same thing for a list of polynomial sets. That is, output the set of
│ │ │ │  sets of leading terms.
│ │ │ │  
│ │ │ │ -using temporary file /tmp/M2-16949-0/200
│ │ │ │ - -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-16949-0/202
│ │ │ │ +using temporary file /tmp/M2-22465-0/200
│ │ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-22465-0/202
│ │ │ │  This program marks a set of polynomials with respect to the vector given at the
│ │ │ │  end of the input, meaning that the largest terms are moved to the front. In
│ │ │ │  case of a tie the lexicographic term order with $a>b>c...$ is used to break it.
│ │ │ │  Options:
│ │ │ │ -using temporary file /tmp/M2-16949-0/202
│ │ │ │ - -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-16949-0/204
│ │ │ │ +using temporary file /tmp/M2-22465-0/202
│ │ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-22465-0/204
│ │ │ │  This is a program for computing the normal fan of the Minkowski sum of the
│ │ │ │  Newton polytopes of a list of polynomials.
│ │ │ │  Options:
│ │ │ │  --symmetry:
│ │ │ │   Tells the program to read in generators for a group of symmetries (subgroup of
│ │ │ │  $S_n$) after having read in the ideal. The program checks that the ideal stays
│ │ │ │  fixed when permuting the variables with respect to elements in the group. The
│ │ │ │ @@ -338,16 +338,16 @@
│ │ │ │  --disableSymmetryTest:
│ │ │ │   When using --symmetry this option will disable the check that the group read
│ │ │ │  off from the input actually is a symmetry group with respect to the input
│ │ │ │  ideal.
│ │ │ │  
│ │ │ │  --nocones:
│ │ │ │   Tell the program to not list cones in the output.
│ │ │ │ -using temporary file /tmp/M2-16949-0/204
│ │ │ │ - -- running: /usr/bin/gfan _minors --help < /tmp/M2-16949-0/206
│ │ │ │ +using temporary file /tmp/M2-22465-0/204
│ │ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-22465-0/206
│ │ │ │  This program will generate the r*r minors of a d*n matrix of indeterminates.
│ │ │ │  Options:
│ │ │ │  -r value:
│ │ │ │   Specify r.
│ │ │ │  -d value:
│ │ │ │   Specify d.
│ │ │ │  -n value:
│ │ │ │ @@ -365,16 +365,16 @@
│ │ │ │   Produces a list of generators for the group of symmetries keeping the set of
│ │ │ │  minors fixed. (Only without --names).
│ │ │ │  --parametrize:
│ │ │ │   Parametrize the set of d times n matrices of Barvinok rank less than or equal
│ │ │ │  to r-1 by a list of tropical polynomials.
│ │ │ │  --ultrametric:
│ │ │ │   Produce tropical equations cutting out the ultrametrics.
│ │ │ │ -using temporary file /tmp/M2-16949-0/206
│ │ │ │ - -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-16949-0/208
│ │ │ │ +using temporary file /tmp/M2-22465-0/206
│ │ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-22465-0/208
│ │ │ │  This program computes the mixed volume of the Newton polytopes of a list of
│ │ │ │  polynomials. The ring is specified on the input. After this follows the list of
│ │ │ │  polynomials.
│ │ │ │  Options:
│ │ │ │  --vectorinput:
│ │ │ │   Read in a list of point configurations instead of a polynomial ring and a list
│ │ │ │  of polynomials.
│ │ │ │ @@ -388,25 +388,25 @@
│ │ │ │   Use Katsura-n example instead of reading input.
│ │ │ │  --gaukwa value:
│ │ │ │   Use Gaukwa-n example instead of reading input.
│ │ │ │  --eco value:
│ │ │ │   Use Eco-n example instead of reading input.
│ │ │ │  -j value:
│ │ │ │   Number of threads
│ │ │ │ -using temporary file /tmp/M2-16949-0/208
│ │ │ │ - -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-16949-0/210
│ │ │ │ +using temporary file /tmp/M2-22465-0/208
│ │ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-22465-0/210
│ │ │ │  This program computes the union of a list of polynomial sets given as input.
│ │ │ │  The polynomials must all belong to the same ring. The ring is specified on the
│ │ │ │  input. After this follows the list of polynomial sets.
│ │ │ │  Options:
│ │ │ │  -s:
│ │ │ │   Sort output by degree.
│ │ │ │  
│ │ │ │ -using temporary file /tmp/M2-16949-0/210
│ │ │ │ - -- running: /usr/bin/gfan _render --help < /tmp/M2-16949-0/212
│ │ │ │ +using temporary file /tmp/M2-22465-0/210
│ │ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-22465-0/212
│ │ │ │  This program renders a Groebner fan as an xfig file. To be more precise, the
│ │ │ │  input is the list of all reduced Groebner bases of an ideal. The output is a
│ │ │ │  drawing of the Groebner fan intersected with a triangle. The corners of the
│ │ │ │  triangle are (1,0,0) to the right, (0,1,0) to the left and (0,0,1) at the top.
│ │ │ │  If there are more than three variables in the ring these coordinates are
│ │ │ │  extended with zeros. It is possible to shift the 1 entry cyclic with the option
│ │ │ │  --shiftVariables.
│ │ │ │ @@ -414,16 +414,16 @@
│ │ │ │  -L:
│ │ │ │   Make the triangle larger so that the shape of the Groebner region appears.
│ │ │ │  --shiftVariables value:
│ │ │ │   Shift the positions of the variables in the drawing. For example with the
│ │ │ │  value equal to 1 the corners will be right: (0,1,0,0,...), left: (0,0,1,0,...)
│ │ │ │  and top: (0,0,0,1,...). The shifting is done modulo the number of variables in
│ │ │ │  the polynomial ring. The default value is 0.
│ │ │ │ -using temporary file /tmp/M2-16949-0/212
│ │ │ │ - -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-16949-0/214
│ │ │ │ +using temporary file /tmp/M2-22465-0/212
│ │ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-22465-0/214
│ │ │ │  This program renders a staircase diagram of a monomial initial ideal to an xfig
│ │ │ │  file. The input is a Groebner basis of a (not necessarily monomial) polynomial
│ │ │ │  ideal. The initial ideal is given by the leading terms in the Groebner basis.
│ │ │ │  Using the -m option it is possible to render more than one staircase diagram.
│ │ │ │  The program only works for ideals in a polynomial ring with three variables.
│ │ │ │  Options:
│ │ │ │  -m:
│ │ │ │ @@ -436,16 +436,16 @@
│ │ │ │  number is large enough to give a correct picture of the standard monomials. The
│ │ │ │  default value is 8.
│ │ │ │  
│ │ │ │  -w value:
│ │ │ │   Width. Specifies the number of staircase diagrams per row in the xfig file.
│ │ │ │  The default value is 5.
│ │ │ │  
│ │ │ │ -using temporary file /tmp/M2-16949-0/214
│ │ │ │ - -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-16949-0/216
│ │ │ │ +using temporary file /tmp/M2-22465-0/214
│ │ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-22465-0/216
│ │ │ │  This program computes the resultant fan as defined in "Computing Tropical
│ │ │ │  Resultants" by Jensen and Yu. The input is a polynomial ring followed by
│ │ │ │  polynomials, whose coefficients are ignored. The output is the fan of
│ │ │ │  coefficients such that the input system has a tropical solution.
│ │ │ │  Options:
│ │ │ │  --codimension:
│ │ │ │   Compute only the codimension of the resultant fan and return.
│ │ │ │ @@ -473,28 +473,28 @@
│ │ │ │  of polynomials.
│ │ │ │  
│ │ │ │  --projection:
│ │ │ │   Use the projection method to compute the resultant fan. This works only if the
│ │ │ │  resultant fan is a hypersurface. If this option is combined with --special,
│ │ │ │  then the output fan lives in the subspace of the non-specialized coordinates.
│ │ │ │  
│ │ │ │ -using temporary file /tmp/M2-16949-0/216
│ │ │ │ - -- running: /usr/bin/gfan _saturation --help < /tmp/M2-16949-0/218
│ │ │ │ +using temporary file /tmp/M2-22465-0/216
│ │ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-22465-0/218
│ │ │ │  This program computes the saturation of the input ideal with the product of the
│ │ │ │  variables x_1,...,x_n. The ideal does not have to be homogeneous.
│ │ │ │  Options:
│ │ │ │  -h:
│ │ │ │   Tell the program that the input is a homogeneous ideal (with homogeneous
│ │ │ │  generators).
│ │ │ │  
│ │ │ │  --noideal:
│ │ │ │   Do not treat input as an ideal but just factor out common monomial factors of
│ │ │ │  the input polynomials.
│ │ │ │ -using temporary file /tmp/M2-16949-0/218
│ │ │ │ - -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-16949-0/220
│ │ │ │ +using temporary file /tmp/M2-22465-0/218
│ │ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-22465-0/220
│ │ │ │  This program computes the secondary fan of a vector configuration. The
│ │ │ │  configuration is given as an ordered list of vectors. In order to compute the
│ │ │ │  secondary fan of a point configuration an additional coordinate of ones must be
│ │ │ │  added. For example {(1,0),(1,1),(1,2),(1,3)}.
│ │ │ │  Options:
│ │ │ │  --unimodular:
│ │ │ │   Use heuristics to search for unimodular triangulation rather than computing
│ │ │ │ @@ -523,103 +523,103 @@
│ │ │ │   Tells the program not to output the CONES and MAXIMAL_CONES sections, but
│ │ │ │  still output CONES_COMPRESSED and MAXIMAL_CONES_COMPRESSED if --symmetry is
│ │ │ │  used.
│ │ │ │  --interrupt value:
│ │ │ │   Interrupt the enumeration after a specified number of facets have been
│ │ │ │  computed (works for usual symmetric traversals, but may not work in general for
│ │ │ │  non-symmetric traversals or for traversals restricted to fans).
│ │ │ │ -using temporary file /tmp/M2-16949-0/220
│ │ │ │ - -- running: /usr/bin/gfan _stats --help < /tmp/M2-16949-0/222
│ │ │ │ +using temporary file /tmp/M2-22465-0/220
│ │ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-22465-0/222
│ │ │ │  This program takes a list of reduced Groebner bases for the same ideal and
│ │ │ │  computes various statistics. The following information is listed: the number of
│ │ │ │  bases in the input, the number of variables, the dimension of the homogeneity
│ │ │ │  space, the maximal total degree of any polynomial in the input and the minimal
│ │ │ │  total degree of any basis in the input, the maximal number of polynomials and
│ │ │ │  terms in a basis in the input.
│ │ │ │  Options:
│ │ │ │ -using temporary file /tmp/M2-16949-0/222
│ │ │ │ - -- running: /usr/bin/gfan _substitute --help < /tmp/M2-16949-0/224
│ │ │ │ +using temporary file /tmp/M2-22465-0/222
│ │ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-22465-0/224
│ │ │ │  This program changes the variable names of a polynomial ring. The input is a
│ │ │ │  polynomial ring, a polynomial set in the ring and a new polynomial ring with
│ │ │ │  the same coefficient field but different variable names. The output is the
│ │ │ │  polynomial set written with the variable names of the second polynomial ring.
│ │ │ │  Example:
│ │ │ │  Input:
│ │ │ │  Q[a,b,c,d]{2a-3b,c+d}Q[b,a,c,x]
│ │ │ │  Output:
│ │ │ │  Q[b,a,c,x]{2*b-3*a,c+x}
│ │ │ │  Options:
│ │ │ │ -using temporary file /tmp/M2-16949-0/224
│ │ │ │ - -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-16949-0/226
│ │ │ │ +using temporary file /tmp/M2-22465-0/224
│ │ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-22465-0/226
│ │ │ │  This program converts ASCII math to TeX math. The data-type is specified by the
│ │ │ │  options.
│ │ │ │  Options:
│ │ │ │  -h:
│ │ │ │   Add a header to the output. Using this option the output will be LaTeXable
│ │ │ │  right away.
│ │ │ │  --polynomialset_:
│ │ │ │   The data to be converted is a list of polynomials.
│ │ │ │  --polynomialsetlist_:
│ │ │ │   The data to be converted is a list of lists of polynomials.
│ │ │ │ -using temporary file /tmp/M2-16949-0/226
│ │ │ │ - -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-16949-0/228
│ │ │ │ +using temporary file /tmp/M2-22465-0/226
│ │ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-22465-0/228
│ │ │ │  This program takes a list of reduced Groebner bases and produces the fan of all
│ │ │ │  faces of these. In this way by giving the complete list of reduced Groebner
│ │ │ │  bases, the Groebner fan can be computed as a polyhedral complex. The option --
│ │ │ │  restrict lets the user choose between computing the Groebner fan or the
│ │ │ │  restricted Groebner fan.
│ │ │ │  Options:
│ │ │ │  --restrict:
│ │ │ │   Add an inequality for each coordinate, so that the the cones are restricted to
│ │ │ │  the non-negative orthant.
│ │ │ │  --symmetry:
│ │ │ │   Tell the program to read in generators for a group of symmetries (subgroup of
│ │ │ │  $S_n$) after having read in the ring. The output is grouped according to these
│ │ │ │  symmetries. Only one representative for each orbit is needed on the input.
│ │ │ │  
│ │ │ │ -using temporary file /tmp/M2-16949-0/228
│ │ │ │ - -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-16949-0/230
│ │ │ │ +using temporary file /tmp/M2-22465-0/228
│ │ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-22465-0/230
│ │ │ │  This program computes a tropical basis for an ideal defining a tropical curve.
│ │ │ │  Defining a tropical curve means that the Krull dimension of R/I is at most 1 +
│ │ │ │  the dimension of the homogeneity space of I where R is the polynomial ring. The
│ │ │ │  input is a generating set for the ideal. If the input is not homogeneous option
│ │ │ │  -h must be used.
│ │ │ │  Options:
│ │ │ │  -h:
│ │ │ │   Homogenise the input before computing a tropical basis and dehomogenise the
│ │ │ │  output. This is needed if the input generators are not already homogeneous.
│ │ │ │ -using temporary file /tmp/M2-16949-0/230
│ │ │ │ - -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-16949-0/232
│ │ │ │ +using temporary file /tmp/M2-22465-0/230
│ │ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-22465-0/232
│ │ │ │  This program takes a marked reduced Groebner basis for a homogeneous ideal and
│ │ │ │  computes the tropical variety of the ideal as a subfan of the Groebner fan. The
│ │ │ │  program is slow but works for any homogeneous ideal. If you know that your
│ │ │ │  ideal is prime over the complex numbers or you simply know that its tropical
│ │ │ │  variety is pure and connected in codimension one then use
│ │ │ │  gfan_tropicalstartingcone and gfan_tropicaltraverse instead.
│ │ │ │  Options:
│ │ │ │ -using temporary file /tmp/M2-16949-0/232
│ │ │ │ - -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-16949-0/234
│ │ │ │ +using temporary file /tmp/M2-22465-0/232
│ │ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-22465-0/234
│ │ │ │  This program evaluates a tropical polynomial function in a given set of points.
│ │ │ │  Options:
│ │ │ │ -using temporary file /tmp/M2-16949-0/234
│ │ │ │ - -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-16949-0/236
│ │ │ │ +using temporary file /tmp/M2-22465-0/234
│ │ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-22465-0/236
│ │ │ │  This program takes a polynomial and tropicalizes it. The output is piecewise
│ │ │ │  linear function represented by a fan whose cones are the linear regions. Each
│ │ │ │  ray of the fan gets the value of the tropical function assigned to it. In other
│ │ │ │  words this program computes the normal fan of the Newton polytope of the input
│ │ │ │  polynomial with additional information.Options:
│ │ │ │  --exponents:
│ │ │ │   Tell program to read a list of exponent vectors instead.
│ │ │ │ -using temporary file /tmp/M2-16949-0/236
│ │ │ │ - -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-16949-0/238
│ │ │ │ +using temporary file /tmp/M2-22465-0/236
│ │ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-22465-0/238
│ │ │ │  This program computes the tropical hypersurface defined by a principal ideal.
│ │ │ │  The input is the polynomial ring followed by a set containing just a generator
│ │ │ │  of the ideal.Options:
│ │ │ │ -using temporary file /tmp/M2-16949-0/238
│ │ │ │ - -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-16949-0/240
│ │ │ │ +using temporary file /tmp/M2-22465-0/238
│ │ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-22465-0/240
│ │ │ │  This program computes the set theoretical intersection of a set of tropical
│ │ │ │  hypersurfaces (or to be precise, their common refinement as a fan). The input
│ │ │ │  is a list of polynomials with each polynomial defining a hypersurface.
│ │ │ │  Considering tropical hypersurfaces as fans, the intersection can be computed as
│ │ │ │  the common refinement of these. Thus the output is a fan whose support is the
│ │ │ │  intersection of the tropical hypersurfaces.
│ │ │ │  Options:
│ │ │ │ @@ -656,16 +656,16 @@
│ │ │ │  --stable:
│ │ │ │   Find the stable intersection of the input polynomials using tropical
│ │ │ │  intersection theory. This can be slow. Most other options are ignored.
│ │ │ │  --parameters value:
│ │ │ │   With this option you can specify how many variables to treat as parameters
│ │ │ │  instead of variables. This makes it possible to do computations where the
│ │ │ │  coefficient field is the field of rational functions in the parameters.
│ │ │ │ -using temporary file /tmp/M2-16949-0/240
│ │ │ │ - -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-16949-0/242
│ │ │ │ +using temporary file /tmp/M2-22465-0/240
│ │ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-22465-0/242
│ │ │ │  This program is part of the Puiseux lifting algorithm implemented in Gfan and
│ │ │ │  Singular. The Singular part of the implementation can be found in:
│ │ │ │  
│ │ │ │  Anders Nedergaard Jensen, Hannah Markwig, Thomas Markwig:
│ │ │ │   tropical.lib. A SINGULAR 3.0 library for computations in tropical geometry,
│ │ │ │  2007
│ │ │ │  
│ │ │ │ @@ -693,54 +693,54 @@
│ │ │ │  Options:
│ │ │ │  --noMult:
│ │ │ │   Disable the multiplicity computation.
│ │ │ │  -n value:
│ │ │ │   Number of variables that should have negative weight.
│ │ │ │  -c:
│ │ │ │   Only output a list of vectors being the possible choices.
│ │ │ │ -using temporary file /tmp/M2-16949-0/242
│ │ │ │ - -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-16949-0/244
│ │ │ │ +using temporary file /tmp/M2-22465-0/242
│ │ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-22465-0/244
│ │ │ │  This program generates tropical equations for a tropical linear space in the
│ │ │ │  Speyer sense given the tropical Pluecker coordinates as input.
│ │ │ │  Options:
│ │ │ │  -d value:
│ │ │ │   Specify d.
│ │ │ │  -n value:
│ │ │ │   Specify n.
│ │ │ │  --trees:
│ │ │ │   list the boundary trees (assumes d=3)
│ │ │ │ -using temporary file /tmp/M2-16949-0/244
│ │ │ │ - -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-16949-0/246
│ │ │ │ +using temporary file /tmp/M2-22465-0/244
│ │ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-22465-0/246
│ │ │ │  This program computes the multiplicity of a tropical cone given a marked
│ │ │ │  reduced Groebner basis for its initial ideal.
│ │ │ │  Options:
│ │ │ │ -using temporary file /tmp/M2-16949-0/246
│ │ │ │ - -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-16949-0/248
│ │ │ │ +using temporary file /tmp/M2-22465-0/246
│ │ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-22465-0/248
│ │ │ │  This program will compute the tropical rank of matrix given as input. Tropical
│ │ │ │  addition is MAXIMUM.
│ │ │ │  Options:
│ │ │ │  --kapranov:
│ │ │ │   Compute Kapranov rank instead of tropical rank.
│ │ │ │  --determinant:
│ │ │ │   Compute the tropical determinant instead.
│ │ │ │ -using temporary file /tmp/M2-16949-0/248
│ │ │ │ - -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-16949-0/250
│ │ │ │ +using temporary file /tmp/M2-22465-0/248
│ │ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-22465-0/250
│ │ │ │  This program computes a starting pair of marked reduced Groebner bases to be
│ │ │ │  used as input for gfan_tropicaltraverse. The input is a homogeneous ideal whose
│ │ │ │  tropical variety is a pure d-dimensional polyhedral complex.
│ │ │ │  Options:
│ │ │ │  -g:
│ │ │ │   Tell the program that the input is already a reduced Groebner basis.
│ │ │ │  -d:
│ │ │ │   Output dimension information to standard error.
│ │ │ │  --stable:
│ │ │ │   Find starting cone in the stable intersection or, equivalently, pretend that
│ │ │ │  the coefficients are genereric.
│ │ │ │ -using temporary file /tmp/M2-16949-0/250
│ │ │ │ - -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-16949-0/252
│ │ │ │ +using temporary file /tmp/M2-22465-0/250
│ │ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-22465-0/252
│ │ │ │  This program computes a polyhedral fan representation of the tropical variety
│ │ │ │  of a homogeneous prime ideal $I$. Let $d$ be the Krull dimension of $I$ and let
│ │ │ │  $\omega$ be a relative interior point of $d$-dimensional Groebner cone
│ │ │ │  contained in the tropical variety. The input for this program is a pair of
│ │ │ │  marked reduced Groebner bases with respect to the term order represented by
│ │ │ │  $\omega$, tie-broken in some way. The first one is for the initial ideal
│ │ │ │  $in_\omega(I)$ the second one for $I$ itself. The pair is the starting point
│ │ │ │ @@ -770,27 +770,27 @@
│ │ │ │  --stable:
│ │ │ │   Traverse the stable intersection or, equivalently, pretend that the
│ │ │ │  coefficients are genereric.
│ │ │ │  --interrupt value:
│ │ │ │   Interrupt the enumeration after a specified number of facets have been
│ │ │ │  computed (works for usual symmetric traversals, but may not work in general for
│ │ │ │  non-symmetric traversals or for traversals restricted to fans).
│ │ │ │ -using temporary file /tmp/M2-16949-0/252
│ │ │ │ - -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-16949-0/254
│ │ │ │ +using temporary file /tmp/M2-22465-0/252
│ │ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-22465-0/254
│ │ │ │  This program computes the tropical Weil divisor of piecewise linear (or
│ │ │ │  tropical rational) function on a tropical k-cycle. See the Gfan manual for more
│ │ │ │  information.
│ │ │ │  Options:
│ │ │ │  -i1 value:
│ │ │ │   Specify the name of the Polymake input file containing the k-cycle.
│ │ │ │  -i2 value:
│ │ │ │   Specify the name of the Polymake input file containing the piecewise linear
│ │ │ │  function.
│ │ │ │ -using temporary file /tmp/M2-16949-0/254
│ │ │ │ - -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-16949-0/256
│ │ │ │ +using temporary file /tmp/M2-22465-0/254
│ │ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-22465-0/256
│ │ │ │  This program is an experimental implementation of Groebner bases for ideals in
│ │ │ │  Z[x_1,...,x_n].
│ │ │ │  Several operations are supported by specifying the appropriate option:
│ │ │ │   (1) computation of the reduced Groebner basis with respect to a given vector
│ │ │ │  (tiebroken lexicographically),
│ │ │ │   (2) computation of an initial ideal,
│ │ │ │   (3) computation of the Groebner fan,
│ │ │ │ @@ -825,23 +825,23 @@
│ │ │ │   For the operations taking a vector as input, read in a list of vectors
│ │ │ │  instead, and perform the operation for each vector in the list.
│ │ │ │  -g:
│ │ │ │   Tells the program that the input is already a Groebner basis (with the initial
│ │ │ │  term of each polynomial being the first ones listed). Use this option if the
│ │ │ │  usual --groebnerFan is too slow.
│ │ │ │  
│ │ │ │ -using temporary file /tmp/M2-16949-0/256
│ │ │ │ +using temporary file /tmp/M2-22465-0/256
│ │ │ │  i6 : QQ[x,y];
│ │ │ │  i7 : gfan {x,y};
│ │ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-16949-0/258
│ │ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-22465-0/258
│ │ │ │  Q[x1,x2]
│ │ │ │  {{
│ │ │ │  x2,
│ │ │ │  x1}
│ │ │ │  }
│ │ │ │ -using temporary file /tmp/M2-16949-0/258
│ │ │ │ +using temporary file /tmp/M2-22465-0/258
│ │ │ │  Finally, if you want to be able to render Groebner fans and monomial staircases
│ │ │ │  to .png files, you should install fig2dev. If it is installed in a non-standard
│ │ │ │  location, then you may specify its path using _p_r_o_g_r_a_m_P_a_t_h_s.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/gfanInterface.m2:2630:0.
│ │ ├── ./usr/share/info/AInfinity.info.gz
│ │ │ ├── AInfinity.info
│ │ │ │ @@ -6133,16 +6133,16 @@
│ │ │ │  00017f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017f70: 2b0a 7c69 3320 3a20 656c 6170 7365 6454  +.|i3 : elapsedT
│ │ │ │  00017f80: 696d 6520 6275 726b 6552 6573 6f6c 7574  ime burkeResolut
│ │ │ │  00017f90: 696f 6e28 4d2c 2037 2c20 4368 6563 6b20  ion(M, 7, Check 
│ │ │ │  00017fa0: 3d3e 2066 616c 7365 2920 2020 2020 2020  => false)       
│ │ │ │ -00017fb0: 2020 2020 7c0a 7c20 2d2d 2031 2e37 3131      |.| -- 1.711
│ │ │ │ -00017fc0: 3338 7320 656c 6170 7365 6420 2020 2020  38s elapsed     
│ │ │ │ +00017fb0: 2020 2020 7c0a 7c20 2d2d 2031 2e33 3238      |.| -- 1.328
│ │ │ │ +00017fc0: 3133 7320 656c 6170 7365 6420 2020 2020  13s 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 2032 2e31 3536 3935 7320 656c 6170  -- 2.15695s elap
│ │ │ │ +00018260: 2d2d 2031 2e37 3238 3732 7320 656c 6170  -- 1.72872s 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 3232 3837     |.| -- .02287
│ │ │ │ -00002ec0: 3931 7320 656c 6170 7365 6420 2020 2020  91s elapsed     
│ │ │ │ +00002eb0: 2020 207c 0a7c 202d 2d20 2e30 3237 3135     |.| -- .02715
│ │ │ │ +00002ec0: 3932 7320 656c 6170 7365 6420 2020 2020  92s elapsed     
│ │ │ │  00002ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002ef0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00002f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002f30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ @@ -1596,15 +1596,15 @@
│ │ │ │  000063b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000063c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000063d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  000063e0: 7c69 3135 203a 2065 6c61 7073 6564 5469  |i15 : elapsedTi
│ │ │ │  000063f0: 6d65 2062 6574 7469 2849 273d 7472 696d  me betti(I'=trim
│ │ │ │  00006400: 206b 6572 2070 6869 2920 2020 2020 2020   ker phi)       
│ │ │ │  00006410: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ -00006420: 2e36 3730 3236 3373 2065 6c61 7073 6564  .670263s elapsed
│ │ │ │ +00006420: 2e35 3230 3535 3673 2065 6c61 7073 6564  .520556s elapsed
│ │ │ │  00006430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006450: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00006460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006480: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00006490: 0a7c 2020 2020 2020 2020 2020 2020 2030  .|             0
│ │ │ │ @@ -1651,15 +1651,15 @@
│ │ │ │  00006720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00006730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00006740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00006750: 2d2d 2d2b 0a7c 6931 3720 3a20 656c 6170  ---+.|i17 : elap
│ │ │ │  00006760: 7365 6454 696d 6520 6261 7365 5074 733d  sedTime basePts=
│ │ │ │  00006770: 7072 696d 6172 7944 6563 6f6d 706f 7369  primaryDecomposi
│ │ │ │  00006780: 7469 6f6e 2069 6465 616c 2048 3b20 7c0a  tion ideal H; |.
│ │ │ │ -00006790: 7c20 2d2d 2035 2e37 3737 3937 7320 656c  | -- 5.77797s el
│ │ │ │ +00006790: 7c20 2d2d 2034 2e38 3233 3534 7320 656c  | -- 4.82354s el
│ │ │ │  000067a0: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │  000067b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000067c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  000067d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000067e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000067f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00006800: 2d2d 2d2d 2b0a 7c69 3138 203a 2074 616c  ----+.|i18 : tal
│ │ │ │ @@ -2608,15 +2608,15 @@
│ │ │ │  0000a2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  0000a320: 3134 203a 2065 6c61 7073 6564 5469 6d65  14 : elapsedTime
│ │ │ │  0000a330: 2073 7562 2849 2c48 2920 2020 2020 2020   sub(I,H)       
│ │ │ │  0000a340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a350: 2020 2020 2020 207c 0a7c 202d 2d20 2e30         |.| -- .0
│ │ │ │ -0000a360: 3133 3636 3338 7320 656c 6170 7365 6420  136638s elapsed 
│ │ │ │ +0000a360: 3135 3539 3732 7320 656c 6170 7365 6420  155972s elapsed 
│ │ │ │  0000a370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a390: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0000a3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a3c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0000a3d0: 6f31 3420 3d20 6964 6561 6c20 2830 2c20  o14 = ideal (0, 
│ │ │ │ @@ -2648,15 +2648,15 @@
│ │ │ │  0000a570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a5a0: 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20 656c  -----+.|i16 : el
│ │ │ │  0000a5b0: 6170 7365 6454 696d 6520 6265 7474 6928  apsedTime betti(
│ │ │ │  0000a5c0: 4927 3d74 7269 6d20 6b65 7220 7068 6929  I'=trim ker phi)
│ │ │ │  0000a5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a5e0: 7c0a 7c20 2d2d 202e 3035 3633 3733 3173  |.| -- .0563731s
│ │ │ │ +0000a5e0: 7c0a 7c20 2d2d 202e 3036 3638 3932 3673  |.| -- .0668926s
│ │ │ │  0000a5f0: 2065 6c61 7073 6564 2020 2020 2020 2020   elapsed        
│ │ │ │  0000a600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a610: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0000a620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a650: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ @@ -2700,15 +2700,15 @@
│ │ │ │  0000a8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000a8e0: 0a7c 6931 3820 3a20 656c 6170 7365 6454  .|i18 : elapsedT
│ │ │ │  0000a8f0: 696d 6520 6261 7365 5074 733d 7072 696d  ime basePts=prim
│ │ │ │  0000a900: 6172 7944 6563 6f6d 706f 7369 7469 6f6e  aryDecomposition
│ │ │ │  0000a910: 2069 6465 616c 2048 3b20 7c0a 7c20 2d2d   ideal H; |.| --
│ │ │ │ -0000a920: 2031 2e38 3334 3635 7320 656c 6170 7365   1.83465s elapse
│ │ │ │ +0000a920: 2031 2e34 3935 3937 7320 656c 6170 7365   1.49597s elapse
│ │ │ │  0000a930: 6420 2020 2020 2020 2020 2020 2020 2020  d               
│ │ │ │  0000a940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a950: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  0000a960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a990: 2b0a 7c69 3139 203a 2074 616c 6c79 2061  +.|i19 : tally a
│ │ ├── ./usr/share/info/BGG.info.gz
│ │ │ ├── BGG.info
│ │ │ │ @@ -4338,16 +4338,16 @@
│ │ │ │  00010f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  00010f40: 3134 203a 2074 696d 6520 6265 7474 6920  14 : time betti 
│ │ │ │  00010f50: 2846 203d 2070 7572 6552 6573 6f6c 7574  (F = pureResolut
│ │ │ │  00010f60: 696f 6e28 4d2c 7b30 2c32 2c34 7d29 2920  ion(M,{0,2,4})) 
│ │ │ │  00010f70: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ -00010f80: 302e 3434 3039 3235 7320 2863 7075 293b  0.440925s (cpu);
│ │ │ │ -00010f90: 2030 2e33 3637 3030 3473 2028 7468 7265   0.367004s (thre
│ │ │ │ +00010f80: 302e 3439 3633 3034 7320 2863 7075 293b  0.496304s (cpu);
│ │ │ │ +00010f90: 2030 2e34 3132 3534 3373 2028 7468 7265   0.412543s (thre
│ │ │ │  00010fa0: 6164 293b 2030 7320 2867 6329 7c0a 7c20  ad); 0s (gc)|.| 
│ │ │ │  00010fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010fe0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00010ff0: 2020 2020 3020 3120 3220 2020 2020 2020      0 1 2       
│ │ │ │  00011000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4403,16 +4403,16 @@
│ │ │ │  00011320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  00011350: 0a7c 6931 3520 3a20 7469 6d65 2062 6574  .|i15 : time bet
│ │ │ │  00011360: 7469 2028 4620 3d20 7075 7265 5265 736f  ti (F = pureReso
│ │ │ │  00011370: 6c75 7469 6f6e 2831 312c 342c 7b30 2c32  lution(11,4,{0,2
│ │ │ │  00011380: 2c34 7d29 2920 207c 0a7c 202d 2d20 7573  ,4}))  |.| -- us
│ │ │ │ -00011390: 6564 2030 2e34 3737 3333 3473 2028 6370  ed 0.477334s (cp
│ │ │ │ -000113a0: 7529 3b20 302e 3430 3035 3337 7320 2874  u); 0.400537s (t
│ │ │ │ +00011390: 6564 2030 2e35 3438 3337 3573 2028 6370  ed 0.548375s (cp
│ │ │ │ +000113a0: 7529 3b20 302e 3436 3833 3633 7320 2874  u); 0.468363s (t
│ │ │ │  000113b0: 6872 6561 6429 3b20 3073 2028 6763 297c  hread); 0s (gc)|
│ │ │ │  000113c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000113d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000113e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000113f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00011400: 2020 2020 2020 2030 2031 2032 2020 2020         0 1 2    
│ │ │ │  00011410: 2020 2020 2020 2020 2020 2020 2020 2020
│ │ ├── ./usr/share/info/Benchmark.info.gz
│ │ │ ├── Benchmark.info
│ │ │ │ @@ -200,71 +200,76 @@
│ │ │ │  00000c70: 2d2d 2d2d 2b0a 7c69 3120 3a20 7275 6e42  ----+.|i1 : runB
│ │ │ │  00000c80: 656e 6368 6d61 726b 7320 2272 6573 3339  enchmarks "res39
│ │ │ │  00000c90: 2220 2020 2020 2020 2020 2020 2020 2020  "               
│ │ │ │  00000ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000cc0: 2020 2020 7c0a 7c2d 2d20 6265 6769 6e6e      |.|-- beginn
│ │ │ │  00000cd0: 696e 6720 636f 6d70 7574 6174 696f 6e20  ing computation 
│ │ │ │ -00000ce0: 5375 6e20 4465 6320 3134 2031 353a 3331  Sun Dec 14 15:31
│ │ │ │ -00000cf0: 3a34 3220 5554 4320 3230 3235 2020 2020  :42 UTC 2025    
│ │ │ │ +00000ce0: 5468 7520 4a61 6e20 2031 2031 313a 3036  Thu Jan  1 11:06
│ │ │ │ +00000cf0: 3a32 3420 5554 4320 3230 3236 2020 2020  :24 UTC 2026    
│ │ │ │  00000d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000d10: 2020 2020 7c0a 7c2d 2d20 4c69 6e75 7820      |.|-- Linux 
│ │ │ │  00000d20: 7362 7569 6c64 2036 2e31 322e 3537 2b64  sbuild 6.12.57+d
│ │ │ │ -00000d30: 6562 3133 2d61 6d64 3634 2023 3120 534d  eb13-amd64 #1 SM
│ │ │ │ -00000d40: 5020 5052 4545 4d50 545f 4459 4e41 4d49  P PREEMPT_DYNAMI
│ │ │ │ -00000d50: 4320 4465 6269 616e 2036 2e31 322e 3537  C Debian 6.12.57
│ │ │ │ -00000d60: 2d31 2020 7c0a 7c2d 2d20 414d 4420 4550  -1  |.|-- AMD EP
│ │ │ │ -00000d70: 5943 2037 3730 3250 2036 342d 436f 7265  YC 7702P 64-Core
│ │ │ │ -00000d80: 2050 726f 6365 7373 6f72 2020 4175 7468   Processor  Auth
│ │ │ │ -00000d90: 656e 7469 6341 4d44 2020 6370 7520 4d48  enticAMD  cpu MH
│ │ │ │ -00000da0: 7a20 3139 3936 2e32 3439 2020 2020 2020  z 1996.249      
│ │ │ │ +00000d30: 6562 3133 2d63 6c6f 7564 2d61 6d64 3634  eb13-cloud-amd64
│ │ │ │ +00000d40: 2023 3120 534d 5020 5052 4545 4d50 545f   #1 SMP PREEMPT_
│ │ │ │ +00000d50: 4459 4e41 4d49 4320 4465 6269 616e 2020  DYNAMIC Debian  
│ │ │ │ +00000d60: 2020 2020 7c0a 7c2d 2d20 496e 7465 6c20      |.|-- Intel 
│ │ │ │ +00000d70: 5865 6f6e 2050 726f 6365 7373 6f72 2028  Xeon Processor (
│ │ │ │ +00000d80: 536b 796c 616b 652c 2049 4252 5329 2020  Skylake, IBRS)  
│ │ │ │ +00000d90: 4765 6e75 696e 6549 6e74 656c 2020 6370  GenuineIntel  cp
│ │ │ │ +00000da0: 7520 4d48 7a20 3230 3939 2e39 3938 2020  u MHz 2099.998  
│ │ │ │  00000db0: 2020 2020 7c0a 7c2d 2d20 4d61 6361 756c      |.|-- Macaul
│ │ │ │  00000dc0: 6179 3220 312e 3235 2e31 312c 2063 6f6d  ay2 1.25.11, com
│ │ │ │  00000dd0: 7069 6c65 6420 7769 7468 2067 6363 2031  piled with gcc 1
│ │ │ │  00000de0: 352e 322e 3020 2020 2020 2020 2020 2020  5.2.0           
│ │ │ │  00000df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000e00: 2020 2020 7c0a 7c2d 2d20 7265 7333 393a      |.|-- res39:
│ │ │ │  00000e10: 2072 6573 206f 6620 6120 6765 6e65 7269   res of a generi
│ │ │ │  00000e20: 6320 3320 6279 2039 206d 6174 7269 7820  c 3 by 9 matrix 
│ │ │ │ -00000e30: 6f76 6572 205a 5a2f 3130 313a 202e 3135  over ZZ/101: .15
│ │ │ │ -00000e40: 3332 3135 2073 6563 6f6e 6473 2020 2020  3215 seconds    
│ │ │ │ +00000e30: 6f76 6572 205a 5a2f 3130 313a 202e 3139  over ZZ/101: .19
│ │ │ │ +00000e40: 3237 3833 2073 6563 6f6e 6473 2020 2020  2783 seconds    
│ │ │ │  00000e50: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │  00000e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00000e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00000e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00000e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00000ea0: 2d2d 2d2d 7c0a 7c28 3230 3235 2d31 312d  ----|.|(2025-11-
│ │ │ │ -00000eb0: 3035 2920 7838 365f 3634 2047 4e55 2f4c  05) x86_64 GNU/L
│ │ │ │ -00000ec0: 696e 7578 2020 2020 2020 2020 2020 2020  inux            
│ │ │ │ +00000ea0: 2d2d 2d2d 7c0a 7c36 2e31 322e 3537 2d31  ----|.|6.12.57-1
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│ │ │ │ +00000ec0: 365f 3634 2047 4e55 2f4c 696e 7578 2020  6_64 GNU/Linux  
│ │ │ │  00000ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00000ef0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00000f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00000f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00000f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00000f40: 2d2d 2d2d 2b0a 0a46 6f72 2074 6865 2070  ----+..For the p
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│ │ │ │ -00000f90: 6e42 656e 6368 6d61 726b 732c 2069 7320  nBenchmarks, is 
│ │ │ │ -00000fa0: 6120 2a6e 6f74 6520 636f 6d6d 616e 643a  a *note command:
│ │ │ │ -00000fb0: 0a28 4d61 6361 756c 6179 3244 6f63 2943  .(Macaulay2Doc)C
│ │ │ │ -00000fc0: 6f6d 6d61 6e64 2c2e 0a0a 2d2d 2d2d 2d2d  ommand,...------
│ │ │ │ -00000fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00000fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00000ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00001000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00001010: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073  ---------..The s
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│ │ │ │ -00001040: 696c 642f 7265 7072 6f64 7563 6962 6c65  ild/reproducible
│ │ │ │ -00001050: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ -00001060: 312e 3235 2e31 312b 6473 2f4d 322f 4d61  1.25.11+ds/M2/Ma
│ │ │ │ -00001070: 6361 756c 6179 322f 7061 636b 6167 6573  caulay2/packages
│ │ │ │ -00001080: 2f42 656e 6368 6d61 726b 2e0a 6d32 3a32  /Benchmark..m2:2
│ │ │ │ -00001090: 3937 3a30 2e0a 1f0a 5461 6720 5461 626c  97:0....Tag Tabl
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│ │ │ │ +00000f40: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
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│ │ │ │ +00000f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00000f90: 2d2d 2d2d 2b0a 0a46 6f72 2074 6865 2070  ----+..For the p
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│ │ │ │ +00000fc0: 6520 6f62 6a65 6374 202a 6e6f 7465 2072  e object *note r
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│ │ │ │ +00001010: 6f6d 6d61 6e64 2c2e 0a0a 2d2d 2d2d 2d2d  ommand,...------
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│ │ │ │ +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
│ │ │ │ @@ -2253,16 +2253,16 @@
│ │ │ │  00008cc0: 616c 206e 756d 6265 720a 2020 2020 2020  al number.      
│ │ │ │  00008cd0: 2020 6f72 2072 616e 646f 6d20 636f 6d70    or random comp
│ │ │ │  00008ce0: 6c65 7820 6e75 6d62 6572 0a20 2020 2020  lex number.     
│ │ │ │  00008cf0: 202a 202a 6e6f 7465 2054 6f70 4469 7265   * *note TopDire
│ │ │ │  00008d00: 6374 6f72 793a 2054 6f70 4469 7265 6374  ctory: TopDirect
│ │ │ │  00008d10: 6f72 792c 203d 3e20 2e2e 2e2c 2064 6566  ory, => ..., def
│ │ │ │  00008d20: 6175 6c74 2076 616c 7565 0a20 2020 2020  ault value.     
│ │ │ │ -00008d30: 2020 2022 2f74 6d70 2f4d 322d 3238 3730     "/tmp/M2-2870
│ │ │ │ -00008d40: 362d 302f 3022 2c20 4f70 7469 6f6e 2074  6-0/0", Option t
│ │ │ │ +00008d30: 2020 2022 2f74 6d70 2f4d 322d 3431 3033     "/tmp/M2-4103
│ │ │ │ +00008d40: 372d 302f 3022 2c20 4f70 7469 6f6e 2074  7-0/0", Option t
│ │ │ │  00008d50: 6f20 6368 616e 6765 2064 6972 6563 746f  o change directo
│ │ │ │  00008d60: 7279 2066 6f72 2066 696c 6520 7374 6f72  ry for file stor
│ │ │ │  00008d70: 6167 652e 0a20 2020 2020 202a 202a 6e6f  age..      * *no
│ │ │ │  00008d80: 7465 2056 6572 626f 7365 3a20 6265 7274  te Verbose: bert
│ │ │ │  00008d90: 696e 6954 7261 636b 486f 6d6f 746f 7079  iniTrackHomotopy
│ │ │ │  00008da0: 5f6c 705f 7064 5f70 645f 7064 5f63 6d56  _lp_pd_pd_pd_cmV
│ │ │ │  00008db0: 6572 626f 7365 3d3e 5f70 645f 7064 5f70  erbose=>_pd_pd_p
│ │ │ │ @@ -4971,15 +4971,15 @@
│ │ │ │  000136a0: 6e74 6174 696f 6e29 203d 3e20 2e2e 2e2c  ntation) => ...,
│ │ │ │  000136b0: 2064 6566 6175 6c74 2076 616c 7565 207b   default value {
│ │ │ │  000136c0: 7d2c 200a 2020 2020 2020 2a20 2a6e 6f74  }, .      * *not
│ │ │ │  000136d0: 6520 546f 7044 6972 6563 746f 7279 3a20  e TopDirectory: 
│ │ │ │  000136e0: 546f 7044 6972 6563 746f 7279 2c20 3d3e  TopDirectory, =>
│ │ │ │  000136f0: 202e 2e2e 2c20 6465 6661 756c 7420 7661   ..., default va
│ │ │ │  00013700: 6c75 650a 2020 2020 2020 2020 222f 746d  lue.        "/tm
│ │ │ │ -00013710: 702f 4d32 2d32 3837 3036 2d30 2f30 222c  p/M2-28706-0/0",
│ │ │ │ +00013710: 702f 4d32 2d34 3130 3337 2d30 2f30 222c  p/M2-41037-0/0",
│ │ │ │  00013720: 204f 7074 696f 6e20 746f 2063 6861 6e67   Option to chang
│ │ │ │  00013730: 6520 6469 7265 6374 6f72 7920 666f 7220  e directory for 
│ │ │ │  00013740: 6669 6c65 2073 746f 7261 6765 2e0a 2020  file storage..  
│ │ │ │  00013750: 2020 2020 2a20 2a6e 6f74 6520 5665 7262      * *note Verb
│ │ │ │  00013760: 6f73 653a 2062 6572 7469 6e69 5472 6163  ose: bertiniTrac
│ │ │ │  00013770: 6b48 6f6d 6f74 6f70 795f 6c70 5f70 645f  kHomotopy_lp_pd_
│ │ │ │  00013780: 7064 5f70 645f 636d 5665 7262 6f73 653d  pd_pd_cmVerbose=
│ │ │ │ @@ -5472,16 +5472,16 @@
│ │ │ │  000155f0: 6561 6c20 6e75 6d62 6572 0a20 2020 2020  eal number.     
│ │ │ │  00015600: 2020 206f 7220 7261 6e64 6f6d 2063 6f6d     or random com
│ │ │ │  00015610: 706c 6578 206e 756d 6265 720a 2020 2020  plex number.    
│ │ │ │  00015620: 2020 2a20 2a6e 6f74 6520 546f 7044 6972    * *note TopDir
│ │ │ │  00015630: 6563 746f 7279 3a20 546f 7044 6972 6563  ectory: TopDirec
│ │ │ │  00015640: 746f 7279 2c20 3d3e 202e 2e2e 2c20 6465  tory, => ..., de
│ │ │ │  00015650: 6661 756c 7420 7661 6c75 650a 2020 2020  fault value.    
│ │ │ │ -00015660: 2020 2020 222f 746d 702f 4d32 2d32 3837      "/tmp/M2-287
│ │ │ │ -00015670: 3036 2d30 2f30 222c 204f 7074 696f 6e20  06-0/0", Option 
│ │ │ │ +00015660: 2020 2020 222f 746d 702f 4d32 2d34 3130      "/tmp/M2-410
│ │ │ │ +00015670: 3337 2d30 2f30 222c 204f 7074 696f 6e20  37-0/0", Option 
│ │ │ │  00015680: 746f 2063 6861 6e67 6520 6469 7265 6374  to change direct
│ │ │ │  00015690: 6f72 7920 666f 7220 6669 6c65 2073 746f  ory for file sto
│ │ │ │  000156a0: 7261 6765 2e0a 2020 2020 2020 2a20 5573  rage..      * Us
│ │ │ │  000156b0: 6552 6567 656e 6572 6174 696f 6e20 286d  eRegeneration (m
│ │ │ │  000156c0: 6973 7369 6e67 2064 6f63 756d 656e 7461  issing documenta
│ │ │ │  000156d0: 7469 6f6e 2920 3d3e 202e 2e2e 2c20 6465  tion) => ..., de
│ │ │ │  000156e0: 6661 756c 7420 7661 6c75 6520 2d31 2c20  fault value -1,
│ │ ├── ./usr/share/info/BettiCharacters.info.gz
│ │ │ ├── BettiCharacters.info
│ │ │ │ @@ -12972,15 +12972,15 @@
│ │ │ │  00032ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  00032ad0: 0a7c 6939 203a 2065 6c61 7073 6564 5469  .|i9 : elapsedTi
│ │ │ │  00032ae0: 6d65 2063 203d 2063 6861 7261 6374 6572  me c = character
│ │ │ │  00032af0: 2041 2020 2020 2020 2020 2020 2020 2020   A              
│ │ │ │  00032b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032b10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00032b20: 0a7c 202d 2d20 2e35 3539 3133 3573 2065  .| -- .559135s e
│ │ │ │ +00032b20: 0a7c 202d 2d20 2e33 3934 3834 3773 2065  .| -- .394847s e
│ │ │ │  00032b30: 6c61 7073 6564 2020 2020 2020 2020 2020  lapsed          
│ │ │ │  00032b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032b60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00032b70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00032b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -14183,15 +14183,15 @@
│ │ │ │  00037660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037680: 2b0a 7c69 3720 3a20 656c 6170 7365 6454  +.|i7 : elapsedT
│ │ │ │  00037690: 696d 6520 633d 6368 6172 6163 7465 7220  ime c=character 
│ │ │ │  000376a0: 4120 2020 2020 2020 2020 2020 2020 2020  A               
│ │ │ │  000376b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000376c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000376d0: 7c0a 7c20 2d2d 202e 3437 3439 3136 7320  |.| -- .474916s 
│ │ │ │ +000376d0: 7c0a 7c20 2d2d 202e 3435 3138 3233 7320  |.| -- .451823s 
│ │ │ │  000376e0: 656c 6170 7365 6420 2020 2020 2020 2020  elapsed         
│ │ │ │  000376f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037720: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00037730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -15614,16 +15614,16 @@
│ │ │ │  0003cfd0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  0003cfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003cff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d010: 2d2b 0a7c 6932 3020 3a20 656c 6170 7365  -+.|i20 : elapse
│ │ │ │  0003d020: 6454 696d 6520 6131 203d 2063 6861 7261  dTime a1 = chara
│ │ │ │  0003d030: 6374 6572 2041 3120 2020 2020 2020 2020  cter A1         
│ │ │ │ -0003d040: 2020 2020 2020 7c0a 7c20 2d2d 202e 3835        |.| -- .85
│ │ │ │ -0003d050: 3237 3632 7320 656c 6170 7365 6420 2020  2762s elapsed   
│ │ │ │ +0003d040: 2020 2020 2020 7c0a 7c20 2d2d 202e 3639        |.| -- .69
│ │ │ │ +0003d050: 3237 3033 7320 656c 6170 7365 6420 2020  2703s elapsed   
│ │ │ │  0003d060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d070: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0003d080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d0b0: 7c0a 7c6f 3230 203d 2043 6861 7261 6374  |.|o20 = Charact
│ │ │ │  0003d0c0: 6572 206f 7665 7220 5220 2020 2020 2020  er over R       
│ │ │ │ @@ -15654,16 +15654,16 @@
│ │ │ │  0003d250: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │  0003d260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  0003d290: 6932 3120 3a20 656c 6170 7365 6454 696d  i21 : elapsedTim
│ │ │ │  0003d2a0: 6520 6132 203d 2063 6861 7261 6374 6572  e a2 = character
│ │ │ │  0003d2b0: 2041 3220 2020 2020 2020 2020 2020 2020   A2             
│ │ │ │ -0003d2c0: 2020 7c0a 7c20 2d2d 2033 342e 3334 3473    |.| -- 34.344s
│ │ │ │ -0003d2d0: 2065 6c61 7073 6564 2020 2020 2020 2020   elapsed        
│ │ │ │ +0003d2c0: 2020 7c0a 7c20 2d2d 2032 352e 3434 3031    |.| -- 25.4401
│ │ │ │ +0003d2d0: 7320 656c 6170 7365 6420 2020 2020 2020  s elapsed       
│ │ │ │  0003d2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d2f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0003d300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d320: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │  0003d330: 3231 203d 2043 6861 7261 6374 6572 206f  21 = Character o
│ │ │ │  0003d340: 7665 7220 5220 2020 2020 2020 2020 2020  ver R           
│ │ │ │ @@ -16112,16 +16112,16 @@
│ │ │ │  0003eef0: 6f33 3120 3a20 4163 7469 6f6e 4f6e 4772  o31 : ActionOnGr
│ │ │ │  0003ef00: 6164 6564 4d6f 6475 6c65 2020 2020 2020  adedModule      
│ │ │ │  0003ef10: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  0003ef20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ef30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  0003ef40: 6933 3220 3a20 656c 6170 7365 6454 696d  i32 : elapsedTim
│ │ │ │  0003ef50: 6520 6220 3d20 6368 6172 6163 7465 7228  e b = character(
│ │ │ │ -0003ef60: 422c 3231 297c 0a7c 202d 2d20 3134 2e32  B,21)|.| -- 14.2
│ │ │ │ -0003ef70: 3631 3773 2065 6c61 7073 6564 2020 2020  617s elapsed    
│ │ │ │ +0003ef60: 422c 3231 297c 0a7c 202d 2d20 3131 2e34  B,21)|.| -- 11.4
│ │ │ │ +0003ef70: 3530 3473 2065 6c61 7073 6564 2020 2020  504s elapsed    
│ │ │ │  0003ef80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0003ef90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003efa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003efb0: 2020 2020 207c 0a7c 6f33 3220 3d20 4368       |.|o32 = Ch
│ │ │ │  0003efc0: 6172 6163 7465 7220 6f76 6572 2052 2020  aracter over R  
│ │ │ │  0003efd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0003efe0: 2020 2020 2020 2020 2020 2020 2020 2020
│ │ ├── ./usr/share/info/Bruns.info.gz
│ │ │ ├── Bruns.info
│ │ │ │ @@ -1095,18 +1095,18 @@
│ │ │ │  00004460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  00004480: 6932 3320 3a20 7469 6d65 206a 3d62 7275  i23 : time j=bru
│ │ │ │  00004490: 6e73 2046 2e64 645f 333b 2020 2020 2020  ns F.dd_3;      
│ │ │ │  000044a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000044b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000044c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000044d0: 202d 2d20 7573 6564 2030 2e33 3534 3231   -- used 0.35421
│ │ │ │ -000044e0: 3873 2028 6370 7529 3b20 302e 3238 3930  8s (cpu); 0.2890
│ │ │ │ -000044f0: 3037 7320 2874 6872 6561 6429 3b20 3073  07s (thread); 0s
│ │ │ │ -00004500: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │ +000044d0: 202d 2d20 7573 6564 2030 2e32 3635 3534   -- used 0.26554
│ │ │ │ +000044e0: 3373 2028 6370 7529 3b20 302e 3139 3838  3s (cpu); 0.1988
│ │ │ │ +000044f0: 3973 2028 7468 7265 6164 293b 2030 7320  9s (thread); 0s 
│ │ │ │ +00004500: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │  00004510: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00004520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004560: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00004570: 6f32 3320 3a20 4964 6561 6c20 6f66 2053  o23 : Ideal of S
│ │ ├── ./usr/share/info/CellularResolutions.info.gz
│ │ │ ├── CellularResolutions.info
│ │ │ │ @@ -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                  
│ │ │ │ @@ -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  ----------------
│ │ │ │ @@ -4011,17 +4011,17 @@
│ │ │ │  0000faa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fab0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  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
│ │ │ │ +0000fb10: 7769 7468 206c 6162 656c 2078 2c20 4365  with label x, 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 207a 7d7d  0 with label z}}
│ │ │ │  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 782c 2043 656c 6c20 6f66 2064 696d  l x, 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,    |.|   
│ │ │ │ +00010ca0: 6265 6c20 7a2c 2020 2020 7c0a 7c20 2020  bel z,    |.|   
│ │ │ │  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 
│ │ │ │ @@ -8240,21 +8240,21 @@
│ │ │ │  000202f0: 6c4c 6162 656c 2863 2920 2020 2020 2020  lLabel(c)       
│ │ │ │  00020300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020310: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00020320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020360: 2020 7c0a 7c20 2020 2020 2020 2032 2020    |.|        2  
│ │ │ │ -00020370: 2020 2020 3220 2020 3220 2020 2020 2020      2   2       
│ │ │ │ +00020360: 2020 7c0a 7c20 2020 2020 2020 2020 2032    |.|          2
│ │ │ │ +00020370: 2020 2032 2020 2020 2020 2020 3220 2020     2        2   
│ │ │ │  00020380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000203a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000203b0: 2020 7c0a 7c6f 3133 203d 207b 6120 622c    |.|o13 = {a b,
│ │ │ │ -000203c0: 2062 2a63 202c 2062 202c 2061 2a63 7d20   b*c , b , a*c} 
│ │ │ │ +000203b0: 2020 7c0a 7c6f 3133 203d 207b 622a 6320    |.|o13 = {b*c 
│ │ │ │ +000203c0: 2c20 6220 2c20 612a 632c 2061 2062 7d20  , b , a*c, a b} 
│ │ │ │  000203d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000203e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000203f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020400: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00020410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -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 2020 2020 2032 2020 2020 2032 2020         2     2  
│ │ │ │ +000205b0: 2020 3220 2020 3220 2020 3220 3220 2020    2   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 612a 622a 6320 2c20 612a 6220 632c  , a*b*c , a*b c,
│ │ │ │ +00020600: 2061 2062 2a63 202c 2061 2062 207d 2020   a b*c , a b }  
│ │ │ │  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
│ │ │ │ @@ -4819,17 +4819,17 @@
│ │ │ │  00012d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  00012d50: 3133 203a 2074 696d 6520 6d20 3d20 6d69  13 : time m = mi
│ │ │ │  00012d60: 6e69 6d69 7a65 2028 455b 315d 293b 2020  nimize (E[1]);  
│ │ │ │  00012d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012d80: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ -00012d90: 302e 3239 3737 3232 7320 2863 7075 293b  0.297722s (cpu);
│ │ │ │ -00012da0: 2030 2e32 3339 3538 3473 2028 7468 7265   0.239584s (thre
│ │ │ │ -00012db0: 6164 293b 2030 7320 2867 6329 7c0a 2b2d  ad); 0s (gc)|.+-
│ │ │ │ +00012d90: 302e 3333 3839 7320 2863 7075 293b 2030  0.3389s (cpu); 0
│ │ │ │ +00012da0: 2e32 3633 3935 3973 2028 7468 7265 6164  .263959s (thread
│ │ │ │ +00012db0: 293b 2030 7320 2867 6329 2020 7c0a 2b2d  ); 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               
│ │ │ │  00012e20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ @@ -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 3638 3333  -- used 0.096833
│ │ │ │ -00019ba0: 3673 2028 6370 7529 3b20 302e 3039 3638  6s (cpu); 0.0968
│ │ │ │ -00019bb0: 3332 3773 2028 7468 7265 6164 293b 2030  327s (thread); 0
│ │ │ │ -00019bc0: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │ +00019b90: 2d2d 2075 7365 6420 302e 3130 3335 3738  -- used 0.103578
│ │ │ │ +00019ba0: 7320 2863 7075 293b 2030 2e31 3033 3537  s (cpu); 0.10357
│ │ │ │ +00019bb0: 3973 2028 7468 7265 6164 293b 2030 7320  9s (thread); 0s 
│ │ │ │ +00019bc0: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │  00019bd0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  00019be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  00019c30: 3920 3a20 7469 6d65 206e 203d 2063 6172  9 : time n = car
│ │ │ │  00019c40: 7461 6e45 696c 656e 6265 7267 5265 736f  tanEilenbergReso
│ │ │ │  00019c50: 6c75 7469 6f6e 2043 3b20 2020 2020 2020  lution C;       
│ │ │ │  00019c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019c70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00019c80: 2d2d 2075 7365 6420 302e 3232 3630 3032  -- used 0.226002
│ │ │ │ -00019c90: 7320 2863 7075 293b 2030 2e31 3536 3739  s (cpu); 0.15679
│ │ │ │ -00019ca0: 3673 2028 7468 7265 6164 293b 2030 7320  6s (thread); 0s 
│ │ │ │ -00019cb0: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │ +00019c80: 2d2d 2075 7365 6420 302e 3234 3038 3773  -- used 0.24087s
│ │ │ │ +00019c90: 2028 6370 7529 3b20 302e 3136 3438 3734   (cpu); 0.164874
│ │ │ │ +00019ca0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ +00019cb0: 6763 2920 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 2e32 3436 3137 3473 2028  used 0.246174s (
│ │ │ │ -00004c60: 6370 7529 3b20 302e 3136 3436 3234 7320  cpu); 0.164624s 
│ │ │ │ +00004c50: 7573 6564 2030 2e32 3434 3139 3273 2028  used 0.244192s (
│ │ │ │ +00004c60: 6370 7529 3b20 302e 3136 3330 3534 7320  cpu); 0.163054s 
│ │ │ │  00004c70: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │  00004c80: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  00004c90: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00004ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -1300,16 +1300,16 @@
│ │ │ │  00005130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00005140: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a  ---------+.|i4 :
│ │ │ │  00005150: 2074 696d 6520 4353 4d28 552c 4368 6563   time CSM(U,Chec
│ │ │ │  00005160: 6b53 6d6f 6f74 683d 3e66 616c 7365 2920  kSmooth=>false) 
│ │ │ │  00005170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005190: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ -000051a0: 7573 6564 2030 2e33 3734 3331 3273 2028  used 0.374312s (
│ │ │ │ -000051b0: 6370 7529 3b20 302e 3239 3531 3235 7320  cpu); 0.295125s 
│ │ │ │ +000051a0: 7573 6564 2030 2e34 3133 3030 3773 2028  used 0.413007s (
│ │ │ │ +000051b0: 6370 7529 3b20 302e 3331 3330 3632 7320  cpu); 0.313062s 
│ │ │ │  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                  
│ │ │ │ @@ -4341,17 +4341,17 @@
│ │ │ │  00010f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010f60: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a  ---------+.|i5 :
│ │ │ │  00010f70: 2074 696d 6520 4353 4d28 492c 436f 6d70   time CSM(I,Comp
│ │ │ │  00010f80: 4d65 7468 6f64 3d3e 5072 6f6a 6563 7469  Method=>Projecti
│ │ │ │  00010f90: 7665 4465 6772 6565 2920 2020 2020 2020  veDegree)       
│ │ │ │  00010fa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00010fb0: 2d2d 2075 7365 6420 302e 3632 3933 3032  -- used 0.629302
│ │ │ │ -00010fc0: 7320 2863 7075 293b 2030 2e33 3031 3631  s (cpu); 0.30161
│ │ │ │ -00010fd0: 3373 2028 7468 7265 6164 293b 2030 7320  3s (thread); 0s 
│ │ │ │ +00010fb0: 2d2d 2075 7365 6420 302e 3836 3637 3033  -- used 0.866703
│ │ │ │ +00010fc0: 7320 2863 7075 293b 2030 2e33 3736 3936  s (cpu); 0.37696
│ │ │ │ +00010fd0: 3573 2028 7468 7265 6164 293b 2030 7320  5s (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   
│ │ │ │  00011040: 2020 2034 2020 2020 2020 3320 2020 2020     4      3     
│ │ │ │ @@ -4400,17 +4400,17 @@
│ │ │ │  000112f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011310: 2d2d 2d2b 0a7c 6936 203a 2074 696d 6520  ---+.|i6 : time 
│ │ │ │  00011320: 4353 4d28 492c 436f 6d70 4d65 7468 6f64  CSM(I,CompMethod
│ │ │ │  00011330: 3d3e 506e 5265 7369 6475 616c 2920 2020  =>PnResidual)   
│ │ │ │  00011340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011350: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ -00011360: 6420 322e 3133 3132 3773 2028 6370 7529  d 2.13127s (cpu)
│ │ │ │ -00011370: 3b20 312e 3832 3435 3573 2028 7468 7265  ; 1.82455s (thre
│ │ │ │ -00011380: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │ +00011360: 6420 322e 3432 3232 7320 2863 7075 293b  d 2.4222s (cpu);
│ │ │ │ +00011370: 2032 2e31 3434 3532 7320 2874 6872 6561   2.14452s (threa
│ │ │ │ +00011380: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 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  
│ │ │ │  000113f0: 2020 2020 3320 2020 2020 2032 2020 2020      3      2    
│ │ │ │ @@ -4488,16 +4488,16 @@
│ │ │ │  00011870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011890: 2d2d 2b0a 7c69 3130 203a 2074 696d 6520  --+.|i10 : time 
│ │ │ │  000118a0: 4353 4d28 4b2c 436f 6d70 4d65 7468 6f64  CSM(K,CompMethod
│ │ │ │  000118b0: 3d3e 5072 6f6a 6563 7469 7665 4465 6772  =>ProjectiveDegr
│ │ │ │  000118c0: 6565 2920 2020 2020 2020 2020 2020 2020  ee)             
│ │ │ │  000118d0: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -000118e0: 2030 2e32 3739 3130 3673 2028 6370 7529   0.279106s (cpu)
│ │ │ │ -000118f0: 3b20 302e 3139 3532 3732 7320 2874 6872  ; 0.195272s (thr
│ │ │ │ +000118e0: 2030 2e33 3432 3030 3973 2028 6370 7529   0.342009s (cpu)
│ │ │ │ +000118f0: 3b20 302e 3235 3135 3235 7320 2874 6872  ; 0.251525s (thr
│ │ │ │  00011900: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │  00011910: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00011920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011950: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00011960: 2020 2020 2020 3320 2020 2020 3220 2020        3     2   
│ │ │ │ @@ -4546,18 +4546,18 @@
│ │ │ │  00011c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  00011c40: 3131 203a 2074 696d 6520 4353 4d28 4b2c  11 : time CSM(K,
│ │ │ │  00011c50: 436f 6d70 4d65 7468 6f64 3d3e 506e 5265  CompMethod=>PnRe
│ │ │ │  00011c60: 7369 6475 616c 2920 2020 2020 2020 2020  sidual)         
│ │ │ │  00011c70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00011c80: 0a7c 202d 2d20 7573 6564 2030 2e30 3832  .| -- used 0.082
│ │ │ │ -00011c90: 3531 3433 7320 2863 7075 293b 2030 2e30  5143s (cpu); 0.0
│ │ │ │ -00011ca0: 3832 3532 3039 7320 2874 6872 6561 6429  825209s (thread)
│ │ │ │ -00011cb0: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ +00011c80: 0a7c 202d 2d20 7573 6564 2030 2e31 3032  .| -- used 0.102
│ │ │ │ +00011c90: 3134 3873 2028 6370 7529 3b20 302e 3130  148s (cpu); 0.10
│ │ │ │ +00011ca0: 3231 3773 2028 7468 7265 6164 293b 2030  217s (thread); 0
│ │ │ │ +00011cb0: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │  00011cc0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00011cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011d00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00011d10: 3320 2020 2020 3220 2020 2020 2020 2020  3     2         
│ │ │ │  00011d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -5446,6793 +5446,6788 @@
│ │ │ │  00015450: 2072 6574 7572 6e65 6420 696e 2074 6865   returned in the
│ │ │ │  00015460: 2073 616d 6520 7269 6e67 2e20 5765 206d   same ring. We m
│ │ │ │  00015470: 6179 2061 6c73 6f20 7265 7475 726e 2061  ay also return a
│ │ │ │  00015480: 0a4d 7574 6162 6c65 4861 7368 5461 626c  .MutableHashTabl
│ │ │ │  00015490: 652e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  e...+-----------
│ │ │ │  000154a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000154b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000154c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131  ----------+.|i11
│ │ │ │ -000154d0: 203a 2052 3d4d 756c 7469 5072 6f6a 436f   : R=MultiProjCo
│ │ │ │ -000154e0: 6f72 6452 696e 6728 7b32 2c32 7d29 2020  ordRing({2,2})  
│ │ │ │ +000154c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120  ---------+.|i11 
│ │ │ │ +000154d0: 3a20 523d 4d75 6c74 6950 726f 6a43 6f6f  : R=MultiProjCoo
│ │ │ │ +000154e0: 7264 5269 6e67 287b 322c 327d 2920 2020  rdRing({2,2})   
│ │ │ │  000154f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015500: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00015500: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00015510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015530: 2020 2020 2020 2020 2020 7c0a 7c6f 3131            |.|o11
│ │ │ │ -00015540: 203d 2052 2020 2020 2020 2020 2020 2020   = R            
│ │ │ │ +00015530: 2020 2020 2020 207c 0a7c 6f31 3120 3d20         |.|o11 = 
│ │ │ │ +00015540: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │  00015550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015570: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00015560: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015570: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00015580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000155a0: 2020 2020 2020 2020 2020 7c0a 7c6f 3131            |.|o11
│ │ │ │ -000155b0: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e   : PolynomialRin
│ │ │ │ -000155c0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ -000155d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000155e0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +000155a0: 2020 2020 207c 0a7c 6f31 3120 3a20 506f       |.|o11 : Po
│ │ │ │ +000155b0: 6c79 6e6f 6d69 616c 5269 6e67 2020 2020  lynomialRing    
│ │ │ │ +000155c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000155d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +000155e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000155f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015610: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3132  ----------+.|i12
│ │ │ │ -00015620: 203a 2041 3d43 686f 7752 696e 6728 5229   : A=ChowRing(R)
│ │ │ │ +00015610: 2d2d 2d2b 0a7c 6931 3220 3a20 413d 4368  ---+.|i12 : A=Ch
│ │ │ │ +00015620: 6f77 5269 6e67 2852 2920 2020 2020 2020  owRing(R)       
│ │ │ │  00015630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015650: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00015640: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00015650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015680: 2020 2020 2020 2020 2020 7c0a 7c6f 3132            |.|o12
│ │ │ │ -00015690: 203d 2041 2020 2020 2020 2020 2020 2020   = A            
│ │ │ │ +00015680: 207c 0a7c 6f31 3220 3d20 4120 2020 2020   |.|o12 = A     
│ │ │ │ +00015690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000156a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000156b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000156c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000156b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000156c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000156d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000156e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000156f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3132            |.|o12
│ │ │ │ -00015700: 203a 2051 756f 7469 656e 7452 696e 6720   : QuotientRing 
│ │ │ │ +000156e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000156f0: 0a7c 6f31 3220 3a20 5175 6f74 6965 6e74  .|o12 : Quotient
│ │ │ │ +00015700: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │  00015710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015730: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00015720: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00015730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015760: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3133  ----------+.|i13
│ │ │ │ -00015770: 203a 2072 3d67 656e 7320 5220 2020 2020   : r=gens R     
│ │ │ │ +00015750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00015760: 6931 3320 3a20 723d 6765 6e73 2052 2020  i13 : r=gens R  
│ │ │ │ +00015770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000157a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00015790: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000157a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000157b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000157c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000157d0: 2020 2020 2020 2020 2020 7c0a 7c6f 3133            |.|o13
│ │ │ │ -000157e0: 203d 207b 7820 2c20 7820 2c20 7820 2c20   = {x , x , x , 
│ │ │ │ -000157f0: 7820 2c20 7820 2c20 7820 7d20 2020 2020  x , x , x }     
│ │ │ │ -00015800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015810: 2020 7c0a 7c20 2020 2020 2020 2030 2020    |.|        0  
│ │ │ │ -00015820: 2031 2020 2032 2020 2033 2020 2034 2020   1   2   3   4  
│ │ │ │ -00015830: 2035 2020 2020 2020 2020 2020 2020 2020   5              
│ │ │ │ -00015840: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000157c0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +000157d0: 3320 3d20 7b78 202c 2078 202c 2078 202c  3 = {x , x , x ,
│ │ │ │ +000157e0: 2078 202c 2078 202c 2078 207d 2020 2020   x , x , x }    
│ │ │ │ +000157f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015800: 2020 7c0a 7c20 2020 2020 2020 2030 2020    |.|        0  
│ │ │ │ +00015810: 2031 2020 2032 2020 2033 2020 2034 2020   1   2   3   4  
│ │ │ │ +00015820: 2035 2020 2020 2020 2020 2020 2020 2020   5              
│ │ │ │ +00015830: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00015840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015880: 2020 7c0a 7c6f 3133 203a 204c 6973 7420    |.|o13 : List 
│ │ │ │ +00015870: 7c0a 7c6f 3133 203a 204c 6973 7420 2020  |.|o13 : List   
│ │ │ │ +00015880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000158a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000158b0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +000158a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000158b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000158c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000158d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000158e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000158f0: 2d2d 2b0a 7c69 3134 203a 204b 3d69 6465  --+.|i14 : K=ide
│ │ │ │ -00015900: 616c 2872 5f30 5e32 2a72 5f33 2d72 5f34  al(r_0^2*r_3-r_4
│ │ │ │ -00015910: 2a72 5f31 2a72 5f32 2c72 5f32 5e32 2a72  *r_1*r_2,r_2^2*r
│ │ │ │ -00015920: 5f35 2920 2020 2020 2020 7c0a 7c20 2020  _5)       |.|   
│ │ │ │ +000158d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000158e0: 7c69 3134 203a 204b 3d69 6465 616c 2872  |i14 : K=ideal(r
│ │ │ │ +000158f0: 5f30 5e32 2a72 5f33 2d72 5f34 2a72 5f31  _0^2*r_3-r_4*r_1
│ │ │ │ +00015900: 2a72 5f32 2c72 5f32 5e32 2a72 5f35 2920  *r_2,r_2^2*r_5) 
│ │ │ │ +00015910: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00015920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015960: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00015970: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -00015980: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -00015990: 2020 2020 2020 2020 2020 7c0a 7c6f 3134            |.|o14
│ │ │ │ -000159a0: 203d 2069 6465 616c 2028 7820 7820 202d   = ideal (x x  -
│ │ │ │ -000159b0: 2078 2078 2078 202c 2078 2078 2029 2020   x x x , x x )  
│ │ │ │ -000159c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000159d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -000159e0: 2020 2030 2033 2020 2020 3120 3220 3420     0 3    1 2 4 
│ │ │ │ -000159f0: 2020 3220 3520 2020 2020 2020 2020 2020    2 5           
│ │ │ │ -00015a00: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00015940: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00015950: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +00015960: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +00015970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015980: 2020 207c 0a7c 6f31 3420 3d20 6964 6561     |.|o14 = idea
│ │ │ │ +00015990: 6c20 2878 2078 2020 2d20 7820 7820 7820  l (x x  - x x x 
│ │ │ │ +000159a0: 2c20 7820 7820 2920 2020 2020 2020 2020  , x x )         
│ │ │ │ +000159b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000159c0: 2020 2020 2020 2020 2020 2030 2033 2020             0 3  
│ │ │ │ +000159d0: 2020 3120 3220 3420 2020 3220 3520 2020    1 2 4   2 5   
│ │ │ │ +000159e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000159f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00015a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a40: 2020 7c0a 7c6f 3134 203a 2049 6465 616c    |.|o14 : Ideal
│ │ │ │ -00015a50: 206f 6620 5220 2020 2020 2020 2020 2020   of R           
│ │ │ │ -00015a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a70: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00015a20: 2020 2020 2020 2020 7c0a 7c6f 3134 203a          |.|o14 :
│ │ │ │ +00015a30: 2049 6465 616c 206f 6620 5220 2020 2020   Ideal of R     
│ │ │ │ +00015a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015a50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00015a60: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00015a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015ab0: 2d2d 2b0a 7c69 3135 203a 2074 696d 6520  --+.|i15 : time 
│ │ │ │ -00015ac0: 6373 6d4b 3d43 534d 2841 2c4b 2920 2020  csmK=CSM(A,K)   
│ │ │ │ -00015ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015ae0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ -00015af0: 2075 7365 6420 302e 3938 3435 3736 7320   used 0.984576s 
│ │ │ │ -00015b00: 2863 7075 293b 2030 2e34 3633 3332 3673  (cpu); 0.463326s
│ │ │ │ -00015b10: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ -00015b20: 6329 7c0a 7c20 2020 2020 2020 2020 2020  c)|.|           
│ │ │ │ -00015b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015b50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00015b60: 2020 2020 2032 2032 2020 2020 2032 2020       2 2     2  
│ │ │ │ -00015b70: 2020 2020 2020 2032 2020 2020 3220 2020         2    2   
│ │ │ │ -00015b80: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ -00015b90: 2020 7c0a 7c6f 3135 203d 2037 6820 6820    |.|o15 = 7h h 
│ │ │ │ -00015ba0: 202b 2035 6820 6820 202b 2034 6820 6820   + 5h h  + 4h h 
│ │ │ │ -00015bb0: 202b 2068 2020 2b20 3368 2068 2020 2b20   + h  + 3h h  + 
│ │ │ │ -00015bc0: 6820 2020 2020 2020 2020 7c0a 7c20 2020  h         |.|   
│ │ │ │ -00015bd0: 2020 2020 2031 2032 2020 2020 2031 2032       1 2     1 2
│ │ │ │ -00015be0: 2020 2020 2031 2032 2020 2020 3120 2020       1 2    1   
│ │ │ │ -00015bf0: 2020 3120 3220 2020 2032 2020 2020 2020    1 2    2      
│ │ │ │ -00015c00: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00015c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015c30: 2020 2020 2020 2020 2020 7c0a 7c6f 3135            |.|o15
│ │ │ │ -00015c40: 203a 2041 2020 2020 2020 2020 2020 2020   : A            
│ │ │ │ -00015c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015c70: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -00015c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3136  ----------+.|i16
│ │ │ │ -00015cb0: 203a 2063 736d 4b48 6173 683d 2043 534d   : csmKHash= CSM
│ │ │ │ -00015cc0: 2841 2c4b 2c4f 7574 7075 743d 3e48 6173  (A,K,Output=>Has
│ │ │ │ -00015cd0: 6846 6f72 6d29 2020 2020 2020 2020 2020  hForm)          
│ │ │ │ -00015ce0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00015cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015d10: 2020 2020 2020 2020 2020 7c0a 7c6f 3136            |.|o16
│ │ │ │ -00015d20: 203d 204d 7574 6162 6c65 4861 7368 5461   = MutableHashTa
│ │ │ │ -00015d30: 626c 657b 2e2e 2e34 2e2e 2e7d 2020 2020  ble{...4...}    
│ │ │ │ +00015a90: 2d2d 2d2d 2d2d 2b0a 7c69 3135 203a 2074  ------+.|i15 : t
│ │ │ │ +00015aa0: 696d 6520 6373 6d4b 3d43 534d 2841 2c4b  ime csmK=CSM(A,K
│ │ │ │ +00015ab0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +00015ac0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00015ad0: 202d 2d20 7573 6564 2031 2e32 3335 3534   -- used 1.23554
│ │ │ │ +00015ae0: 7320 2863 7075 293b 2030 2e34 3134 3835  s (cpu); 0.41485
│ │ │ │ +00015af0: 3973 2028 7468 7265 6164 293b 2030 7320  9s (thread); 0s 
│ │ │ │ +00015b00: 2867 6329 7c0a 7c20 2020 2020 2020 2020  (gc)|.|         
│ │ │ │ +00015b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015b30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00015b40: 2020 2020 2020 3220 3220 2020 2020 3220        2 2     2 
│ │ │ │ +00015b50: 2020 2020 2020 2020 3220 2020 2032 2020          2    2  
│ │ │ │ +00015b60: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +00015b70: 2020 7c0a 7c6f 3135 203d 2037 6820 6820    |.|o15 = 7h h 
│ │ │ │ +00015b80: 202b 2035 6820 6820 202b 2034 6820 6820   + 5h h  + 4h h 
│ │ │ │ +00015b90: 202b 2068 2020 2b20 3368 2068 2020 2b20   + h  + 3h h  + 
│ │ │ │ +00015ba0: 6820 2020 2020 2020 207c 0a7c 2020 2020  h        |.|    
│ │ │ │ +00015bb0: 2020 2020 3120 3220 2020 2020 3120 3220      1 2     1 2 
│ │ │ │ +00015bc0: 2020 2020 3120 3220 2020 2031 2020 2020      1 2    1    
│ │ │ │ +00015bd0: 2031 2032 2020 2020 3220 2020 2020 2020   1 2    2       
│ │ │ │ +00015be0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00015bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015c10: 2020 2020 2020 207c 0a7c 6f31 3520 3a20         |.|o15 : 
│ │ │ │ +00015c20: 4120 2020 2020 2020 2020 2020 2020 2020  A               
│ │ │ │ +00015c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015c40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015c50: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00015c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015c80: 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20 6373  -----+.|i16 : cs
│ │ │ │ +00015c90: 6d4b 4861 7368 3d20 4353 4d28 412c 4b2c  mKHash= CSM(A,K,
│ │ │ │ +00015ca0: 4f75 7470 7574 3d3e 4861 7368 466f 726d  Output=>HashForm
│ │ │ │ +00015cb0: 2920 2020 2020 2020 2020 2020 7c0a 7c20  )           |.| 
│ │ │ │ +00015cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015cf0: 2020 207c 0a7c 6f31 3620 3d20 4d75 7461     |.|o16 = Muta
│ │ │ │ +00015d00: 626c 6548 6173 6854 6162 6c65 7b2e 2e2e  bleHashTable{...
│ │ │ │ +00015d10: 342e 2e2e 7d20 2020 2020 2020 2020 2020  4...}           
│ │ │ │ +00015d20: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00015d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015d50: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00015d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015d80: 2020 2020 2020 2020 2020 7c0a 7c6f 3136            |.|o16
│ │ │ │ -00015d90: 203a 204d 7574 6162 6c65 4861 7368 5461   : MutableHashTa
│ │ │ │ -00015da0: 626c 6520 2020 2020 2020 2020 2020 2020  ble             
│ │ │ │ -00015db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015dc0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -00015dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3137  ----------+.|i17
│ │ │ │ -00015e00: 203a 2063 736d 4b3d 3d63 736d 4b48 6173   : csmK==csmKHas
│ │ │ │ -00015e10: 6823 2243 534d 2220 2020 2020 2020 2020  h#"CSM"         
│ │ │ │ +00015d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015d60: 207c 0a7c 6f31 3620 3a20 4d75 7461 626c   |.|o16 : Mutabl
│ │ │ │ +00015d70: 6548 6173 6854 6162 6c65 2020 2020 2020  eHashTable      
│ │ │ │ +00015d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015d90: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00015da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00015dd0: 0a7c 6931 3720 3a20 6373 6d4b 3d3d 6373  .|i17 : csmK==cs
│ │ │ │ +00015de0: 6d4b 4861 7368 2322 4353 4d22 2020 2020  mKHash#"CSM"    
│ │ │ │ +00015df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015e00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00015e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015e30: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00015e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015e30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00015e40: 6f31 3720 3d20 7472 7565 2020 2020 2020  o17 = true      
│ │ │ │  00015e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015e60: 2020 2020 2020 2020 2020 7c0a 7c6f 3137            |.|o17
│ │ │ │ -00015e70: 203d 2074 7275 6520 2020 2020 2020 2020   = true         
│ │ │ │ -00015e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015ea0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -00015eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138  ----------+.|i18
│ │ │ │ -00015ee0: 203a 2043 534d 2841 2c69 6465 616c 284b   : CSM(A,ideal(K
│ │ │ │ -00015ef0: 5f30 2929 3d3d 6373 6d4b 4861 7368 237b  _0))==csmKHash#{
│ │ │ │ -00015f00: 307d 2020 2020 2020 2020 2020 2020 2020  0}              
│ │ │ │ -00015f10: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00015f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015e70: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00015e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00015eb0: 3820 3a20 4353 4d28 412c 6964 6561 6c28  8 : CSM(A,ideal(
│ │ │ │ +00015ec0: 4b5f 3029 293d 3d63 736d 4b48 6173 6823  K_0))==csmKHash#
│ │ │ │ +00015ed0: 7b30 7d20 2020 2020 2020 2020 2020 2020  {0}             
│ │ │ │ +00015ee0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00015ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015f10: 2020 2020 2020 2020 207c 0a7c 6f31 3820           |.|o18 
│ │ │ │ +00015f20: 3d20 7472 7565 2020 2020 2020 2020 2020  = true          
│ │ │ │  00015f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015f40: 2020 2020 2020 2020 2020 7c0a 7c6f 3138            |.|o18
│ │ │ │ -00015f50: 203d 2074 7275 6520 2020 2020 2020 2020   = true         
│ │ │ │ -00015f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015f80: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -00015f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 7570  ----------+..Sup
│ │ │ │ -00015fc0: 706f 7365 2077 6520 6861 7665 2061 6c72  pose we have alr
│ │ │ │ -00015fd0: 6561 6479 2063 6f6d 7075 7465 6420 736f  eady computed so
│ │ │ │ -00015fe0: 6d65 206f 6620 4353 4d20 636c 6173 7365  me of CSM classe
│ │ │ │ -00015ff0: 7320 6f66 2068 7970 6572 7375 7266 6163  s of hypersurfac
│ │ │ │ -00016000: 6573 2069 6e76 6f6c 7665 640a 696e 2074  es involved.in t
│ │ │ │ -00016010: 6865 2069 6e63 6c75 7369 6f6e 2d65 7863  he inclusion-exc
│ │ │ │ -00016020: 6c75 7369 6f6e 2070 726f 6365 6475 7265  lusion procedure
│ │ │ │ -00016030: 2c20 7468 656e 2077 6520 6d61 7920 696e  , then we may in
│ │ │ │ -00016040: 7075 7420 7468 6573 6520 746f 2062 6520  put these to be 
│ │ │ │ -00016050: 7573 6564 2062 7920 7468 650a 4353 4d20  used by the.CSM 
│ │ │ │ -00016060: 6675 6e63 7469 6f6e 2e20 496e 2074 6865  function. In the
│ │ │ │ -00016070: 2065 7861 6d70 6c65 2062 656c 6f77 2077   example below w
│ │ │ │ -00016080: 6520 696e 7075 7420 7468 6520 4353 4d20  e input the CSM 
│ │ │ │ -00016090: 636c 6173 7320 6f66 2056 284b 5f30 2920  class of V(K_0) 
│ │ │ │ -000160a0: 2874 6861 7420 6973 206f 660a 7468 6520  (that is of.the 
│ │ │ │ -000160b0: 6879 7065 7273 7572 6661 6365 2064 6566  hypersurface def
│ │ │ │ -000160c0: 696e 6564 2062 7920 7468 6520 6669 7273  ined by the firs
│ │ │ │ -000160d0: 7420 706f 6c79 6e6f 6d69 616c 2067 656e  t polynomial gen
│ │ │ │ -000160e0: 6572 6174 696e 6720 4b29 2061 6e64 2074  erating K) and t
│ │ │ │ -000160f0: 6865 2043 534d 0a63 6c61 7373 206f 6620  he CSM.class of 
│ │ │ │ -00016100: 7468 6520 6879 7065 7273 7572 6661 6365  the hypersurface
│ │ │ │ -00016110: 2064 6566 696e 6564 2062 7920 7468 6520   defined by the 
│ │ │ │ -00016120: 7072 6f64 7563 7420 6f66 2074 6865 2067  product of the g
│ │ │ │ -00016130: 656e 6572 6174 6f72 7320 6f66 204b 2e0a  enerators of K..
│ │ │ │ -00016140: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00016150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00016180: 6931 3920 3a20 6d3d 6e65 7720 4d75 7461  i19 : m=new Muta
│ │ │ │ -00016190: 626c 6548 6173 6854 6162 6c65 3b20 2020  bleHashTable;   
│ │ │ │ -000161a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000161b0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -000161c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000161d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000161e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000161f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3020  ---------+.|i20 
│ │ │ │ -00016200: 3a20 6d23 7b30 7d3d 6373 6d4b 4861 7368  : m#{0}=csmKHash
│ │ │ │ -00016210: 237b 307d 2020 2020 2020 2020 2020 2020  #{0}            
│ │ │ │ +00015f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015f50: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00015f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015f80: 2d2d 2d2d 2d2d 2d2b 0a0a 5375 7070 6f73  -------+..Suppos
│ │ │ │ +00015f90: 6520 7765 2068 6176 6520 616c 7265 6164  e we have alread
│ │ │ │ +00015fa0: 7920 636f 6d70 7574 6564 2073 6f6d 6520  y computed some 
│ │ │ │ +00015fb0: 6f66 2043 534d 2063 6c61 7373 6573 206f  of CSM classes o
│ │ │ │ +00015fc0: 6620 6879 7065 7273 7572 6661 6365 7320  f hypersurfaces 
│ │ │ │ +00015fd0: 696e 766f 6c76 6564 0a69 6e20 7468 6520  involved.in the 
│ │ │ │ +00015fe0: 696e 636c 7573 696f 6e2d 6578 636c 7573  inclusion-exclus
│ │ │ │ +00015ff0: 696f 6e20 7072 6f63 6564 7572 652c 2074  ion procedure, t
│ │ │ │ +00016000: 6865 6e20 7765 206d 6179 2069 6e70 7574  hen we may input
│ │ │ │ +00016010: 2074 6865 7365 2074 6f20 6265 2075 7365   these to be use
│ │ │ │ +00016020: 6420 6279 2074 6865 0a43 534d 2066 756e  d by the.CSM fun
│ │ │ │ +00016030: 6374 696f 6e2e 2049 6e20 7468 6520 6578  ction. In the ex
│ │ │ │ +00016040: 616d 706c 6520 6265 6c6f 7720 7765 2069  ample below we i
│ │ │ │ +00016050: 6e70 7574 2074 6865 2043 534d 2063 6c61  nput the CSM cla
│ │ │ │ +00016060: 7373 206f 6620 5628 4b5f 3029 2028 7468  ss of V(K_0) (th
│ │ │ │ +00016070: 6174 2069 7320 6f66 0a74 6865 2068 7970  at is of.the hyp
│ │ │ │ +00016080: 6572 7375 7266 6163 6520 6465 6669 6e65  ersurface define
│ │ │ │ +00016090: 6420 6279 2074 6865 2066 6972 7374 2070  d by the first p
│ │ │ │ +000160a0: 6f6c 796e 6f6d 6961 6c20 6765 6e65 7261  olynomial genera
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│ │ │ │ +000160f0: 6475 6374 206f 6620 7468 6520 6765 6e65  duct of the gene
│ │ │ │ +00016100: 7261 746f 7273 206f 6620 4b2e 0a0a 2b2d  rators of K...+-
│ │ │ │ +00016110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016140: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3139  ----------+.|i19
│ │ │ │ +00016150: 203a 206d 3d6e 6577 204d 7574 6162 6c65   : m=new Mutable
│ │ │ │ +00016160: 4861 7368 5461 626c 653b 2020 2020 2020  HashTable;      
│ │ │ │ +00016170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016180: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00016190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000161a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000161b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000161c0: 2d2d 2d2d 2d2d 2b0a 7c69 3230 203a 206d  ------+.|i20 : m
│ │ │ │ +000161d0: 237b 307d 3d63 736d 4b48 6173 6823 7b30  #{0}=csmKHash#{0
│ │ │ │ +000161e0: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +000161f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016200: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00016210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016230: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00016240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016270: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00016280: 3220 3220 2020 2020 3220 2020 2020 2020  2 2     2       
│ │ │ │ -00016290: 2020 3220 2020 2020 3220 2020 2020 2020    2     2       
│ │ │ │ -000162a0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -000162b0: 2020 207c 0a7c 6f32 3020 3d20 3868 2068     |.|o20 = 8h h
│ │ │ │ -000162c0: 2020 2b20 3768 2068 2020 2b20 3668 2068    + 7h h  + 6h h
│ │ │ │ -000162d0: 2020 2b20 3268 2020 2b20 3568 2068 2020    + 2h  + 5h h  
│ │ │ │ -000162e0: 2b20 3268 2020 2b20 3268 2020 2b20 6820  + 2h  + 2h  + h 
│ │ │ │ -000162f0: 207c 0a7c 2020 2020 2020 2020 3120 3220   |.|        1 2 
│ │ │ │ -00016300: 2020 2020 3120 3220 2020 2020 3120 3220      1 2     1 2 
│ │ │ │ -00016310: 2020 2020 3120 2020 2020 3120 3220 2020      1     1 2   
│ │ │ │ -00016320: 2020 3220 2020 2020 3120 2020 2032 207c    2     1    2 |
│ │ │ │ -00016330: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00016340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016240: 2020 7c0a 7c20 2020 2020 2020 2032 2032    |.|        2 2
│ │ │ │ +00016250: 2020 2020 2032 2020 2020 2020 2020 2032       2         2
│ │ │ │ +00016260: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00016270: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00016280: 7c0a 7c6f 3230 203d 2038 6820 6820 202b  |.|o20 = 8h h  +
│ │ │ │ +00016290: 2037 6820 6820 202b 2036 6820 6820 202b   7h h  + 6h h  +
│ │ │ │ +000162a0: 2032 6820 202b 2035 6820 6820 202b 2032   2h  + 5h h  + 2
│ │ │ │ +000162b0: 6820 202b 2032 6820 202b 2068 2020 7c0a  h  + 2h  + h  |.
│ │ │ │ +000162c0: 7c20 2020 2020 2020 2031 2032 2020 2020  |        1 2    
│ │ │ │ +000162d0: 2031 2032 2020 2020 2031 2032 2020 2020   1 2     1 2    
│ │ │ │ +000162e0: 2031 2020 2020 2031 2032 2020 2020 2032   1     1 2     2
│ │ │ │ +000162f0: 2020 2020 2031 2020 2020 3220 7c0a 7c20       1    2 |.| 
│ │ │ │ +00016300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016330: 2020 2020 2020 2020 2020 7c0a 7c6f 3230            |.|o20
│ │ │ │ +00016340: 203a 2041 2020 2020 2020 2020 2020 2020   : A            
│ │ │ │  00016350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016360: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00016370: 6f32 3020 3a20 4120 2020 2020 2020 2020  o20 : A         
│ │ │ │ -00016380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000163a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -000163b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000163c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000163d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000163e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3120  ---------+.|i21 
│ │ │ │ -000163f0: 3a20 6d23 7b30 2c31 7d3d 6373 6d4b 4861  : m#{0,1}=csmKHa
│ │ │ │ -00016400: 7368 237b 302c 317d 2020 2020 2020 2020  sh#{0,1}        
│ │ │ │ +00016360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016370: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00016380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000163a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000163b0: 2d2d 2d2d 2d2d 2b0a 7c69 3231 203a 206d  ------+.|i21 : m
│ │ │ │ +000163c0: 237b 302c 317d 3d63 736d 4b48 6173 6823  #{0,1}=csmKHash#
│ │ │ │ +000163d0: 7b30 2c31 7d20 2020 2020 2020 2020 2020  {0,1}           
│ │ │ │ +000163e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000163f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00016400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016420: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00016430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016460: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00016470: 3220 3220 2020 2020 3220 2020 2020 2020  2 2     2       
│ │ │ │ -00016480: 2020 3220 2020 2020 3220 2020 2020 2020    2     2       
│ │ │ │ -00016490: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -000164a0: 2020 207c 0a7c 6f32 3120 3d20 3968 2068     |.|o21 = 9h h
│ │ │ │ -000164b0: 2020 2b20 3968 2068 2020 2b20 3968 2068    + 9h h  + 9h h
│ │ │ │ -000164c0: 2020 2b20 3368 2020 2b20 3768 2068 2020    + 3h  + 7h h  
│ │ │ │ -000164d0: 2b20 3368 2020 2b20 3368 2020 2b20 3268  + 3h  + 3h  + 2h
│ │ │ │ -000164e0: 207c 0a7c 2020 2020 2020 2020 3120 3220   |.|        1 2 
│ │ │ │ -000164f0: 2020 2020 3120 3220 2020 2020 3120 3220      1 2     1 2 
│ │ │ │ -00016500: 2020 2020 3120 2020 2020 3120 3220 2020      1     1 2   
│ │ │ │ -00016510: 2020 3220 2020 2020 3120 2020 2020 327c    2     1     2|
│ │ │ │ -00016520: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00016530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016430: 2020 7c0a 7c20 2020 2020 2020 2032 2032    |.|        2 2
│ │ │ │ +00016440: 2020 2020 2032 2020 2020 2020 2020 2032       2         2
│ │ │ │ +00016450: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00016460: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00016470: 7c0a 7c6f 3231 203d 2039 6820 6820 202b  |.|o21 = 9h h  +
│ │ │ │ +00016480: 2039 6820 6820 202b 2039 6820 6820 202b   9h h  + 9h h  +
│ │ │ │ +00016490: 2033 6820 202b 2037 6820 6820 202b 2033   3h  + 7h h  + 3
│ │ │ │ +000164a0: 6820 202b 2033 6820 202b 2032 6820 7c0a  h  + 3h  + 2h |.
│ │ │ │ +000164b0: 7c20 2020 2020 2020 2031 2032 2020 2020  |        1 2    
│ │ │ │ +000164c0: 2031 2032 2020 2020 2031 2032 2020 2020   1 2     1 2    
│ │ │ │ +000164d0: 2031 2020 2020 2031 2032 2020 2020 2032   1     1 2     2
│ │ │ │ +000164e0: 2020 2020 2031 2020 2020 2032 7c0a 7c20       1     2|.| 
│ │ │ │ +000164f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016520: 2020 2020 2020 2020 2020 7c0a 7c6f 3231            |.|o21
│ │ │ │ +00016530: 203a 2041 2020 2020 2020 2020 2020 2020   : A            
│ │ │ │  00016540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016550: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00016560: 6f32 3120 3a20 4120 2020 2020 2020 2020  o21 : A         
│ │ │ │ -00016570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016590: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -000165a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000165b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000165c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000165d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3220  ---------+.|i22 
│ │ │ │ -000165e0: 3a20 7469 6d65 2043 534d 2841 2c4b 2c6d  : time CSM(A,K,m
│ │ │ │ -000165f0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -00016600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016610: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -00016620: 6564 2030 2e31 3131 3430 3373 2028 6370  ed 0.111403s (cp
│ │ │ │ -00016630: 7529 3b20 302e 3035 3830 3932 3573 2028  u); 0.0580925s (
│ │ │ │ -00016640: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │ -00016650: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00016660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016690: 2020 207c 0a7c 2020 2020 2020 2020 3220     |.|        2 
│ │ │ │ -000166a0: 3220 2020 2020 3220 2020 2020 2020 2020  2     2         
│ │ │ │ -000166b0: 3220 2020 2032 2020 2020 2020 2020 2020  2    2          
│ │ │ │ -000166c0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -000166d0: 207c 0a7c 6f32 3220 3d20 3768 2068 2020   |.|o22 = 7h h  
│ │ │ │ -000166e0: 2b20 3568 2068 2020 2b20 3468 2068 2020  + 5h h  + 4h h  
│ │ │ │ -000166f0: 2b20 6820 202b 2033 6820 6820 202b 2068  + h  + 3h h  + h
│ │ │ │ -00016700: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016710: 0a7c 2020 2020 2020 2020 3120 3220 2020  .|        1 2   
│ │ │ │ -00016720: 2020 3120 3220 2020 2020 3120 3220 2020    1 2     1 2   
│ │ │ │ -00016730: 2031 2020 2020 2031 2032 2020 2020 3220   1     1 2    2 
│ │ │ │ -00016740: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00016750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016560: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00016570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000165a0: 2d2d 2d2d 2d2d 2b0a 7c69 3232 203a 2074  ------+.|i22 : t
│ │ │ │ +000165b0: 696d 6520 4353 4d28 412c 4b2c 6d29 2020  ime CSM(A,K,m)  
│ │ │ │ +000165c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000165d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000165e0: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ +000165f0: 302e 3130 3738 3037 7320 2863 7075 293b  0.107807s (cpu);
│ │ │ │ +00016600: 2030 2e30 3733 3435 3032 7320 2874 6872   0.0734502s (thr
│ │ │ │ +00016610: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │ +00016620: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00016630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016660: 7c0a 7c20 2020 2020 2020 2032 2032 2020  |.|        2 2  
│ │ │ │ +00016670: 2020 2032 2020 2020 2020 2020 2032 2020     2         2  
│ │ │ │ +00016680: 2020 3220 2020 2020 2020 2020 2020 2032    2            2
│ │ │ │ +00016690: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000166a0: 7c6f 3232 203d 2037 6820 6820 202b 2035  |o22 = 7h h  + 5
│ │ │ │ +000166b0: 6820 6820 202b 2034 6820 6820 202b 2068  h h  + 4h h  + h
│ │ │ │ +000166c0: 2020 2b20 3368 2068 2020 2b20 6820 2020    + 3h h  + h   
│ │ │ │ +000166d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000166e0: 2020 2020 2020 2031 2032 2020 2020 2031         1 2     1
│ │ │ │ +000166f0: 2032 2020 2020 2031 2032 2020 2020 3120   2     1 2    1 
│ │ │ │ +00016700: 2020 2020 3120 3220 2020 2032 2020 2020      1 2    2    
│ │ │ │ +00016710: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00016720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016750: 2020 2020 2020 2020 7c0a 7c6f 3232 203a          |.|o22 :
│ │ │ │ +00016760: 2041 2020 2020 2020 2020 2020 2020 2020   A              
│ │ │ │  00016770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016780: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -00016790: 3220 3a20 4120 2020 2020 2020 2020 2020  2 : A           
│ │ │ │ -000167a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000167b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000167c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -000167d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000167e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000167f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016800: 2d2d 2d2d 2d2d 2d2b 0a0a 496e 2074 6865  -------+..In the
│ │ │ │ -00016810: 2063 6173 6520 7768 6572 6520 7468 6520   case where the 
│ │ │ │ -00016820: 616d 6269 656e 7420 7370 6163 6520 6973  ambient space is
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│ │ │ │ -00016840: 2077 6869 6368 2069 7320 6e6f 7420 6120   which is not a 
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│ │ │ │ -00016870: 6d75 7374 206c 6f61 6420 7468 6520 4e6f  must load the No
│ │ │ │ -00016880: 726d 616c 546f 7269 6356 6172 6965 7469  rmalToricVarieti
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│ │ │ │ -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
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│ │ │ │ -000168e0: 7563 7420 6f66 2070 726f 6a65 6374 6976  uct of projectiv
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│ │ │ │ -00016940: 7269 6574 7920 666f 7220 6566 6669 6369  riety for effici
│ │ │ │ -00016950: 656e 6379 2072 6561 736f 6e73 2e0a 0a2b  ency reasons...+
│ │ │ │ +00016780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016790: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000167a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000167b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000167c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000167d0: 2d2d 2d2d 2b0a 0a49 6e20 7468 6520 6361  ----+..In the ca
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│ │ │ │ +00016800: 746f 7269 6320 7661 7269 6574 7920 7768  toric variety wh
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│ │ │ │ +00016820: 6475 6374 0a6f 6620 7072 6f6a 6563 7469  duct.of projecti
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│ │ │ │ +00016840: 7420 6c6f 6164 2074 6865 204e 6f72 6d61  t load the Norma
│ │ │ │ +00016850: 6c54 6f72 6963 5661 7269 6574 6965 7320  lToricVarieties 
│ │ │ │ +00016860: 7061 636b 6167 6520 616e 6420 6d75 7374  package and must
│ │ │ │ +00016870: 0a61 6c73 6f20 696e 7075 7420 7468 6520  .also input the 
│ │ │ │ +00016880: 746f 7269 6320 7661 7269 6574 792e 2049  toric variety. I
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│ │ │ │ +000168a0: 6574 7920 6973 2061 2070 726f 6475 6374  ety is a product
│ │ │ │ +000168b0: 206f 6620 7072 6f6a 6563 7469 7665 0a73   of projective.s
│ │ │ │ +000168c0: 7061 6365 2069 7420 6973 2072 6563 6f6d  pace it is recom
│ │ │ │ +000168d0: 6d65 6e64 2074 6f20 7573 6520 7468 6520  mend to use the 
│ │ │ │ +000168e0: 666f 726d 2061 626f 7665 2072 6174 6865  form above rathe
│ │ │ │ +000168f0: 7220 7468 616e 2069 6e70 7574 7469 6e67  r than inputting
│ │ │ │ +00016900: 2074 6865 2074 6f72 6963 0a76 6172 6965   the toric.varie
│ │ │ │ +00016910: 7479 2066 6f72 2065 6666 6963 6965 6e63  ty for efficienc
│ │ │ │ +00016920: 7920 7265 6173 6f6e 732e 0a0a 2b2d 2d2d  y reasons...+---
│ │ │ │ +00016930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000169a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3233 203a  --------+.|i23 :
│ │ │ │ -000169b0: 206e 6565 6473 5061 636b 6167 6520 224e   needsPackage "N
│ │ │ │ -000169c0: 6f72 6d61 6c54 6f72 6963 5661 7269 6574  ormalToricVariet
│ │ │ │ -000169d0: 6965 7322 2020 2020 2020 2020 2020 2020  ies"            
│ │ │ │ +00016970: 2d2d 2d2d 2d2b 0a7c 6932 3320 3a20 6e65  -----+.|i23 : ne
│ │ │ │ +00016980: 6564 7350 6163 6b61 6765 2022 4e6f 726d  edsPackage "Norm
│ │ │ │ +00016990: 616c 546f 7269 6356 6172 6965 7469 6573  alToricVarieties
│ │ │ │ +000169a0: 2220 2020 2020 2020 2020 2020 2020 2020  "               
│ │ │ │ +000169b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000169c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000169d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000169e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000169f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00016a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016a30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00016a40: 7c6f 3233 203d 204e 6f72 6d61 6c54 6f72  |o23 = NormalTor
│ │ │ │ -00016a50: 6963 5661 7269 6574 6965 7320 2020 2020  icVarieties     
│ │ │ │ +000169f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016a00: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00016a10: 3320 3d20 4e6f 726d 616c 546f 7269 6356  3 = NormalToricV
│ │ │ │ +00016a20: 6172 6965 7469 6573 2020 2020 2020 2020  arieties        
│ │ │ │ +00016a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016a50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00016a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016a80: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00016a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016aa0: 207c 0a7c 6f32 3320 3a20 5061 636b 6167   |.|o23 : Packag
│ │ │ │ +00016ab0: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │  00016ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ad0: 2020 2020 7c0a 7c6f 3233 203a 2050 6163      |.|o23 : Pac
│ │ │ │ -00016ae0: 6b61 6765 2020 2020 2020 2020 2020 2020  kage            
│ │ │ │ -00016af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016b10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016b20: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00016b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3234  ----------+.|i24
│ │ │ │ -00016b70: 203a 2052 686f 203d 207b 7b31 2c30 2c30   : Rho = {{1,0,0
│ │ │ │ -00016b80: 7d2c 7b30 2c31 2c30 7d2c 7b30 2c30 2c31  },{0,1,0},{0,0,1
│ │ │ │ -00016b90: 7d2c 7b2d 312c 2d31 2c30 7d2c 7b30 2c30  },{-1,-1,0},{0,0
│ │ │ │ -00016ba0: 2c2d 317d 7d20 2020 2020 2020 2020 2020  ,-1}}           
│ │ │ │ -00016bb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00016bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016c00: 7c0a 7c6f 3234 203d 207b 7b31 2c20 302c  |.|o24 = {{1, 0,
│ │ │ │ -00016c10: 2030 7d2c 207b 302c 2031 2c20 307d 2c20   0}, {0, 1, 0}, 
│ │ │ │ -00016c20: 7b30 2c20 302c 2031 7d2c 207b 2d31 2c20  {0, 0, 1}, {-1, 
│ │ │ │ -00016c30: 2d31 2c20 307d 2c20 7b30 2c20 302c 202d  -1, 0}, {0, 0, -
│ │ │ │ -00016c40: 317d 7d20 2020 2020 2020 207c 0a7c 2020  1}}        |.|  
│ │ │ │ +00016ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016ae0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00016af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016b30: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3420 3a20  -------+.|i24 : 
│ │ │ │ +00016b40: 5268 6f20 3d20 7b7b 312c 302c 307d 2c7b  Rho = {{1,0,0},{
│ │ │ │ +00016b50: 302c 312c 307d 2c7b 302c 302c 317d 2c7b  0,1,0},{0,0,1},{
│ │ │ │ +00016b60: 2d31 2c2d 312c 307d 2c7b 302c 302c 2d31  -1,-1,0},{0,0,-1
│ │ │ │ +00016b70: 7d7d 2020 2020 2020 2020 2020 2020 2020  }}              
│ │ │ │ +00016b80: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00016b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016bc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00016bd0: 6f32 3420 3d20 7b7b 312c 2030 2c20 307d  o24 = {{1, 0, 0}
│ │ │ │ +00016be0: 2c20 7b30 2c20 312c 2030 7d2c 207b 302c  , {0, 1, 0}, {0,
│ │ │ │ +00016bf0: 2030 2c20 317d 2c20 7b2d 312c 202d 312c   0, 1}, {-1, -1,
│ │ │ │ +00016c00: 2030 7d2c 207b 302c 2030 2c20 2d31 7d7d   0}, {0, 0, -1}}
│ │ │ │ +00016c10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00016c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016c60: 2020 207c 0a7c 6f32 3420 3a20 4c69 7374     |.|o24 : List
│ │ │ │  00016c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016c90: 2020 2020 2020 7c0a 7c6f 3234 203a 204c        |.|o24 : L
│ │ │ │ -00016ca0: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ -00016cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ce0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -00016cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00016d30: 3235 203a 2053 6967 6d61 203d 207b 7b30  25 : Sigma = {{0
│ │ │ │ -00016d40: 2c31 2c32 7d2c 7b31 2c32 2c33 7d2c 7b30  ,1,2},{1,2,3},{0
│ │ │ │ -00016d50: 2c32 2c33 7d2c 7b30 2c31 2c34 7d2c 7b31  ,2,3},{0,1,4},{1
│ │ │ │ -00016d60: 2c33 2c34 7d2c 7b30 2c33 2c34 7d7d 2020  ,3,4},{0,3,4}}  
│ │ │ │ -00016d70: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00016d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016dc0: 2020 7c0a 7c6f 3235 203d 207b 7b30 2c20    |.|o25 = {{0, 
│ │ │ │ -00016dd0: 312c 2032 7d2c 207b 312c 2032 2c20 337d  1, 2}, {1, 2, 3}
│ │ │ │ -00016de0: 2c20 7b30 2c20 322c 2033 7d2c 207b 302c  , {0, 2, 3}, {0,
│ │ │ │ -00016df0: 2031 2c20 347d 2c20 7b31 2c20 332c 2034   1, 4}, {1, 3, 4
│ │ │ │ -00016e00: 7d2c 207b 302c 2033 2c20 347d 7d7c 0a7c  }, {0, 3, 4}}|.|
│ │ │ │ +00016c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016ca0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016cb0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00016cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016cf0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3520  ---------+.|i25 
│ │ │ │ +00016d00: 3a20 5369 676d 6120 3d20 7b7b 302c 312c  : Sigma = {{0,1,
│ │ │ │ +00016d10: 327d 2c7b 312c 322c 337d 2c7b 302c 322c  2},{1,2,3},{0,2,
│ │ │ │ +00016d20: 337d 2c7b 302c 312c 347d 2c7b 312c 332c  3},{0,1,4},{1,3,
│ │ │ │ +00016d30: 347d 2c7b 302c 332c 347d 7d20 2020 2020  4},{0,3,4}}     
│ │ │ │ +00016d40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00016d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016d80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00016d90: 0a7c 6f32 3520 3d20 7b7b 302c 2031 2c20  .|o25 = {{0, 1, 
│ │ │ │ +00016da0: 327d 2c20 7b31 2c20 322c 2033 7d2c 207b  2}, {1, 2, 3}, {
│ │ │ │ +00016db0: 302c 2032 2c20 337d 2c20 7b30 2c20 312c  0, 2, 3}, {0, 1,
│ │ │ │ +00016dc0: 2034 7d2c 207b 312c 2033 2c20 347d 2c20   4}, {1, 3, 4}, 
│ │ │ │ +00016dd0: 7b30 2c20 332c 2034 7d7d 7c0a 7c20 2020  {0, 3, 4}}|.|   
│ │ │ │ +00016de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016e20: 2020 2020 207c 0a7c 6f32 3520 3a20 4c69       |.|o25 : Li
│ │ │ │ +00016e30: 7374 2020 2020 2020 2020 2020 2020 2020  st              
│ │ │ │  00016e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e50: 2020 2020 2020 2020 7c0a 7c6f 3235 203a          |.|o25 :
│ │ │ │ -00016e60: 204c 6973 7420 2020 2020 2020 2020 2020   List           
│ │ │ │ -00016e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ea0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -00016eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00016ef0: 7c69 3236 203a 2058 203d 206e 6f72 6d61  |i26 : X = norma
│ │ │ │ -00016f00: 6c54 6f72 6963 5661 7269 6574 7928 5268  lToricVariety(Rh
│ │ │ │ -00016f10: 6f2c 5369 676d 612c 436f 6566 6669 6369  o,Sigma,Coeffici
│ │ │ │ -00016f20: 656e 7452 696e 6720 3d3e 5a5a 2f33 3237  entRing =>ZZ/327
│ │ │ │ -00016f30: 3439 2920 2020 2020 207c 0a7c 2020 2020  49)      |.|    
│ │ │ │ +00016e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016e70: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00016e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +00016ec0: 3620 3a20 5820 3d20 6e6f 726d 616c 546f  6 : X = normalTo
│ │ │ │ +00016ed0: 7269 6356 6172 6965 7479 2852 686f 2c53  ricVariety(Rho,S
│ │ │ │ +00016ee0: 6967 6d61 2c43 6f65 6666 6963 6965 6e74  igma,Coefficient
│ │ │ │ +00016ef0: 5269 6e67 203d 3e5a 5a2f 3332 3734 3929  Ring =>ZZ/32749)
│ │ │ │ +00016f00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00016f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016f50: 207c 0a7c 6f32 3620 3d20 5820 2020 2020   |.|o26 = X     
│ │ │ │  00016f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016f80: 2020 2020 7c0a 7c6f 3236 203d 2058 2020      |.|o26 = X  
│ │ │ │ -00016f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016f90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00016fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016fc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016fd0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00016fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017010: 2020 2020 2020 2020 2020 7c0a 7c6f 3236            |.|o26
│ │ │ │ -00017020: 203a 204e 6f72 6d61 6c54 6f72 6963 5661   : NormalToricVa
│ │ │ │ -00017030: 7269 6574 7920 2020 2020 2020 2020 2020  riety           
│ │ │ │ -00017040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017060: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00017070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000170a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000170b0: 2b0a 7c69 3237 203a 2063 736d 583d 4353  +.|i27 : csmX=CS
│ │ │ │ -000170c0: 4d20 5820 2020 2020 2020 2020 2020 2020  M X             
│ │ │ │ +00016fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016fe0: 2020 2020 2020 207c 0a7c 6f32 3620 3a20         |.|o26 : 
│ │ │ │ +00016ff0: 4e6f 726d 616c 546f 7269 6356 6172 6965  NormalToricVarie
│ │ │ │ +00017000: 7479 2020 2020 2020 2020 2020 2020 2020  ty              
│ │ │ │ +00017010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017030: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00017040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00017080: 6932 3720 3a20 6373 6d58 3d43 534d 2058  i27 : csmX=CSM X
│ │ │ │ +00017090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000170a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000170b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000170c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  000170d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000170e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000170f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000170f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017110: 2020 207c 0a7c 2020 2020 2020 2020 3220     |.|        2 
│ │ │ │ +00017120: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  00017130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017140: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00017150: 2032 2020 2020 2020 2032 2020 2020 2020   2       2      
│ │ │ │ -00017160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017190: 207c 0a7c 6f32 3720 3d20 3678 2078 2020   |.|o27 = 6x x  
│ │ │ │ -000171a0: 2b20 3378 2020 2b20 3678 2078 2020 2b20  + 3x  + 6x x  + 
│ │ │ │ -000171b0: 3378 2020 2b20 3278 2020 2b20 3120 2020  3x  + 2x  + 1   
│ │ │ │ -000171c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000171d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000171e0: 2020 2020 2020 2033 2034 2020 2020 2033         3 4     3
│ │ │ │ -000171f0: 2020 2020 2033 2034 2020 2020 2033 2020       3 4     3  
│ │ │ │ -00017200: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ +00017140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017150: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00017160: 7c6f 3237 203d 2036 7820 7820 202b 2033  |o27 = 6x x  + 3
│ │ │ │ +00017170: 7820 202b 2036 7820 7820 202b 2033 7820  x  + 6x x  + 3x 
│ │ │ │ +00017180: 202b 2032 7820 202b 2031 2020 2020 2020   + 2x  + 1      
│ │ │ │ +00017190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000171a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000171b0: 2020 2020 3320 3420 2020 2020 3320 2020      3 4     3   
│ │ │ │ +000171c0: 2020 3320 3420 2020 2020 3320 2020 2020    3 4     3     
│ │ │ │ +000171d0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ +000171e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000171f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00017200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017220: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00017230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017270: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00017280: 2020 2020 2020 2020 2020 205a 5a5b 7820             ZZ[x 
│ │ │ │ -00017290: 2e2e 7820 5d20 2020 2020 2020 2020 2020  ..x ]           
│ │ │ │ -000172a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000172b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00017220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017230: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00017240: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00017250: 2020 2020 2020 2020 5a5a 5b78 202e 2e78          ZZ[x ..x
│ │ │ │ +00017260: 205d 2020 2020 2020 2020 2020 2020 2020   ]              
│ │ │ │ +00017270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017280: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00017290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000172a0: 2020 2020 2020 2030 2020 2034 2020 2020         0   4    
│ │ │ │ +000172b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000172c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000172d0: 2020 2020 2020 2020 2020 3020 2020 3420            0   4 
│ │ │ │ -000172e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000172f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017300: 2020 2020 2020 2020 7c0a 7c6f 3237 203a          |.|o27 :
│ │ │ │ -00017310: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   ---------------
│ │ │ │ -00017320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017330: 2d2d 2d2d 2d2d 2d2d 2d2d 2020 2020 2020  ----------      
│ │ │ │ -00017340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017350: 2020 207c 0a7c 2020 2020 2020 2878 2078     |.|      (x x
│ │ │ │ -00017360: 202c 2078 2078 2078 202c 2078 2020 2d20   , x x x , x  - 
│ │ │ │ -00017370: 7820 2c20 7820 202d 2078 202c 2078 2020  x , x  - x , x  
│ │ │ │ -00017380: 2d20 7820 2920 2020 2020 2020 2020 2020  - x )           
│ │ │ │ -00017390: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000173a0: 7c20 2020 2020 2020 2032 2034 2020 2030  |        2 4   0
│ │ │ │ -000173b0: 2031 2033 2020 2030 2020 2020 3320 2020   1 3   0    3   
│ │ │ │ -000173c0: 3120 2020 2033 2020 2032 2020 2020 3420  1    3   2    4 
│ │ │ │ -000173d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000173e0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +000172d0: 2020 2020 207c 0a7c 6f32 3720 3a20 2d2d       |.|o27 : --
│ │ │ │ +000172e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000172f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017300: 2d2d 2d2d 2d2d 2d20 2020 2020 2020 2020  -------         
│ │ │ │ +00017310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017320: 7c0a 7c20 2020 2020 2028 7820 7820 2c20  |.|      (x x , 
│ │ │ │ +00017330: 7820 7820 7820 2c20 7820 202d 2078 202c  x x x , x  - x ,
│ │ │ │ +00017340: 2078 2020 2d20 7820 2c20 7820 202d 2078   x  - x , x  - x
│ │ │ │ +00017350: 2029 2020 2020 2020 2020 2020 2020 2020   )              
│ │ │ │ +00017360: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00017370: 2020 2020 2020 3220 3420 2020 3020 3120        2 4   0 1 
│ │ │ │ +00017380: 3320 2020 3020 2020 2033 2020 2031 2020  3   0    3   1  
│ │ │ │ +00017390: 2020 3320 2020 3220 2020 2034 2020 2020    3   2    4    
│ │ │ │ +000173a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000173b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000173c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000173d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000173e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000173f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017430: 2d2d 2d2d 2b0a 7c69 3238 203a 2043 683d  ----+.|i28 : Ch=
│ │ │ │ -00017440: 7269 6e67 2063 736d 5820 2020 2020 2020  ring csmX       
│ │ │ │ +00017400: 2d2b 0a7c 6932 3820 3a20 4368 3d72 696e  -+.|i28 : Ch=rin
│ │ │ │ +00017410: 6720 6373 6d58 2020 2020 2020 2020 2020  g csmX          
│ │ │ │ +00017420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017440: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017470: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00017480: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00017490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000174a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017490: 2020 2020 2020 207c 0a7c 6f32 3820 3d20         |.|o28 = 
│ │ │ │ +000174a0: 4368 2020 2020 2020 2020 2020 2020 2020  Ch              
│ │ │ │  000174b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000174c0: 2020 2020 2020 2020 2020 7c0a 7c6f 3238            |.|o28
│ │ │ │ -000174d0: 203d 2043 6820 2020 2020 2020 2020 2020   = Ch           
│ │ │ │ -000174e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000174c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000174d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000174e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000174f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017510: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00017520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017520: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00017530: 6f32 3820 3a20 5175 6f74 6965 6e74 5269  o28 : QuotientRi
│ │ │ │ +00017540: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │  00017550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017560: 7c0a 7c6f 3238 203a 2051 756f 7469 656e  |.|o28 : Quotien
│ │ │ │ -00017570: 7452 696e 6720 2020 2020 2020 2020 2020  tRing           
│ │ │ │ -00017580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000175a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00017560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017570: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00017580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000175a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000175b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000175c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000175d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000175e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000175f0: 2d2d 2d2d 2d2d 2b0a 7c69 3239 203a 2043  ------+.|i29 : C
│ │ │ │ -00017600: 6865 636b 546f 7269 6356 6172 6965 7479  heckToricVariety
│ │ │ │ -00017610: 5661 6c69 6428 5829 2020 2020 2020 2020  Valid(X)        
│ │ │ │ +000175c0: 2d2d 2d2b 0a7c 6932 3920 3a20 4368 6563  ---+.|i29 : Chec
│ │ │ │ +000175d0: 6b54 6f72 6963 5661 7269 6574 7956 616c  kToricVarietyVal
│ │ │ │ +000175e0: 6964 2858 2920 2020 2020 2020 2020 2020  id(X)           
│ │ │ │ +000175f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017600: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00017610: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00017620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017640: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00017650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017650: 2020 2020 2020 2020 207c 0a7c 6f32 3920           |.|o29 
│ │ │ │ +00017660: 3d20 7472 7565 2020 2020 2020 2020 2020  = true          
│ │ │ │  00017670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017680: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00017690: 3239 203d 2074 7275 6520 2020 2020 2020  29 = true       
│ │ │ │ -000176a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000176b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000176c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000176d0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -000176e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000176f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017720: 2d2d 2b0a 7c69 3330 203a 2052 3d72 696e  --+.|i30 : R=rin
│ │ │ │ -00017730: 6728 5829 2020 2020 2020 2020 2020 2020  g(X)            
│ │ │ │ +00017680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000176a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000176b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000176c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000176d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000176e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +000176f0: 0a7c 6933 3020 3a20 523d 7269 6e67 2858  .|i30 : R=ring(X
│ │ │ │ +00017700: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +00017710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017730: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00017740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017760: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00017760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017780: 2020 2020 207c 0a7c 6f33 3020 3d20 5220       |.|o30 = R 
│ │ │ │  00017790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000177a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000177b0: 2020 2020 2020 2020 7c0a 7c6f 3330 203d          |.|o30 =
│ │ │ │ -000177c0: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ -000177d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000177b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000177c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000177d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000177e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000177f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017800: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00017810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017840: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00017850: 7c6f 3330 203a 2050 6f6c 796e 6f6d 6961  |o30 : Polynomia
│ │ │ │ -00017860: 6c52 696e 6720 2020 2020 2020 2020 2020  lRing           
│ │ │ │ -00017870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017890: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00017800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017810: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ +00017820: 3020 3a20 506f 6c79 6e6f 6d69 616c 5269  0 : PolynomialRi
│ │ │ │ +00017830: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ +00017840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017860: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00017870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000178a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000178b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000178c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000178d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000178e0: 2d2d 2d2d 2b0a 7c69 3331 203a 2049 3d69  ----+.|i31 : I=i
│ │ │ │ -000178f0: 6465 616c 2852 5f30 5e34 2a52 5f31 2c52  deal(R_0^4*R_1,R
│ │ │ │ -00017900: 5f30 2a52 5f33 2a52 5f34 2a52 5f32 2d52  _0*R_3*R_4*R_2-R
│ │ │ │ -00017910: 5f32 5e32 2a52 5f30 5e32 2920 2020 2020  _2^2*R_0^2)     
│ │ │ │ -00017920: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00017930: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00017940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017970: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00017980: 2020 2020 2020 2020 2020 2034 2020 2020             4    
│ │ │ │ -00017990: 2020 2032 2032 2020 2020 2020 2020 2020     2 2          
│ │ │ │ -000179a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000179b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000179c0: 2020 2020 207c 0a7c 6f33 3120 3d20 6964       |.|o31 = id
│ │ │ │ -000179d0: 6561 6c20 2878 2078 202c 202d 2078 2078  eal (x x , - x x
│ │ │ │ -000179e0: 2020 2b20 7820 7820 7820 7820 2920 2020    + x x x x )   
│ │ │ │ -000179f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017a10: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00017a20: 2030 2031 2020 2020 2030 2032 2020 2020   0 1     0 2    
│ │ │ │ -00017a30: 3020 3220 3320 3420 2020 2020 2020 2020  0 2 3 4         
│ │ │ │ +000178b0: 2d2b 0a7c 6933 3120 3a20 493d 6964 6561  -+.|i31 : I=idea
│ │ │ │ +000178c0: 6c28 525f 305e 342a 525f 312c 525f 302a  l(R_0^4*R_1,R_0*
│ │ │ │ +000178d0: 525f 332a 525f 342a 525f 322d 525f 325e  R_3*R_4*R_2-R_2^
│ │ │ │ +000178e0: 322a 525f 305e 3229 2020 2020 2020 2020  2*R_0^2)        
│ │ │ │ +000178f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00017900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017940: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00017950: 2020 2020 2020 2020 3420 2020 2020 2020          4       
│ │ │ │ +00017960: 3220 3220 2020 2020 2020 2020 2020 2020  2 2             
│ │ │ │ +00017970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017990: 2020 7c0a 7c6f 3331 203d 2069 6465 616c    |.|o31 = ideal
│ │ │ │ +000179a0: 2028 7820 7820 2c20 2d20 7820 7820 202b   (x x , - x x  +
│ │ │ │ +000179b0: 2078 2078 2078 2078 2029 2020 2020 2020   x x x x )      
│ │ │ │ +000179c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000179d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000179e0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +000179f0: 3120 2020 2020 3020 3220 2020 2030 2032  1     0 2    0 2
│ │ │ │ +00017a00: 2033 2034 2020 2020 2020 2020 2020 2020   3 4            
│ │ │ │ +00017a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017a20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00017a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017a50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00017a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017a70: 2020 207c 0a7c 6f33 3120 3a20 4964 6561     |.|o31 : Idea
│ │ │ │ +00017a80: 6c20 6f66 2052 2020 2020 2020 2020 2020  l of R          
│ │ │ │  00017a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017aa0: 2020 2020 2020 7c0a 7c6f 3331 203a 2049        |.|o31 : I
│ │ │ │ -00017ab0: 6465 616c 206f 6620 5220 2020 2020 2020  deal of R       
│ │ │ │ -00017ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017af0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -00017b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00017b40: 3332 203a 2043 534d 2858 2c49 2920 2020  32 : CSM(X,I)   
│ │ │ │ -00017b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017ab0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00017ac0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00017ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017b00: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 3220  ---------+.|i32 
│ │ │ │ +00017b10: 3a20 4353 4d28 582c 4929 2020 2020 2020  : CSM(X,I)      
│ │ │ │ +00017b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017b50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00017b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017b80: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00017b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017b90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00017ba0: 0a7c 2020 2020 2020 2020 3220 2020 2020  .|        2     
│ │ │ │ +00017bb0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  00017bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017bd0: 2020 7c0a 7c20 2020 2020 2020 2032 2020    |.|        2  
│ │ │ │ -00017be0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -00017bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017c10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00017c20: 6f33 3220 3d20 3578 2078 2020 2b20 3378  o32 = 5x x  + 3x
│ │ │ │ -00017c30: 2020 2b20 3478 2078 2020 2b20 7820 2020    + 4x x  + x   
│ │ │ │ -00017c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017c60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00017c70: 2020 2033 2034 2020 2020 2033 2020 2020     3 4     3    
│ │ │ │ -00017c80: 2033 2034 2020 2020 3320 2020 2020 2020   3 4    3       
│ │ │ │ +00017bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017be0: 2020 2020 2020 2020 2020 7c0a 7c6f 3332            |.|o32
│ │ │ │ +00017bf0: 203d 2035 7820 7820 202b 2033 7820 202b   = 5x x  + 3x  +
│ │ │ │ +00017c00: 2034 7820 7820 202b 2078 2020 2020 2020   4x x  + x      
│ │ │ │ +00017c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017c30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00017c40: 3320 3420 2020 2020 3320 2020 2020 3320  3 4     3     3 
│ │ │ │ +00017c50: 3420 2020 2033 2020 2020 2020 2020 2020  4    3          
│ │ │ │ +00017c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017c80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00017c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017cb0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00017cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017cc0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00017cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017cf0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00017d00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00017d10: 2020 2020 2020 205a 5a5b 7820 2e2e 7820         ZZ[x ..x 
│ │ │ │ -00017d20: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │ -00017d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017d40: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00017ce0: 2020 2020 5a5a 5b78 202e 2e78 205d 2020      ZZ[x ..x ]  
│ │ │ │ +00017cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017d10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017d30: 2020 2030 2020 2034 2020 2020 2020 2020     0   4        
│ │ │ │ +00017d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017d60: 2020 2020 2020 3020 2020 3420 2020 2020        0   4     
│ │ │ │ -00017d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017d90: 2020 2020 7c0a 7c6f 3332 203a 202d 2d2d      |.|o32 : ---
│ │ │ │ -00017da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017dc0: 2d2d 2d2d 2d2d 2020 2020 2020 2020 2020  ------          
│ │ │ │ -00017dd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00017de0: 0a7c 2020 2020 2020 2878 2078 202c 2078  .|      (x x , x
│ │ │ │ -00017df0: 2078 2078 202c 2078 2020 2d20 7820 2c20   x x , x  - x , 
│ │ │ │ -00017e00: 7820 202d 2078 202c 2078 2020 2d20 7820  x  - x , x  - x 
│ │ │ │ -00017e10: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -00017e20: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00017e30: 2020 2020 2032 2034 2020 2030 2031 2033       2 4   0 1 3
│ │ │ │ -00017e40: 2020 2030 2020 2020 3320 2020 3120 2020     0    3   1   
│ │ │ │ -00017e50: 2033 2020 2032 2020 2020 3420 2020 2020   3   2    4     
│ │ │ │ -00017e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017e70: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00017e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017ec0: 2b0a 7c69 3333 203a 2043 534d 2843 682c  +.|i33 : CSM(Ch,
│ │ │ │ -00017ed0: 582c 4929 2020 2020 2020 2020 2020 2020  X,I)            
│ │ │ │ +00017d60: 207c 0a7c 6f33 3220 3a20 2d2d 2d2d 2d2d   |.|o32 : ------
│ │ │ │ +00017d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017d90: 2d2d 2d20 2020 2020 2020 2020 2020 2020  ---             
│ │ │ │ +00017da0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00017db0: 2020 2020 2028 7820 7820 2c20 7820 7820       (x x , x x 
│ │ │ │ +00017dc0: 7820 2c20 7820 202d 2078 202c 2078 2020  x , x  - x , x  
│ │ │ │ +00017dd0: 2d20 7820 2c20 7820 202d 2078 2029 2020  - x , x  - x )  
│ │ │ │ +00017de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017df0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00017e00: 2020 3220 3420 2020 3020 3120 3320 2020    2 4   0 1 3   
│ │ │ │ +00017e10: 3020 2020 2033 2020 2031 2020 2020 3320  0    3   1    3 
│ │ │ │ +00017e20: 2020 3220 2020 2034 2020 2020 2020 2020    2    4        
│ │ │ │ +00017e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017e40: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00017e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00017e90: 6933 3320 3a20 4353 4d28 4368 2c58 2c49  i33 : CSM(Ch,X,I
│ │ │ │ +00017ea0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +00017eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017ed0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00017ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017f00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00017f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017f20: 2020 207c 0a7c 2020 2020 2020 2020 3220     |.|        2 
│ │ │ │ +00017f30: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  00017f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017f50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00017f60: 2032 2020 2020 2020 2032 2020 2020 2020   2       2      
│ │ │ │ -00017f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017f60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00017f70: 7c6f 3333 203d 2038 7820 7820 202b 2033  |o33 = 8x x  + 3
│ │ │ │ +00017f80: 7820 202b 2035 7820 7820 202b 2078 2020  x  + 5x x  + x  
│ │ │ │  00017f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017fa0: 207c 0a7c 6f33 3320 3d20 3878 2078 2020   |.|o33 = 8x x  
│ │ │ │ -00017fb0: 2b20 3378 2020 2b20 3578 2078 2020 2b20  + 3x  + 5x x  + 
│ │ │ │ -00017fc0: 7820 2020 2020 2020 2020 2020 2020 2020  x               
│ │ │ │ -00017fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017fe0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00017ff0: 2020 2020 2020 2033 2034 2020 2020 2033         3 4     3
│ │ │ │ -00018000: 2020 2020 2033 2034 2020 2020 3320 2020       3 4    3   
│ │ │ │ +00017fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017fb0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00017fc0: 2020 2020 3320 3420 2020 2020 3320 2020      3 4     3   
│ │ │ │ +00017fd0: 2020 3320 3420 2020 2033 2020 2020 2020    3 4    3      
│ │ │ │ +00017fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018000: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00018010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018030: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00018040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018040: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00018050: 0a7c 6f33 3320 3a20 4368 2020 2020 2020  .|o33 : Ch      
│ │ │ │  00018060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018080: 2020 7c0a 7c6f 3333 203a 2043 6820 2020    |.|o33 : Ch   
│ │ │ │ -00018090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000180a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000180b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000180c0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00018080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018090: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +000180a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000180b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000180c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000180d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000180e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000180f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018110: 2d2d 2d2d 2d2d 2d2d 2b0a 0a54 6869 7320  --------+..This 
│ │ │ │ -00018120: 6675 6e63 7469 6f6e 206d 6179 2061 6c73  function may als
│ │ │ │ -00018130: 6f20 636f 6d70 7574 6520 7468 6520 4353  o compute the CS
│ │ │ │ -00018140: 4d20 636c 6173 7320 6f66 2061 206e 6f72  M class of a nor
│ │ │ │ -00018150: 6d61 6c20 746f 7269 6320 7661 7269 6574  mal toric variet
│ │ │ │ -00018160: 7920 6465 6669 6e65 640a 6279 2061 2066  y defined.by a f
│ │ │ │ -00018170: 616e 2e20 496e 2074 6869 7320 6361 7365  an. In this case
│ │ │ │ -00018180: 2061 2063 6f6d 6269 6e61 746f 7269 616c   a combinatorial
│ │ │ │ -00018190: 206d 6574 686f 6420 6973 2075 7365 642e   method is used.
│ │ │ │ -000181a0: 2054 6869 7320 6d65 7468 6f64 2069 7320   This method is 
│ │ │ │ -000181b0: 6163 6365 7373 6564 0a77 6974 6820 7468  accessed.with th
│ │ │ │ -000181c0: 6520 7573 7561 6c20 4353 4d20 636f 6d6d  e usual CSM comm
│ │ │ │ -000181d0: 616e 6420 7769 7468 2065 6974 6865 7220  and with either 
│ │ │ │ -000181e0: 6f6e 6c79 2061 2074 6f72 6963 2076 6172  only a toric var
│ │ │ │ -000181f0: 6965 7479 206f 7220 6120 746f 7269 6320  iety or a toric 
│ │ │ │ -00018200: 7661 7269 6574 790a 616e 6420 6120 4368  variety.and a Ch
│ │ │ │ -00018210: 6f77 2072 696e 6720 6173 2069 6e70 7574  ow ring as input
│ │ │ │ -00018220: 2e20 496e 2074 6869 7320 6361 7365 2077  . In this case w
│ │ │ │ -00018230: 6520 6f6e 6c79 2072 6571 7569 7265 2074  e only require t
│ │ │ │ -00018240: 6861 7420 7468 6520 696e 7075 7420 746f  hat the input to
│ │ │ │ -00018250: 7269 630a 7661 7269 6574 7920 6973 2063  ric.variety is c
│ │ │ │ -00018260: 6f6d 706c 6574 6520 616e 6420 7369 6d70  omplete and simp
│ │ │ │ -00018270: 6c69 6369 616c 2028 696e 2070 6172 7469  licial (in parti
│ │ │ │ -00018280: 6375 6c61 7220 7765 2064 6f20 6e6f 7420  cular we do not 
│ │ │ │ -00018290: 6e65 6564 2069 7420 746f 2062 650a 736d  need it to be.sm
│ │ │ │ -000182a0: 6f6f 7468 292e 0a0a 2b2d 2d2d 2d2d 2d2d  ooth)...+-------
│ │ │ │ -000182b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000182c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000182d0: 2d2d 2d2b 0a7c 6933 3420 3a20 6e65 6564  ---+.|i34 : need
│ │ │ │ -000182e0: 7350 6163 6b61 6765 2022 4e6f 726d 616c  sPackage "Normal
│ │ │ │ -000182f0: 546f 7269 6356 6172 6965 7469 6573 2220  ToricVarieties" 
│ │ │ │ -00018300: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00018310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018320: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00018330: 6f33 3420 3d20 4e6f 726d 616c 546f 7269  o34 = NormalTori
│ │ │ │ -00018340: 6356 6172 6965 7469 6573 2020 2020 2020  cVarieties      
│ │ │ │ -00018350: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00018360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000180e0: 2d2d 2d2d 2d2b 0a0a 5468 6973 2066 756e  -----+..This fun
│ │ │ │ +000180f0: 6374 696f 6e20 6d61 7920 616c 736f 2063  ction may also c
│ │ │ │ +00018100: 6f6d 7075 7465 2074 6865 2043 534d 2063  ompute the CSM c
│ │ │ │ +00018110: 6c61 7373 206f 6620 6120 6e6f 726d 616c  lass of a normal
│ │ │ │ +00018120: 2074 6f72 6963 2076 6172 6965 7479 2064   toric variety d
│ │ │ │ +00018130: 6566 696e 6564 0a62 7920 6120 6661 6e2e  efined.by a fan.
│ │ │ │ +00018140: 2049 6e20 7468 6973 2063 6173 6520 6120   In this case a 
│ │ │ │ +00018150: 636f 6d62 696e 6174 6f72 6961 6c20 6d65  combinatorial me
│ │ │ │ +00018160: 7468 6f64 2069 7320 7573 6564 2e20 5468  thod is used. Th
│ │ │ │ +00018170: 6973 206d 6574 686f 6420 6973 2061 6363  is method is acc
│ │ │ │ +00018180: 6573 7365 640a 7769 7468 2074 6865 2075  essed.with the u
│ │ │ │ +00018190: 7375 616c 2043 534d 2063 6f6d 6d61 6e64  sual CSM command
│ │ │ │ +000181a0: 2077 6974 6820 6569 7468 6572 206f 6e6c   with either onl
│ │ │ │ +000181b0: 7920 6120 746f 7269 6320 7661 7269 6574  y a toric variet
│ │ │ │ +000181c0: 7920 6f72 2061 2074 6f72 6963 2076 6172  y or a toric var
│ │ │ │ +000181d0: 6965 7479 0a61 6e64 2061 2043 686f 7720  iety.and a Chow 
│ │ │ │ +000181e0: 7269 6e67 2061 7320 696e 7075 742e 2049  ring as input. I
│ │ │ │ +000181f0: 6e20 7468 6973 2063 6173 6520 7765 206f  n this case we o
│ │ │ │ +00018200: 6e6c 7920 7265 7175 6972 6520 7468 6174  nly require that
│ │ │ │ +00018210: 2074 6865 2069 6e70 7574 2074 6f72 6963   the input toric
│ │ │ │ +00018220: 0a76 6172 6965 7479 2069 7320 636f 6d70  .variety is comp
│ │ │ │ +00018230: 6c65 7465 2061 6e64 2073 696d 706c 6963  lete and simplic
│ │ │ │ +00018240: 6961 6c20 2869 6e20 7061 7274 6963 756c  ial (in particul
│ │ │ │ +00018250: 6172 2077 6520 646f 206e 6f74 206e 6565  ar we do not nee
│ │ │ │ +00018260: 6420 6974 2074 6f20 6265 0a73 6d6f 6f74  d it to be.smoot
│ │ │ │ +00018270: 6829 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  h)...+----------
│ │ │ │ +00018280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000182a0: 2b0a 7c69 3334 203a 206e 6565 6473 5061  +.|i34 : needsPa
│ │ │ │ +000182b0: 636b 6167 6520 224e 6f72 6d61 6c54 6f72  ckage "NormalTor
│ │ │ │ +000182c0: 6963 5661 7269 6574 6965 7322 207c 0a7c  icVarieties" |.|
│ │ │ │ +000182d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000182e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000182f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3334            |.|o34
│ │ │ │ +00018300: 203d 204e 6f72 6d61 6c54 6f72 6963 5661   = NormalToricVa
│ │ │ │ +00018310: 7269 6574 6965 7320 2020 2020 2020 2020  rieties         
│ │ │ │ +00018320: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00018330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018350: 2020 2020 7c0a 7c6f 3334 203a 2050 6163      |.|o34 : Pac
│ │ │ │ +00018360: 6b61 6765 2020 2020 2020 2020 2020 2020  kage            
│ │ │ │  00018370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018380: 2020 2020 2020 207c 0a7c 6f33 3420 3a20         |.|o34 : 
│ │ │ │ -00018390: 5061 636b 6167 6520 2020 2020 2020 2020  Package         
│ │ │ │ -000183a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000183b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -000183c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000183d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000183e0: 2d2b 0a7c 6933 3520 3a20 5520 3d20 6869  -+.|i35 : U = hi
│ │ │ │ -000183f0: 727a 6562 7275 6368 5375 7266 6163 6520  rzebruchSurface 
│ │ │ │ -00018400: 3720 2020 2020 2020 2020 2020 2020 7c0a  7             |.
│ │ │ │ -00018410: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00018380: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00018390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000183a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000183b0: 7c69 3335 203a 2055 203d 2068 6972 7a65  |i35 : U = hirze
│ │ │ │ +000183c0: 6272 7563 6853 7572 6661 6365 2037 2020  bruchSurface 7  
│ │ │ │ +000183d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000183e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000183f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018400: 2020 2020 2020 2020 7c0a 7c6f 3335 203d          |.|o35 =
│ │ │ │ +00018410: 2055 2020 2020 2020 2020 2020 2020 2020   U              
│ │ │ │  00018420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018430: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ -00018440: 3520 3d20 5520 2020 2020 2020 2020 2020  5 = U           
│ │ │ │ +00018430: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00018440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018460: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00018470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018490: 2020 2020 207c 0a7c 6f33 3520 3a20 4e6f       |.|o35 : No
│ │ │ │ -000184a0: 726d 616c 546f 7269 6356 6172 6965 7479  rmalToricVariety
│ │ │ │ -000184b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000184c0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -000184d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000184e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000184f0: 0a7c 6933 3620 3a20 4368 3d54 6f72 6963  .|i36 : Ch=Toric
│ │ │ │ -00018500: 4368 6f77 5269 6e67 2855 2920 2020 2020  ChowRing(U)     
│ │ │ │ -00018510: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00018520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018460: 2020 7c0a 7c6f 3335 203a 204e 6f72 6d61    |.|o35 : Norma
│ │ │ │ +00018470: 6c54 6f72 6963 5661 7269 6574 7920 2020  lToricVariety   
│ │ │ │ +00018480: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00018490: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000184a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000184b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +000184c0: 3336 203a 2043 683d 546f 7269 6343 686f  36 : Ch=ToricCho
│ │ │ │ +000184d0: 7752 696e 6728 5529 2020 2020 2020 2020  wRing(U)        
│ │ │ │ +000184e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000184f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018510: 2020 2020 2020 7c0a 7c6f 3336 203d 2043        |.|o36 = C
│ │ │ │ +00018520: 6820 2020 2020 2020 2020 2020 2020 2020  h               
│ │ │ │  00018530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018540: 2020 2020 2020 2020 207c 0a7c 6f33 3620           |.|o36 
│ │ │ │ -00018550: 3d20 4368 2020 2020 2020 2020 2020 2020  = Ch            
│ │ │ │ +00018540: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00018550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018570: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00018580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000185a0: 2020 207c 0a7c 6f33 3620 3a20 5175 6f74     |.|o36 : Quot
│ │ │ │ -000185b0: 6965 6e74 5269 6e67 2020 2020 2020 2020  ientRing        
│ │ │ │ -000185c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000185d0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -000185e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000185f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00018600: 6933 3720 3a20 4353 4d20 5520 2020 2020  i37 : CSM U     
│ │ │ │ +00018570: 7c0a 7c6f 3336 203a 2051 756f 7469 656e  |.|o36 : Quotien
│ │ │ │ +00018580: 7452 696e 6720 2020 2020 2020 2020 2020  tRing           
│ │ │ │ +00018590: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000185a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000185b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000185c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3337  ----------+.|i37
│ │ │ │ +000185d0: 203a 2043 534d 2055 2020 2020 2020 2020   : CSM U        
│ │ │ │ +000185e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000185f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00018600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018620: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00018630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018620: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00018630: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  00018640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018650: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00018660: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -00018670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018680: 2020 2020 7c0a 7c6f 3337 203d 202d 2033      |.|o37 = - 3
│ │ │ │ -00018690: 7820 7820 202b 2078 2020 2d20 3578 2020  x x  + x  - 5x  
│ │ │ │ -000186a0: 2b20 3278 2020 2b20 3120 2020 2020 2020  + 2x  + 1       
│ │ │ │ -000186b0: 207c 0a7c 2020 2020 2020 2020 2020 3220   |.|          2 
│ │ │ │ -000186c0: 3320 2020 2033 2020 2020 2032 2020 2020  3    3     2    
│ │ │ │ -000186d0: 2033 2020 2020 2020 2020 2020 2020 7c0a   3            |.
│ │ │ │ -000186e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000186f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018700: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00018710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018720: 205a 5a5b 7820 2e2e 7820 5d20 2020 2020   ZZ[x ..x ]     
│ │ │ │ -00018730: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00018740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018750: 2020 3020 2020 3320 2020 2020 2020 2020    0   3         
│ │ │ │ -00018760: 2020 2020 207c 0a7c 6f33 3720 3a20 2d2d       |.|o37 : --
│ │ │ │ -00018770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018790: 2d2d 7c0a 7c20 2020 2020 2028 7820 7820  --|.|      (x x 
│ │ │ │ -000187a0: 2c20 7820 7820 2c20 7820 202d 2078 202c  , x x , x  - x ,
│ │ │ │ -000187b0: 2078 2020 2b20 3778 2020 2d20 7820 297c   x  + 7x  - x )|
│ │ │ │ -000187c0: 0a7c 2020 2020 2020 2020 3020 3220 2020  .|        0 2   
│ │ │ │ -000187d0: 3120 3320 2020 3020 2020 2032 2020 2031  1 3   0    2   1
│ │ │ │ -000187e0: 2020 2020 2032 2020 2020 3320 7c0a 2b2d       2    3 |.+-
│ │ │ │ -000187f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018810: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 3820  ---------+.|i38 
│ │ │ │ -00018820: 3a20 6373 6d31 3d43 534d 2843 682c 5529  : csm1=CSM(Ch,U)
│ │ │ │ +00018650: 207c 0a7c 6f33 3720 3d20 2d20 3378 2078   |.|o37 = - 3x x
│ │ │ │ +00018660: 2020 2b20 7820 202d 2035 7820 202b 2032    + x  - 5x  + 2
│ │ │ │ +00018670: 7820 202b 2031 2020 2020 2020 2020 7c0a  x  + 1        |.
│ │ │ │ +00018680: 7c20 2020 2020 2020 2020 2032 2033 2020  |          2 3  
│ │ │ │ +00018690: 2020 3320 2020 2020 3220 2020 2020 3320    3     2     3 
│ │ │ │ +000186a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000186b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000186c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000186d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000186e0: 2020 2020 2020 2020 2020 2020 2020 5a5a                ZZ
│ │ │ │ +000186f0: 5b78 202e 2e78 205d 2020 2020 2020 2020  [x ..x ]        
│ │ │ │ +00018700: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00018710: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +00018720: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ +00018730: 2020 7c0a 7c6f 3337 203a 202d 2d2d 2d2d    |.|o37 : -----
│ │ │ │ +00018740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00018760: 0a7c 2020 2020 2020 2878 2078 202c 2078  .|      (x x , x
│ │ │ │ +00018770: 2078 202c 2078 2020 2d20 7820 2c20 7820   x , x  - x , x 
│ │ │ │ +00018780: 202b 2037 7820 202d 2078 2029 7c0a 7c20   + 7x  - x )|.| 
│ │ │ │ +00018790: 2020 2020 2020 2030 2032 2020 2031 2033         0 2   1 3
│ │ │ │ +000187a0: 2020 2030 2020 2020 3220 2020 3120 2020     0    2   1   
│ │ │ │ +000187b0: 2020 3220 2020 2033 207c 0a2b 2d2d 2d2d    2    3 |.+----
│ │ │ │ +000187c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000187d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000187e0: 2d2d 2d2d 2d2d 2b0a 7c69 3338 203a 2063  ------+.|i38 : c
│ │ │ │ +000187f0: 736d 313d 4353 4d28 4368 2c55 2920 2020  sm1=CSM(Ch,U)   
│ │ │ │ +00018800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018810: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00018820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018840: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00018850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018870: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00018880: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -00018890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000188a0: 7c0a 7c6f 3338 203d 202d 2033 7820 7820  |.|o38 = - 3x x 
│ │ │ │ -000188b0: 202b 2078 2020 2d20 3578 2020 2b20 3278   + x  - 5x  + 2x
│ │ │ │ -000188c0: 2020 2b20 3120 2020 2020 2020 207c 0a7c    + 1        |.|
│ │ │ │ -000188d0: 2020 2020 2020 2020 2020 3220 3320 2020            2 3   
│ │ │ │ -000188e0: 2033 2020 2020 2032 2020 2020 2033 2020   3     2     3  
│ │ │ │ -000188f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00018840: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00018850: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00018860: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00018870: 6f33 3820 3d20 2d20 3378 2078 2020 2b20  o38 = - 3x x  + 
│ │ │ │ +00018880: 7820 202d 2035 7820 202b 2032 7820 202b  x  - 5x  + 2x  +
│ │ │ │ +00018890: 2031 2020 2020 2020 2020 7c0a 7c20 2020   1        |.|   
│ │ │ │ +000188a0: 2020 2020 2020 2032 2033 2020 2020 3320         2 3    3 
│ │ │ │ +000188b0: 2020 2020 3220 2020 2020 3320 2020 2020      2     3     
│ │ │ │ +000188c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000188d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000188e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000188f0: 2020 2020 7c0a 7c6f 3338 203a 2043 6820      |.|o38 : Ch 
│ │ │ │  00018900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018920: 2020 2020 2020 207c 0a7c 6f33 3820 3a20         |.|o38 : 
│ │ │ │ -00018930: 4368 2020 2020 2020 2020 2020 2020 2020  Ch              
│ │ │ │ -00018940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018950: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -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
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│ │ │ │ -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
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│ │ │ │ -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
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│ │ │ │ -00018a40: 6420 4265 7274 696e 6920 6973 202a 6e6f  d Bertini is *no
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│ │ │ │ -00018a60: 2063 6f6e 6669 6775 7265 643a 0a63 6f6e   configured:.con
│ │ │ │ -00018a70: 6669 6775 7269 6e67 2042 6572 7469 6e69  figuring Bertini
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│ │ │ │ -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
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│ │ │ │ -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  ================
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│ │ │ │ -00018d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00019790: 7375 6273 6574 7320 6f66 2074 6865 2067  subsets of the g
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│ │ │ │ -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
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│ │ │ │ -00019800: 7375 7266 6163 6520 6769 7665 6e20 6279  surface given by
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│ │ │ │ -00019820: 616c 6c20 706f 6c79 6e6f 6d69 616c 7320  all polynomials 
│ │ │ │ -00019830: 696e 2074 6865 0a20 2020 2020 2020 2063  in the.        c
│ │ │ │ -00019840: 6f72 7265 7370 6f6e 6469 6e67 2073 6574  orresponding set
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│ │ │ │ -00019870: 6973 206e 6f20 6578 7472 6120 636f 7374  is no extra cost
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│ │ │ │ -000198d0: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
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│ │ │ │ -00019920: 6520 4469 7265 6374 436f 6d70 6c65 7465  e DirectComplete
│ │ │ │ -00019930: 496e 740a 2020 2020 2020 2020 4d65 7468  Int.        Meth
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│ │ │ │ -00019960: 696e 666f 726d 6174 696f 6e20 6162 6f75  information abou
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│ │ │ │ -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
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│ │ │ │ +00019590: 2020 2020 202a 204f 7574 7075 7420 3d3e       * Output =>
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│ │ │ │ +000195e0: 2020 2072 6574 7572 6e20 7468 6520 6465     return the de
│ │ │ │ +000195f0: 6661 756c 7420 6f75 7470 7574 2069 7320  fault output is 
│ │ │ │ +00019600: 616e 2069 6e74 6567 6572 0a20 2020 2020  an integer.     
│ │ │ │ +00019610: 202a 204f 7574 7075 7420 3d3e 202e 2e2e   * Output => ...
│ │ │ │ +00019620: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ +00019630: 4368 6f77 5269 6e67 456c 656d 656e 742c  ChowRingElement,
│ │ │ │ +00019640: 2048 6173 6846 6f72 6d2c 2074 6865 2074   HashForm, the t
│ │ │ │ +00019650: 7970 6520 6f66 0a20 2020 2020 2020 206f  ype of.        o
│ │ │ │ +00019660: 7574 7075 7420 746f 2072 6574 7572 6e2c  utput to return,
│ │ │ │ +00019670: 2048 6173 6846 6f72 6d20 7265 7475 726e   HashForm return
│ │ │ │ +00019680: 7320 6120 4d75 7461 626c 6548 6173 6854  s a MutableHashT
│ │ │ │ +00019690: 6162 6c65 2063 6f6e 7461 696e 696e 6720  able containing 
│ │ │ │ +000196a0: 7468 650a 2020 2020 2020 2020 6b65 7920  the.        key 
│ │ │ │ +000196b0: 2243 534d 2220 2874 6865 2043 534d 2063  "CSM" (the CSM c
│ │ │ │ +000196c0: 6c61 7373 292c 2061 6e64 206b 6579 7320  lass), and keys 
│ │ │ │ +000196d0: 6f66 2074 6865 2066 6f72 6d0a 2020 2020  of the form.    
│ │ │ │ +000196e0: 2020 2020 5c7b 305c 7d2c 5c7b 315c 7d2c      \{0\},\{1\},
│ │ │ │ +000196f0: 5c7b 325c 7d2c 2e2e 2e2c 5c7b 302c 315c  \{2\},...,\{0,1\
│ │ │ │ +00019700: 7d2c 5c7b 302c 325c 7d20 2e2e 2e2e 5c7b  },\{0,2\} ....\{
│ │ │ │ +00019710: 302c 312c 325c 7d2e 2e2e 2061 6e64 2073  0,1,2\}... and s
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│ │ │ │ +00019760: 7365 7473 206f 6620 7468 6520 6765 6e65  sets of the gene
│ │ │ │ +00019770: 7261 746f 7273 206f 660a 2020 2020 2020  rators of.      
│ │ │ │ +00019780: 2020 7468 6520 696e 7075 7420 6964 6561    the input idea
│ │ │ │ +00019790: 6c2c 2066 6f72 2065 6163 6820 7365 7420  l, for each set 
│ │ │ │ +000197a0: 6f66 2069 6e64 6963 6573 2074 6865 2043  of indices the C
│ │ │ │ +000197b0: 534d 2063 6c61 7373 206f 6620 7468 650a  SM class of the.
│ │ │ │ +000197c0: 2020 2020 2020 2020 6879 7065 7273 7572          hypersur
│ │ │ │ +000197d0: 6661 6365 2067 6976 656e 2062 7920 7468  face given by th
│ │ │ │ +000197e0: 6520 7072 6f64 7563 7420 6f66 2061 6c6c  e product of all
│ │ │ │ +000197f0: 2070 6f6c 796e 6f6d 6961 6c73 2069 6e20   polynomials in 
│ │ │ │ +00019800: 7468 650a 2020 2020 2020 2020 636f 7272  the.        corr
│ │ │ │ +00019810: 6573 706f 6e64 696e 6720 7365 7420 6f66  esponding set of
│ │ │ │ +00019820: 2067 656e 6572 6174 6f72 7320 6973 2073   generators is s
│ │ │ │ +00019830: 746f 7265 642c 2074 6865 7265 2069 7320  tored, there is 
│ │ │ │ +00019840: 6e6f 2065 7874 7261 2063 6f73 7420 746f  no extra cost to
│ │ │ │ +00019850: 0a20 2020 2020 2020 2075 7369 6e67 2074  .        using t
│ │ │ │ +00019860: 6869 7320 6f70 7469 6f6e 0a20 2020 2020  his option.     
│ │ │ │ +00019870: 202a 2049 6e64 734f 6653 6d6f 6f74 6820   * IndsOfSmooth 
│ │ │ │ +00019880: 286d 6973 7369 6e67 2064 6f63 756d 656e  (missing documen
│ │ │ │ +00019890: 7461 7469 6f6e 2920 3d3e 202e 2e2e 2c20  tation) => ..., 
│ │ │ │ +000198a0: 6465 6661 756c 7420 7661 6c75 6520 7b7d  default value {}
│ │ │ │ +000198b0: 2c20 7468 6973 0a20 2020 2020 2020 206f  , this.        o
│ │ │ │ +000198c0: 7074 696f 6e20 6d61 7920 7370 6565 6420  ption may speed 
│ │ │ │ +000198d0: 7570 2074 6865 2072 756e 2074 696d 6520  up the run time 
│ │ │ │ +000198e0: 7768 656e 2075 7369 6e67 2074 6865 2044  when using the D
│ │ │ │ +000198f0: 6972 6563 7443 6f6d 706c 6574 6549 6e74  irectCompleteInt
│ │ │ │ +00019900: 0a20 2020 2020 2020 204d 6574 686f 6420  .        Method 
│ │ │ │ +00019910: 6966 2074 6865 2075 7365 7220 6b6e 6f77  if the user know
│ │ │ │ +00019920: 7320 6164 6469 7469 6f6e 616c 2069 6e66  s additional inf
│ │ │ │ +00019930: 6f72 6d61 7469 6f6e 2061 626f 7574 2074  ormation about t
│ │ │ │ +00019940: 6865 2069 6e70 7574 2069 6465 616c 2c0a  he input ideal,.
│ │ │ │ +00019950: 2020 2020 2020 2020 7365 6520 2a6e 6f74          see *not
│ │ │ │ +00019960: 6520 496e 6473 4f66 536d 6f6f 7468 3a20  e IndsOfSmooth: 
│ │ │ │ +00019970: 496e 6473 4f66 536d 6f6f 7468 2c0a 2020  IndsOfSmooth,.  
│ │ │ │ +00019980: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ +00019990: 202a 2061 202a 6e6f 7465 2072 696e 6720   * a *note ring 
│ │ │ │ +000199a0: 656c 656d 656e 743a 2028 4d61 6361 756c  element: (Macaul
│ │ │ │ +000199b0: 6179 3244 6f63 2952 696e 6745 6c65 6d65  ay2Doc)RingEleme
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│ │ │ │ +000199d0: 2020 2020 2020 2063 6861 7261 6374 6572         character
│ │ │ │ +000199e0: 6973 7469 630a 0a44 6573 6372 6970 7469  istic..Descripti
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│ │ │ │ +00019a00: 466f 7220 6120 7375 6273 6368 656d 6520  For a subscheme 
│ │ │ │ +00019a10: 5620 6f66 2061 6e20 6170 706c 6963 6162  V of an applicab
│ │ │ │ +00019a20: 6c65 2074 6f72 6963 2076 6172 6965 7479  le toric variety
│ │ │ │ +00019a30: 2058 2c20 7468 6973 2063 6f6d 6d61 6e64   X, this command
│ │ │ │ +00019a40: 2063 6f6d 7075 7465 7320 7468 650a 4575   computes the.Eu
│ │ │ │ +00019a50: 6c65 7220 6368 6172 6163 7465 7269 7374  ler characterist
│ │ │ │ +00019a60: 6963 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  ic..+-----------
│ │ │ │ +00019a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019ae0: 2d2d 2d2d 2d2b 0a7c 6931 203a 206b 6b3d  -----+.|i1 : kk=
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│ │ │ │ -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                  
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│ │ │ │ +00019b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  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  ----------------
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│ │ │ │ +00019b60: 5f30 2e2e 785f 345d 2020 2020 2020 2020  _0..x_4]        
│ │ │ │ +00019b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00019c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019c20: 2020 2020 207c 0a7c 6f32 203d 2052 2020       |.|o2 = R  
│ │ │ │ +00019c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00019c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00019cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00019d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00019d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00019e80: 3837 7820 202b 2033 3738 3378 202c 202d  87x  + 3783x , -
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│ │ │ │ -00019ea0: 7820 202b 207c 0a7c 2020 2020 2020 2020  x  + |.|        
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│ │ │ │ -00019ee0: 2020 2020 2020 3020 2020 2020 2020 2030        0        0
│ │ │ │ -00019ef0: 2031 2020 207c 0a7c 2020 2020 202d 2d2d   1   |.|     ---
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│ │ │ │ +00019e10: 2020 2032 2020 2020 2020 2020 2020 2020     2            
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│ │ │ │ +00019e80: 2020 2020 2030 2020 2020 2020 2020 3120       0        1 
│ │ │ │ +00019e90: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +00019ea0: 3320 2020 2020 2020 2034 2020 2020 2020  3        4      
│ │ │ │ +00019eb0: 2020 2030 2020 2020 2020 2020 3020 3120     0        0 1 
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│ │ │ │ -00019f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00019fe0: 3978 2078 207c 0a7c 2020 2020 2020 2020  9x x |.|        
│ │ │ │ -00019ff0: 2020 2031 2020 2020 2020 2020 2030 2032     1         0 2
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│ │ │ │ -0001a010: 2020 2032 2020 2020 2020 2020 3020 3320     2        0 3 
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│ │ │ │ +00019f90: 2020 2b20 3834 3434 7820 7820 202d 2038    + 8444x x  - 8
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│ │ │ │ +00019fb0: 7820 7c0a 7c20 2020 2020 2020 2020 2020  x |.|           
│ │ │ │ +00019fc0: 3120 2020 2020 2020 2020 3020 3220 2020  1         0 2   
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│ │ │ │ +00019fe0: 3220 2020 2020 2020 2030 2033 2020 2020  2        0 3    
│ │ │ │ +00019ff0: 2020 2020 3120 3320 2020 2020 2020 2032      1 3        2
│ │ │ │ +0001a000: 2033 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d   3|.|     ------
│ │ │ │ +0001a010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a080: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ -0001a090: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -0001a0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a0c0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -0001a0d0: 2020 2020 207c 0a7c 2020 2020 202d 2031       |.|     - 1
│ │ │ │ -0001a0e0: 3432 3534 7820 202d 2031 3132 3236 7820  4254x  - 11226x 
│ │ │ │ -0001a0f0: 7820 202b 2032 3635 3378 2078 2020 2b20  x  + 2653x x  + 
│ │ │ │ -0001a100: 3132 3336 3578 2078 2020 2d20 3130 3232  12365x x  - 1022
│ │ │ │ -0001a110: 3678 2078 2020 2d20 3132 3639 3678 2029  6x x  - 12696x )
│ │ │ │ -0001a120: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001a130: 2020 2020 2033 2020 2020 2020 2020 2030       3         0
│ │ │ │ -0001a140: 2034 2020 2020 2020 2020 3120 3420 2020   4        1 4   
│ │ │ │ -0001a150: 2020 2020 2020 3220 3420 2020 2020 2020        2 4       
│ │ │ │ -0001a160: 2020 3320 3420 2020 2020 2020 2020 3420    3 4         4 
│ │ │ │ -0001a170: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001a050: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020  --|.|           
│ │ │ │ +0001a060: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0001a070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a090: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +0001a0a0: 2020 7c0a 7c20 2020 2020 2d20 3134 3235    |.|     - 1425
│ │ │ │ +0001a0b0: 3478 2020 2d20 3131 3232 3678 2078 2020  4x  - 11226x x  
│ │ │ │ +0001a0c0: 2b20 3236 3533 7820 7820 202b 2031 3233  + 2653x x  + 123
│ │ │ │ +0001a0d0: 3635 7820 7820 202d 2031 3032 3236 7820  65x x  - 10226x 
│ │ │ │ +0001a0e0: 7820 202d 2031 3236 3936 7820 2920 2020  x  - 12696x )   
│ │ │ │ +0001a0f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001a100: 2020 3320 2020 2020 2020 2020 3020 3420    3         0 4 
│ │ │ │ +0001a110: 2020 2020 2020 2031 2034 2020 2020 2020         1 4      
│ │ │ │ +0001a120: 2020 2032 2034 2020 2020 2020 2020 2033     2 4         3
│ │ │ │ +0001a130: 2034 2020 2020 2020 2020 2034 2020 2020   4         4    
│ │ │ │ +0001a140: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001a150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a190: 2020 7c0a 7c6f 3320 3a20 4964 6561 6c20    |.|o3 : Ideal 
│ │ │ │ +0001a1a0: 6f66 2052 2020 2020 2020 2020 2020 2020  of R            
│ │ │ │  0001a1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a1c0: 2020 2020 207c 0a7c 6f33 203a 2049 6465       |.|o3 : Ide
│ │ │ │ -0001a1d0: 616c 206f 6620 5220 2020 2020 2020 2020  al of R         
│ │ │ │ -0001a1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a210: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001a1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a1e0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0001a1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a260: 2d2d 2d2d 2d2b 0a7c 6934 203a 2074 696d  -----+.|i4 : tim
│ │ │ │ -0001a270: 6520 4575 6c65 7228 492c 496e 7075 7449  e Euler(I,InputI
│ │ │ │ -0001a280: 7353 6d6f 6f74 683d 3e74 7275 6529 2020  sSmooth=>true)  
│ │ │ │ -0001a290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a2b0: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -0001a2c0: 2030 2e30 3537 3639 3539 7320 2863 7075   0.0576959s (cpu
│ │ │ │ -0001a2d0: 293b 2030 2e30 3335 3439 3238 7320 2874  ); 0.0354928s (t
│ │ │ │ -0001a2e0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ +0001a230: 2d2d 2b0a 7c69 3420 3a20 7469 6d65 2045  --+.|i4 : time E
│ │ │ │ +0001a240: 756c 6572 2849 2c49 6e70 7574 4973 536d  uler(I,InputIsSm
│ │ │ │ +0001a250: 6f6f 7468 3d3e 7472 7565 2920 2020 2020  ooth=>true)     
│ │ │ │ +0001a260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a280: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e    |.| -- used 0.
│ │ │ │ +0001a290: 3036 3737 3437 3473 2028 6370 7529 3b20  0677474s (cpu); 
│ │ │ │ +0001a2a0: 302e 3034 3438 3739 3873 2028 7468 7265  0.0448798s (thre
│ │ │ │ +0001a2b0: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │ +0001a2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a2d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001a2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a300: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001a300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a320: 2020 7c0a 7c6f 3420 3d20 3420 2020 2020    |.|o4 = 4     
│ │ │ │  0001a330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a350: 2020 2020 207c 0a7c 6f34 203d 2034 2020       |.|o4 = 4  
│ │ │ │ +0001a350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a3a0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001a370: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0001a380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a3f0: 2d2d 2d2d 2d2b 0a7c 6935 203a 2074 696d  -----+.|i5 : tim
│ │ │ │ -0001a400: 6520 4575 6c65 7220 4920 2020 2020 2020  e Euler I       
│ │ │ │ -0001a410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a440: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -0001a450: 2030 2e32 3534 3131 3473 2028 6370 7529   0.254114s (cpu)
│ │ │ │ -0001a460: 3b20 302e 3134 3638 3333 7320 2874 6872  ; 0.146833s (thr
│ │ │ │ -0001a470: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │ +0001a3c0: 2d2d 2b0a 7c69 3520 3a20 7469 6d65 2045  --+.|i5 : time E
│ │ │ │ +0001a3d0: 756c 6572 2049 2020 2020 2020 2020 2020  uler I          
│ │ │ │ +0001a3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a410: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e    |.| -- used 0.
│ │ │ │ +0001a420: 3330 3630 3232 7320 2863 7075 293b 2030  306022s (cpu); 0
│ │ │ │ +0001a430: 2e31 3834 3437 3973 2028 7468 7265 6164  .184479s (thread
│ │ │ │ +0001a440: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ +0001a450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a460: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001a470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a490: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001a490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a4b0: 2020 7c0a 7c6f 3520 3d20 3420 2020 2020    |.|o5 = 4     
│ │ │ │  0001a4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a4e0: 2020 2020 207c 0a7c 6f35 203d 2034 2020       |.|o5 = 4  
│ │ │ │ +0001a4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a530: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001a500: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0001a510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a580: 2d2d 2d2d 2d2b 0a7c 6936 203a 2045 756c  -----+.|i6 : Eul
│ │ │ │ -0001a590: 6572 4948 6173 683d 4575 6c65 7228 492c  erIHash=Euler(I,
│ │ │ │ -0001a5a0: 4f75 7470 7574 3d3e 4861 7368 466f 726d  Output=>HashForm
│ │ │ │ -0001a5b0: 293b 2020 2020 2020 2020 2020 2020 2020  );              
│ │ │ │ -0001a5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a5d0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001a550: 2d2d 2b0a 7c69 3620 3a20 4575 6c65 7249  --+.|i6 : EulerI
│ │ │ │ +0001a560: 4861 7368 3d45 756c 6572 2849 2c4f 7574  Hash=Euler(I,Out
│ │ │ │ +0001a570: 7075 743d 3e48 6173 6846 6f72 6d29 3b20  put=>HashForm); 
│ │ │ │ +0001a580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a5a0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0001a5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a620: 2d2d 2d2d 2d2b 0a7c 6937 203a 2041 3d72  -----+.|i7 : A=r
│ │ │ │ -0001a630: 696e 6720 4575 6c65 7249 4861 7368 2322  ing EulerIHash#"
│ │ │ │ -0001a640: 4353 4d22 2020 2020 2020 2020 2020 2020  CSM"            
│ │ │ │ +0001a5f0: 2d2d 2b0a 7c69 3720 3a20 413d 7269 6e67  --+.|i7 : A=ring
│ │ │ │ +0001a600: 2045 756c 6572 4948 6173 6823 2243 534d   EulerIHash#"CSM
│ │ │ │ +0001a610: 2220 2020 2020 2020 2020 2020 2020 2020  "               
│ │ │ │ +0001a620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a640: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001a650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a670: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001a670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a690: 2020 7c0a 7c6f 3720 3d20 4120 2020 2020    |.|o7 = A     
│ │ │ │  0001a6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a6c0: 2020 2020 207c 0a7c 6f37 203d 2041 2020       |.|o7 = A  
│ │ │ │ +0001a6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a6e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001a6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a710: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001a710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a730: 2020 7c0a 7c6f 3720 3a20 5175 6f74 6965    |.|o7 : Quotie
│ │ │ │ +0001a740: 6e74 5269 6e67 2020 2020 2020 2020 2020  ntRing          
│ │ │ │  0001a750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a760: 2020 2020 207c 0a7c 6f37 203a 2051 756f       |.|o7 : Quo
│ │ │ │ -0001a770: 7469 656e 7452 696e 6720 2020 2020 2020  tientRing       
│ │ │ │ -0001a780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a7b0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001a760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a780: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0001a790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a800: 2d2d 2d2d 2d2b 0a7c 6938 203a 2045 756c  -----+.|i8 : Eul
│ │ │ │ -0001a810: 6572 4948 6173 6823 7b30 2c31 7d3d 3d43  erIHash#{0,1}==C
│ │ │ │ -0001a820: 534d 2841 2c69 6465 616c 2849 5f30 2a49  SM(A,ideal(I_0*I
│ │ │ │ -0001a830: 5f31 2929 2020 2020 2020 2020 2020 2020  _1))            
│ │ │ │ +0001a7d0: 2d2d 2b0a 7c69 3820 3a20 4575 6c65 7249  --+.|i8 : EulerI
│ │ │ │ +0001a7e0: 4861 7368 237b 302c 317d 3d3d 4353 4d28  Hash#{0,1}==CSM(
│ │ │ │ +0001a7f0: 412c 6964 6561 6c28 495f 302a 495f 3129  A,ideal(I_0*I_1)
│ │ │ │ +0001a800: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +0001a810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a820: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001a830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a850: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001a850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a870: 2020 7c0a 7c6f 3820 3d20 7472 7565 2020    |.|o8 = true  
│ │ │ │  0001a880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a8a0: 2020 2020 207c 0a7c 6f38 203d 2074 7275       |.|o8 = tru
│ │ │ │ -0001a8b0: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ -0001a8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a8f0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001a8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a8c0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0001a8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0001a950: 2b69 6465 616c 2878 5f30 2a78 5f32 2d78  +ideal(x_0*x_2-x
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│ │ │ │ +0001a910: 2d2d 2b0a 7c69 3920 3a20 4a3d 492b 6964  --+.|i9 : J=I+id
│ │ │ │ +0001a920: 6561 6c28 785f 302a 785f 322d 785f 332a  eal(x_0*x_2-x_3*
│ │ │ │ +0001a930: 785f 3029 2020 2020 2020 2020 2020 2020  x_0)            
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│ │ │ │  0001a970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a990: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001a990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a9b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001a9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a9e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001a9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa20: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -0001aa30: 2020 2020 207c 0a7c 6f39 203d 2069 6465       |.|o9 = ide
│ │ │ │ -0001aa40: 616c 2028 3130 3778 2020 2b20 3433 3736  al (107x  + 4376
│ │ │ │ -0001aa50: 7820 202d 2036 3331 3678 2020 2b20 3331  x  - 6316x  + 31
│ │ │ │ -0001aa60: 3837 7820 202b 2033 3738 3378 202c 202d  87x  + 3783x , -
│ │ │ │ -0001aa70: 2036 3035 3378 2020 2b20 3835 3730 7820   6053x  + 8570x 
│ │ │ │ -0001aa80: 7820 202b 207c 0a7c 2020 2020 2020 2020  x  + |.|        
│ │ │ │ -0001aa90: 2020 2020 2020 2020 3020 2020 2020 2020          0       
│ │ │ │ -0001aaa0: 2031 2020 2020 2020 2020 3220 2020 2020   1        2     
│ │ │ │ -0001aab0: 2020 2033 2020 2020 2020 2020 3420 2020     3        4   
│ │ │ │ -0001aac0: 2020 2020 2020 3020 2020 2020 2020 2030        0        0
│ │ │ │ -0001aad0: 2031 2020 207c 0a7c 2020 2020 202d 2d2d   1   |.|     ---
│ │ │ │ +0001a9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a9f0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +0001aa00: 2020 7c0a 7c6f 3920 3d20 6964 6561 6c20    |.|o9 = ideal 
│ │ │ │ +0001aa10: 2831 3037 7820 202b 2034 3337 3678 2020  (107x  + 4376x  
│ │ │ │ +0001aa20: 2d20 3633 3136 7820 202b 2033 3138 3778  - 6316x  + 3187x
│ │ │ │ +0001aa30: 2020 2b20 3337 3833 7820 2c20 2d20 3630    + 3783x , - 60
│ │ │ │ +0001aa40: 3533 7820 202b 2038 3537 3078 2078 2020  53x  + 8570x x  
│ │ │ │ +0001aa50: 2b20 7c0a 7c20 2020 2020 2020 2020 2020  + |.|           
│ │ │ │ +0001aa60: 2020 2020 2030 2020 2020 2020 2020 3120       0        1 
│ │ │ │ +0001aa70: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +0001aa80: 3320 2020 2020 2020 2034 2020 2020 2020  3        4      
│ │ │ │ +0001aa90: 2020 2030 2020 2020 2020 2020 3020 3120     0        0 1 
│ │ │ │ +0001aaa0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d    |.|     ------
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│ │ │ │ +0001aac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001aad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0001aaf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0001ab40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001ab60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001ab80: 3539 7820 202d 2031 3630 3930 7820 7820  59x  - 16090x x 
│ │ │ │ -0001ab90: 202d 2038 3231 3078 2078 2020 2b20 3530   - 8210x x  + 50
│ │ │ │ -0001aba0: 3731 7820 202b 2038 3434 3478 2078 2020  71x  + 8444x x  
│ │ │ │ -0001abb0: 2d20 3839 3937 7820 7820 202d 2036 3934  - 8997x x  - 694
│ │ │ │ -0001abc0: 3978 2078 207c 0a7c 2020 2020 2020 2020  9x x |.|        
│ │ │ │ -0001abd0: 2020 2031 2020 2020 2020 2020 2030 2032     1         0 2
│ │ │ │ -0001abe0: 2020 2020 2020 2020 3120 3220 2020 2020          1 2     
│ │ │ │ -0001abf0: 2020 2032 2020 2020 2020 2020 3020 3320     2        0 3 
│ │ │ │ -0001ac00: 2020 2020 2020 2031 2033 2020 2020 2020         1 3      
│ │ │ │ -0001ac10: 2020 3220 337c 0a7c 2020 2020 202d 2d2d    2 3|.|     ---
│ │ │ │ +0001aaf0: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020  --|.|           
│ │ │ │ +0001ab00: 3220 2020 2020 2020 2020 2020 2020 2020  2               
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│ │ │ │ +0001ab30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ab40: 2020 7c0a 7c20 2020 2020 3130 3335 3978    |.|     10359x
│ │ │ │ +0001ab50: 2020 2d20 3136 3039 3078 2078 2020 2d20    - 16090x x  - 
│ │ │ │ +0001ab60: 3832 3130 7820 7820 202b 2035 3037 3178  8210x x  + 5071x
│ │ │ │ +0001ab70: 2020 2b20 3834 3434 7820 7820 202d 2038    + 8444x x  - 8
│ │ │ │ +0001ab80: 3939 3778 2078 2020 2d20 3639 3439 7820  997x x  - 6949x 
│ │ │ │ +0001ab90: 7820 7c0a 7c20 2020 2020 2020 2020 2020  x |.|           
│ │ │ │ +0001aba0: 3120 2020 2020 2020 2020 3020 3220 2020  1         0 2   
│ │ │ │ +0001abb0: 2020 2020 2031 2032 2020 2020 2020 2020       1 2        
│ │ │ │ +0001abc0: 3220 2020 2020 2020 2030 2033 2020 2020  2        0 3    
│ │ │ │ +0001abd0: 2020 2020 3120 3320 2020 2020 2020 2032      1 3        2
│ │ │ │ +0001abe0: 2033 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d   3|.|     ------
│ │ │ │ +0001abf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0001ac70: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -0001ac80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ac90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aca0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -0001acb0: 2020 2020 207c 0a7c 2020 2020 202d 2031       |.|     - 1
│ │ │ │ -0001acc0: 3432 3534 7820 202d 2031 3132 3236 7820  4254x  - 11226x 
│ │ │ │ -0001acd0: 7820 202b 2032 3635 3378 2078 2020 2b20  x  + 2653x x  + 
│ │ │ │ -0001ace0: 3132 3336 3578 2078 2020 2d20 3130 3232  12365x x  - 1022
│ │ │ │ -0001acf0: 3678 2078 2020 2d20 3132 3639 3678 202c  6x x  - 12696x ,
│ │ │ │ -0001ad00: 2078 2078 207c 0a7c 2020 2020 2020 2020   x x |.|        
│ │ │ │ -0001ad10: 2020 2020 2033 2020 2020 2020 2020 2030       3         0
│ │ │ │ -0001ad20: 2034 2020 2020 2020 2020 3120 3420 2020   4        1 4   
│ │ │ │ -0001ad30: 2020 2020 2020 3220 3420 2020 2020 2020        2 4       
│ │ │ │ -0001ad40: 2020 3320 3420 2020 2020 2020 2020 3420    3 4         4 
│ │ │ │ -0001ad50: 2020 3020 327c 0a7c 2020 2020 202d 2d2d    0 2|.|     ---
│ │ │ │ +0001ac30: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020  --|.|           
│ │ │ │ +0001ac40: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0001ac50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001ac70: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +0001ac80: 2020 7c0a 7c20 2020 2020 2d20 3134 3235    |.|     - 1425
│ │ │ │ +0001ac90: 3478 2020 2d20 3131 3232 3678 2078 2020  4x  - 11226x x  
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│ │ │ │ +0001acc0: 7820 202d 2031 3236 3936 7820 2c20 7820  x  - 12696x , x 
│ │ │ │ +0001acd0: 7820 7c0a 7c20 2020 2020 2020 2020 2020  x |.|           
│ │ │ │ +0001ace0: 2020 3320 2020 2020 2020 2020 3020 3420    3         0 4 
│ │ │ │ +0001acf0: 2020 2020 2020 2031 2034 2020 2020 2020         1 4      
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│ │ │ │ +0001ad10: 2034 2020 2020 2020 2020 2034 2020 2030   4         4   0
│ │ │ │ +0001ad20: 2032 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d   2|.|     ------
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│ │ │ │ -0001ad70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0001ad90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ada0: 2d2d 2d2d 2d7c 0a7c 2020 2020 202d 2078  -----|.|     - x
│ │ │ │ -0001adb0: 2078 2029 2020 2020 2020 2020 2020 2020   x )            
│ │ │ │ -0001adc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ad70: 2d2d 7c0a 7c20 2020 2020 2d20 7820 7820  --|.|     - x x 
│ │ │ │ +0001ad80: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +0001ad90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ada0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001adb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001adc0: 2020 7c0a 7c20 2020 2020 2020 2030 2033    |.|        0 3
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│ │ │ │  0001ade0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001adf0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001ae00: 3020 3320 2020 2020 2020 2020 2020 2020  0 3             
│ │ │ │ -0001ae10: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001ae00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0001ae20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001ae70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ae60: 2020 7c0a 7c6f 3920 3a20 4964 6561 6c20    |.|o9 : Ideal 
│ │ │ │ +0001ae70: 6f66 2052 2020 2020 2020 2020 2020 2020  of R            
│ │ │ │  0001ae80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ae90: 2020 2020 207c 0a7c 6f39 203a 2049 6465       |.|o9 : Ide
│ │ │ │ -0001aea0: 616c 206f 6620 5220 2020 2020 2020 2020  al of R         
│ │ │ │ -0001aeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aee0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001ae90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001aea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001aeb0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0001aec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001aed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001aee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001aef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001af00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001af10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001af20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001af30: 2d2d 2d2d 2d2b 0a0a 4e6f 7465 2074 6861  -----+..Note tha
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│ │ │ │ -0001af50: 6f76 6520 6973 2061 2063 6f6d 706c 6574  ove is a complet
│ │ │ │ -0001af60: 6520 696e 7465 7273 6563 7469 6f6e 2c20  e intersection, 
│ │ │ │ -0001af70: 7468 7573 2077 6520 6d61 7920 6368 616e  thus we may chan
│ │ │ │ -0001af80: 6765 2074 6865 0a6d 6574 686f 6420 6f70  ge the.method op
│ │ │ │ -0001af90: 7469 6f6e 2077 6869 6368 206d 6179 2073  tion which may s
│ │ │ │ -0001afa0: 7065 6564 2063 6f6d 7075 7461 7469 6f6e  peed computation
│ │ │ │ -0001afb0: 2069 6e20 736f 6d65 2063 6173 6573 2e20   in some cases. 
│ │ │ │ -0001afc0: 5765 206d 6179 2061 6c73 6f20 6e6f 7465  We may also note
│ │ │ │ -0001afd0: 2074 6861 740a 7468 6520 6964 6561 6c20   that.the ideal 
│ │ │ │ -0001afe0: 6765 6e65 7261 7465 6420 6279 2074 6865  generated by the
│ │ │ │ -0001aff0: 2066 6972 7374 2032 2067 656e 6572 6174   first 2 generat
│ │ │ │ -0001b000: 6f72 7320 6f66 2049 2064 6566 696e 6573  ors of I defines
│ │ │ │ -0001b010: 2061 2073 6d6f 6f74 6820 7363 6865 6d65   a smooth scheme
│ │ │ │ -0001b020: 2061 6e64 0a69 6e70 7574 2074 6869 7320   and.input this 
│ │ │ │ -0001b030: 696e 666f 726d 6174 696f 6e20 696e 746f  information into
│ │ │ │ -0001b040: 2074 6865 206d 6574 686f 642e 2054 6869   the method. Thi
│ │ │ │ -0001b050: 7320 6d61 7920 616c 736f 2069 6d70 726f  s may also impro
│ │ │ │ -0001b060: 7665 2063 6f6d 7075 7461 7469 6f6e 0a73  ve computation.s
│ │ │ │ -0001b070: 7065 6564 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  peed...+--------
│ │ │ │ -0001b080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b0b0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020  ---------+.|i10 
│ │ │ │ -0001b0c0: 3a20 7469 6d65 2045 756c 6572 284a 2c4d  : time Euler(J,M
│ │ │ │ -0001b0d0: 6574 686f 643d 3e44 6972 6563 7443 6f6d  ethod=>DirectCom
│ │ │ │ -0001b0e0: 706c 6574 6549 6e74 2920 2020 2020 2020  pleteInt)       
│ │ │ │ -0001b0f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001b100: 202d 2d20 7573 6564 2030 2e31 3832 3936   -- used 0.18296
│ │ │ │ -0001b110: 3773 2028 6370 7529 3b20 302e 3036 3930  7s (cpu); 0.0690
│ │ │ │ -0001b120: 3139 3973 2028 7468 7265 6164 293b 2030  199s (thread); 0
│ │ │ │ -0001b130: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │ -0001b140: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0001b150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001af00: 2d2d 2b0a 0a4e 6f74 6520 7468 6174 2074  --+..Note that t
│ │ │ │ +0001af10: 6865 2069 6465 616c 204a 2061 626f 7665  he ideal J above
│ │ │ │ +0001af20: 2069 7320 6120 636f 6d70 6c65 7465 2069   is a complete i
│ │ │ │ +0001af30: 6e74 6572 7365 6374 696f 6e2c 2074 6875  ntersection, thu
│ │ │ │ +0001af40: 7320 7765 206d 6179 2063 6861 6e67 6520  s we may change 
│ │ │ │ +0001af50: 7468 650a 6d65 7468 6f64 206f 7074 696f  the.method optio
│ │ │ │ +0001af60: 6e20 7768 6963 6820 6d61 7920 7370 6565  n which may spee
│ │ │ │ +0001af70: 6420 636f 6d70 7574 6174 696f 6e20 696e  d computation in
│ │ │ │ +0001af80: 2073 6f6d 6520 6361 7365 732e 2057 6520   some cases. We 
│ │ │ │ +0001af90: 6d61 7920 616c 736f 206e 6f74 6520 7468  may also note th
│ │ │ │ +0001afa0: 6174 0a74 6865 2069 6465 616c 2067 656e  at.the ideal gen
│ │ │ │ +0001afb0: 6572 6174 6564 2062 7920 7468 6520 6669  erated by the fi
│ │ │ │ +0001afc0: 7273 7420 3220 6765 6e65 7261 746f 7273  rst 2 generators
│ │ │ │ +0001afd0: 206f 6620 4920 6465 6669 6e65 7320 6120   of I defines a 
│ │ │ │ +0001afe0: 736d 6f6f 7468 2073 6368 656d 6520 616e  smooth scheme an
│ │ │ │ +0001aff0: 640a 696e 7075 7420 7468 6973 2069 6e66  d.input this inf
│ │ │ │ +0001b000: 6f72 6d61 7469 6f6e 2069 6e74 6f20 7468  ormation into th
│ │ │ │ +0001b010: 6520 6d65 7468 6f64 2e20 5468 6973 206d  e method. This m
│ │ │ │ +0001b020: 6179 2061 6c73 6f20 696d 7072 6f76 6520  ay also improve 
│ │ │ │ +0001b030: 636f 6d70 7574 6174 696f 6e0a 7370 6565  computation.spee
│ │ │ │ +0001b040: 642e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  d...+-----------
│ │ │ │ +0001b050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b080: 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a 2074  ------+.|i10 : t
│ │ │ │ +0001b090: 696d 6520 4575 6c65 7228 4a2c 4d65 7468  ime Euler(J,Meth
│ │ │ │ +0001b0a0: 6f64 3d3e 4469 7265 6374 436f 6d70 6c65  od=>DirectComple
│ │ │ │ +0001b0b0: 7465 496e 7429 2020 2020 2020 2020 2020  teInt)          
│ │ │ │ +0001b0c0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ +0001b0d0: 2075 7365 6420 302e 3134 3534 3132 7320   used 0.145412s 
│ │ │ │ +0001b0e0: 2863 7075 293b 2030 2e30 3935 3037 3631  (cpu); 0.0950761
│ │ │ │ +0001b0f0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ +0001b100: 6763 2920 2020 2020 2020 2020 2020 7c0a  gc)           |.
│ │ │ │ +0001b110: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0001b120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b150: 2020 7c0a 7c6f 3130 203d 2032 2020 2020    |.|o10 = 2    
│ │ │ │  0001b160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b180: 2020 2020 207c 0a7c 6f31 3020 3d20 3220       |.|o10 = 2 
│ │ │ │ -0001b190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b1c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -0001b1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -0001b210: 6931 3120 3a20 7469 6d65 2045 756c 6572  i11 : time Euler
│ │ │ │ -0001b220: 284a 2c4d 6574 686f 643d 3e44 6972 6563  (J,Method=>Direc
│ │ │ │ -0001b230: 7443 6f6d 706c 6574 6549 6e74 2c49 6e64  tCompleteInt,Ind
│ │ │ │ -0001b240: 734f 6653 6d6f 6f74 683d 3e7b 302c 317d  sOfSmooth=>{0,1}
│ │ │ │ -0001b250: 297c 0a7c 202d 2d20 7573 6564 2030 2e31  )|.| -- used 0.1
│ │ │ │ -0001b260: 3930 3537 3573 2028 6370 7529 3b20 302e  90575s (cpu); 0.
│ │ │ │ -0001b270: 3038 3435 3639 3373 2028 7468 7265 6164  0845693s (thread
│ │ │ │ -0001b280: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ -0001b290: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001b2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b190: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001b1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131  ----------+.|i11
│ │ │ │ +0001b1e0: 203a 2074 696d 6520 4575 6c65 7228 4a2c   : time Euler(J,
│ │ │ │ +0001b1f0: 4d65 7468 6f64 3d3e 4469 7265 6374 436f  Method=>DirectCo
│ │ │ │ +0001b200: 6d70 6c65 7465 496e 742c 496e 6473 4f66  mpleteInt,IndsOf
│ │ │ │ +0001b210: 536d 6f6f 7468 3d3e 7b30 2c31 7d29 7c0a  Smooth=>{0,1})|.
│ │ │ │ +0001b220: 7c20 2d2d 2075 7365 6420 302e 3238 3632  | -- used 0.2862
│ │ │ │ +0001b230: 3436 7320 2863 7075 293b 2030 2e31 3332  46s (cpu); 0.132
│ │ │ │ +0001b240: 3138 7320 2874 6872 6561 6429 3b20 3073  18s (thread); 0s
│ │ │ │ +0001b250: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │ +0001b260: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001b270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b2a0: 2020 2020 2020 7c0a 7c6f 3131 203d 2032        |.|o11 = 2
│ │ │ │  0001b2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b2d0: 2020 2020 2020 2020 207c 0a7c 6f31 3120           |.|o11 
│ │ │ │ -0001b2e0: 3d20 3220 2020 2020 2020 2020 2020 2020  = 2             
│ │ │ │ -0001b2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b310: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -0001b320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b360: 2d2b 0a0a 4e6f 7720 636f 6e73 6964 6572  -+..Now consider
│ │ │ │ -0001b370: 2061 6e20 6578 616d 706c 6520 696e 205c   an example in \
│ │ │ │ -0001b380: 5050 5e32 205c 7469 6d65 7320 5c50 505e  PP^2 \times \PP^
│ │ │ │ -0001b390: 322e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  2...+-----------
│ │ │ │ -0001b3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b3c0: 2d2d 2d2b 0a7c 6931 3220 3a20 523d 4d75  ---+.|i12 : R=Mu
│ │ │ │ -0001b3d0: 6c74 6950 726f 6a43 6f6f 7264 5269 6e67  ltiProjCoordRing
│ │ │ │ -0001b3e0: 287b 322c 327d 2920 2020 2020 2020 2020  ({2,2})         
│ │ │ │ -0001b3f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001b2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b2e0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0001b2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0001b330: 0a4e 6f77 2063 6f6e 7369 6465 7220 616e  .Now consider an
│ │ │ │ +0001b340: 2065 7861 6d70 6c65 2069 6e20 5c50 505e   example in \PP^
│ │ │ │ +0001b350: 3220 5c74 696d 6573 205c 5050 5e32 2e0a  2 \times \PP^2..
│ │ │ │ +0001b360: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0001b370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b390: 2b0a 7c69 3132 203a 2052 3d4d 756c 7469  +.|i12 : R=Multi
│ │ │ │ +0001b3a0: 5072 6f6a 436f 6f72 6452 696e 6728 7b32  ProjCoordRing({2
│ │ │ │ +0001b3b0: 2c32 7d29 2020 2020 2020 2020 2020 2020  ,2})            
│ │ │ │ +0001b3c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001b3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b3f0: 2020 7c0a 7c6f 3132 203d 2052 2020 2020    |.|o12 = R    
│ │ │ │  0001b400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b420: 2020 2020 207c 0a7c 6f31 3220 3d20 5220       |.|o12 = R 
│ │ │ │ +0001b420: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0001b430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b450: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001b460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b450: 2020 2020 7c0a 7c6f 3132 203a 2050 6f6c      |.|o12 : Pol
│ │ │ │ +0001b460: 796e 6f6d 6961 6c52 696e 6720 2020 2020  ynomialRing     
│ │ │ │  0001b470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b480: 2020 2020 2020 207c 0a7c 6f31 3220 3a20         |.|o12 : 
│ │ │ │ -0001b490: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020  PolynomialRing  
│ │ │ │ -0001b4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b4b0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -0001b4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b4e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320  ---------+.|i13 
│ │ │ │ -0001b4f0: 3a20 723d 6765 6e73 2052 2020 2020 2020  : r=gens R      
│ │ │ │ +0001b480: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001b490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b4b0: 2d2d 2d2d 2d2d 2b0a 7c69 3133 203a 2072  ------+.|i13 : r
│ │ │ │ +0001b4c0: 3d67 656e 7320 5220 2020 2020 2020 2020  =gens R         
│ │ │ │ +0001b4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b4e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001b4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b510: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001b520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b540: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -0001b550: 3320 3d20 7b78 202c 2078 202c 2078 202c  3 = {x , x , x ,
│ │ │ │ -0001b560: 2078 202c 2078 202c 2078 207d 2020 2020   x , x , x }    
│ │ │ │ -0001b570: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001b580: 2020 2020 2020 2030 2020 2031 2020 2032         0   1   2
│ │ │ │ -0001b590: 2020 2033 2020 2034 2020 2035 2020 2020     3   4   5    
│ │ │ │ -0001b5a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001b5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b510: 2020 2020 2020 2020 7c0a 7c6f 3133 203d          |.|o13 =
│ │ │ │ +0001b520: 207b 7820 2c20 7820 2c20 7820 2c20 7820   {x , x , x , x 
│ │ │ │ +0001b530: 2c20 7820 2c20 7820 7d20 2020 2020 2020  , x , x }       
│ │ │ │ +0001b540: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001b550: 2020 2020 3020 2020 3120 2020 3220 2020      0   1   2   
│ │ │ │ +0001b560: 3320 2020 3420 2020 3520 2020 2020 2020  3   4   5       
│ │ │ │ +0001b570: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001b580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b5a0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0001b5b0: 3320 3a20 4c69 7374 2020 2020 2020 2020  3 : List        
│ │ │ │  0001b5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b5d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001b5e0: 7c6f 3133 203a 204c 6973 7420 2020 2020  |o13 : List     
│ │ │ │ -0001b5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b600: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001b610: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -0001b620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b640: 2b0a 7c69 3134 203a 204b 3d69 6465 616c  +.|i14 : K=ideal
│ │ │ │ -0001b650: 2872 5f30 5e32 2a72 5f33 2d72 5f34 2a72  (r_0^2*r_3-r_4*r
│ │ │ │ -0001b660: 5f31 2a72 5f32 2c72 5f32 5e32 2a72 5f35  _1*r_2,r_2^2*r_5
│ │ │ │ -0001b670: 297c 0a7c 2020 2020 2020 2020 2020 2020  )|.|            
│ │ │ │ -0001b680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b5d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0001b5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0001b610: 6931 3420 3a20 4b3d 6964 6561 6c28 725f  i14 : K=ideal(r_
│ │ │ │ +0001b620: 305e 322a 725f 332d 725f 342a 725f 312a  0^2*r_3-r_4*r_1*
│ │ │ │ +0001b630: 725f 322c 725f 325e 322a 725f 3529 7c0a  r_2,r_2^2*r_5)|.
│ │ │ │ +0001b640: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0001b650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b660: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001b670: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001b680: 3220 2020 2020 2020 2020 2020 2020 2032  2              2
│ │ │ │  0001b690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b6a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0001b6b0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -0001b6c0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -0001b6d0: 2020 207c 0a7c 6f31 3420 3d20 6964 6561     |.|o14 = idea
│ │ │ │ -0001b6e0: 6c20 2878 2078 2020 2d20 7820 7820 7820  l (x x  - x x x 
│ │ │ │ -0001b6f0: 2c20 7820 7820 2920 2020 2020 2020 2020  , x x )         
│ │ │ │ -0001b700: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001b710: 2020 2020 2030 2033 2020 2020 3120 3220       0 3    1 2 
│ │ │ │ -0001b720: 3420 2020 3220 3520 2020 2020 2020 2020  4   2 5         
│ │ │ │ -0001b730: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001b740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b6a0: 7c0a 7c6f 3134 203d 2069 6465 616c 2028  |.|o14 = ideal (
│ │ │ │ +0001b6b0: 7820 7820 202d 2078 2078 2078 202c 2078  x x  - x x x , x
│ │ │ │ +0001b6c0: 2078 2029 2020 2020 2020 2020 2020 2020   x )            
│ │ │ │ +0001b6d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001b6e0: 2020 3020 3320 2020 2031 2032 2034 2020    0 3    1 2 4  
│ │ │ │ +0001b6f0: 2032 2035 2020 2020 2020 2020 2020 2020   2 5            
│ │ │ │ +0001b700: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001b710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b730: 2020 207c 0a7c 6f31 3420 3a20 4964 6561     |.|o14 : Idea
│ │ │ │ +0001b740: 6c20 6f66 2052 2020 2020 2020 2020 2020  l of R          
│ │ │ │  0001b750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b760: 2020 2020 2020 7c0a 7c6f 3134 203a 2049        |.|o14 : I
│ │ │ │ -0001b770: 6465 616c 206f 6620 5220 2020 2020 2020  deal of R       
│ │ │ │ -0001b780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b790: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -0001b7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b7c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3135 203a  --------+.|i15 :
│ │ │ │ -0001b7d0: 2045 756c 6572 4b3d 4575 6c65 7228 4b29   EulerK=Euler(K)
│ │ │ │ +0001b760: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0001b770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b790: 2d2d 2d2d 2d2b 0a7c 6931 3520 3a20 4575  -----+.|i15 : Eu
│ │ │ │ +0001b7a0: 6c65 724b 3d45 756c 6572 284b 2920 2020  lerK=Euler(K)   
│ │ │ │ +0001b7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b7c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001b7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b7f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0001b800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b7f0: 2020 2020 2020 207c 0a7c 6f31 3520 3d20         |.|o15 = 
│ │ │ │ +0001b800: 3720 2020 2020 2020 2020 2020 2020 2020  7               
│ │ │ │  0001b810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b820: 2020 2020 2020 2020 2020 7c0a 7c6f 3135            |.|o15
│ │ │ │ -0001b830: 203d 2037 2020 2020 2020 2020 2020 2020   = 7            
│ │ │ │ -0001b840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b850: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -0001b860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0001b890: 3136 203a 2063 736d 4b3d 2043 534d 284b  16 : csmK= CSM(K
│ │ │ │ -0001b8a0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -0001b8b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001b8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b8e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001b8f0: 7c20 2020 2020 2020 2032 2032 2020 2020  |        2 2    
│ │ │ │ -0001b900: 2032 2020 2020 2020 2020 2032 2020 2020   2         2    
│ │ │ │ -0001b910: 3220 2020 2020 2020 2020 2020 2032 207c  2            2 |
│ │ │ │ -0001b920: 0a7c 6f31 3620 3d20 3768 2068 2020 2b20  .|o16 = 7h h  + 
│ │ │ │ -0001b930: 3568 2068 2020 2b20 3468 2068 2020 2b20  5h h  + 4h h  + 
│ │ │ │ -0001b940: 6820 202b 2033 6820 6820 202b 2068 2020  h  + 3h h  + h  
│ │ │ │ -0001b950: 7c0a 7c20 2020 2020 2020 2031 2032 2020  |.|        1 2  
│ │ │ │ -0001b960: 2020 2031 2032 2020 2020 2031 2032 2020     1 2     1 2  
│ │ │ │ -0001b970: 2020 3120 2020 2020 3120 3220 2020 2032    1     1 2    2
│ │ │ │ -0001b980: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0001b990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b820: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001b830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b850: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3620  ---------+.|i16 
│ │ │ │ +0001b860: 3a20 6373 6d4b 3d20 4353 4d28 4b29 2020  : csmK= CSM(K)  
│ │ │ │ +0001b870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b880: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001b890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b8b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001b8c0: 2020 2020 2020 3220 3220 2020 2020 3220        2 2     2 
│ │ │ │ +0001b8d0: 2020 2020 2020 2020 3220 2020 2032 2020          2    2  
│ │ │ │ +0001b8e0: 2020 2020 2020 2020 2020 3220 7c0a 7c6f            2 |.|o
│ │ │ │ +0001b8f0: 3136 203d 2037 6820 6820 202b 2035 6820  16 = 7h h  + 5h 
│ │ │ │ +0001b900: 6820 202b 2034 6820 6820 202b 2068 2020  h  + 4h h  + h  
│ │ │ │ +0001b910: 2b20 3368 2068 2020 2b20 6820 207c 0a7c  + 3h h  + h  |.|
│ │ │ │ +0001b920: 2020 2020 2020 2020 3120 3220 2020 2020          1 2     
│ │ │ │ +0001b930: 3120 3220 2020 2020 3120 3220 2020 2031  1 2     1 2    1
│ │ │ │ +0001b940: 2020 2020 2031 2032 2020 2020 3220 7c0a       1 2    2 |.
│ │ │ │ +0001b950: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0001b960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b970: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001b980: 0a7c 2020 2020 2020 5a5a 5b68 202e 2e68  .|      ZZ[h ..h
│ │ │ │ +0001b990: 205d 2020 2020 2020 2020 2020 2020 2020   ]              
│ │ │ │  0001b9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b9b0: 2020 7c0a 7c20 2020 2020 205a 5a5b 6820    |.|      ZZ[h 
│ │ │ │ -0001b9c0: 2e2e 6820 5d20 2020 2020 2020 2020 2020  ..h ]           
│ │ │ │ +0001b9b0: 7c0a 7c20 2020 2020 2020 2020 2031 2020  |.|          1  
│ │ │ │ +0001b9c0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  0001b9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b9e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0001b9f0: 3120 2020 3220 2020 2020 2020 2020 2020  1   2           
│ │ │ │ +0001b9e0: 207c 0a7c 6f31 3620 3a20 2d2d 2d2d 2d2d   |.|o16 : ------
│ │ │ │ +0001b9f0: 2d2d 2d2d 2020 2020 2020 2020 2020 2020  ----            
│ │ │ │  0001ba00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ba10: 2020 2020 7c0a 7c6f 3136 203a 202d 2d2d      |.|o16 : ---
│ │ │ │ -0001ba20: 2d2d 2d2d 2d2d 2d20 2020 2020 2020 2020  -------         
│ │ │ │ +0001ba10: 2020 7c0a 7c20 2020 2020 2020 2020 3320    |.|         3 
│ │ │ │ +0001ba20: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │  0001ba30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ba40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001ba50: 2033 2020 2033 2020 2020 2020 2020 2020   3   3          
│ │ │ │ +0001ba40: 2020 207c 0a7c 2020 2020 2020 2028 6820     |.|       (h 
│ │ │ │ +0001ba50: 2c20 6820 2920 2020 2020 2020 2020 2020  , h )           
│ │ │ │  0001ba60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ba70: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001ba80: 2868 202c 2068 2029 2020 2020 2020 2020  (h , h )        
│ │ │ │ +0001ba70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001ba80: 3120 2020 3220 2020 2020 2020 2020 2020  1   2           
│ │ │ │  0001ba90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001baa0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0001bab0: 2020 2031 2020 2032 2020 2020 2020 2020     1   2        
│ │ │ │ -0001bac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bad0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -0001bae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001baf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bb00: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3720  ---------+.|i17 
│ │ │ │ -0001bb10: 3a20 4575 6c65 724b 3d3d 4575 6c65 7228  : EulerK==Euler(
│ │ │ │ -0001bb20: 6373 6d4b 2920 2020 2020 2020 2020 2020  csmK)           
│ │ │ │ -0001bb30: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001bb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001baa0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001bab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bad0: 2d2d 2d2d 2d2d 2b0a 7c69 3137 203a 2045  ------+.|i17 : E
│ │ │ │ +0001bae0: 756c 6572 4b3d 3d45 756c 6572 2863 736d  ulerK==Euler(csm
│ │ │ │ +0001baf0: 4b29 2020 2020 2020 2020 2020 2020 2020  K)              
│ │ │ │ +0001bb00: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001bb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bb30: 2020 2020 2020 2020 7c0a 7c6f 3137 203d          |.|o17 =
│ │ │ │ +0001bb40: 2074 7275 6520 2020 2020 2020 2020 2020   true           
│ │ │ │  0001bb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bb60: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -0001bb70: 3720 3d20 7472 7565 2020 2020 2020 2020  7 = true        
│ │ │ │ -0001bb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bb90: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -0001bba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ -0001bbd0: 496e 2074 6865 2063 6173 6520 7768 6572  In the case wher
│ │ │ │ -0001bbe0: 6520 7468 6520 616d 6269 656e 7420 7370  e the ambient sp
│ │ │ │ -0001bbf0: 6163 6520 6973 2061 2074 6f72 6963 2076  ace is a toric v
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│ │ │ │ -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
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│ │ │ │ -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
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│ │ │ │ -0001bd00: 746f 7269 630a 7661 7269 6574 7920 666f  toric.variety fo
│ │ │ │ -0001bd10: 7220 6566 6669 6369 656e 6379 2072 6561  r efficiency rea
│ │ │ │ -0001bd20: 736f 6e73 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  sons...+--------
│ │ │ │ -0001bd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bd70: 2b0a 7c69 3138 203a 206e 6565 6473 5061  +.|i18 : needsPa
│ │ │ │ -0001bd80: 636b 6167 6520 224e 6f72 6d61 6c54 6f72  ckage "NormalTor
│ │ │ │ -0001bd90: 6963 5661 7269 6574 6965 7322 2020 2020  icVarieties"    
│ │ │ │ +0001bb60: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001bb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a49 6e20  ----------+..In 
│ │ │ │ +0001bba0: 7468 6520 6361 7365 2077 6865 7265 2074  the case where t
│ │ │ │ +0001bbb0: 6865 2061 6d62 6965 6e74 2073 7061 6365  he ambient space
│ │ │ │ +0001bbc0: 2069 7320 6120 746f 7269 6320 7661 7269   is a toric vari
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│ │ │ │ +0001bbe0: 2061 2070 726f 6475 6374 0a6f 6620 7072   a product.of pr
│ │ │ │ +0001bbf0: 6f6a 6563 7469 7665 2073 7061 6365 7320  ojective spaces 
│ │ │ │ +0001bc00: 7765 206d 7573 7420 6c6f 6164 2074 6865  we must load the
│ │ │ │ +0001bc10: 204e 6f72 6d61 6c54 6f72 6963 5661 7269   NormalToricVari
│ │ │ │ +0001bc20: 6574 6965 7320 7061 636b 6167 6520 616e  eties package an
│ │ │ │ +0001bc30: 6420 6d75 7374 0a61 6c73 6f20 696e 7075  d must.also inpu
│ │ │ │ +0001bc40: 7420 7468 6520 746f 7269 6320 7661 7269  t the toric vari
│ │ │ │ +0001bc50: 6574 792e 2049 6620 7468 6520 746f 7269  ety. If the tori
│ │ │ │ +0001bc60: 6320 7661 7269 6574 7920 6973 2061 2070  c variety is a p
│ │ │ │ +0001bc70: 726f 6475 6374 206f 6620 7072 6f6a 6563  roduct of projec
│ │ │ │ +0001bc80: 7469 7665 0a73 7061 6365 2069 7420 6973  tive.space it is
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│ │ │ │ +0001bca0: 7573 6520 7468 6520 666f 726d 2061 626f  use the form abo
│ │ │ │ +0001bcb0: 7665 2072 6174 6865 7220 7468 616e 2069  ve rather than i
│ │ │ │ +0001bcc0: 6e70 7574 7469 6e67 2074 6865 2074 6f72  nputting the tor
│ │ │ │ +0001bcd0: 6963 0a76 6172 6965 7479 2066 6f72 2065  ic.variety for e
│ │ │ │ +0001bce0: 6666 6963 6965 6e63 7920 7265 6173 6f6e  fficiency reason
│ │ │ │ +0001bcf0: 732e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  s...+-----------
│ │ │ │ +0001bd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0001bd40: 6931 3820 3a20 6e65 6564 7350 6163 6b61  i18 : needsPacka
│ │ │ │ +0001bd50: 6765 2022 4e6f 726d 616c 546f 7269 6356  ge "NormalToricV
│ │ │ │ +0001bd60: 6172 6965 7469 6573 2220 2020 2020 2020  arieties"       
│ │ │ │ +0001bd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bd80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001bd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bdb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001bdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bdd0: 2020 207c 0a7c 6f31 3820 3d20 4e6f 726d     |.|o18 = Norm
│ │ │ │ +0001bde0: 616c 546f 7269 6356 6172 6965 7469 6573  alToricVarieties
│ │ │ │  0001bdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001be00: 2020 2020 2020 7c0a 7c6f 3138 203d 204e        |.|o18 = N
│ │ │ │ -0001be10: 6f72 6d61 6c54 6f72 6963 5661 7269 6574  ormalToricVariet
│ │ │ │ -0001be20: 6965 7320 2020 2020 2020 2020 2020 2020  ies             
│ │ │ │ +0001be00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001be10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001be20: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0001be30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001be40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001be50: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0001be60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001be70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001be50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001be60: 2020 2020 2020 2020 207c 0a7c 6f31 3820           |.|o18 
│ │ │ │ +0001be70: 3a20 5061 636b 6167 6520 2020 2020 2020  : Package       
│ │ │ │  0001be80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001be90: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0001bea0: 3138 203a 2050 6163 6b61 6765 2020 2020  18 : Package    
│ │ │ │ -0001beb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bee0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -0001bef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bf00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bf10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bf20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bf30: 2d2d 2b0a 7c69 3139 203a 2052 686f 203d  --+.|i19 : Rho =
│ │ │ │ -0001bf40: 207b 7b31 2c30 2c30 7d2c 7b30 2c31 2c30   {{1,0,0},{0,1,0
│ │ │ │ -0001bf50: 7d2c 7b30 2c30 2c31 7d2c 7b2d 312c 2d31  },{0,0,1},{-1,-1
│ │ │ │ -0001bf60: 2c30 7d2c 7b30 2c30 2c2d 317d 7d20 2020  ,0},{0,0,-1}}   
│ │ │ │ -0001bf70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001be90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001beb0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0001bec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0001bf00: 0a7c 6931 3920 3a20 5268 6f20 3d20 7b7b  .|i19 : Rho = {{
│ │ │ │ +0001bf10: 312c 302c 307d 2c7b 302c 312c 307d 2c7b  1,0,0},{0,1,0},{
│ │ │ │ +0001bf20: 302c 302c 317d 2c7b 2d31 2c2d 312c 307d  0,0,1},{-1,-1,0}
│ │ │ │ +0001bf30: 2c7b 302c 302c 2d31 7d7d 2020 2020 2020  ,{0,0,-1}}      
│ │ │ │ +0001bf40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001bf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bfc0: 2020 2020 2020 2020 7c0a 7c6f 3139 203d          |.|o19 =
│ │ │ │ -0001bfd0: 207b 7b31 2c20 302c 2030 7d2c 207b 302c   {{1, 0, 0}, {0,
│ │ │ │ -0001bfe0: 2031 2c20 307d 2c20 7b30 2c20 302c 2031   1, 0}, {0, 0, 1
│ │ │ │ -0001bff0: 7d2c 207b 2d31 2c20 2d31 2c20 307d 2c20  }, {-1, -1, 0}, 
│ │ │ │ -0001c000: 7b30 2c20 302c 202d 317d 7d20 2020 2020  {0, 0, -1}}     
│ │ │ │ -0001c010: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0001c020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bf90: 2020 2020 207c 0a7c 6f31 3920 3d20 7b7b       |.|o19 = {{
│ │ │ │ +0001bfa0: 312c 2030 2c20 307d 2c20 7b30 2c20 312c  1, 0, 0}, {0, 1,
│ │ │ │ +0001bfb0: 2030 7d2c 207b 302c 2030 2c20 317d 2c20   0}, {0, 0, 1}, 
│ │ │ │ +0001bfc0: 7b2d 312c 202d 312c 2030 7d2c 207b 302c  {-1, -1, 0}, {0,
│ │ │ │ +0001bfd0: 2030 2c20 2d31 7d7d 2020 2020 2020 2020   0, -1}}        
│ │ │ │ +0001bfe0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0001bff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c020: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0001c030: 3920 3a20 4c69 7374 2020 2020 2020 2020  9 : List        
│ │ │ │  0001c040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c050: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001c060: 7c6f 3139 203a 204c 6973 7420 2020 2020  |o19 : List     
│ │ │ │ -0001c070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c0a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001c050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c070: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001c080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c0f0: 2d2d 2d2d 2b0a 7c69 3230 203a 2053 6967  ----+.|i20 : Sig
│ │ │ │ -0001c100: 6d61 203d 207b 7b30 2c31 2c32 7d2c 7b31  ma = {{0,1,2},{1
│ │ │ │ -0001c110: 2c32 2c33 7d2c 7b30 2c32 2c33 7d2c 7b30  ,2,3},{0,2,3},{0
│ │ │ │ -0001c120: 2c31 2c34 7d2c 7b31 2c33 2c34 7d2c 7b30  ,1,4},{1,3,4},{0
│ │ │ │ -0001c130: 2c33 2c34 7d7d 2020 2020 2020 2020 207c  ,3,4}}         |
│ │ │ │ -0001c140: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0001c150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c180: 2020 2020 2020 2020 2020 7c0a 7c6f 3230            |.|o20
│ │ │ │ -0001c190: 203d 207b 7b30 2c20 312c 2032 7d2c 207b   = {{0, 1, 2}, {
│ │ │ │ -0001c1a0: 312c 2032 2c20 337d 2c20 7b30 2c20 322c  1, 2, 3}, {0, 2,
│ │ │ │ -0001c1b0: 2033 7d2c 207b 302c 2031 2c20 347d 2c20   3}, {0, 1, 4}, 
│ │ │ │ -0001c1c0: 7b31 2c20 332c 2034 7d2c 207b 302c 2033  {1, 3, 4}, {0, 3
│ │ │ │ -0001c1d0: 2c20 347d 7d7c 0a7c 2020 2020 2020 2020  , 4}}|.|        
│ │ │ │ -0001c1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c0c0: 2d2b 0a7c 6932 3020 3a20 5369 676d 6120  -+.|i20 : Sigma 
│ │ │ │ +0001c0d0: 3d20 7b7b 302c 312c 327d 2c7b 312c 322c  = {{0,1,2},{1,2,
│ │ │ │ +0001c0e0: 337d 2c7b 302c 322c 337d 2c7b 302c 312c  3},{0,2,3},{0,1,
│ │ │ │ +0001c0f0: 347d 2c7b 312c 332c 347d 2c7b 302c 332c  4},{1,3,4},{0,3,
│ │ │ │ +0001c100: 347d 7d20 2020 2020 2020 2020 7c0a 7c20  4}}         |.| 
│ │ │ │ +0001c110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c150: 2020 2020 2020 207c 0a7c 6f32 3020 3d20         |.|o20 = 
│ │ │ │ +0001c160: 7b7b 302c 2031 2c20 327d 2c20 7b31 2c20  {{0, 1, 2}, {1, 
│ │ │ │ +0001c170: 322c 2033 7d2c 207b 302c 2032 2c20 337d  2, 3}, {0, 2, 3}
│ │ │ │ +0001c180: 2c20 7b30 2c20 312c 2034 7d2c 207b 312c  , {0, 1, 4}, {1,
│ │ │ │ +0001c190: 2033 2c20 347d 2c20 7b30 2c20 332c 2034   3, 4}, {0, 3, 4
│ │ │ │ +0001c1a0: 7d7d 7c0a 7c20 2020 2020 2020 2020 2020  }}|.|           
│ │ │ │ +0001c1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c1e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001c1f0: 6f32 3020 3a20 4c69 7374 2020 2020 2020  o20 : List      
│ │ │ │  0001c200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c220: 7c0a 7c6f 3230 203a 204c 6973 7420 2020  |.|o20 : List   
│ │ │ │ -0001c230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c260: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0001c220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c230: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001c240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c2b0: 2d2d 2d2d 2d2d 2b0a 7c69 3231 203a 2058  ------+.|i21 : X
│ │ │ │ -0001c2c0: 203d 206e 6f72 6d61 6c54 6f72 6963 5661   = normalToricVa
│ │ │ │ -0001c2d0: 7269 6574 7928 5268 6f2c 5369 676d 612c  riety(Rho,Sigma,
│ │ │ │ -0001c2e0: 436f 6566 6669 6369 656e 7452 696e 6720  CoefficientRing 
│ │ │ │ -0001c2f0: 3d3e 5a5a 2f33 3237 3439 2920 2020 2020  =>ZZ/32749)     
│ │ │ │ -0001c300: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0001c310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c280: 2d2d 2d2b 0a7c 6932 3120 3a20 5820 3d20  ---+.|i21 : X = 
│ │ │ │ +0001c290: 6e6f 726d 616c 546f 7269 6356 6172 6965  normalToricVarie
│ │ │ │ +0001c2a0: 7479 2852 686f 2c53 6967 6d61 2c43 6f65  ty(Rho,Sigma,Coe
│ │ │ │ +0001c2b0: 6666 6963 6965 6e74 5269 6e67 203d 3e5a  fficientRing =>Z
│ │ │ │ +0001c2c0: 5a2f 3332 3734 3929 2020 2020 2020 7c0a  Z/32749)      |.
│ │ │ │ +0001c2d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0001c2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c310: 2020 2020 2020 2020 207c 0a7c 6f32 3120           |.|o21 
│ │ │ │ +0001c320: 3d20 5820 2020 2020 2020 2020 2020 2020  = X             
│ │ │ │  0001c330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c340: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0001c350: 3231 203d 2058 2020 2020 2020 2020 2020  21 = X          
│ │ │ │ -0001c360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c360: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0001c370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c390: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0001c3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c3a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001c3b0: 0a7c 6f32 3120 3a20 4e6f 726d 616c 546f  .|o21 : NormalTo
│ │ │ │ +0001c3c0: 7269 6356 6172 6965 7479 2020 2020 2020  ricVariety      
│ │ │ │  0001c3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c3e0: 2020 7c0a 7c6f 3231 203a 204e 6f72 6d61    |.|o21 : Norma
│ │ │ │ -0001c3f0: 6c54 6f72 6963 5661 7269 6574 7920 2020  lToricVariety   
│ │ │ │ -0001c400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c420: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0001c3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c3f0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0001c400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c470: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3232 203a  --------+.|i22 :
│ │ │ │ -0001c480: 2043 6865 636b 546f 7269 6356 6172 6965   CheckToricVarie
│ │ │ │ -0001c490: 7479 5661 6c69 6428 5829 2020 2020 2020  tyValid(X)      
│ │ │ │ +0001c440: 2d2d 2d2d 2d2b 0a7c 6932 3220 3a20 4368  -----+.|i22 : Ch
│ │ │ │ +0001c450: 6563 6b54 6f72 6963 5661 7269 6574 7956  eckToricVarietyV
│ │ │ │ +0001c460: 616c 6964 2858 2920 2020 2020 2020 2020  alid(X)         
│ │ │ │ +0001c470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c490: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0001c4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c4c0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0001c4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c4d0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0001c4e0: 3220 3d20 7472 7565 2020 2020 2020 2020  2 = true        
│ │ │ │  0001c4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c500: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001c510: 7c6f 3232 203d 2074 7275 6520 2020 2020  |o22 = true     
│ │ │ │ -0001c520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c550: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001c500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c520: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001c530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c5a0: 2d2d 2d2d 2b0a 7c69 3233 203a 2052 3d72  ----+.|i23 : R=r
│ │ │ │ -0001c5b0: 696e 6728 5829 2020 2020 2020 2020 2020  ing(X)          
│ │ │ │ +0001c570: 2d2b 0a7c 6932 3320 3a20 523d 7269 6e67  -+.|i23 : R=ring
│ │ │ │ +0001c580: 2858 2920 2020 2020 2020 2020 2020 2020  (X)             
│ │ │ │ +0001c590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c5b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001c5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c5e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001c5f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0001c600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c600: 2020 2020 2020 207c 0a7c 6f32 3320 3d20         |.|o23 = 
│ │ │ │ +0001c610: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │  0001c620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c630: 2020 2020 2020 2020 2020 7c0a 7c6f 3233            |.|o23
│ │ │ │ -0001c640: 203d 2052 2020 2020 2020 2020 2020 2020   = R            
│ │ │ │ -0001c650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c650: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001c660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c680: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001c690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c690: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001c6a0: 6f32 3320 3a20 506f 6c79 6e6f 6d69 616c  o23 : Polynomial
│ │ │ │ +0001c6b0: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │  0001c6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c6d0: 7c0a 7c6f 3233 203a 2050 6f6c 796e 6f6d  |.|o23 : Polynom
│ │ │ │ -0001c6e0: 6961 6c52 696e 6720 2020 2020 2020 2020  ialRing         
│ │ │ │ -0001c6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c710: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0001c6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c6e0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001c6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c760: 2d2d 2d2d 2d2d 2b0a 7c69 3234 203a 2049  ------+.|i24 : I
│ │ │ │ -0001c770: 3d69 6465 616c 2852 5f30 5e34 2a52 5f31  =ideal(R_0^4*R_1
│ │ │ │ -0001c780: 2c52 5f30 2a52 5f33 2a52 5f34 2a52 5f32  ,R_0*R_3*R_4*R_2
│ │ │ │ -0001c790: 2d52 5f32 5e32 2a52 5f30 5e32 2920 2020  -R_2^2*R_0^2)   
│ │ │ │ +0001c730: 2d2d 2d2b 0a7c 6932 3420 3a20 493d 6964  ---+.|i24 : I=id
│ │ │ │ +0001c740: 6561 6c28 525f 305e 342a 525f 312c 525f  eal(R_0^4*R_1,R_
│ │ │ │ +0001c750: 302a 525f 332a 525f 342a 525f 322d 525f  0*R_3*R_4*R_2-R_
│ │ │ │ +0001c760: 325e 322a 525f 305e 3229 2020 2020 2020  2^2*R_0^2)      
│ │ │ │ +0001c770: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001c780: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0001c790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c7b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0001c7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c7f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001c800: 2020 2020 2020 2020 2020 2020 2034 2020               4  
│ │ │ │ -0001c810: 2020 2020 2032 2032 2020 2020 2020 2020       2 2        
│ │ │ │ -0001c820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c840: 2020 2020 2020 207c 0a7c 6f32 3420 3d20         |.|o24 = 
│ │ │ │ -0001c850: 6964 6561 6c20 2878 2078 202c 202d 2078  ideal (x x , - x
│ │ │ │ -0001c860: 2078 2020 2b20 7820 7820 7820 7820 2920   x  + x x x x ) 
│ │ │ │ -0001c870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c890: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0001c8a0: 2020 2030 2031 2020 2020 2030 2032 2020     0 1     0 2  
│ │ │ │ -0001c8b0: 2020 3020 3220 3320 3420 2020 2020 2020    0 2 3 4       
│ │ │ │ +0001c7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c7c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001c7d0: 2020 2020 2020 2020 2020 3420 2020 2020            4     
│ │ │ │ +0001c7e0: 2020 3220 3220 2020 2020 2020 2020 2020    2 2           
│ │ │ │ +0001c7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c810: 2020 2020 7c0a 7c6f 3234 203d 2069 6465      |.|o24 = ide
│ │ │ │ +0001c820: 616c 2028 7820 7820 2c20 2d20 7820 7820  al (x x , - x x 
│ │ │ │ +0001c830: 202b 2078 2078 2078 2078 2029 2020 2020   + x x x x )    
│ │ │ │ +0001c840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c850: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001c860: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001c870: 3020 3120 2020 2020 3020 3220 2020 2030  0 1     0 2    0
│ │ │ │ +0001c880: 2032 2033 2034 2020 2020 2020 2020 2020   2 3 4          
│ │ │ │ +0001c890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c8a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001c8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c8d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001c8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c8f0: 2020 2020 207c 0a7c 6f32 3420 3a20 4964       |.|o24 : Id
│ │ │ │ +0001c900: 6561 6c20 6f66 2052 2020 2020 2020 2020  eal of R        
│ │ │ │  0001c910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c920: 2020 2020 2020 2020 7c0a 7c6f 3234 203a          |.|o24 :
│ │ │ │ -0001c930: 2049 6465 616c 206f 6620 5220 2020 2020   Ideal of R     
│ │ │ │ -0001c940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c970: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -0001c980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0001c9c0: 7c69 3235 203a 2063 736d 493d 4353 4d28  |i25 : csmI=CSM(
│ │ │ │ -0001c9d0: 582c 4929 2020 2020 2020 2020 2020 2020  X,I)            
│ │ │ │ +0001c920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c940: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0001c950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +0001c990: 3520 3a20 6373 6d49 3d43 534d 2858 2c49  5 : csmI=CSM(X,I
│ │ │ │ +0001c9a0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +0001c9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c9d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001c9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ca00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001ca00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ca10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ca20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ca30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ca20: 207c 0a7c 2020 2020 2020 2020 3220 2020   |.|        2   
│ │ │ │ +0001ca30: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  0001ca40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ca50: 2020 2020 7c0a 7c20 2020 2020 2020 2032      |.|        2
│ │ │ │ -0001ca60: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -0001ca70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ca80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ca90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001caa0: 0a7c 6f32 3520 3d20 3578 2078 2020 2b20  .|o25 = 5x x  + 
│ │ │ │ -0001cab0: 3378 2020 2b20 3478 2078 2020 2b20 7820  3x  + 4x x  + x 
│ │ │ │ -0001cac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cae0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001caf0: 2020 2020 2033 2034 2020 2020 2033 2020       3 4     3  
│ │ │ │ -0001cb00: 2020 2033 2034 2020 2020 3320 2020 2020     3 4    3     
│ │ │ │ +0001ca50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ca60: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0001ca70: 3235 203d 2035 7820 7820 202b 2033 7820  25 = 5x x  + 3x 
│ │ │ │ +0001ca80: 202b 2034 7820 7820 202b 2078 2020 2020   + 4x x  + x    
│ │ │ │ +0001ca90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001caa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cab0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001cac0: 2020 3320 3420 2020 2020 3320 2020 2020    3 4     3     
│ │ │ │ +0001cad0: 3320 3420 2020 2033 2020 2020 2020 2020  3 4    3        
│ │ │ │ +0001cae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001caf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cb00: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001cb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cb30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001cb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cb40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0001cb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cb60: 2020 2020 2020 5a5a 5b78 202e 2e78 205d        ZZ[x ..x ]
│ │ │ │  0001cb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cb80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0001cb90: 2020 2020 2020 2020 205a 5a5b 7820 2e2e           ZZ[x ..
│ │ │ │ -0001cba0: 7820 5d20 2020 2020 2020 2020 2020 2020  x ]             
│ │ │ │ -0001cbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cbc0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001cb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cb90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001cba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cbb0: 2020 2020 2030 2020 2034 2020 2020 2020       0   4      
│ │ │ │ +0001cbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cbe0: 2020 2020 2020 2020 3020 2020 3420 2020          0   4   
│ │ │ │ -0001cbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cc10: 2020 2020 2020 7c0a 7c6f 3235 203a 202d        |.|o25 : -
│ │ │ │ -0001cc20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cc40: 2d2d 2d2d 2d2d 2d2d 2020 2020 2020 2020  --------        
│ │ │ │ -0001cc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cc60: 207c 0a7c 2020 2020 2020 2878 2078 202c   |.|      (x x ,
│ │ │ │ -0001cc70: 2078 2078 2078 202c 2078 2020 2d20 7820   x x x , x  - x 
│ │ │ │ -0001cc80: 2c20 7820 202d 2078 202c 2078 2020 2d20  , x  - x , x  - 
│ │ │ │ -0001cc90: 7820 2920 2020 2020 2020 2020 2020 2020  x )             
│ │ │ │ -0001cca0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001ccb0: 2020 2020 2020 2032 2034 2020 2030 2031         2 4   0 1
│ │ │ │ -0001ccc0: 2033 2020 2030 2020 2020 3320 2020 3120   3   0    3   1 
│ │ │ │ -0001ccd0: 2020 2033 2020 2032 2020 2020 3420 2020     3   2    4   
│ │ │ │ -0001cce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ccf0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -0001cd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cd40: 2d2d 2b0a 7c69 3236 203a 2045 756c 6572  --+.|i26 : Euler
│ │ │ │ -0001cd50: 493d 4575 6c65 7228 582c 4929 2020 2020  I=Euler(X,I)    
│ │ │ │ +0001cbe0: 2020 207c 0a7c 6f32 3520 3a20 2d2d 2d2d     |.|o25 : ----
│ │ │ │ +0001cbf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001cc00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001cc10: 2d2d 2d2d 2d20 2020 2020 2020 2020 2020  -----           
│ │ │ │ +0001cc20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001cc30: 7c20 2020 2020 2028 7820 7820 2c20 7820  |      (x x , x 
│ │ │ │ +0001cc40: 7820 7820 2c20 7820 202d 2078 202c 2078  x x , x  - x , x
│ │ │ │ +0001cc50: 2020 2d20 7820 2c20 7820 202d 2078 2029    - x , x  - x )
│ │ │ │ +0001cc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cc70: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001cc80: 2020 2020 3220 3420 2020 3020 3120 3320      2 4   0 1 3 
│ │ │ │ +0001cc90: 2020 3020 2020 2033 2020 2031 2020 2020    0    3   1    
│ │ │ │ +0001cca0: 3320 2020 3220 2020 2034 2020 2020 2020  3   2    4      
│ │ │ │ +0001ccb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ccc0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0001ccd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001cce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ccf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001cd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0001cd10: 0a7c 6932 3620 3a20 4575 6c65 7249 3d45  .|i26 : EulerI=E
│ │ │ │ +0001cd20: 756c 6572 2858 2c49 2920 2020 2020 2020  uler(X,I)       
│ │ │ │ +0001cd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cd50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0001cd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cd80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001cd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cda0: 2020 2020 207c 0a7c 6f32 3620 3d20 3520       |.|o26 = 5 
│ │ │ │  0001cdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cdd0: 2020 2020 2020 2020 7c0a 7c6f 3236 203d          |.|o26 =
│ │ │ │ -0001cde0: 2035 2020 2020 2020 2020 2020 2020 2020   5              
│ │ │ │ -0001cdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ce00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ce10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ce20: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -0001ce30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ce40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ce50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ce60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0001ce70: 7c69 3237 203a 2045 756c 6572 2863 736d  |i27 : Euler(csm
│ │ │ │ -0001ce80: 4929 3d3d 4575 6c65 7249 2020 2020 2020  I)==EulerI      
│ │ │ │ +0001cdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cdf0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0001ce00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ce10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ce20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ce30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +0001ce40: 3720 3a20 4575 6c65 7228 6373 6d49 293d  7 : Euler(csmI)=
│ │ │ │ +0001ce50: 3d45 756c 6572 4920 2020 2020 2020 2020  =EulerI         
│ │ │ │ +0001ce60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ce70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ce80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001ce90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ceb0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001ceb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ced0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ced0: 207c 0a7c 6f32 3720 3d20 7472 7565 2020   |.|o27 = true  
│ │ │ │  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
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│ │ │ │ -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,...----------
│ │ │ │ +0001cf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cf10: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0001cf20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001cf30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001cf40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001cf50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001cf60: 2d2d 2d2d 2d2d 2d2b 0a0a 416c 6c20 7468  -------+..All th
│ │ │ │ +0001cf70: 6520 6578 616d 706c 6573 2077 6572 6520  e examples were 
│ │ │ │ +0001cf80: 646f 6e65 2075 7369 6e67 2073 796d 626f  done using symbo
│ │ │ │ +0001cf90: 6c69 6320 636f 6d70 7574 6174 696f 6e73  lic computations
│ │ │ │ +0001cfa0: 2077 6974 6820 4772 5c22 6f62 6e65 7220   with Gr\"obner 
│ │ │ │ +0001cfb0: 6261 7365 732e 0a43 6861 6e67 696e 6720  bases..Changing 
│ │ │ │ +0001cfc0: 7468 6520 6f70 7469 6f6e 202a 6e6f 7465  the option *note
│ │ │ │ +0001cfd0: 2043 6f6d 704d 6574 686f 643a 2043 6f6d   CompMethod: Com
│ │ │ │ +0001cfe0: 704d 6574 686f 642c 2074 6f20 6265 7274  pMethod, to bert
│ │ │ │ +0001cff0: 696e 6920 7769 6c6c 2064 6f20 7468 6520  ini will do the 
│ │ │ │ +0001d000: 6d61 696e 0a63 6f6d 7075 7461 7469 6f6e  main.computation
│ │ │ │ +0001d010: 7320 6e75 6d65 7269 6361 6c6c 792c 2070  s numerically, p
│ │ │ │ +0001d020: 726f 7669 6465 6420 4265 7274 696e 6920  rovided Bertini 
│ │ │ │ +0001d030: 6973 202a 6e6f 7465 2069 6e73 7461 6c6c  is *note install
│ │ │ │ +0001d040: 6564 2061 6e64 2063 6f6e 6669 6775 7265  ed and configure
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│ │ │ │ -0001d640: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
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│ │ │ │ +0001d5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
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│ │ │ │ +0001d630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0001d670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d680: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001d690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d6c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0001d6c0: 2020 2020 2020 207c 0a7c 6f32 203d 2052         |.|o2 = R
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│ │ │ │  0001d6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d6f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
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│ │ │ │ +0001d700: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001d710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d730: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001d740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d750: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001d830: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0001d770: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
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│ │ │ │ +0001d7d0: 2d31 2920 2020 2020 2020 2020 2020 2020  -1)             
│ │ │ │ +0001d7e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
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│ │ │ │ +0001d800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d820: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0001d830: 2020 3220 2020 2032 2020 2020 3220 2020    2    2    2   
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│ │ │ │ -0001d850: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001d860: 2020 2020 2032 2020 2020 3220 2020 2032       2    2    2
│ │ │ │ -0001d870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d850: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001d860: 6f33 203d 2069 6465 616c 2878 2020 2b20  o3 = ideal(x  + 
│ │ │ │ +0001d870: 7820 202b 2078 2020 2d20 3129 2020 2020  x  + x  - 1)    
│ │ │ │  0001d880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d890: 7c0a 7c6f 3320 3d20 6964 6561 6c28 7820  |.|o3 = ideal(x 
│ │ │ │ -0001d8a0: 202b 2078 2020 2b20 7820 202d 2031 2920   + x  + x  - 1) 
│ │ │ │ -0001d8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d8c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001d8d0: 2020 2020 2020 2020 2031 2020 2020 3220           1    2 
│ │ │ │ -0001d8e0: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ +0001d890: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001d8a0: 2020 2020 2020 3120 2020 2032 2020 2020        1    2    
│ │ │ │ +0001d8b0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +0001d8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d8d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001d8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d900: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001d910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d900: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ +0001d910: 203a 2049 6465 616c 206f 6620 5220 2020   : Ideal of R   
│ │ │ │  0001d920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d930: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001d940: 7c6f 3320 3a20 4964 6561 6c20 6f66 2052  |o3 : Ideal of R
│ │ │ │ -0001d950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d970: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -0001d980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0001d9e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001d9f0: 2d2d 2075 7365 6420 302e 3038 3734 3133  -- used 0.087413
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│ │ │ │ -0001da20: 7320 2867 6329 7c0a 7c20 2020 2020 2020  s (gc)|.|       
│ │ │ │ -0001da30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d940: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001d950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0001d980: 0a7c 6934 203a 2074 696d 6520 4575 6c65  .|i4 : time Eule
│ │ │ │ +0001d990: 7241 6666 696e 6520 4920 2020 2020 2020  rAffine I       
│ │ │ │ +0001d9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d9b0: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ +0001d9c0: 7573 6564 2030 2e30 3735 3738 3535 7320  used 0.0757855s 
│ │ │ │ +0001d9d0: 2863 7075 293b 2030 2e30 3631 3833 3635  (cpu); 0.0618365
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│ │ │ │ +0001d9f0: 6763 297c 0a7c 2020 2020 2020 2020 2020  gc)|.|          
│ │ │ │ +0001da00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001da10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001da20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001da30: 6f34 203d 2032 2020 2020 2020 2020 2020  o4 = 2          
│ │ │ │  0001da40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001da50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001da60: 7c0a 7c6f 3420 3d20 3220 2020 2020 2020  |.|o4 = 2       
│ │ │ │ -0001da70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001da80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001da90: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
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│ │ │ │ -0001dac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0001da70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001da80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001da90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0001dec0: 6365 7320 666f 7220 7468 6520 6765 6e65  ces for the gene
│ │ │ │ +0001ded0: 7261 746f 7273 206f 6620 7468 6520 696e  rators of the in
│ │ │ │ +0001dee0: 7075 7420 6964 6561 6c20 7468 6174 2c0a  put ideal that,.
│ │ │ │ +0001def0: 7768 656e 2074 616b 656e 2074 6f67 6574  when taken toget
│ │ │ │ +0001df00: 6865 722c 2064 6566 696e 6520 6120 736d  her, define a sm
│ │ │ │ +0001df10: 6f6f 7468 2073 7562 7363 6865 6d65 206f  ooth subscheme o
│ │ │ │ +0001df20: 6620 7468 6520 616d 6269 656e 7420 7370  f the ambient sp
│ │ │ │ +0001df30: 6163 652e 2054 6869 730a 6f70 7469 6f6e  ace. This.option
│ │ │ │ +0001df40: 2077 696c 6c20 6265 2069 676e 6f72 6564   will be ignored
│ │ │ │ +0001df50: 206f 7468 6572 7769 7365 2e0a 0a2b 2d2d   otherwise...+--
│ │ │ │ +0001df60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001df70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001df80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001df90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001dfa0: 2d2d 2b0a 7c69 3120 3a20 5220 3d20 4d75  --+.|i1 : R = Mu
│ │ │ │ +0001dfb0: 6c74 6950 726f 6a43 6f6f 7264 5269 6e67  ltiProjCoordRing
│ │ │ │ +0001dfc0: 287b 322c 327d 2920 2020 2020 2020 2020  ({2,2})         
│ │ │ │ +0001dfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001dfe0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001dff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e010: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +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                  
│ │ │ │ +0001e030: 7c0a 7c6f 3120 3d20 5220 2020 2020 2020  |.|o1 = R       
│ │ │ │  0001e040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e060: 2020 207c 0a7c 6f31 203d 2052 2020 2020     |.|o1 = R    
│ │ │ │ -0001e070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e070: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0001e080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e0a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001e0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e0b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001e0c0: 7c6f 3120 3a20 506f 6c79 6e6f 6d69 616c  |o1 : Polynomial
│ │ │ │ +0001e0d0: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │  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));|.|       
│ │ │ │ -0001e1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e100: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001e110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0001e150: 3220 3a20 493d 6964 6561 6c28 525f 302a  2 : I=ideal(R_0*
│ │ │ │ +0001e160: 525f 312a 525f 332d 525f 305e 322a 525f  R_1*R_3-R_0^2*R_
│ │ │ │ +0001e170: 332c 7261 6e64 6f6d 287b 302c 317d 2c52  3,random({0,1},R
│ │ │ │ +0001e180: 292c 7261 6e64 6f6d 287b 312c 327d 2c52  ),random({1,2},R
│ │ │ │ +0001e190: 2929 3b7c 0a7c 2020 2020 2020 2020 2020  ));|.|          
│ │ │ │ +0001e1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e1d0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ +0001e1e0: 3a20 4964 6561 6c20 6f66 2052 2020 2020  : Ideal of R    
│ │ │ │  0001e1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e200: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001e210: 6f32 203a 2049 6465 616c 206f 6620 5220  o2 : Ideal of R 
│ │ │ │ -0001e220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e250: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -0001e260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │ -0001e2a0: 203a 2074 696d 6520 4353 4d28 492c 4d65   : time CSM(I,Me
│ │ │ │ -0001e2b0: 7468 6f64 3d3e 4469 7265 6374 436f 6d70  thod=>DirectComp
│ │ │ │ -0001e2c0: 6c65 7449 6e74 2920 2020 2020 2020 2020  letInt)         
│ │ │ │ -0001e2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e2e0: 2020 7c0a 7c20 2d2d 2075 7365 6420 322e    |.| -- used 2.
│ │ │ │ -0001e2f0: 3538 3339 3773 2028 6370 7529 3b20 312e  58397s (cpu); 1.
│ │ │ │ -0001e300: 3132 3030 3373 2028 7468 7265 6164 293b  12003s (thread);
│ │ │ │ -0001e310: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │ -0001e320: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0001e330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e220: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0001e230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e260: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ +0001e270: 7469 6d65 2043 534d 2849 2c4d 6574 686f  time CSM(I,Metho
│ │ │ │ +0001e280: 643d 3e44 6972 6563 7443 6f6d 706c 6574  d=>DirectComplet
│ │ │ │ +0001e290: 496e 7429 2020 2020 2020 2020 2020 2020  Int)            
│ │ │ │ +0001e2a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001e2b0: 0a7c 202d 2d20 7573 6564 2036 2e32 3531  .| -- used 6.251
│ │ │ │ +0001e2c0: 3839 7320 2863 7075 293b 2031 2e34 3036  89s (cpu); 1.406
│ │ │ │ +0001e2d0: 3233 7320 2874 6872 6561 6429 3b20 3073  23s (thread); 0s
│ │ │ │ +0001e2e0: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │ +0001e2f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001e300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e330: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001e340: 2020 2020 2020 2032 2032 2020 2020 2032         2 2     2
│ │ │ │ +0001e350: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │  0001e360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e370: 7c0a 7c20 2020 2020 2020 3220 3220 2020  |.|       2 2   
│ │ │ │ -0001e380: 2020 3220 2020 2020 2020 2020 3220 2020    2         2   
│ │ │ │ -0001e390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e380: 2020 2020 7c0a 7c6f 3320 3d20 3268 2068      |.|o3 = 2h h
│ │ │ │ +0001e390: 2020 2b20 3268 2068 2020 2b20 3568 2068    + 2h h  + 5h h
│ │ │ │  0001e3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e3b0: 2020 2020 2020 207c 0a7c 6f33 203d 2032         |.|o3 = 2
│ │ │ │ -0001e3c0: 6820 6820 202b 2032 6820 6820 202b 2035  h h  + 2h h  + 5
│ │ │ │ -0001e3d0: 6820 6820 2020 2020 2020 2020 2020 2020  h h             
│ │ │ │ -0001e3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e3f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001e400: 7c20 2020 2020 2020 3120 3220 2020 2020  |       1 2     
│ │ │ │ -0001e410: 3120 3220 2020 2020 3120 3220 2020 2020  1 2     1 2     
│ │ │ │ +0001e3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e3c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001e3d0: 2020 2020 2031 2032 2020 2020 2031 2032       1 2     1 2
│ │ │ │ +0001e3e0: 2020 2020 2031 2032 2020 2020 2020 2020       1 2        
│ │ │ │ +0001e3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e410: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001e420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e440: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001e450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e450: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001e460: 205a 5a5b 6820 2e2e 6820 5d20 2020 2020   ZZ[h ..h ]     
│ │ │ │  0001e470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e480: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001e490: 2020 2020 5a5a 5b68 202e 2e68 205d 2020      ZZ[h ..h ]  
│ │ │ │ -0001e4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e4a0: 7c0a 7c20 2020 2020 2020 2020 3120 2020  |.|         1   
│ │ │ │ +0001e4b0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  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                  
│ │ │ │ +0001e4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e4e0: 2020 2020 2020 207c 0a7c 6f33 203a 202d         |.|o3 : -
│ │ │ │ +0001e4f0: 2d2d 2d2d 2d2d 2d2d 2d20 2020 2020 2020  ---------       
│ │ │ │  0001e500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e510: 2020 2020 2020 2020 2020 7c0a 7c6f 3320            |.|o3 
│ │ │ │ -0001e520: 3a20 2d2d 2d2d 2d2d 2d2d 2d2d 2020 2020  : ----------    
│ │ │ │ -0001e530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e520: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001e530: 7c20 2020 2020 2020 2033 2020 2033 2020  |        3   3  
│ │ │ │  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                  
│ │ │ │ +0001e560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e570: 2020 2020 207c 0a7c 2020 2020 2020 2868       |.|      (h
│ │ │ │ +0001e580: 202c 2068 2029 2020 2020 2020 2020 2020   , h )          
│ │ │ │  0001e590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e5a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001e5b0: 2028 6820 2c20 6820 2920 2020 2020 2020   (h , h )       
│ │ │ │ -0001e5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e5b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001e5c0: 2020 2020 2020 2031 2020 2032 2020 2020         1   2    
│ │ │ │  0001e5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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: 322e 3733 3530 3573 2028 6370 7529 3b20  2.73505s (cpu); 
│ │ │ │ -0001e6e0: 312e 3234 3237 7320 2874 6872 6561 6429  1.2427s (thread)
│ │ │ │ -0001e6f0: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ -0001e700: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001e710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e600: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0001e610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e640: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420  ----------+.|i4 
│ │ │ │ +0001e650: 3a20 7469 6d65 2043 534d 2849 2c4d 6574  : time CSM(I,Met
│ │ │ │ +0001e660: 686f 643d 3e44 6972 6563 7443 6f6d 706c  hod=>DirectCompl
│ │ │ │ +0001e670: 6574 496e 742c 496e 6473 4f66 536d 6f6f  etInt,IndsOfSmoo
│ │ │ │ +0001e680: 7468 3d3e 7b31 2c32 7d29 2020 2020 2020  th=>{1,2})      
│ │ │ │ +0001e690: 207c 0a7c 202d 2d20 7573 6564 2036 2e31   |.| -- used 6.1
│ │ │ │ +0001e6a0: 3137 3533 7320 2863 7075 293b 2031 2e34  1753s (cpu); 1.4
│ │ │ │ +0001e6b0: 3139 3434 7320 2874 6872 6561 6429 3b20  1944s (thread); 
│ │ │ │ +0001e6c0: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │ +0001e6d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001e6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e710: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001e720: 0a7c 2020 2020 2020 2032 2032 2020 2020  .|       2 2    
│ │ │ │ +0001e730: 2032 2020 2020 2020 2020 2032 2020 2020   2         2    
│ │ │ │  0001e740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e750: 2020 7c0a 7c20 2020 2020 2020 3220 3220    |.|       2 2 
│ │ │ │ -0001e760: 2020 2020 3220 2020 2020 2020 2020 3220      2         2 
│ │ │ │ -0001e770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e790: 2020 2020 2020 2020 207c 0a7c 6f34 203d           |.|o4 =
│ │ │ │ -0001e7a0: 2032 6820 6820 202b 2032 6820 6820 202b   2h h  + 2h h  +
│ │ │ │ -0001e7b0: 2035 6820 6820 2020 2020 2020 2020 2020   5h h           
│ │ │ │ -0001e7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e760: 2020 2020 2020 7c0a 7c6f 3420 3d20 3268        |.|o4 = 2h
│ │ │ │ +0001e770: 2068 2020 2b20 3268 2068 2020 2b20 3568   h  + 2h h  + 5h
│ │ │ │ +0001e780: 2068 2020 2020 2020 2020 2020 2020 2020   h              
│ │ │ │ +0001e790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e7a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001e7b0: 2020 2020 2020 2031 2032 2020 2020 2031         1 2     1
│ │ │ │ +0001e7c0: 2032 2020 2020 2031 2032 2020 2020 2020   2     1 2      
│ │ │ │  0001e7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e7e0: 7c0a 7c20 2020 2020 2020 3120 3220 2020  |.|       1 2   
│ │ │ │ -0001e7f0: 2020 3120 3220 2020 2020 3120 3220 2020    1 2     1 2   
│ │ │ │ +0001e7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e7f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  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         |.|      
│ │ │ │ -0001e830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e820: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0001ee70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ee80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0001eea0: 2f33 3237 3439 5b78 5f30 2e2e 785f 345d  /32749[x_0..x_4]
│ │ │ │ -0001eeb0: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ -0001eec0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -0001eed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0001eef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0001ef40: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
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│ │ │ │ +0001ed70: 2043 534d 3a20 4353 4d2c 2c20 616e 640a   CSM: CSM,, and.
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│ │ │ │ +0001ee30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ee40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ee50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
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│ │ │ │ +0001ee70: 3439 5b78 5f30 2e2e 785f 345d 3b20 2020  49[x_0..x_4];   
│ │ │ │ +0001ee80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ee90: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0001eea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001eeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0001eef0: 6f6d 2832 2c52 292c 7261 6e64 6f6d 2831  om(2,R),random(1
│ │ │ │ +0001ef00: 2c52 2929 3b20 2020 207c 0a7c 2020 2020  ,R));    |.|    
│ │ │ │ +0001ef10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ef20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ef30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ef40: 2020 7c0a 7c6f 3220 3a20 4964 6561 6c20    |.|o2 : Ideal 
│ │ │ │ +0001ef50: 6f66 2052 2020 2020 2020 2020 2020 2020  of R            
│ │ │ │  0001ef60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ef70: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ -0001ef80: 3a20 4964 6561 6c20 6f66 2052 2020 2020  : Ideal of R    
│ │ │ │ -0001ef90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001efa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001efb0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -0001efc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001efd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001efe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0001eff0: 7c69 3320 3a20 7469 6d65 2043 534d 2049  |i3 : time CSM I
│ │ │ │ -0001f000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f020: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ -0001f030: 7365 6420 302e 3835 3232 3433 7320 2863  sed 0.852243s (c
│ │ │ │ -0001f040: 7075 293b 2030 2e34 3338 3333 3273 2028  pu); 0.438332s (
│ │ │ │ -0001f050: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │ -0001f060: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001ef70: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0001ef80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ef90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001efa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001efb0: 2d2d 2d2d 2b0a 7c69 3320 3a20 7469 6d65  ----+.|i3 : time
│ │ │ │ +0001efc0: 2043 534d 2049 2020 2020 2020 2020 2020   CSM I          
│ │ │ │ +0001efd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001efe0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001eff0: 202d 2d20 7573 6564 2031 2e30 3234 3235   -- used 1.02425
│ │ │ │ +0001f000: 7320 2863 7075 293b 2030 2e35 3036 3734  s (cpu); 0.50674
│ │ │ │ +0001f010: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ +0001f020: 6763 2920 2020 7c0a 7c20 2020 2020 2020  gc)   |.|       
│ │ │ │ +0001f030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f050: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001f060: 0a7c 2020 2020 2020 2033 2020 2020 2020  .|       3      
│ │ │ │  0001f070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f090: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001f0a0: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ +0001f090: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20          |.|o3 = 
│ │ │ │ +0001f0a0: 3468 2020 2020 2020 2020 2020 2020 2020  4h              
│ │ │ │  0001f0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f0d0: 2020 2020 2020 7c0a 7c6f 3320 3d20 3468        |.|o3 = 4h
│ │ │ │ +0001f0d0: 207c 0a7c 2020 2020 2020 2031 2020 2020   |.|       1    
│ │ │ │  0001f0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f110: 7c0a 7c20 2020 2020 2020 3120 2020 2020  |.|       1     
│ │ │ │ +0001f100: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001f110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f140: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001f150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f140: 2020 207c 0a7c 2020 2020 205a 5a5b 6820     |.|     ZZ[h 
│ │ │ │ +0001f150: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │  0001f160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f180: 2020 2020 7c0a 7c20 2020 2020 5a5a 5b68      |.|     ZZ[h
│ │ │ │ -0001f190: 205d 2020 2020 2020 2020 2020 2020 2020   ]              
│ │ │ │ +0001f170: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001f180: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ +0001f190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f1b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001f1c0: 7c20 2020 2020 2020 2020 3120 2020 2020  |         1     
│ │ │ │ +0001f1b0: 2020 2020 207c 0a7c 6f33 203a 202d 2d2d       |.|o3 : ---
│ │ │ │ +0001f1c0: 2d2d 2d20 2020 2020 2020 2020 2020 2020  ---             
│ │ │ │  0001f1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f1f0: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20          |.|o3 : 
│ │ │ │ -0001f200: 2d2d 2d2d 2d2d 2020 2020 2020 2020 2020  ------          
│ │ │ │ +0001f1e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001f1f0: 7c20 2020 2020 2020 2035 2020 2020 2020  |        5      
│ │ │ │ +0001f200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f230: 2020 7c0a 7c20 2020 2020 2020 2035 2020    |.|        5  
│ │ │ │ +0001f220: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001f230: 2068 2020 2020 2020 2020 2020 2020 2020   h              
│ │ │ │  0001f240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f260: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001f270: 2020 2020 2020 6820 2020 2020 2020 2020        h         
│ │ │ │ +0001f260: 7c0a 7c20 2020 2020 2020 2031 2020 2020  |.|        1    
│ │ │ │ +0001f270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f2a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001f2b0: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ -0001f2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f2e0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -0001f2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f310: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420  ----------+.|i4 
│ │ │ │ -0001f320: 3a20 7469 6d65 2043 534d 2849 2c49 6e70  : time CSM(I,Inp
│ │ │ │ -0001f330: 7574 4973 536d 6f6f 7468 3d3e 7472 7565  utIsSmooth=>true
│ │ │ │ -0001f340: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -0001f350: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ -0001f360: 302e 3039 3131 3232 3873 2028 6370 7529  0.0911228s (cpu)
│ │ │ │ -0001f370: 3b20 302e 3033 3136 3439 3573 2028 7468  ; 0.0316495s (th
│ │ │ │ -0001f380: 7265 6164 293b 2030 7320 2867 6329 7c0a  read); 0s (gc)|.
│ │ │ │ -0001f390: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0001f290: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001f2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f2d0: 2d2d 2b0a 7c69 3420 3a20 7469 6d65 2043  --+.|i4 : time C
│ │ │ │ +0001f2e0: 534d 2849 2c49 6e70 7574 4973 536d 6f6f  SM(I,InputIsSmoo
│ │ │ │ +0001f2f0: 7468 3d3e 7472 7565 2920 2020 2020 2020  th=>true)       
│ │ │ │ +0001f300: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ +0001f310: 2d20 7573 6564 2030 2e30 3737 3539 3532  - used 0.0775952
│ │ │ │ +0001f320: 7320 2863 7075 293b 2030 2e30 3431 3132  s (cpu); 0.04112
│ │ │ │ +0001f330: 3173 2028 7468 7265 6164 293b 2030 7320  1s (thread); 0s 
│ │ │ │ +0001f340: 2867 6329 7c0a 7c20 2020 2020 2020 2020  (gc)|.|         
│ │ │ │ +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 207c 0a7c               |.|
│ │ │ │ +0001f380: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ +0001f390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f3c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001f3d0: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ -0001f3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f400: 2020 7c0a 7c6f 3420 3d20 3468 2020 2020    |.|o4 = 4h    
│ │ │ │ +0001f3b0: 2020 2020 2020 7c0a 7c6f 3420 3d20 3468        |.|o4 = 4h
│ │ │ │ +0001f3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f3e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001f3f0: 0a7c 2020 2020 2020 2031 2020 2020 2020  .|       1      
│ │ │ │ +0001f400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f430: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001f440: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │ +0001f420: 2020 2020 2020 2020 7c0a 7c20 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 2020 2020 2020 2020                  
│ │ │ │ -0001f460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f470: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001f460: 207c 0a7c 2020 2020 205a 5a5b 6820 5d20   |.|     ZZ[h ] 
│ │ │ │ +0001f470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f4b0: 7c0a 7c20 2020 2020 5a5a 5b68 205d 2020  |.|     ZZ[h ]  
│ │ │ │ +0001f490: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001f4a0: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │ +0001f4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f4e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001f4f0: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │ -0001f500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f520: 2020 2020 7c0a 7c6f 3420 3a20 2d2d 2d2d      |.|o4 : ----
│ │ │ │ -0001f530: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │ -0001f540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f550: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001f560: 7c20 2020 2020 2020 2035 2020 2020 2020  |        5      
│ │ │ │ -0001f570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f590: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001f5a0: 2020 6820 2020 2020 2020 2020 2020 2020    h             
│ │ │ │ -0001f5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f5d0: 2020 7c0a 7c20 2020 2020 2020 2031 2020    |.|        1  
│ │ │ │ -0001f5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f600: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -0001f610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f640: 2d2d 2d2d 2d2d 2b0a 0a4e 6f74 6520 7468  ------+..Note th
│ │ │ │ -0001f650: 6174 206f 6e65 2063 6f75 6c64 2c20 6571  at one could, eq
│ │ │ │ -0001f660: 7569 7661 6c65 6e74 6c79 2c20 7573 6520  uivalently, use 
│ │ │ │ -0001f670: 7468 6520 636f 6d6d 616e 6420 2a6e 6f74  the command *not
│ │ │ │ -0001f680: 6520 4368 6572 6e3a 2043 6865 726e 2c20  e Chern: Chern, 
│ │ │ │ -0001f690: 696e 7374 6561 640a 696e 2074 6869 7320  instead.in this 
│ │ │ │ -0001f6a0: 6361 7365 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  case...+--------
│ │ │ │ -0001f6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0001f6e0: 0a7c 6935 203a 2074 696d 6520 4368 6572  .|i5 : time Cher
│ │ │ │ -0001f6f0: 6e20 4920 2020 2020 2020 2020 2020 2020  n I             
│ │ │ │ +0001f4d0: 2020 207c 0a7c 6f34 203a 202d 2d2d 2d2d     |.|o4 : -----
│ │ │ │ +0001f4e0: 2d20 2020 2020 2020 2020 2020 2020 2020  -               
│ │ │ │ +0001f4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f500: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001f510: 2020 2020 2020 2035 2020 2020 2020 2020         5        
│ │ │ │ +0001f520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f540: 2020 2020 207c 0a7c 2020 2020 2020 2068       |.|       h
│ │ │ │ +0001f550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f570: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001f580: 7c20 2020 2020 2020 2031 2020 2020 2020  |        1      
│ │ │ │ +0001f590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f5b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0001f5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f5f0: 2b0a 0a4e 6f74 6520 7468 6174 206f 6e65  +..Note that one
│ │ │ │ +0001f600: 2063 6f75 6c64 2c20 6571 7569 7661 6c65   could, equivale
│ │ │ │ +0001f610: 6e74 6c79 2c20 7573 6520 7468 6520 636f  ntly, use the co
│ │ │ │ +0001f620: 6d6d 616e 6420 2a6e 6f74 6520 4368 6572  mmand *note Cher
│ │ │ │ +0001f630: 6e3a 2043 6865 726e 2c20 696e 7374 6561  n: Chern, instea
│ │ │ │ +0001f640: 640a 696e 2074 6869 7320 6361 7365 2e0a  d.in this case..
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│ │ │ │ +0001f660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f680: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a  ---------+.|i5 :
│ │ │ │ +0001f690: 2074 696d 6520 4368 6572 6e20 4920 2020   time Chern I   
│ │ │ │ +0001f6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f6c0: 2020 207c 0a7c 202d 2d20 7573 6564 2030     |.| -- used 0
│ │ │ │ +0001f6d0: 2e30 3635 3638 3936 7320 2863 7075 293b  .0656896s (cpu);
│ │ │ │ +0001f6e0: 2030 2e30 3430 3631 3237 7320 2874 6872   0.0406127s (thr
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│ │ │ │  0001f700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f710: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ -0001f720: 7573 6564 2030 2e30 3735 3131 3431 7320  used 0.0751141s 
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│ │ │ │ +0001f710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f730: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001f740: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ +0001f750: 2020 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 207c 0a7c               |.|
│ │ │ │ -0001f790: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ -0001f7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f7c0: 2020 2020 2020 207c 0a7c 6f35 203d 2034         |.|o5 = 4
│ │ │ │ -0001f7d0: 6820 2020 2020 2020 2020 2020 2020 2020  h               
│ │ │ │ -0001f7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f770: 207c 0a7c 6f35 203d 2034 6820 2020 2020   |.|o5 = 4h     
│ │ │ │ +0001f780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f7a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001f7b0: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +0001f7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f7e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0001f7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f800: 207c 0a7c 2020 2020 2020 2031 2020 2020   |.|       1    
│ │ │ │ -0001f810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f830: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001f800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f810: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001f820: 0a7c 2020 2020 205a 5a5b 6820 5d20 2020  .|     ZZ[h ]   
│ │ │ │ +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 2020 207c 0a7c 2020 2020 205a 5a5b       |.|     ZZ[
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│ │ │ │ -00020180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000201a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
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│ │ │ │ -000201d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000201e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001fda0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a  ***********..  *
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│ │ │ │ +0001fe50: 6d75 6c74 692d 6772 6164 6564 2072 696e  multi-graded rin
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│ │ │ │ +0001fea0: 2065 6c65 6d65 6e74 2069 6e20 610a 2020   element in a.  
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│ │ │ │ +00020130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00020190: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
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│ │ │ │ +000201c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00020230: 3120 3d20 5220 2020 2020 2020 2020 2020  1 = R           
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│ │ │ │ -00020270: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00020250: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
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│ │ │ │ -000202e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00020380: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00020380: 202c 2078 207d 2020 2020 2020 2020 2020   , x }          
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│ │ │ │ -000203a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000203c0: 7b78 202c 2078 202c 2078 202c 2078 202c  {x , x , x , x ,
│ │ │ │ -000203d0: 2078 202c 2078 202c 2078 207d 2020 2020   x , x , x }    
│ │ │ │ -000203e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000203f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00020400: 2020 2020 3020 2020 3120 2020 3220 2020      0   1   2   
│ │ │ │ -00020410: 3320 2020 3420 2020 3520 2020 3620 2020  3   4   5   6   
│ │ │ │ -00020420: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000203a0: 2020 2020 7c0a 7c20 2020 2020 2020 3020      |.|       0 
│ │ │ │ +000203b0: 2020 3120 2020 3220 2020 3320 2020 3420    1   2   3   4 
│ │ │ │ +000203c0: 2020 3520 2020 3620 2020 2020 2020 2020    5   6         
│ │ │ │ +000203d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000203e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000203f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00020430: 4c69 7374 2020 2020 2020 2020 2020 2020  List            
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│ │ │ │ -00020490: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000204d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000204e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000204f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00020510: 6c28 785f 305e 322a 785f 332d 785f 312a  l(x_0^2*x_3-x_1*
│ │ │ │ -00020520: 785f 302a 785f 342c 785f 365e 3329 2020  x_0*x_4,x_6^3)  
│ │ │ │ -00020530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020540: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00020460: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00020470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000204a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +000204b0: 3320 3a20 493d 6964 6561 6c28 785f 305e  3 : I=ideal(x_0^
│ │ │ │ +000204c0: 322a 785f 332d 785f 312a 785f 302a 785f  2*x_3-x_1*x_0*x_
│ │ │ │ +000204d0: 342c 785f 365e 3329 2020 2020 2020 2020  4,x_6^3)        
│ │ │ │ +000204e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000204f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00020500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020520: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00020540: 3220 2020 2020 2020 2020 2020 2020 2033  2              3
│ │ │ │  00020550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020580: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00020590: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -000205a0: 2020 2020 2033 2020 2020 2020 2020 2020       3          
│ │ │ │ -000205b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000205c0: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20          |.|o3 = 
│ │ │ │ -000205d0: 6964 6561 6c20 2878 2078 2020 2d20 7820  ideal (x x  - x 
│ │ │ │ -000205e0: 7820 7820 2c20 7820 2920 2020 2020 2020  x x , x )       
│ │ │ │ -000205f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020600: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00020610: 2020 2020 2020 2020 2020 3020 3320 2020            0 3   
│ │ │ │ -00020620: 2030 2031 2034 2020 2036 2020 2020 2020   0 1 4   6      
│ │ │ │ -00020630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020640: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00020570: 2020 7c0a 7c6f 3320 3d20 6964 6561 6c20    |.|o3 = ideal 
│ │ │ │ +00020580: 2878 2078 2020 2d20 7820 7820 7820 2c20  (x x  - x x x , 
│ │ │ │ +00020590: 7820 2920 2020 2020 2020 2020 2020 2020  x )             
│ │ │ │ +000205a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000205b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000205c0: 2020 2020 3020 3320 2020 2030 2031 2034      0 3    0 1 4
│ │ │ │ +000205d0: 2020 2036 2020 2020 2020 2020 2020 2020     6            
│ │ │ │ +000205e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000205f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00020600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020620: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00020640: 4964 6561 6c20 6f66 2052 2020 2020 2020  Ideal of R      
│ │ │ │  00020650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020680: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00020690: 7c6f 3320 3a20 4964 6561 6c20 6f66 2052  |o3 : Ideal of R
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│ │ │ │ -000206b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000206d0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -000206e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000206f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00020720: 6948 6f6d 6f67 656e 656f 7573 2049 2020  iHomogeneous I  
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│ │ │ │ +00020680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000206a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000206b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +000206c0: 3420 3a20 6973 4d75 6c74 6948 6f6d 6f67  4 : isMultiHomog
│ │ │ │ +000206d0: 656e 656f 7573 2049 2020 2020 2020 2020  eneous I        
│ │ │ │ +000206e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000206f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020700: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00020710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020730: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00020770: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00020790: 2020 2020 2020 7c0a 7c6f 3420 3d20 7472        |.|o4 = tr
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│ │ │ │ +00020880: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
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│ │ │ │  000208d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000208e0: 7c0a 7c20 2020 3020 3120 3420 2020 2030  |.|   0 1 4    0
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│ │ │ │ -00020900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020920: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000208e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000208f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020900: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00020910: 7c6f 3520 3d20 6661 6c73 6520 2020 2020  |o5 = false     
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│ │ │ │  00020930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020960: 2020 2020 7c0a 7c6f 3520 3d20 6661 6c73      |.|o5 = fals
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│ │ │ │ -00020990: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000209d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00020a90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ -00020f80: 6d70 4d65 7468 6f64 2c20 506e 5265 7369  mpMethod, PnResi
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│ │ │ │ -00020fa0: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │ -00020fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00020fe0: 203a 2052 203d 205a 5a2f 3332 3734 395b   : R = ZZ/32749[
│ │ │ │ -00020ff0: 785f 302e 2e78 5f36 5d20 2020 2020 2020  x_0..x_6]       
│ │ │ │ -00021000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021010: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00020950: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00020960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00020a20: 7469 486f 6d6f 6765 6e65 6f75 733a 0a3d  tiHomogeneous:.=
│ │ │ │ +00020a30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00020a40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +00020a50: 2020 2a20 2269 734d 756c 7469 486f 6d6f    * "isMultiHomo
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│ │ │ │ +00020b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00020bf0: 6374 6572 6973 7469 6343 6c61 7373 6573  cteristicClasses
│ │ │ │ +00020c00: 2e6d 323a 3230 3132 3a30 2e0a 1f0a 4669  .m2:2012:0....Fi
│ │ │ │ +00020c10: 6c65 3a20 4368 6172 6163 7465 7269 7374  le: Characterist
│ │ │ │ +00020c20: 6963 436c 6173 7365 732e 696e 666f 2c20  icClasses.info, 
│ │ │ │ +00020c30: 4e6f 6465 3a20 4d65 7468 6f64 2c20 4e65  Node: Method, Ne
│ │ │ │ +00020c40: 7874 3a20 4d75 6c74 6950 726f 6a43 6f6f  xt: MultiProjCoo
│ │ │ │ +00020c50: 7264 5269 6e67 2c20 5072 6576 3a20 6973  rdRing, Prev: is
│ │ │ │ +00020c60: 4d75 6c74 6948 6f6d 6f67 656e 656f 7573  MultiHomogeneous
│ │ │ │ +00020c70: 2c20 5570 3a20 546f 700a 0a4d 6574 686f  , Up: Top..Metho
│ │ │ │ +00020c80: 640a 2a2a 2a2a 2a2a 0a0a 4465 7363 7269  d.******..Descri
│ │ │ │ +00020c90: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ +00020ca0: 3d0a 0a54 6865 206f 7074 696f 6e20 4d65  =..The option Me
│ │ │ │ +00020cb0: 7468 6f64 2069 7320 6f6e 6c79 2075 7365  thod is only use
│ │ │ │ +00020cc0: 6420 6279 2074 6865 2063 6f6d 6d61 6e64  d by the command
│ │ │ │ +00020cd0: 7320 2a6e 6f74 6520 4353 4d3a 2043 534d  s *note CSM: CSM
│ │ │ │ +00020ce0: 2c20 616e 6420 2a6e 6f74 6520 4575 6c65  , and *note Eule
│ │ │ │ +00020cf0: 723a 0a45 756c 6572 2c20 616e 6420 6f6e  r:.Euler, and on
│ │ │ │ +00020d00: 6c79 2069 6e20 636f 6d62 696e 6174 696f  ly in combinatio
│ │ │ │ +00020d10: 6e20 7769 7468 202a 6e6f 7465 2043 6f6d  n with *note Com
│ │ │ │ +00020d20: 704d 6574 686f 643a 0a43 6f6d 704d 6574  pMethod:.CompMet
│ │ │ │ +00020d30: 686f 642c 3d3e 5072 6f6a 6563 7469 7665  hod,=>Projective
│ │ │ │ +00020d40: 4465 6772 6565 2e20 5468 6520 4d65 7468  Degree. The Meth
│ │ │ │ +00020d50: 6f64 2049 6e63 6c75 7369 6f6e 4578 636c  od InclusionExcl
│ │ │ │ +00020d60: 7573 696f 6e20 7769 6c6c 2061 6c77 6179  usion will alway
│ │ │ │ +00020d70: 7320 6265 0a75 7365 6420 7769 7468 202a  s be.used with *
│ │ │ │ +00020d80: 6e6f 7465 2043 6f6d 704d 6574 686f 643a  note CompMethod:
│ │ │ │ +00020d90: 2043 6f6d 704d 6574 686f 642c 2050 6e52   CompMethod, PnR
│ │ │ │ +00020da0: 6573 6964 7561 6c20 6f72 2062 6572 7469  esidual or berti
│ │ │ │ +00020db0: 6e69 2e20 5768 656e 2074 6865 2069 6e70  ni. When the inp
│ │ │ │ +00020dc0: 7574 0a69 6465 616c 2069 7320 6120 636f  ut.ideal is a co
│ │ │ │ +00020dd0: 6d70 6c65 7465 2069 6e74 6572 7365 6374  mplete intersect
│ │ │ │ +00020de0: 696f 6e20 6f6e 6520 6d61 792c 2070 6f74  ion one may, pot
│ │ │ │ +00020df0: 656e 7469 616c 6c79 2c20 7370 6565 6420  entially, speed 
│ │ │ │ +00020e00: 7570 2074 6865 2063 6f6d 7075 7461 7469  up the computati
│ │ │ │ +00020e10: 6f6e 0a62 7920 7365 7474 696e 6720 4d65  on.by setting Me
│ │ │ │ +00020e20: 7468 6f64 3d3e 2044 6972 6563 7443 6f6d  thod=> DirectCom
│ │ │ │ +00020e30: 706c 6574 6549 6e74 2e20 5468 6520 6f70  pleteInt. The op
│ │ │ │ +00020e40: 7469 6f6e 204d 6574 686f 6420 6973 206f  tion Method is o
│ │ │ │ +00020e50: 6e6c 7920 7573 6564 2062 7920 7468 650a  nly used by the.
│ │ │ │ +00020e60: 636f 6d6d 616e 6473 202a 6e6f 7465 2043  commands *note C
│ │ │ │ +00020e70: 534d 3a20 4353 4d2c 2061 6e64 202a 6e6f  SM: CSM, and *no
│ │ │ │ +00020e80: 7465 2045 756c 6572 3a20 4575 6c65 722c  te Euler: Euler,
│ │ │ │ +00020e90: 2061 6e64 206f 6e6c 7920 696e 2063 6f6d   and only in com
│ │ │ │ +00020ea0: 6269 6e61 7469 6f6e 2077 6974 680a 2a6e  bination with.*n
│ │ │ │ +00020eb0: 6f74 6520 436f 6d70 4d65 7468 6f64 3a20  ote CompMethod: 
│ │ │ │ +00020ec0: 436f 6d70 4d65 7468 6f64 2c3d 3e50 726f  CompMethod,=>Pro
│ │ │ │ +00020ed0: 6a65 6374 6976 6544 6567 7265 652e 2054  jectiveDegree. T
│ │ │ │ +00020ee0: 6865 204d 6574 686f 6420 496e 636c 7573  he Method Inclus
│ │ │ │ +00020ef0: 696f 6e45 7863 6c75 7369 6f6e 0a77 696c  ionExclusion.wil
│ │ │ │ +00020f00: 6c20 616c 7761 7973 2062 6520 7573 6564  l always be used
│ │ │ │ +00020f10: 2077 6974 6820 2a6e 6f74 6520 436f 6d70   with *note Comp
│ │ │ │ +00020f20: 4d65 7468 6f64 3a20 436f 6d70 4d65 7468  Method: CompMeth
│ │ │ │ +00020f30: 6f64 2c20 506e 5265 7369 6475 616c 206f  od, PnResidual o
│ │ │ │ +00020f40: 7220 6265 7274 696e 692e 0a0a 2b2d 2d2d  r bertini...+---
│ │ │ │ +00020f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020f80: 2d2d 2d2d 2d2b 0a7c 6931 203a 2052 203d  -----+.|i1 : R =
│ │ │ │ +00020f90: 205a 5a2f 3332 3734 395b 785f 302e 2e78   ZZ/32749[x_0..x
│ │ │ │ +00020fa0: 5f36 5d20 2020 2020 2020 2020 2020 2020  _6]             
│ │ │ │ +00020fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020fc0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00020fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020ff0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00021000: 203d 2052 2020 2020 2020 2020 2020 2020   = R            
│ │ │ │ +00021010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021030: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00021040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021050: 207c 0a7c 6f31 203d 2052 2020 2020 2020   |.|o1 = R      
│ │ │ │ +00021050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021080: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00021070: 207c 0a7c 6f31 203a 2050 6f6c 796e 6f6d   |.|o1 : Polynom
│ │ │ │ +00021080: 6961 6c52 696e 6720 2020 2020 2020 2020  ialRing         
│ │ │ │  00021090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000210a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000210b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000210c0: 2020 2020 2020 207c 0a7c 6f31 203a 2050         |.|o1 : P
│ │ │ │ -000210d0: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020  olynomialRing   
│ │ │ │ -000210e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000210f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021100: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -00021110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00021140: 6932 203a 2049 3d69 6465 616c 2872 616e  i2 : I=ideal(ran
│ │ │ │ -00021150: 646f 6d28 322c 5229 2c72 616e 646f 6d28  dom(2,R),random(
│ │ │ │ -00021160: 312c 5229 2c52 5f30 2a52 5f31 2a52 5f36  1,R),R_0*R_1*R_6
│ │ │ │ -00021170: 2d52 5f30 5e33 293b 7c0a 7c20 2020 2020  -R_0^3);|.|     
│ │ │ │ +000210a0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +000210b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000210c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000210d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000210e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2049  -------+.|i2 : I
│ │ │ │ +000210f0: 3d69 6465 616c 2872 616e 646f 6d28 322c  =ideal(random(2,
│ │ │ │ +00021100: 5229 2c72 616e 646f 6d28 312c 5229 2c52  R),random(1,R),R
│ │ │ │ +00021110: 5f30 2a52 5f31 2a52 5f36 2d52 5f30 5e33  _0*R_1*R_6-R_0^3
│ │ │ │ +00021120: 293b 7c0a 7c20 2020 2020 2020 2020 2020  );|.|           
│ │ │ │ +00021130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021150: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00021160: 6f32 203a 2049 6465 616c 206f 6620 5220  o2 : Ideal of R 
│ │ │ │ +00021170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000211a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000211b0: 2020 207c 0a7c 6f32 203a 2049 6465 616c     |.|o2 : Ideal
│ │ │ │ -000211c0: 206f 6620 5220 2020 2020 2020 2020 2020   of R           
│ │ │ │ -000211d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000211e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000211f0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -00021200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021220: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ -00021230: 2074 696d 6520 4353 4d20 4920 2020 2020   time CSM I     
│ │ │ │ -00021240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021190: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +000211a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000211b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000211c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000211d0: 2d2d 2d2b 0a7c 6933 203a 2074 696d 6520  ---+.|i3 : time 
│ │ │ │ +000211e0: 4353 4d20 4920 2020 2020 2020 2020 2020  CSM I           
│ │ │ │ +000211f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021200: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00021210: 7c20 2d2d 2075 7365 6420 322e 3734 3834  | -- used 2.7484
│ │ │ │ +00021220: 3673 2028 6370 7529 3b20 312e 3037 3232  6s (cpu); 1.0722
│ │ │ │ +00021230: 3273 2028 7468 7265 6164 293b 2030 7320  2s (thread); 0s 
│ │ │ │ +00021240: 2867 6329 2020 2020 207c 0a7c 2020 2020  (gc)     |.|    
│ │ │ │  00021250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021260: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ -00021270: 312e 3731 3030 3273 2028 6370 7529 3b20  1.71002s (cpu); 
│ │ │ │ -00021280: 302e 3938 3437 3132 7320 2874 6872 6561  0.984712s (threa
│ │ │ │ -00021290: 6429 3b20 3073 2028 6763 2920 2020 207c  d); 0s (gc)    |
│ │ │ │ -000212a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000212b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000212c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000212d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -000212e0: 2020 2020 2035 2020 2020 2020 3420 2020       5      4   
│ │ │ │ -000212f0: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ -00021300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021310: 2020 2020 207c 0a7c 6f33 203d 2031 3268       |.|o3 = 12h
│ │ │ │ -00021320: 2020 2b20 3130 6820 202b 2036 6820 2020    + 10h  + 6h   
│ │ │ │ -00021330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021280: 2020 2020 7c0a 7c20 2020 2020 2020 2035      |.|        5
│ │ │ │ +00021290: 2020 2020 2020 3420 2020 2020 3320 2020        4     3   
│ │ │ │ +000212a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000212b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000212c0: 0a7c 6f33 203d 2031 3268 2020 2b20 3130  .|o3 = 12h  + 10
│ │ │ │ +000212d0: 6820 202b 2036 6820 2020 2020 2020 2020  h  + 6h         
│ │ │ │ +000212e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000212f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00021300: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ +00021310: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │ +00021320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021330: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00021340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021350: 7c0a 7c20 2020 2020 2020 2031 2020 2020  |.|        1    
│ │ │ │ -00021360: 2020 3120 2020 2020 3120 2020 2020 2020    1     1       
│ │ │ │ -00021370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021380: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00021350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021370: 7c0a 7c20 2020 2020 5a5a 5b68 205d 2020  |.|     ZZ[h ]  
│ │ │ │ +00021380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000213a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000213b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000213c0: 2020 2020 2020 7c0a 7c20 2020 2020 5a5a        |.|     ZZ
│ │ │ │ -000213d0: 5b68 205d 2020 2020 2020 2020 2020 2020  [h ]            
│ │ │ │ -000213e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000213f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021400: 207c 0a7c 2020 2020 2020 2020 2031 2020   |.|         1  
│ │ │ │ +000213a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000213b0: 2020 2020 2020 2031 2020 2020 2020 2020         1        
│ │ │ │ +000213c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000213d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000213e0: 2020 2020 2020 7c0a 7c6f 3320 3a20 2d2d        |.|o3 : --
│ │ │ │ +000213f0: 2d2d 2d2d 2020 2020 2020 2020 2020 2020  ----            
│ │ │ │ +00021400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021430: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00021440: 3320 3a20 2d2d 2d2d 2d2d 2020 2020 2020  3 : ------      
│ │ │ │ -00021450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021470: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00021480: 2020 3720 2020 2020 2020 2020 2020 2020    7             
│ │ │ │ -00021490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000214a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000214b0: 2020 7c0a 7c20 2020 2020 2020 6820 2020    |.|       h   
│ │ │ │ +00021420: 207c 0a7c 2020 2020 2020 2020 3720 2020   |.|        7   
│ │ │ │ +00021430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021450: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00021460: 2020 2020 2020 6820 2020 2020 2020 2020        h         
│ │ │ │ +00021470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021490: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000214a0: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │ +000214b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000214c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000214d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000214e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000214f0: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ -00021500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021520: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -00021530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021560: 2d2d 2d2b 0a7c 6934 203a 2074 696d 6520  ---+.|i4 : time 
│ │ │ │ -00021570: 4353 4d28 492c 4d65 7468 6f64 3d3e 4469  CSM(I,Method=>Di
│ │ │ │ -00021580: 7265 6374 436f 6d70 6c65 7465 496e 7429  rectCompleteInt)
│ │ │ │ -00021590: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000215a0: 7c20 2d2d 2075 7365 6420 302e 3433 3836  | -- used 0.4386
│ │ │ │ -000215b0: 3032 7320 2863 7075 293b 2030 2e32 3039  02s (cpu); 0.209
│ │ │ │ -000215c0: 3233 3273 2028 7468 7265 6164 293b 2030  232s (thread); 0
│ │ │ │ -000215d0: 7320 2867 6329 2020 207c 0a7c 2020 2020  s (gc)   |.|    
│ │ │ │ +000214d0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +000214e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000214f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00021510: 6934 203a 2074 696d 6520 4353 4d28 492c  i4 : time CSM(I,
│ │ │ │ +00021520: 4d65 7468 6f64 3d3e 4469 7265 6374 436f  Method=>DirectCo
│ │ │ │ +00021530: 6d70 6c65 7465 496e 7429 2020 2020 2020  mpleteInt)      
│ │ │ │ +00021540: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ +00021550: 7365 6420 302e 3637 3231 7320 2863 7075  sed 0.6721s (cpu
│ │ │ │ +00021560: 293b 2030 2e32 3631 3932 3873 2028 7468  ); 0.261928s (th
│ │ │ │ +00021570: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │ +00021580: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00021590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000215a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000215b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000215c0: 7c20 2020 2020 2020 2035 2020 2020 2020  |        5      
│ │ │ │ +000215d0: 3420 2020 2020 3320 2020 2020 2020 2020  4     3         
│ │ │ │  000215e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000215f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021610: 2020 2020 7c0a 7c20 2020 2020 2020 2035      |.|        5
│ │ │ │ -00021620: 2020 2020 2020 3420 2020 2020 3320 2020        4     3   
│ │ │ │ -00021630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021640: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021650: 0a7c 6f34 203d 2031 3268 2020 2b20 3130  .|o4 = 12h  + 10
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│ │ │ │ -00021680: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00021690: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
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│ │ │ │ -000216b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00021620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021630: 2020 2020 7c0a 7c20 2020 2020 2020 2031      |.|        1
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│ │ │ │ +00021650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021660: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00021670: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
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│ │ │ │ +00021690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000216a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000216b0: 2020 5a5a 5b68 205d 2020 2020 2020 2020    ZZ[h ]        
│ │ │ │ +000216c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000216d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000216e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000216f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021700: 7c0a 7c20 2020 2020 5a5a 5b68 205d 2020  |.|     ZZ[h ]  
│ │ │ │ +000216e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000216f0: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ +00021700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021730: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00021740: 2020 2020 2020 2031 2020 2020 2020 2020         1        
│ │ │ │ -00021750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021770: 2020 2020 2020 7c0a 7c6f 3420 3a20 2d2d        |.|o4 : --
│ │ │ │ -00021780: 2d2d 2d2d 2020 2020 2020 2020 2020 2020  ----            
│ │ │ │ -00021790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000217a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000217b0: 207c 0a7c 2020 2020 2020 2020 3720 2020   |.|        7   
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│ │ │ │ +00021730: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00021750: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00021760: 2020 2020 2020 3720 2020 2020 2020 2020        7         
│ │ │ │ +00021770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021790: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000217a0: 6820 2020 2020 2020 2020 2020 2020 2020  h               
│ │ │ │ +000217b0: 2020 2020 2020 2020 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 7c0a 7c20              |.| 
│ │ │ │ -000217f0: 2020 2020 2020 6820 2020 2020 2020 2020        h         
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│ │ │ │ -00021820: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00021830: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │ -00021840: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00021970: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00021980: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ +000219d0: 2054 6865 0a20 2020 2043 6865 726e 2d53   The.    Chern-S
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│ │ │ │ +00021b40: 2f4d 322f 4d61 6361 756c 6179 322f 7061  /M2/Macaulay2/pa
│ │ │ │ +00021b50: 636b 6167 6573 2f0a 4368 6172 6163 7465  ckages/.Characte
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│ │ │ │ +00021bd0: 642c 2055 703a 2054 6f70 0a0a 4d75 6c74  d, Up: Top..Mult
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│ │ │ │ +00021c40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00021c50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00021c60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00021c70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00021c80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00021c90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20  *************.. 
│ │ │ │ +00021ca0: 202a 2055 7361 6765 3a20 0a20 2020 2020   * Usage: .     
│ │ │ │ +00021cb0: 2020 204d 756c 7469 5072 6f6a 436f 6f72     MultiProjCoor
│ │ │ │ +00021cc0: 6452 696e 6720 4469 6d73 0a20 2020 2020  dRing Dims.     
│ │ │ │ +00021cd0: 2020 204d 756c 7469 5072 6f6a 436f 6f72     MultiProjCoor
│ │ │ │ +00021ce0: 6452 696e 6720 2843 6f65 6666 5269 6e67  dRing (CoeffRing
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│ │ │ │ +00021d10: 6720 2876 6172 2c44 696d 7329 0a20 2020  g (var,Dims).   
│ │ │ │ +00021d20: 2020 2020 204d 756c 7469 5072 6f6a 436f       MultiProjCo
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│ │ │ │ +00021d50: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ +00021d60: 2044 696d 732c 2061 202a 6e6f 7465 206c   Dims, a *note l
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│ │ │ │ +00021e10: 2020 2020 2a20 436f 6566 6652 696e 672c      * CoeffRing,
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│ │ │ │ +00021e60: 2020 2020 7468 6520 6772 6164 6564 2070      the graded p
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│ │ │ │ +00021ec0: 2020 2020 202a 2076 6172 2c20 6120 2a6e       * var, a *n
│ │ │ │ +00021ed0: 6f74 6520 7379 6d62 6f6c 3a20 284d 6163  ote symbol: (Mac
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│ │ │ │ +00021f70: 2072 696e 673a 2028 4d61 6361 756c 6179   ring: (Macaulay
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│ │ │ │ +00022020: 3d3d 3d3d 3d3d 0a0a 436f 6d70 7574 6573  ======..Computes
│ │ │ │ +00022030: 2074 6865 2067 7261 6465 6420 636f 6f72   the graded coor
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│ │ │ │ +00022060: 2e2e 2078 205c 5050 5e7b 6e5f 6d7d 2077  .. x \PP^{n_m} w
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│ │ │ │ +00022090: 6c69 7374 206f 6620 6469 6d65 6e73 696f  list of dimensio
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│ │ │ │ +00022130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022160: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2053  -------+.|i1 : S
│ │ │ │ +00022170: 3d4d 756c 7469 5072 6f6a 436f 6f72 6452  =MultiProjCoordR
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│ │ │ │ +000221a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000221b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000221c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000221d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000221e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000221f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022200: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022200: 2020 2020 2020 207c 0a7c 6f31 203d 2053         |.|o1 = S
│ │ │ │  00022210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022230: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00022250: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022260: 6f31 203d 2053 2020 2020 2020 2020 2020  o1 = S          
│ │ │ │ +00022250: 2020 2020 2020 207c 0a7c 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 2020 2020 2020                  
│ │ │ │  00022290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000222a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000222b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000222a0: 2020 2020 2020 207c 0a7c 6f31 203a 2050         |.|o1 : P
│ │ │ │ +000222b0: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020  olynomialRing   
│ │ │ │  000222c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000222d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000222e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000222f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022300: 6f31 203a 2050 6f6c 796e 6f6d 6961 6c52  o1 : PolynomialR
│ │ │ │ -00022310: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ -00022320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022340: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00022350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -000223a0: 6932 203a 2064 6567 7265 6573 2053 2020  i2 : degrees S  
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│ │ │ │ +00022310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022340: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2064  -------+.|i2 : d
│ │ │ │ +00022350: 6567 7265 6573 2053 2020 2020 2020 2020  egrees S        
│ │ │ │ +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                  
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│ │ │ │  000223c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000223d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000223e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000223f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022430: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022440: 6f32 203d 207b 7b31 2c20 302c 2030 7d2c  o2 = {{1, 0, 0},
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│ │ │ │ -00022480: 2031 2c20 307d 2c20 7b30 2c20 207c 0a7c   1, 0}, {0,  |.|
│ │ │ │ -00022490: 2020 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d       -----------
│ │ │ │ -000224a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000224b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00022500: 2c20 7b30 2c20 302c 2031 7d7d 2020 2020  , {0, 0, 1}}    
│ │ │ │ +000223e0: 2020 2020 2020 207c 0a7c 6f32 203d 207b         |.|o2 = {
│ │ │ │ +000223f0: 7b31 2c20 302c 2030 7d2c 207b 312c 2030  {1, 0, 0}, {1, 0
│ │ │ │ +00022400: 2c20 307d 2c20 7b30 2c20 312c 2030 7d2c  , 0}, {0, 1, 0},
│ │ │ │ +00022410: 207b 302c 2031 2c20 307d 2c20 7b30 2c20   {0, 1, 0}, {0, 
│ │ │ │ +00022420: 312c 2030 7d2c 207b 302c 2031 2c20 307d  1, 0}, {0, 1, 0}
│ │ │ │ +00022430: 2c20 7b30 2c20 207c 0a7c 2020 2020 202d  , {0,  |.|     -
│ │ │ │ +00022440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022480: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2030  -------|.|     0
│ │ │ │ +00022490: 2c20 317d 2c20 7b30 2c20 302c 2031 7d2c  , 1}, {0, 0, 1},
│ │ │ │ +000224a0: 207b 302c 2030 2c20 317d 2c20 7b30 2c20   {0, 0, 1}, {0, 
│ │ │ │ +000224b0: 302c 2031 7d7d 2020 2020 2020 2020 2020  0, 1}}          
│ │ │ │ +000224c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000224d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000224e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000224f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022520: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022520: 2020 2020 2020 207c 0a7c 6f32 203a 204c         |.|o2 : L
│ │ │ │ +00022530: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │  00022540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022570: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022580: 6f32 203a 204c 6973 7420 2020 2020 2020  o2 : List       
│ │ │ │ -00022590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000225a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000225b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000225c0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -000225d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000225e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000225f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00022620: 6933 203a 2052 3d4d 756c 7469 5072 6f6a  i3 : R=MultiProj
│ │ │ │ -00022630: 436f 6f72 6452 696e 6720 7b32 2c33 7d20  CoordRing {2,3} 
│ │ │ │ +00022570: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00022580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000225a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000225b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000225c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2052  -------+.|i3 : R
│ │ │ │ +000225d0: 3d4d 756c 7469 5072 6f6a 436f 6f72 6452  =MultiProjCoordR
│ │ │ │ +000225e0: 696e 6720 7b32 2c33 7d20 2020 2020 2020  ing {2,3}       
│ │ │ │ +000225f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022610: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00022620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022660: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022660: 2020 2020 2020 207c 0a7c 6f33 203d 2052         |.|o3 = R
│ │ │ │  00022670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000226a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000226b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000226c0: 6f33 203d 2052 2020 2020 2020 2020 2020  o3 = R          
│ │ │ │ +000226b0: 2020 2020 2020 207c 0a7c 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 2020 2020 2020                  
│ │ │ │  000226f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022700: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022700: 2020 2020 2020 207c 0a7c 6f33 203a 2050         |.|o3 : P
│ │ │ │ +00022710: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020  olynomialRing   
│ │ │ │  00022720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022750: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022760: 6f33 203a 2050 6f6c 796e 6f6d 6961 6c52  o3 : PolynomialR
│ │ │ │ -00022770: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ -00022780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000227a0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -000227b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000227c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000227d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000227e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000227f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00022800: 6934 203a 2063 6f65 6666 6963 6965 6e74  i4 : coefficient
│ │ │ │ -00022810: 5269 6e67 2052 2020 2020 2020 2020 2020  Ring R          
│ │ │ │ +00022750: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00022760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000227a0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2063  -------+.|i4 : c
│ │ │ │ +000227b0: 6f65 6666 6963 6965 6e74 5269 6e67 2052  oefficientRing R
│ │ │ │ +000227c0: 2020 2020 2020 2020 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 207c 0a7c 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 2020 2020 2020 2020                  
│ │ │ │ -00022840: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022840: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00022850: 205a 5a20 2020 2020 2020 2020 2020 2020   ZZ             
│ │ │ │  00022860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022890: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000228a0: 2020 2020 2020 205a 5a20 2020 2020 2020         ZZ       
│ │ │ │ +00022890: 2020 2020 2020 207c 0a7c 6f34 203d 202d         |.|o4 = -
│ │ │ │ +000228a0: 2d2d 2d2d 2020 2020 2020 2020 2020 2020  ----            
│ │ │ │  000228b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000228c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000228d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000228e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000228f0: 6f34 203d 202d 2d2d 2d2d 2020 2020 2020  o4 = -----      
│ │ │ │ +000228e0: 2020 2020 2020 207c 0a7c 2020 2020 2033         |.|     3
│ │ │ │ +000228f0: 3237 3439 2020 2020 2020 2020 2020 2020  2749            
│ │ │ │  00022900: 2020 2020 2020 2020 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 207c 0a7c               |.|
│ │ │ │ -00022940: 2020 2020 2033 3237 3439 2020 2020 2020       32749      
│ │ │ │ +00022930: 2020 2020 2020 207c 0a7c 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 2020 2020 2020                  
│ │ │ │  00022970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022980: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022980: 2020 2020 2020 207c 0a7c 6f34 203a 2051         |.|o4 : Q
│ │ │ │ +00022990: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │  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 207c 0a7c               |.|
│ │ │ │ -000229e0: 6f34 203a 2051 756f 7469 656e 7452 696e  o4 : QuotientRin
│ │ │ │ -000229f0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ -00022a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022a20: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00022a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00022a80: 6935 203a 2064 6573 6372 6962 6520 5220  i5 : describe R 
│ │ │ │ +000229d0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000229e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000229f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022a20: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2064  -------+.|i5 : d
│ │ │ │ +00022a30: 6573 6372 6962 6520 5220 2020 2020 2020  escribe R       
│ │ │ │ +00022a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022a70: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00022a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022a90: 2020 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 207c 0a7c               |.|
│ │ │ │ -00022ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022ac0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00022ad0: 205a 5a20 2020 2020 2020 2020 2020 2020   ZZ             
│ │ │ │  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 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022b20: 2020 2020 2020 205a 5a20 2020 2020 2020         ZZ       
│ │ │ │ -00022b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022b60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022b70: 6f35 203d 202d 2d2d 2d2d 5b78 202e 2e78  o5 = -----[x ..x
│ │ │ │ -00022b80: 202c 2044 6567 7265 6573 203d 3e20 7b33   , Degrees => {3
│ │ │ │ -00022b90: 3a7b 317d 2c20 343a 7b30 7d7d 2c20 4865  :{1}, 4:{0}}, He
│ │ │ │ -00022ba0: 6674 203d 3e20 7b32 3a31 7d5d 2020 2020  ft => {2:1}]    
│ │ │ │ -00022bb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022bc0: 2020 2020 2033 3237 3439 2020 3020 2020       32749  0   
│ │ │ │ -00022bd0: 3620 2020 2020 2020 2020 2020 2020 2020  6               
│ │ │ │ -00022be0: 207b 307d 2020 2020 7b31 7d20 2020 2020   {0}    {1}     
│ │ │ │ -00022bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022c00: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00022c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00022c60: 6936 203a 2041 3d43 686f 7752 696e 6720  i6 : A=ChowRing 
│ │ │ │ -00022c70: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │ +00022b10: 2020 2020 2020 207c 0a7c 6f35 203d 202d         |.|o5 = -
│ │ │ │ +00022b20: 2d2d 2d2d 5b78 202e 2e78 202c 2044 6567  ----[x ..x , Deg
│ │ │ │ +00022b30: 7265 6573 203d 3e20 7b33 3a7b 317d 2c20  rees => {3:{1}, 
│ │ │ │ +00022b40: 343a 7b30 7d7d 2c20 4865 6674 203d 3e20  4:{0}}, Heft => 
│ │ │ │ +00022b50: 7b32 3a31 7d5d 2020 2020 2020 2020 2020  {2:1}]          
│ │ │ │ +00022b60: 2020 2020 2020 207c 0a7c 2020 2020 2033         |.|     3
│ │ │ │ +00022b70: 3237 3439 2020 3020 2020 3620 2020 2020  2749  0   6     
│ │ │ │ +00022b80: 2020 2020 2020 2020 2020 207b 307d 2020             {0}  
│ │ │ │ +00022b90: 2020 7b31 7d20 2020 2020 2020 2020 2020    {1}           
│ │ │ │ +00022ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022bb0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00022bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022c00: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2041  -------+.|i6 : A
│ │ │ │ +00022c10: 3d43 686f 7752 696e 6720 5220 2020 2020  =ChowRing R     
│ │ │ │ +00022c20: 2020 2020 2020 2020 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 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00022c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ca0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022ca0: 2020 2020 2020 207c 0a7c 6f36 203d 2041         |.|o6 = A
│ │ │ │  00022cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022cf0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022d00: 6f36 203d 2041 2020 2020 2020 2020 2020  o6 = A          
│ │ │ │ +00022cf0: 2020 2020 2020 207c 0a7c 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 2020 2020 2020                  
│ │ │ │  00022d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022d40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022d40: 2020 2020 2020 207c 0a7c 6f36 203a 2051         |.|o6 : Q
│ │ │ │ +00022d50: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │  00022d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022d90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022da0: 6f36 203a 2051 756f 7469 656e 7452 696e  o6 : QuotientRin
│ │ │ │ -00022db0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ -00022dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022de0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00022df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00022e40: 6937 203a 2064 6573 6372 6962 6520 4120  i7 : describe A 
│ │ │ │ +00022d90: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00022da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00022e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00022e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022e80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022e80: 2020 2020 2020 207c 0a7c 2020 2020 205a         |.|     Z
│ │ │ │ +00022e90: 5a5b 6820 2e2e 6820 5d20 2020 2020 2020  Z[h ..h ]       
│ │ │ │  00022ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ed0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022ee0: 2020 2020 205a 5a5b 6820 2e2e 6820 5d20       ZZ[h ..h ] 
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│ │ │ │  00022ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022f30: 2020 2020 2020 2020 2031 2020 2032 2020           1   2  
│ │ │ │ +00022f20: 2020 2020 2020 207c 0a7c 6f37 203d 202d         |.|o7 = -
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│ │ │ │  00022f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022f80: 6f37 203d 202d 2d2d 2d2d 2d2d 2d2d 2d20  o7 = ---------- 
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│ │ │ │ +00022f80: 2020 3320 2020 3420 2020 2020 2020 2020    3   4         
│ │ │ │  00022f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022fc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00022fd0: 2020 2020 2020 2020 3320 2020 3420 2020          3   4   
│ │ │ │ +00022fc0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00022fd0: 2868 202c 2068 2029 2020 2020 2020 2020  (h , h )        
│ │ │ │  00022fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023010: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00023020: 2020 2020 2020 2868 202c 2068 2029 2020        (h , h )  
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│ │ │ │ +00023020: 2020 3120 2020 3220 2020 2020 2020 2020    1   2         
│ │ │ │  00023030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023060: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00023070: 2020 2020 2020 2020 3120 2020 3220 2020          1   2   
│ │ │ │ -00023080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023090: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000230b0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
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│ │ │ │ -000230e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +000230e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000230f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00023110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023150: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00023160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000231a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000231b0: 2020 2020 2020 2020 3220 3320 2020 2020          2 3     
│ │ │ │ -000231c0: 3220 3220 2020 2020 2020 3320 2020 2020  2 2       3     
│ │ │ │ -000231d0: 3220 2020 2020 2020 2020 3220 2020 2033  2         2    3
│ │ │ │ -000231e0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -000231f0: 2032 2020 2020 2020 2020 2020 207c 0a7c   2           |.|
│ │ │ │ -00023200: 6f38 203d 2031 3068 2068 2020 2d20 3668  o8 = 10h h  - 6h
│ │ │ │ -00023210: 2068 2020 2d20 3468 2068 2020 2b20 3368   h  - 4h h  + 3h
│ │ │ │ -00023220: 2068 2020 2b20 3368 2068 2020 2b20 6820   h  + 3h h  + h 
│ │ │ │ -00023230: 202d 2068 2020 2d20 3268 2068 2020 2d20   - h  - 2h h  - 
│ │ │ │ -00023240: 6820 202b 2068 2020 2b20 6820 207c 0a7c  h  + h  + h  |.|
│ │ │ │ -00023250: 2020 2020 2020 2020 3120 3220 2020 2020          1 2     
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│ │ │ │ -00023270: 3120 3220 2020 2020 3120 3220 2020 2032  1 2     1 2    2
│ │ │ │ -00023280: 2020 2020 3120 2020 2020 3120 3220 2020      1     1 2   
│ │ │ │ -00023290: 2032 2020 2020 3120 2020 2032 207c 0a7c   2    1    2 |.|
│ │ │ │ +00023150: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00023160: 2020 3220 3320 2020 2020 3220 3220 2020    2 3     2 2   
│ │ │ │ +00023170: 2020 2020 3320 2020 2020 3220 2020 2020      3     2     
│ │ │ │ +00023180: 2020 2020 3220 2020 2033 2020 2020 3220      2    3    2 
│ │ │ │ +00023190: 2020 2020 2020 2020 2020 2032 2020 2020             2    
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│ │ │ │ +000231b0: 3068 2068 2020 2d20 3668 2068 2020 2d20  0h h  - 6h h  - 
│ │ │ │ +000231c0: 3468 2068 2020 2b20 3368 2068 2020 2b20  4h h  + 3h h  + 
│ │ │ │ +000231d0: 3368 2068 2020 2b20 6820 202d 2068 2020  3h h  + h  - h  
│ │ │ │ +000231e0: 2d20 3268 2068 2020 2d20 6820 202b 2068  - 2h h  - h  + h
│ │ │ │ +000231f0: 2020 2b20 6820 207c 0a7c 2020 2020 2020    + h  |.|      
│ │ │ │ +00023200: 2020 3120 3220 2020 2020 3120 3220 2020    1 2     1 2   
│ │ │ │ +00023210: 2020 3120 3220 2020 2020 3120 3220 2020    1 2     1 2   
│ │ │ │ +00023220: 2020 3120 3220 2020 2032 2020 2020 3120    1 2    2    1 
│ │ │ │ +00023230: 2020 2020 3120 3220 2020 2032 2020 2020      1 2    2    
│ │ │ │ +00023240: 3120 2020 2032 207c 0a7c 2020 2020 2020  1    2 |.|      
│ │ │ │ +00023250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023260: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00023280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023290: 2020 2020 2020 207c 0a7c 6f38 203a 2041         |.|o8 : A
│ │ │ │  000232a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000232b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000232c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000232d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000232e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000232f0: 6f38 203a 2041 2020 2020 2020 2020 2020  o8 : A          
│ │ │ │ -00023300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023330: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00023340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00023360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00023310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023330: 2d2d 2d2d 2d2d 2d2b 0a0a 5761 7973 2074  -------+..Ways t
│ │ │ │ +00023340: 6f20 7573 6520 4d75 6c74 6950 726f 6a43  o use MultiProjC
│ │ │ │ +00023350: 6f6f 7264 5269 6e67 3a0a 3d3d 3d3d 3d3d  oordRing:.======
│ │ │ │ +00023360: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00023370: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022  =========..  * "
│ │ │ │ +00023380: 4d75 6c74 6950 726f 6a43 6f6f 7264 5269  MultiProjCoordRi
│ │ │ │ +00023390: 6e67 284c 6973 7429 220a 2020 2a20 224d  ng(List)".  * "M
│ │ │ │ +000233a0: 756c 7469 5072 6f6a 436f 6f72 6452 696e  ultiProjCoordRin
│ │ │ │ +000233b0: 6728 5269 6e67 2c4c 6973 7429 220a 2020  g(Ring,List)".  
│ │ │ │ +000233c0: 2a20 224d 756c 7469 5072 6f6a 436f 6f72  * "MultiProjCoor
│ │ │ │ +000233d0: 6452 696e 6728 5269 6e67 2c53 796d 626f  dRing(Ring,Symbo
│ │ │ │ +000233e0: 6c2c 4c69 7374 2922 0a20 202a 2022 4d75  l,List)".  * "Mu
│ │ │ │ +000233f0: 6c74 6950 726f 6a43 6f6f 7264 5269 6e67  ltiProjCoordRing
│ │ │ │ +00023400: 2853 796d 626f 6c2c 4c69 7374 2922 0a0a  (Symbol,List)"..
│ │ │ │ +00023410: 466f 7220 7468 6520 7072 6f67 7261 6d6d  For the programm
│ │ │ │ +00023420: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  er.=============
│ │ │ │ +00023430: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563  =====..The objec
│ │ │ │ +00023440: 7420 2a6e 6f74 6520 4d75 6c74 6950 726f  t *note MultiPro
│ │ │ │ +00023450: 6a43 6f6f 7264 5269 6e67 3a20 4d75 6c74  jCoordRing: Mult
│ │ │ │ +00023460: 6950 726f 6a43 6f6f 7264 5269 6e67 2c20  iProjCoordRing, 
│ │ │ │ +00023470: 6973 2061 202a 6e6f 7465 206d 6574 686f  is a *note metho
│ │ │ │ +00023480: 640a 6675 6e63 7469 6f6e 3a20 284d 6163  d.function: (Mac
│ │ │ │ +00023490: 6175 6c61 7932 446f 6329 4d65 7468 6f64  aulay2Doc)Method
│ │ │ │ +000234a0: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d  Function,...----
│ │ │ │ +000234b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000234c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000234d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000234e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000234f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
│ │ │ │ +00023500: 2073 6f75 7263 6520 6f66 2074 6869 7320   source of this 
│ │ │ │ +00023510: 646f 6375 6d65 6e74 2069 7320 696e 0a2f  document is in./
│ │ │ │ +00023520: 6275 696c 642f 7265 7072 6f64 7563 6962  build/reproducib
│ │ │ │ +00023530: 6c65 2d70 6174 682f 6d61 6361 756c 6179  le-path/macaulay
│ │ │ │ +00023540: 322d 312e 3235 2e31 312b 6473 2f4d 322f  2-1.25.11+ds/M2/
│ │ │ │ +00023550: 4d61 6361 756c 6179 322f 7061 636b 6167  Macaulay2/packag
│ │ │ │ +00023560: 6573 2f0a 4368 6172 6163 7465 7269 7374  es/.Characterist
│ │ │ │ +00023570: 6963 436c 6173 7365 732e 6d32 3a32 3035  icClasses.m2:205
│ │ │ │ +00023580: 303a 302e 0a1f 0a46 696c 653a 2043 6861  0:0....File: Cha
│ │ │ │ +00023590: 7261 6374 6572 6973 7469 6343 6c61 7373  racteristicClass
│ │ │ │ +000235a0: 6573 2e69 6e66 6f2c 204e 6f64 653a 204f  es.info, Node: O
│ │ │ │ +000235b0: 7574 7075 742c 204e 6578 743a 2070 726f  utput, Next: pro
│ │ │ │ +000235c0: 6261 6269 6c69 7374 6963 2061 6c67 6f72  babilistic algor
│ │ │ │ +000235d0: 6974 686d 2c20 5072 6576 3a20 4d75 6c74  ithm, Prev: Mult
│ │ │ │ +000235e0: 6950 726f 6a43 6f6f 7264 5269 6e67 2c20  iProjCoordRing, 
│ │ │ │ +000235f0: 5570 3a20 546f 700a 0a4f 7574 7075 740a  Up: Top..Output.
│ │ │ │ +00023600: 2a2a 2a2a 2a2a 0a0a 4465 7363 7269 7074  ******..Descript
│ │ │ │ +00023610: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ion.===========.
│ │ │ │ +00023620: 0a54 6865 206f 7074 696f 6e20 4f75 7470  .The option Outp
│ │ │ │ +00023630: 7574 2069 7320 6f6e 6c79 2075 7365 6420  ut is only used 
│ │ │ │ +00023640: 6279 2074 6865 2063 6f6d 6d61 6e64 7320  by the commands 
│ │ │ │ +00023650: 2a6e 6f74 6520 4353 4d3a 2043 534d 2c2c  *note CSM: CSM,,
│ │ │ │ +00023660: 202a 6e6f 7465 2053 6567 7265 3a0a 5365   *note Segre:.Se
│ │ │ │ +00023670: 6772 652c 2c20 2a6e 6f74 6520 4368 6572  gre,, *note Cher
│ │ │ │ +00023680: 6e3a 2043 6865 726e 2c20 616e 6420 2a6e  n: Chern, and *n
│ │ │ │ +00023690: 6f74 6520 4575 6c65 723a 2045 756c 6572  ote Euler: Euler
│ │ │ │ +000236a0: 2c20 746f 2073 7065 6369 6679 2074 6865  , to specify the
│ │ │ │ +000236b0: 2074 7970 6520 6f66 0a6f 7574 7075 7420   type of.output 
│ │ │ │ +000236c0: 746f 2062 6520 7265 7475 726e 6564 2074  to be returned t
│ │ │ │ +000236d0: 6f20 7468 6520 7573 6564 2e20 5468 6973  o the used. This
│ │ │ │ +000236e0: 206f 7074 696f 6e20 7769 6c6c 2062 6520   option will be 
│ │ │ │ +000236f0: 6967 6e6f 7265 6420 7768 656e 2075 7365  ignored when use
│ │ │ │ +00023700: 6420 7769 7468 0a2a 6e6f 7465 2043 6f6d  d with.*note Com
│ │ │ │ +00023710: 704d 6574 686f 643a 2043 6f6d 704d 6574  pMethod: CompMet
│ │ │ │ +00023720: 686f 642c 2050 6e52 6573 6964 7561 6c20  hod, PnResidual 
│ │ │ │ +00023730: 6f72 2062 6572 7469 6e69 2e20 5468 6520  or bertini. The 
│ │ │ │ +00023740: 6f70 7469 6f6e 2077 696c 6c20 616c 736f  option will also
│ │ │ │ +00023750: 2062 650a 6967 6e6f 7265 2077 6865 6e20   be.ignore when 
│ │ │ │ +00023760: 2a6e 6f74 6520 4d65 7468 6f64 3a20 4d65  *note Method: Me
│ │ │ │ +00023770: 7468 6f64 2c3d 3e44 6972 6563 7443 6f6d  thod,=>DirectCom
│ │ │ │ +00023780: 706c 6574 6549 6e74 2069 7320 7573 6564  pleteInt is used
│ │ │ │ +00023790: 2e20 5468 6520 6465 6661 756c 740a 6f75  . The default.ou
│ │ │ │ +000237a0: 7470 7574 2066 6f72 2061 6c6c 2074 6865  tput for all the
│ │ │ │ +000237b0: 7365 206d 6574 686f 6473 2069 7320 4368  se methods is Ch
│ │ │ │ +000237c0: 6f77 5269 6e67 456c 656c 6d65 6e74 2077  owRingElelment w
│ │ │ │ +000237d0: 6869 6368 2077 696c 6c20 7265 7475 726e  hich will return
│ │ │ │ +000237e0: 2061 6e20 656c 656d 656e 740a 6f66 2074   an element.of t
│ │ │ │ +000237f0: 6865 2061 7070 726f 7072 6961 7465 2043  he appropriate C
│ │ │ │ +00023800: 686f 7720 7269 6e67 2e20 416c 6c20 6d65  how ring. All me
│ │ │ │ +00023810: 7468 6f64 7320 616c 736f 2068 6176 6520  thods also have 
│ │ │ │ +00023820: 616e 206f 7074 696f 6e20 4861 7368 466f  an option HashFo
│ │ │ │ +00023830: 726d 2077 6869 6368 0a72 6574 7572 6e73  rm which.returns
│ │ │ │ +00023840: 2061 6464 6974 696f 6e61 6c20 696e 666f   additional info
│ │ │ │ +00023850: 726d 6174 696f 6e20 636f 6d70 7574 6564  rmation computed
│ │ │ │ +00023860: 2062 7920 7468 6520 6d65 7468 6f64 7320   by the methods 
│ │ │ │ +00023870: 6475 7269 6e67 2074 6865 6972 2073 7461  during their sta
│ │ │ │ +00023880: 6e64 6172 640a 6f70 6572 6174 696f 6e2e  ndard.operation.
│ │ │ │ +00023890: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │ +000238a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000238b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000238c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000238d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000238e0: 2b0a 7c69 3120 3a20 5220 3d20 5a5a 2f33  +.|i1 : R = ZZ/3
│ │ │ │ +000238f0: 3237 3439 5b78 5f30 2e2e 785f 365d 2020  2749[x_0..x_6]  
│ │ │ │ +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: 7c0a 7c20 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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023980: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00023980: 7c0a 7c6f 3120 3d20 5220 2020 2020 2020  |.|o1 = R       
│ │ │ │  00023990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000239a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000239b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000239c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000239d0: 2020 2020 2020 7c0a 7c6f 3120 3d20 5220        |.|o1 = R 
│ │ │ │ +000239d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000239e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000239f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023a20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00023a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023a20: 7c0a 7c6f 3120 3a20 506f 6c79 6e6f 6d69  |.|o1 : Polynomi
│ │ │ │ +00023a30: 616c 5269 6e67 2020 2020 2020 2020 2020  alRing          
│ │ │ │  00023a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023a70: 2020 2020 2020 7c0a 7c6f 3120 3a20 506f        |.|o1 : Po
│ │ │ │ -00023a80: 6c79 6e6f 6d69 616c 5269 6e67 2020 2020  lynomialRing    
│ │ │ │ -00023a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023ac0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -00023ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023b10: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 413d  ------+.|i2 : A=
│ │ │ │ -00023b20: 4368 6f77 5269 6e67 2852 2920 2020 2020  ChowRing(R)     
│ │ │ │ +00023a70: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00023a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023ac0: 2b0a 7c69 3220 3a20 413d 4368 6f77 5269  +.|i2 : A=ChowRi
│ │ │ │ +00023ad0: 6e67 2852 2920 2020 2020 2020 2020 2020  ng(R)           
│ │ │ │ +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: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00023b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023b60: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00023b60: 7c0a 7c6f 3220 3d20 4120 2020 2020 2020  |.|o2 = A       
│ │ │ │  00023b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023bb0: 2020 2020 2020 7c0a 7c6f 3220 3d20 4120        |.|o2 = A 
│ │ │ │ +00023bb0: 7c0a 7c20 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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023c00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00023c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023c00: 7c0a 7c6f 3220 3a20 5175 6f74 6965 6e74  |.|o2 : Quotient
│ │ │ │ +00023c10: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │  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 7c0a 7c6f 3220 3a20 5175        |.|o2 : Qu
│ │ │ │ -00023c60: 6f74 6965 6e74 5269 6e67 2020 2020 2020  otientRing      
│ │ │ │ -00023c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023ca0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -00023cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023cf0: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 493d  ------+.|i3 : I=
│ │ │ │ -00023d00: 6964 6561 6c28 7261 6e64 6f6d 2832 2c52  ideal(random(2,R
│ │ │ │ -00023d10: 292c 525f 302a 525f 312a 525f 362d 525f  ),R_0*R_1*R_6-R_
│ │ │ │ -00023d20: 305e 3329 3b20 2020 2020 2020 2020 2020  0^3);           
│ │ │ │ +00023c50: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00023c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023ca0: 2b0a 7c69 3320 3a20 493d 6964 6561 6c28  +.|i3 : I=ideal(
│ │ │ │ +00023cb0: 7261 6e64 6f6d 2832 2c52 292c 525f 302a  random(2,R),R_0*
│ │ │ │ +00023cc0: 525f 312a 525f 362d 525f 305e 3329 3b20  R_1*R_6-R_0^3); 
│ │ │ │ +00023cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023cf0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00023d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023d40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00023d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023d40: 7c0a 7c6f 3320 3a20 4964 6561 6c20 6f66  |.|o3 : Ideal of
│ │ │ │ +00023d50: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │  00023d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023d90: 2020 2020 2020 7c0a 7c6f 3320 3a20 4964        |.|o3 : Id
│ │ │ │ -00023da0: 6561 6c20 6f66 2052 2020 2020 2020 2020  eal of R        
│ │ │ │ -00023db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023de0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -00023df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023e30: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 6373  ------+.|i4 : cs
│ │ │ │ -00023e40: 6d3d 4353 4d28 412c 492c 4f75 7470 7574  m=CSM(A,I,Output
│ │ │ │ -00023e50: 3d3e 4861 7368 466f 726d 2920 2020 2020  =>HashForm)     
│ │ │ │ +00023d90: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00023da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023de0: 2b0a 7c69 3420 3a20 6373 6d3d 4353 4d28  +.|i4 : csm=CSM(
│ │ │ │ +00023df0: 412c 492c 4f75 7470 7574 3d3e 4861 7368  A,I,Output=>Hash
│ │ │ │ +00023e00: 466f 726d 2920 2020 2020 2020 2020 2020  Form)           
│ │ │ │ +00023e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023e30: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00023e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023e80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00023e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023e80: 7c0a 7c6f 3420 3d20 4d75 7461 626c 6548  |.|o4 = MutableH
│ │ │ │ +00023e90: 6173 6854 6162 6c65 7b2e 2e2e 342e 2e2e  ashTable{...4...
│ │ │ │ +00023ea0: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │  00023eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023ed0: 2020 2020 2020 7c0a 7c6f 3420 3d20 4d75        |.|o4 = Mu
│ │ │ │ -00023ee0: 7461 626c 6548 6173 6854 6162 6c65 7b2e  tableHashTable{.
│ │ │ │ -00023ef0: 2e2e 342e 2e2e 7d20 2020 2020 2020 2020  ..4...}         
│ │ │ │ +00023ed0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00023ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023f20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00023f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023f20: 7c0a 7c6f 3420 3a20 4d75 7461 626c 6548  |.|o4 : MutableH
│ │ │ │ +00023f30: 6173 6854 6162 6c65 2020 2020 2020 2020  ashTable        
│ │ │ │  00023f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023f70: 2020 2020 2020 7c0a 7c6f 3420 3a20 4d75        |.|o4 : Mu
│ │ │ │ -00023f80: 7461 626c 6548 6173 6854 6162 6c65 2020  tableHashTable  
│ │ │ │ -00023f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023fc0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -00023fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024010: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 7065  ------+.|i5 : pe
│ │ │ │ -00024020: 656b 2063 736d 2020 2020 2020 2020 2020  ek csm          
│ │ │ │ +00023f70: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00023f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023fc0: 2b0a 7c69 3520 3a20 7065 656b 2063 736d  +.|i5 : peek csm
│ │ │ │ +00023fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024010: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00024020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024060: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00024060: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00024070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000240a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000240b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000240c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000240d0: 2020 2020 2020 2020 2020 2036 2020 2020             6    
│ │ │ │ -000240e0: 2020 3520 2020 2020 2034 2020 2020 2020    5      4      
│ │ │ │ -000240f0: 3320 2020 2020 2032 2020 2020 2020 2020  3      2        
│ │ │ │ -00024100: 2020 2020 2020 7c0a 7c6f 3520 3d20 4d75        |.|o5 = Mu
│ │ │ │ -00024110: 7461 626c 6548 6173 6854 6162 6c65 7b7b  tableHashTable{{
│ │ │ │ -00024120: 302c 2031 7d20 3d3e 2032 6820 202b 2032  0, 1} => 2h  + 2
│ │ │ │ -00024130: 3368 2020 2b20 3332 6820 202b 2033 3368  3h  + 32h  + 33h
│ │ │ │ -00024140: 2020 2b20 3138 6820 202b 2035 6820 7d20    + 18h  + 5h } 
│ │ │ │ -00024150: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00024080: 2020 2020 2036 2020 2020 2020 3520 2020       6      5   
│ │ │ │ +00024090: 2020 2034 2020 2020 2020 3320 2020 2020     4      3     
│ │ │ │ +000240a0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +000240b0: 7c0a 7c6f 3520 3d20 4d75 7461 626c 6548  |.|o5 = MutableH
│ │ │ │ +000240c0: 6173 6854 6162 6c65 7b7b 302c 2031 7d20  ashTable{{0, 1} 
│ │ │ │ +000240d0: 3d3e 2032 6820 202b 2032 3368 2020 2b20  => 2h  + 23h  + 
│ │ │ │ +000240e0: 3332 6820 202b 2033 3368 2020 2b20 3138  32h  + 33h  + 18
│ │ │ │ +000240f0: 6820 202b 2035 6820 7d20 2020 2020 2020  h  + 5h }       
│ │ │ │ +00024100: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00024110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024120: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ +00024130: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ +00024140: 2031 2020 2020 2031 2020 2020 2020 2020   1     1        
│ │ │ │ +00024150: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00024160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024170: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ -00024180: 2020 3120 2020 2020 2031 2020 2020 2020    1      1      
│ │ │ │ -00024190: 3120 2020 2020 2031 2020 2020 2031 2020  1      1     1  
│ │ │ │ -000241a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000241b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000241c0: 2020 2020 2020 2020 2036 2020 2020 2020           6      
│ │ │ │ -000241d0: 3520 2020 2020 2034 2020 2020 2020 3320  5      4      3 
│ │ │ │ -000241e0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -000241f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024200: 2020 2020 2020 2020 2020 2020 2020 2043                 C
│ │ │ │ -00024210: 534d 203d 3e20 3130 6820 202b 2031 3268  SM => 10h  + 12h
│ │ │ │ -00024220: 2020 2b20 3232 6820 202b 2031 3668 2020    + 22h  + 16h  
│ │ │ │ -00024230: 2b20 3668 2020 2020 2020 2020 2020 2020  + 6h            
│ │ │ │ -00024240: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00024170: 2020 2036 2020 2020 2020 3520 2020 2020     6      5     
│ │ │ │ +00024180: 2034 2020 2020 2020 3320 2020 2020 3220   4      3     2 
│ │ │ │ +00024190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000241a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000241b0: 2020 2020 2020 2020 2043 534d 203d 3e20           CSM => 
│ │ │ │ +000241c0: 3130 6820 202b 2031 3268 2020 2b20 3232  10h  + 12h  + 22
│ │ │ │ +000241d0: 6820 202b 2031 3668 2020 2b20 3668 2020  h  + 16h  + 6h  
│ │ │ │ +000241e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000241f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00024200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024210: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ +00024220: 2031 2020 2020 2020 3120 2020 2020 3120   1      1     1 
│ │ │ │ +00024230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024240: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00024250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024260: 2020 2020 2020 2020 2031 2020 2020 2020           1      
│ │ │ │ -00024270: 3120 2020 2020 2031 2020 2020 2020 3120  1      1      1 
│ │ │ │ -00024280: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │ -00024290: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000242a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000242b0: 2020 2020 2020 2020 3620 2020 2020 2035          6      5
│ │ │ │ -000242c0: 2020 2020 2020 3420 2020 2020 2033 2020        4      3  
│ │ │ │ -000242d0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -000242e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000242f0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00024300: 307d 203d 3e20 3668 2020 2b20 3138 6820  0} => 6h  + 18h 
│ │ │ │ -00024310: 202b 2032 3668 2020 2b20 3232 6820 202b   + 26h  + 22h  +
│ │ │ │ -00024320: 2031 3068 2020 2b20 3268 2020 2020 2020   10h  + 2h      
│ │ │ │ -00024330: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00024260: 2020 3620 2020 2020 2035 2020 2020 2020    6      5      
│ │ │ │ +00024270: 3420 2020 2020 2033 2020 2020 2020 3220  4      3      2 
│ │ │ │ +00024280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024290: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000242a0: 2020 2020 2020 2020 207b 307d 203d 3e20           {0} => 
│ │ │ │ +000242b0: 3668 2020 2b20 3138 6820 202b 2032 3668  6h  + 18h  + 26h
│ │ │ │ +000242c0: 2020 2b20 3232 6820 202b 2031 3068 2020    + 22h  + 10h  
│ │ │ │ +000242d0: 2b20 3268 2020 2020 2020 2020 2020 2020  + 2h            
│ │ │ │ +000242e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000242f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024300: 2020 3120 2020 2020 2031 2020 2020 2020    1      1      
│ │ │ │ +00024310: 3120 2020 2020 2031 2020 2020 2020 3120  1      1      1 
│ │ │ │ +00024320: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │ +00024330: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00024340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024350: 2020 2020 2020 2020 3120 2020 2020 2031          1      1
│ │ │ │ -00024360: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ -00024370: 2020 2020 3120 2020 2020 3120 2020 2020      1     1     
│ │ │ │ -00024380: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000243a0: 2020 2020 2020 2020 3620 2020 2020 2035          6      5
│ │ │ │ -000243b0: 2020 2020 2020 3420 2020 2020 2033 2020        4      3  
│ │ │ │ -000243c0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -000243d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000243e0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -000243f0: 317d 203d 3e20 3668 2020 2b20 3137 6820  1} => 6h  + 17h 
│ │ │ │ -00024400: 202b 2032 3868 2020 2b20 3237 6820 202b   + 28h  + 27h  +
│ │ │ │ -00024410: 2031 3468 2020 2b20 3368 2020 2020 2020   14h  + 3h      
│ │ │ │ -00024420: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024440: 2020 2020 2020 2020 3120 2020 2020 2031          1      1
│ │ │ │ -00024450: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ -00024460: 2020 2020 3120 2020 2020 3120 2020 2020      1     1     
│ │ │ │ -00024470: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
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│ │ │ │ -00024490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000244a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000244b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000244c0: 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 4353  ------+.|i6 : CS
│ │ │ │ -000244d0: 4d28 412c 6964 6561 6c20 495f 3029 3d3d  M(A,ideal I_0)==
│ │ │ │ -000244e0: 6373 6d23 7b30 7d20 2020 2020 2020 2020  csm#{0}         
│ │ │ │ +00024350: 2020 3620 2020 2020 2035 2020 2020 2020    6      5      
│ │ │ │ +00024360: 3420 2020 2020 2033 2020 2020 2020 3220  4      3      2 
│ │ │ │ +00024370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024380: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00024390: 2020 2020 2020 2020 207b 317d 203d 3e20           {1} => 
│ │ │ │ +000243a0: 3668 2020 2b20 3137 6820 202b 2032 3868  6h  + 17h  + 28h
│ │ │ │ +000243b0: 2020 2b20 3237 6820 202b 2031 3468 2020    + 27h  + 14h  
│ │ │ │ +000243c0: 2b20 3368 2020 2020 2020 2020 2020 2020  + 3h            
│ │ │ │ +000243d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000243e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000243f0: 2020 3120 2020 2020 2031 2020 2020 2020    1      1      
│ │ │ │ +00024400: 3120 2020 2020 2031 2020 2020 2020 3120  1      1      1 
│ │ │ │ +00024410: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │ +00024420: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00024430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024470: 2b0a 7c69 3620 3a20 4353 4d28 412c 6964  +.|i6 : CSM(A,id
│ │ │ │ +00024480: 6561 6c20 495f 3029 3d3d 6373 6d23 7b30  eal I_0)==csm#{0
│ │ │ │ +00024490: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +000244a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000244b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000244c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000244d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000244e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000244f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024510: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00024510: 7c0a 7c6f 3620 3d20 7472 7565 2020 2020  |.|o6 = true    
│ │ │ │  00024520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024560: 2020 2020 2020 7c0a 7c6f 3620 3d20 7472        |.|o6 = tr
│ │ │ │ -00024570: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ -00024580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000245a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000245b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -000245c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000245d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000245e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000245f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024600: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 4353  ------+.|i7 : CS
│ │ │ │ -00024610: 4d28 412c 6964 6561 6c28 495f 302a 495f  M(A,ideal(I_0*I_
│ │ │ │ -00024620: 3129 293d 3d63 736d 237b 302c 317d 2020  1))==csm#{0,1}  
│ │ │ │ +00024560: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00024570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000245a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000245b0: 2b0a 7c69 3720 3a20 4353 4d28 412c 6964  +.|i7 : CSM(A,id
│ │ │ │ +000245c0: 6561 6c28 495f 302a 495f 3129 293d 3d63  eal(I_0*I_1))==c
│ │ │ │ +000245d0: 736d 237b 302c 317d 2020 2020 2020 2020  sm#{0,1}        
│ │ │ │ +000245e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000245f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024600: 7c0a 7c20 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 2020 2020                  
│ │ │ │ -00024650: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00024650: 7c0a 7c6f 3720 3d20 7472 7565 2020 2020  |.|o7 = true    
│ │ │ │  00024660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000246a0: 2020 2020 2020 7c0a 7c6f 3720 3d20 7472        |.|o7 = tr
│ │ │ │ -000246b0: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ -000246c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000246d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000246e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000246f0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -00024700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024740: 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 633d  ------+.|i8 : c=
│ │ │ │ -00024750: 4368 6572 6e28 2049 2c20 4f75 7470 7574  Chern( I, Output
│ │ │ │ -00024760: 3d3e 4861 7368 466f 726d 2920 2020 2020  =>HashForm)     
│ │ │ │ +000246a0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +000246b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000246c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000246d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000246e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000246f0: 2b0a 7c69 3820 3a20 633d 4368 6572 6e28  +.|i8 : c=Chern(
│ │ │ │ +00024700: 2049 2c20 4f75 7470 7574 3d3e 4861 7368   I, Output=>Hash
│ │ │ │ +00024710: 466f 726d 2920 2020 2020 2020 2020 2020  Form)           
│ │ │ │ +00024720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024740: 7c0a 7c20 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                  
│ │ │ │  00024780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024790: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000247a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000247b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024790: 7c0a 7c6f 3820 3d20 4d75 7461 626c 6548  |.|o8 = MutableH
│ │ │ │ +000247a0: 6173 6854 6162 6c65 7b2e 2e2e 362e 2e2e  ashTable{...6...
│ │ │ │ +000247b0: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │  000247c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000247d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000247e0: 2020 2020 2020 7c0a 7c6f 3820 3d20 4d75        |.|o8 = Mu
│ │ │ │ -000247f0: 7461 626c 6548 6173 6854 6162 6c65 7b2e  tableHashTable{.
│ │ │ │ -00024800: 2e2e 362e 2e2e 7d20 2020 2020 2020 2020  ..6...}         
│ │ │ │ +000247e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000247f0: 2020 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: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024830: 7c0a 7c6f 3820 3a20 4d75 7461 626c 6548  |.|o8 : MutableH
│ │ │ │ +00024840: 6173 6854 6162 6c65 2020 2020 2020 2020  ashTable        
│ │ │ │  00024850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024880: 2020 2020 2020 7c0a 7c6f 3820 3a20 4d75        |.|o8 : Mu
│ │ │ │ -00024890: 7461 626c 6548 6173 6854 6162 6c65 2020  tableHashTable  
│ │ │ │ -000248a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000248b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000248c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000248f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024920: 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20 7065  ------+.|i9 : pe
│ │ │ │ -00024930: 656b 2063 2020 2020 2020 2020 2020 2020  ek c            
│ │ │ │ +00024880: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00024890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000248a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000248b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000248c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000248d0: 2b0a 7c69 3920 3a20 7065 656b 2063 2020  +.|i9 : peek c  
│ │ │ │ +000248e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000248f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024920: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00024930: 2020 2020 2020 2020 2020 2020 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 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00024970: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00024980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000249a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000249b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000249c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000249d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000249e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000249f0: 2020 2020 2032 2020 2020 2020 3320 2020       2      3   
│ │ │ │ -00024a00: 2020 2034 2020 2020 2020 2035 2020 2020     4       5    
│ │ │ │ -00024a10: 2020 2036 2020 7c0a 7c6f 3920 3d20 4d75     6  |.|o9 = Mu
│ │ │ │ -00024a20: 7461 626c 6548 6173 6854 6162 6c65 7b53  tableHashTable{S
│ │ │ │ -00024a30: 6567 7265 4c69 7374 203d 3e20 7b30 2c20  egreList => {0, 
│ │ │ │ -00024a40: 302c 2036 6820 2c20 2d33 3068 202c 2031  0, 6h , -30h , 1
│ │ │ │ -00024a50: 3134 6820 2c20 2d33 3930 6820 2c20 3132  14h , -390h , 12
│ │ │ │ -00024a60: 3636 6820 7d7d 7c0a 7c20 2020 2020 2020  66h }}|.|       
│ │ │ │ +00024990: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +000249a0: 2020 2020 2020 3320 2020 2020 2034 2020        3      4  
│ │ │ │ +000249b0: 2020 2020 2035 2020 2020 2020 2036 2020       5       6  
│ │ │ │ +000249c0: 7c0a 7c6f 3920 3d20 4d75 7461 626c 6548  |.|o9 = MutableH
│ │ │ │ +000249d0: 6173 6854 6162 6c65 7b53 6567 7265 4c69  ashTable{SegreLi
│ │ │ │ +000249e0: 7374 203d 3e20 7b30 2c20 302c 2036 6820  st => {0, 0, 6h 
│ │ │ │ +000249f0: 2c20 2d33 3068 202c 2031 3134 6820 2c20  , -30h , 114h , 
│ │ │ │ +00024a00: 2d33 3930 6820 2c20 3132 3636 6820 7d7d  -390h , 1266h }}
│ │ │ │ +00024a10: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00024a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024a30: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +00024a40: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ +00024a50: 2020 2020 2031 2020 2020 2020 2031 2020       1       1  
│ │ │ │ +00024a60: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00024a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024a90: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ -00024aa0: 2020 2031 2020 2020 2020 2031 2020 2020     1       1    
│ │ │ │ -00024ab0: 2020 2031 2020 7c0a 7c20 2020 2020 2020     1  |.|       
│ │ │ │ -00024ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024ae0: 2020 2032 2020 2020 3320 2020 2034 2020     2    3    4  
│ │ │ │ -00024af0: 2020 3520 2020 2036 2020 2020 2020 2020    5    6        
│ │ │ │ -00024b00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024b10: 2020 2020 2020 2020 2020 2020 2020 2047                 G
│ │ │ │ -00024b20: 6c69 7374 203d 3e20 7b31 2c20 3368 202c  list => {1, 3h ,
│ │ │ │ -00024b30: 2033 6820 2c20 3368 202c 2033 6820 2c20   3h , 3h , 3h , 
│ │ │ │ -00024b40: 3368 202c 2033 6820 7d20 2020 2020 2020  3h , 3h }       
│ │ │ │ -00024b50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00024a80: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +00024a90: 2020 3320 2020 2034 2020 2020 3520 2020    3    4    5   
│ │ │ │ +00024aa0: 2036 2020 2020 2020 2020 2020 2020 2020   6              
│ │ │ │ +00024ab0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00024ac0: 2020 2020 2020 2020 2047 6c69 7374 203d           Glist =
│ │ │ │ +00024ad0: 3e20 7b31 2c20 3368 202c 2033 6820 2c20  > {1, 3h , 3h , 
│ │ │ │ +00024ae0: 3368 202c 2033 6820 2c20 3368 202c 2033  3h , 3h , 3h , 3
│ │ │ │ +00024af0: 6820 7d20 2020 2020 2020 2020 2020 2020  h }             
│ │ │ │ +00024b00: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00024b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024b20: 2020 2020 2020 2020 3120 2020 2031 2020          1    1  
│ │ │ │ +00024b30: 2020 3120 2020 2031 2020 2020 3120 2020    1    1    1   
│ │ │ │ +00024b40: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ +00024b50: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00024b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024b70: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ -00024b80: 2020 2031 2020 2020 3120 2020 2031 2020     1    1    1  
│ │ │ │ -00024b90: 2020 3120 2020 2031 2020 2020 2020 2020    1    1        
│ │ │ │ -00024ba0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024bc0: 2020 2020 2020 2020 2020 2020 2036 2020               6  
│ │ │ │ -00024bd0: 2020 2020 2035 2020 2020 2020 2034 2020       5       4  
│ │ │ │ -00024be0: 2020 2020 3320 2020 2020 3220 2020 2020      3     2     
│ │ │ │ -00024bf0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024c00: 2020 2020 2020 2020 2020 2020 2020 2053                 S
│ │ │ │ -00024c10: 6567 7265 203d 3e20 3132 3636 6820 202d  egre => 1266h  -
│ │ │ │ -00024c20: 2033 3930 6820 202b 2031 3134 6820 202d   390h  + 114h  -
│ │ │ │ -00024c30: 2033 3068 2020 2b20 3668 2020 2020 2020   30h  + 6h      
│ │ │ │ -00024c40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00024b70: 2020 2020 2020 2036 2020 2020 2020 2035         6       5
│ │ │ │ +00024b80: 2020 2020 2020 2034 2020 2020 2020 3320         4      3 
│ │ │ │ +00024b90: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00024ba0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00024bb0: 2020 2020 2020 2020 2053 6567 7265 203d           Segre =
│ │ │ │ +00024bc0: 3e20 3132 3636 6820 202d 2033 3930 6820  > 1266h  - 390h 
│ │ │ │ +00024bd0: 202b 2031 3134 6820 202d 2033 3068 2020   + 114h  - 30h  
│ │ │ │ +00024be0: 2b20 3668 2020 2020 2020 2020 2020 2020  + 6h            
│ │ │ │ +00024bf0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00024c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024c10: 2020 2020 2020 2031 2020 2020 2020 2031         1       1
│ │ │ │ +00024c20: 2020 2020 2020 2031 2020 2020 2020 3120         1      1 
│ │ │ │ +00024c30: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │ +00024c40: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00024c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024c60: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ -00024c70: 2020 2020 2031 2020 2020 2020 2031 2020       1       1  
│ │ │ │ -00024c80: 2020 2020 3120 2020 2020 3120 2020 2020      1     1     
│ │ │ │ -00024c90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024cb0: 2020 2020 2020 2020 2020 2036 2020 2020             6    
│ │ │ │ -00024cc0: 2020 3520 2020 2020 2034 2020 2020 2020    5      4      
│ │ │ │ -00024cd0: 3320 2020 2020 3220 2020 2020 2020 2020  3     2         
│ │ │ │ -00024ce0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024cf0: 2020 2020 2020 2020 2020 2020 2020 2043                 C
│ │ │ │ -00024d00: 6865 726e 203d 3e20 3930 6820 202d 2031  hern => 90h  - 1
│ │ │ │ -00024d10: 3268 2020 2b20 3330 6820 202b 2031 3268  2h  + 30h  + 12h
│ │ │ │ -00024d20: 2020 2b20 3668 2020 2020 2020 2020 2020    + 6h          
│ │ │ │ -00024d30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00024c60: 2020 2020 2036 2020 2020 2020 3520 2020       6      5   
│ │ │ │ +00024c70: 2020 2034 2020 2020 2020 3320 2020 2020     4      3     
│ │ │ │ +00024c80: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00024c90: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00024ca0: 2020 2020 2020 2020 2043 6865 726e 203d           Chern =
│ │ │ │ +00024cb0: 3e20 3930 6820 202d 2031 3268 2020 2b20  > 90h  - 12h  + 
│ │ │ │ +00024cc0: 3330 6820 202b 2031 3268 2020 2b20 3668  30h  + 12h  + 6h
│ │ │ │ +00024cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024ce0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00024cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024d00: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ +00024d10: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ +00024d20: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +00024d30: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00024d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024d50: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ -00024d60: 2020 3120 2020 2020 2031 2020 2020 2020    1      1      
│ │ │ │ -00024d70: 3120 2020 2020 3120 2020 2020 2020 2020  1     1         
│ │ │ │ -00024d80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024da0: 2020 2020 2020 2020 3620 2020 2020 2035          6      5
│ │ │ │ -00024db0: 2020 2020 2020 3420 2020 2020 2033 2020        4      3  
│ │ │ │ -00024dc0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -00024dd0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024de0: 2020 2020 2020 2020 2020 2020 2020 2043                 C
│ │ │ │ -00024df0: 4620 3d3e 2039 3068 2020 2d20 3132 6820  F => 90h  - 12h 
│ │ │ │ -00024e00: 202b 2033 3068 2020 2b20 3132 6820 202b   + 30h  + 12h  +
│ │ │ │ -00024e10: 2036 6820 2020 2020 2020 2020 2020 2020   6h             
│ │ │ │ -00024e20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00024d50: 2020 3620 2020 2020 2035 2020 2020 2020    6      5      
│ │ │ │ +00024d60: 3420 2020 2020 2033 2020 2020 2032 2020  4      3     2  
│ │ │ │ +00024d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024d80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00024d90: 2020 2020 2020 2020 2043 4620 3d3e 2039           CF => 9
│ │ │ │ +00024da0: 3068 2020 2d20 3132 6820 202b 2033 3068  0h  - 12h  + 30h
│ │ │ │ +00024db0: 2020 2b20 3132 6820 202b 2036 6820 2020    + 12h  + 6h   
│ │ │ │ +00024dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024dd0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00024de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024df0: 2020 3120 2020 2020 2031 2020 2020 2020    1      1      
│ │ │ │ +00024e00: 3120 2020 2020 2031 2020 2020 2031 2020  1      1     1  
│ │ │ │ +00024e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024e20: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00024e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024e40: 2020 2020 2020 2020 3120 2020 2020 2031          1      1
│ │ │ │ -00024e50: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ -00024e60: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │ -00024e70: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024e90: 2020 2020 2020 3620 2020 2020 3520 2020        6     5   
│ │ │ │ -00024ea0: 2020 3420 2020 2020 3320 2020 2020 3220    4     3     2 
│ │ │ │ -00024eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024ec0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024ed0: 2020 2020 2020 2020 2020 2020 2020 2047                 G
│ │ │ │ -00024ee0: 203d 3e20 3368 2020 2b20 3368 2020 2b20   => 3h  + 3h  + 
│ │ │ │ -00024ef0: 3368 2020 2b20 3368 2020 2b20 3368 2020  3h  + 3h  + 3h  
│ │ │ │ -00024f00: 2b20 3368 2020 2b20 3120 2020 2020 2020  + 3h  + 1       
│ │ │ │ -00024f10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024f30: 2020 2020 2020 3120 2020 2020 3120 2020        1     1   
│ │ │ │ -00024f40: 2020 3120 2020 2020 3120 2020 2020 3120    1     1     1 
│ │ │ │ -00024f50: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │ -00024f60: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -00024f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024fb0: 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a 2073  ------+.|i10 : s
│ │ │ │ -00024fc0: 6567 3d53 6567 7265 2820 492c 204f 7574  eg=Segre( I, Out
│ │ │ │ -00024fd0: 7075 743d 3e48 6173 6846 6f72 6d29 2020  put=>HashForm)  
│ │ │ │ +00024e40: 3620 2020 2020 3520 2020 2020 3420 2020  6     5     4   
│ │ │ │ +00024e50: 2020 3320 2020 2020 3220 2020 2020 2020    3     2       
│ │ │ │ +00024e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024e70: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00024e80: 2020 2020 2020 2020 2047 203d 3e20 3368           G => 3h
│ │ │ │ +00024e90: 2020 2b20 3368 2020 2b20 3368 2020 2b20    + 3h  + 3h  + 
│ │ │ │ +00024ea0: 3368 2020 2b20 3368 2020 2b20 3368 2020  3h  + 3h  + 3h  
│ │ │ │ +00024eb0: 2b20 3120 2020 2020 2020 2020 2020 2020  + 1             
│ │ │ │ +00024ec0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00024ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024ee0: 3120 2020 2020 3120 2020 2020 3120 2020  1     1     1   
│ │ │ │ +00024ef0: 2020 3120 2020 2020 3120 2020 2020 3120    1     1     1 
│ │ │ │ +00024f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024f10: 7c0a 2b2d 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024f60: 2b0a 7c69 3130 203a 2073 6567 3d53 6567  +.|i10 : seg=Seg
│ │ │ │ +00024f70: 7265 2820 492c 204f 7574 7075 743d 3e48  re( I, Output=>H
│ │ │ │ +00024f80: 6173 6846 6f72 6d29 2020 2020 2020 2020  ashForm)        
│ │ │ │ +00024f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024fb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00024fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025000: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00025010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025000: 7c0a 7c6f 3130 203d 204d 7574 6162 6c65  |.|o10 = Mutable
│ │ │ │ +00025010: 4861 7368 5461 626c 657b 2e2e 2e34 2e2e  HashTable{...4..
│ │ │ │ +00025020: 2e7d 2020 2020 2020 2020 2020 2020 2020  .}              
│ │ │ │  00025030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025050: 2020 2020 2020 7c0a 7c6f 3130 203d 204d        |.|o10 = M
│ │ │ │ -00025060: 7574 6162 6c65 4861 7368 5461 626c 657b  utableHashTable{
│ │ │ │ -00025070: 2e2e 2e34 2e2e 2e7d 2020 2020 2020 2020  ...4...}        
│ │ │ │ +00025050: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00025060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000250a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000250b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000250a0: 7c0a 7c6f 3130 203a 204d 7574 6162 6c65  |.|o10 : Mutable
│ │ │ │ +000250b0: 4861 7368 5461 626c 6520 2020 2020 2020  HashTable       
│ │ │ │  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 7c0a 7c6f 3130 203a 204d        |.|o10 : M
│ │ │ │ -00025100: 7574 6162 6c65 4861 7368 5461 626c 6520  utableHashTable 
│ │ │ │ -00025110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025140: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -00025150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025190: 2d2d 2d2d 2d2d 2b0a 7c69 3131 203a 2070  ------+.|i11 : p
│ │ │ │ -000251a0: 6565 6b20 7365 6720 2020 2020 2020 2020  eek seg         
│ │ │ │ +000250f0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00025100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025140: 2b0a 7c69 3131 203a 2070 6565 6b20 7365  +.|i11 : peek se
│ │ │ │ +00025150: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ +00025160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025190: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000251a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000251b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000251c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000251d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000251e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000251e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000251f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025230: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00025240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025260: 2020 2020 2020 3220 2020 2020 2033 2020        2      3  
│ │ │ │ -00025270: 2020 2020 3420 2020 2020 2020 3520 2020      4       5   
│ │ │ │ -00025280: 2020 2020 2020 7c0a 7c6f 3131 203d 204d        |.|o11 = M
│ │ │ │ -00025290: 7574 6162 6c65 4861 7368 5461 626c 657b  utableHashTable{
│ │ │ │ -000252a0: 5365 6772 654c 6973 7420 3d3e 207b 302c  SegreList => {0,
│ │ │ │ -000252b0: 2030 2c20 3668 202c 202d 3330 6820 2c20   0, 6h , -30h , 
│ │ │ │ -000252c0: 3131 3468 202c 202d 3339 3068 202c 2020  114h , -390h ,  
│ │ │ │ -000252d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00025210: 3220 2020 2020 2033 2020 2020 2020 3420  2      3      4 
│ │ │ │ +00025220: 2020 2020 2020 3520 2020 2020 2020 2020        5         
│ │ │ │ +00025230: 7c0a 7c6f 3131 203d 204d 7574 6162 6c65  |.|o11 = Mutable
│ │ │ │ +00025240: 4861 7368 5461 626c 657b 5365 6772 654c  HashTable{SegreL
│ │ │ │ +00025250: 6973 7420 3d3e 207b 302c 2030 2c20 3668  ist => {0, 0, 6h
│ │ │ │ +00025260: 202c 202d 3330 6820 2c20 3131 3468 202c   , -30h , 114h ,
│ │ │ │ +00025270: 202d 3339 3068 202c 2020 2020 2020 2020   -390h ,        
│ │ │ │ +00025280: 7c0a 7c20 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: 3120 2020 2020 2031 2020 2020 2020 3120  1      1      1 
│ │ │ │ +000252c0: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │ +000252d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000252e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000252f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025300: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ -00025310: 2020 2020 3120 2020 2020 2020 3120 2020      1       1   
│ │ │ │ -00025320: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00025330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025350: 2020 2020 3220 2020 2033 2020 2020 3420      2    3    4 
│ │ │ │ -00025360: 2020 2035 2020 2020 3620 2020 2020 2020     5    6       
│ │ │ │ -00025370: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000252f0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +00025300: 2020 2033 2020 2020 3420 2020 2035 2020     3    4    5  
│ │ │ │ +00025310: 2020 3620 2020 2020 2020 2020 2020 2020    6             
│ │ │ │ +00025320: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00025330: 2020 2020 2020 2020 2020 476c 6973 7420            Glist 
│ │ │ │ +00025340: 3d3e 207b 312c 2033 6820 2c20 3368 202c  => {1, 3h , 3h ,
│ │ │ │ +00025350: 2033 6820 2c20 3368 202c 2033 6820 2c20   3h , 3h , 3h , 
│ │ │ │ +00025360: 3368 207d 2020 2020 2020 2020 2020 2020  3h }            
│ │ │ │ +00025370: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00025380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025390: 476c 6973 7420 3d3e 207b 312c 2033 6820  Glist => {1, 3h 
│ │ │ │ -000253a0: 2c20 3368 202c 2033 6820 2c20 3368 202c  , 3h , 3h , 3h ,
│ │ │ │ -000253b0: 2033 6820 2c20 3368 207d 2020 2020 2020   3h , 3h }      
│ │ │ │ -000253c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00025390: 2020 2020 2020 2020 2031 2020 2020 3120           1    1 
│ │ │ │ +000253a0: 2020 2031 2020 2020 3120 2020 2031 2020     1    1    1  
│ │ │ │ +000253b0: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │ +000253c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000253d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000253e0: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ -000253f0: 2020 2020 3120 2020 2031 2020 2020 3120      1    1    1 
│ │ │ │ -00025400: 2020 2031 2020 2020 3120 2020 2020 2020     1    1       
│ │ │ │ -00025410: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00025420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025430: 2020 2020 2020 2020 2020 2020 2020 3620                6 
│ │ │ │ -00025440: 2020 2020 2020 3520 2020 2020 2020 3420        5       4 
│ │ │ │ -00025450: 2020 2020 2033 2020 2020 2032 2020 2020       3     2    
│ │ │ │ -00025460: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000253e0: 2020 2020 2020 2020 3620 2020 2020 2020          6       
│ │ │ │ +000253f0: 3520 2020 2020 2020 3420 2020 2020 2033  5       4      3
│ │ │ │ +00025400: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00025410: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00025420: 2020 2020 2020 2020 2020 5365 6772 6520            Segre 
│ │ │ │ +00025430: 3d3e 2031 3236 3668 2020 2d20 3339 3068  => 1266h  - 390h
│ │ │ │ +00025440: 2020 2b20 3131 3468 2020 2d20 3330 6820    + 114h  - 30h 
│ │ │ │ +00025450: 202b 2036 6820 2020 2020 2020 2020 2020   + 6h           
│ │ │ │ +00025460: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00025470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025480: 5365 6772 6520 3d3e 2031 3236 3668 2020  Segre => 1266h  
│ │ │ │ -00025490: 2d20 3339 3068 2020 2b20 3131 3468 2020  - 390h  + 114h  
│ │ │ │ -000254a0: 2d20 3330 6820 202b 2036 6820 2020 2020  - 30h  + 6h     
│ │ │ │ -000254b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00025480: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ +00025490: 3120 2020 2020 2020 3120 2020 2020 2031  1       1      1
│ │ │ │ +000254a0: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +000254b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000254c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000254d0: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ -000254e0: 2020 2020 2020 3120 2020 2020 2020 3120        1       1 
│ │ │ │ -000254f0: 2020 2020 2031 2020 2020 2031 2020 2020       1     1    
│ │ │ │ -00025500: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00025510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025520: 2020 2020 2020 2036 2020 2020 2035 2020         6     5  
│ │ │ │ -00025530: 2020 2034 2020 2020 2033 2020 2020 2032     4     3     2
│ │ │ │ -00025540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025550: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000254d0: 2036 2020 2020 2035 2020 2020 2034 2020   6     5     4  
│ │ │ │ +000254e0: 2020 2033 2020 2020 2032 2020 2020 2020     3     2      
│ │ │ │ +000254f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025500: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00025510: 2020 2020 2020 2020 2020 4720 3d3e 2033            G => 3
│ │ │ │ +00025520: 6820 202b 2033 6820 202b 2033 6820 202b  h  + 3h  + 3h  +
│ │ │ │ +00025530: 2033 6820 202b 2033 6820 202b 2033 6820   3h  + 3h  + 3h 
│ │ │ │ +00025540: 202b 2031 2020 2020 2020 2020 2020 2020   + 1            
│ │ │ │ +00025550: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00025560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025570: 4720 3d3e 2033 6820 202b 2033 6820 202b  G => 3h  + 3h  +
│ │ │ │ -00025580: 2033 6820 202b 2033 6820 202b 2033 6820   3h  + 3h  + 3h 
│ │ │ │ -00025590: 202b 2033 6820 202b 2031 2020 2020 2020   + 3h  + 1      
│ │ │ │ -000255a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000255b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000255c0: 2020 2020 2020 2031 2020 2020 2031 2020         1     1  
│ │ │ │ -000255d0: 2020 2031 2020 2020 2031 2020 2020 2031     1     1     1
│ │ │ │ -000255e0: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ -000255f0: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ -00025600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025640: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 3620  ------|.|     6 
│ │ │ │ +00025570: 2031 2020 2020 2031 2020 2020 2031 2020   1     1     1  
│ │ │ │ +00025580: 2020 2031 2020 2020 2031 2020 2020 2031     1     1     1
│ │ │ │ +00025590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000255a0: 7c0a 7c2d 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000255e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000255f0: 7c0a 7c20 2020 2020 3620 2020 2020 2020  |.|     6       
│ │ │ │ +00025600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025640: 7c0a 7c31 3236 3668 207d 7d20 2020 2020  |.|1266h }}     
│ │ │ │  00025650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025690: 2020 2020 2020 7c0a 7c31 3236 3668 207d        |.|1266h }
│ │ │ │ -000256a0: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +00025690: 7c0a 7c20 2020 2020 3120 2020 2020 2020  |.|     1       
│ │ │ │ +000256a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000256b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000256c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000256d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000256e0: 2020 2020 2020 7c0a 7c20 2020 2020 3120        |.|     1 
│ │ │ │ -000256f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025730: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -00025740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025780: 2d2d 2d2d 2d2d 2b0a 7c69 3132 203a 2065  ------+.|i12 : e
│ │ │ │ -00025790: 753d 4575 6c65 7228 2049 2c20 4f75 7470  u=Euler( I, Outp
│ │ │ │ -000257a0: 7574 3d3e 4861 7368 466f 726d 2920 2020  ut=>HashForm)   
│ │ │ │ +000256e0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +000256f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025730: 2b0a 7c69 3132 203a 2065 753d 4575 6c65  +.|i12 : eu=Eule
│ │ │ │ +00025740: 7228 2049 2c20 4f75 7470 7574 3d3e 4861  r( I, Output=>Ha
│ │ │ │ +00025750: 7368 466f 726d 2920 2020 2020 2020 2020  shForm)         
│ │ │ │ +00025760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025780: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00025790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000257a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000257b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000257c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000257d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000257e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000257f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000257d0: 7c0a 7c6f 3132 203d 204d 7574 6162 6c65  |.|o12 = Mutable
│ │ │ │ +000257e0: 4861 7368 5461 626c 657b 2e2e 2e35 2e2e  HashTable{...5..
│ │ │ │ +000257f0: 2e7d 2020 2020 2020 2020 2020 2020 2020  .}              
│ │ │ │  00025800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025820: 2020 2020 2020 7c0a 7c6f 3132 203d 204d        |.|o12 = M
│ │ │ │ -00025830: 7574 6162 6c65 4861 7368 5461 626c 657b  utableHashTable{
│ │ │ │ -00025840: 2e2e 2e35 2e2e 2e7d 2020 2020 2020 2020  ...5...}        
│ │ │ │ +00025820: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00025830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025870: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00025880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025870: 7c0a 7c6f 3132 203a 204d 7574 6162 6c65  |.|o12 : Mutable
│ │ │ │ +00025880: 4861 7368 5461 626c 6520 2020 2020 2020  HashTable       
│ │ │ │  00025890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000258a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000258b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000258c0: 2020 2020 2020 7c0a 7c6f 3132 203a 204d        |.|o12 : M
│ │ │ │ -000258d0: 7574 6162 6c65 4861 7368 5461 626c 6520  utableHashTable 
│ │ │ │ -000258e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000258f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025910: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -00025920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025960: 2d2d 2d2d 2d2d 2b0a 7c69 3133 203a 2070  ------+.|i13 : p
│ │ │ │ -00025970: 6565 6b20 6575 2020 2020 2020 2020 2020  eek eu          
│ │ │ │ +000258c0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +000258d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000258e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000258f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025910: 2b0a 7c69 3133 203a 2070 6565 6b20 6575  +.|i13 : peek eu
│ │ │ │ +00025920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +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: 7c0a 7c20 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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000259a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000259b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000259c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000259d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000259b0: 7c0a 7c6f 3133 203d 204d 7574 6162 6c65  |.|o13 = Mutable
│ │ │ │ +000259c0: 4861 7368 5461 626c 657b 4575 6c65 7220  HashTable{Euler 
│ │ │ │ +000259d0: 3d3e 2031 3020 2020 2020 2020 2020 2020  => 10           
│ │ │ │  000259e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000259f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025a00: 2020 2020 2020 7c0a 7c6f 3133 203d 204d        |.|o13 = M
│ │ │ │ -00025a10: 7574 6162 6c65 4861 7368 5461 626c 657b  utableHashTable{
│ │ │ │ -00025a20: 4575 6c65 7220 3d3e 2031 3020 2020 2020  Euler => 10     
│ │ │ │ -00025a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025a40: 2020 2020 2020 2020 2020 2020 2020 207d                 }
│ │ │ │ -00025a50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00025a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025a70: 2020 2020 2020 2020 2020 2020 3620 2020              6   
│ │ │ │ -00025a80: 2020 2035 2020 2020 2020 3420 2020 2020     5      4     
│ │ │ │ -00025a90: 2033 2020 2020 2020 3220 2020 2020 2020   3      2       
│ │ │ │ -00025aa0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000259f0: 2020 2020 2020 2020 207d 2020 2020 2020           }      
│ │ │ │ +00025a00: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00025a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025a20: 2020 2020 2020 3620 2020 2020 2035 2020        6      5  
│ │ │ │ +00025a30: 2020 2020 3420 2020 2020 2033 2020 2020      4      3    
│ │ │ │ +00025a40: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00025a50: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00025a60: 2020 2020 2020 2020 2020 7b30 2c20 317d            {0, 1}
│ │ │ │ +00025a70: 203d 3e20 3268 2020 2b20 3233 6820 202b   => 2h  + 23h  +
│ │ │ │ +00025a80: 2033 3268 2020 2b20 3333 6820 202b 2031   32h  + 33h  + 1
│ │ │ │ +00025a90: 3868 2020 2b20 3568 2020 2020 2020 2020  8h  + 5h        
│ │ │ │ +00025aa0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00025ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025ac0: 7b30 2c20 317d 203d 3e20 3268 2020 2b20  {0, 1} => 2h  + 
│ │ │ │ -00025ad0: 3233 6820 202b 2033 3268 2020 2b20 3333  23h  + 32h  + 33
│ │ │ │ -00025ae0: 6820 202b 2031 3868 2020 2b20 3568 2020  h  + 18h  + 5h  
│ │ │ │ -00025af0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00025ac0: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ +00025ad0: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ +00025ae0: 2020 3120 2020 2020 3120 2020 2020 2020    1     1       
│ │ │ │ +00025af0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00025b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025b10: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ -00025b20: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ -00025b30: 2031 2020 2020 2020 3120 2020 2020 3120   1      1     1 
│ │ │ │ -00025b40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00025b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025b60: 2020 2020 2020 2020 2020 3620 2020 2020            6     
│ │ │ │ -00025b70: 2035 2020 2020 2020 3420 2020 2020 2033   5      4      3
│ │ │ │ -00025b80: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -00025b90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00025b10: 2020 2020 3620 2020 2020 2035 2020 2020      6      5    
│ │ │ │ +00025b20: 2020 3420 2020 2020 2033 2020 2020 2032    4      3     2
│ │ │ │ +00025b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025b40: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00025b50: 2020 2020 2020 2020 2020 4353 4d20 3d3e            CSM =>
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│ │ │ │ +00025b70: 3268 2020 2b20 3136 6820 202b 2036 6820  2h  + 16h  + 6h 
│ │ │ │ +00025b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025b90: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00025ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025bb0: 4353 4d20 3d3e 2031 3068 2020 2b20 3132  CSM => 10h  + 12
│ │ │ │ -00025bc0: 6820 202b 2032 3268 2020 2b20 3136 6820  h  + 22h  + 16h 
│ │ │ │ -00025bd0: 202b 2036 6820 2020 2020 2020 2020 2020   + 6h           
│ │ │ │ -00025be0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00025bb0: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ +00025bc0: 2020 3120 2020 2020 2031 2020 2020 2031    1      1     1
│ │ │ │ +00025bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025be0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00025bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025c00: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ -00025c10: 2031 2020 2020 2020 3120 2020 2020 2031   1      1      1
│ │ │ │ -00025c20: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ -00025c30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00025c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025c50: 2020 2020 2020 2020 2036 2020 2020 2020           6      
│ │ │ │ -00025c60: 3520 2020 2020 2034 2020 2020 2020 3320  5      4      3 
│ │ │ │ -00025c70: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -00025c80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00025c00: 2020 2036 2020 2020 2020 3520 2020 2020     6      5     
│ │ │ │ +00025c10: 2034 2020 2020 2020 3320 2020 2020 2032   4      3      2
│ │ │ │ +00025c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025c30: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00025c40: 2020 2020 2020 2020 2020 7b30 7d20 3d3e            {0} =>
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│ │ │ │ +00025c60: 6820 202b 2032 3268 2020 2b20 3130 6820  h  + 22h  + 10h 
│ │ │ │ +00025c70: 202b 2032 6820 2020 2020 2020 2020 2020   + 2h           
│ │ │ │ +00025c80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00025c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025ca0: 7b30 7d20 3d3e 2036 6820 202b 2031 3868  {0} => 6h  + 18h
│ │ │ │ -00025cb0: 2020 2b20 3236 6820 202b 2032 3268 2020    + 26h  + 22h  
│ │ │ │ -00025cc0: 2b20 3130 6820 202b 2032 6820 2020 2020  + 10h  + 2h     
│ │ │ │ -00025cd0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00025ca0: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ +00025cb0: 2031 2020 2020 2020 3120 2020 2020 2031   1      1      1
│ │ │ │ +00025cc0: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +00025cd0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00025ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025cf0: 2020 2020 2020 2020 2031 2020 2020 2020           1      
│ │ │ │ -00025d00: 3120 2020 2020 2031 2020 2020 2020 3120  1      1      1 
│ │ │ │ -00025d10: 2020 2020 2031 2020 2020 2031 2020 2020       1     1    
│ │ │ │ -00025d20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00025d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025d40: 2020 2020 2020 2020 2036 2020 2020 2020           6      
│ │ │ │ -00025d50: 3520 2020 2020 2034 2020 2020 2020 3320  5      4      3 
│ │ │ │ -00025d60: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -00025d70: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00025cf0: 2020 2036 2020 2020 2020 3520 2020 2020     6      5     
│ │ │ │ +00025d00: 2034 2020 2020 2020 3320 2020 2020 2032   4      3      2
│ │ │ │ +00025d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025d20: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00025d30: 2020 2020 2020 2020 2020 7b31 7d20 3d3e            {1} =>
│ │ │ │ +00025d40: 2036 6820 202b 2031 3768 2020 2b20 3238   6h  + 17h  + 28
│ │ │ │ +00025d50: 6820 202b 2032 3768 2020 2b20 3134 6820  h  + 27h  + 14h 
│ │ │ │ +00025d60: 202b 2033 6820 2020 2020 2020 2020 2020   + 3h           
│ │ │ │ +00025d70: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00025d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025d90: 7b31 7d20 3d3e 2036 6820 202b 2031 3768  {1} => 6h  + 17h
│ │ │ │ -00025da0: 2020 2b20 3238 6820 202b 2032 3768 2020    + 28h  + 27h  
│ │ │ │ -00025db0: 2b20 3134 6820 202b 2033 6820 2020 2020  + 14h  + 3h     
│ │ │ │ -00025dc0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00025dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025de0: 2020 2020 2020 2020 2031 2020 2020 2020           1      
│ │ │ │ -00025df0: 3120 2020 2020 2031 2020 2020 2020 3120  1      1      1 
│ │ │ │ -00025e00: 2020 2020 2031 2020 2020 2031 2020 2020       1     1    
│ │ │ │ -00025e10: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -00025e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025e60: 2d2d 2d2d 2d2d 2b0a 0a54 6865 204d 7574  ------+..The Mut
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│ │ │ │ -00025e90: 6f70 7469 6f6e 204f 7574 7075 743d 3e48  option Output=>H
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│ │ │ │ -00025ed0: 206f 6e20 7468 6520 6d65 7468 6f64 2077   on the method w
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│ │ │ │ -00025f10: 202a 6e6f 7465 2049 6e70 7574 4973 536d   *note InputIsSm
│ │ │ │ -00025f20: 6f6f 7468 3a20 496e 7075 7449 7353 6d6f  ooth: InputIsSmo
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│ │ │ │ -00025f40: 6e20 7468 650a 6861 7368 2074 6162 6c65  n the.hash table
│ │ │ │ -00025f50: 2072 6574 7572 6e65 6420 6279 2074 6865   returned by the
│ │ │ │ -00025f60: 206d 6574 686f 6473 2045 756c 6572 2061   methods Euler a
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│ │ │ │ -00025fc0: 2c20 2063 6f6d 6d61 6e64 2069 6e20 7468  ,  command in th
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│ │ │ │ -000262a0: 616e 2069 6465 616c 2049 2065 7175 616c  an ideal I equal
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│ │ │ │ -00026410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026450: 2d2d 2d2d 2b0a 7c69 3134 203a 2063 736d  ----+.|i14 : csm
│ │ │ │ -00026460: 584c 6861 7368 3d43 534d 2841 2c49 2c4f  XLhash=CSM(A,I,O
│ │ │ │ -00026470: 7574 7075 743d 3e48 6173 6846 6f72 6d58  utput=>HashFormX
│ │ │ │ -00026480: 4c29 2020 2020 2020 2020 2020 2020 2020  L)              
│ │ │ │ -00026490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000264a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000264b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000264c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025d90: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ +00025da0: 2031 2020 2020 2020 3120 2020 2020 2031   1      1      1
│ │ │ │ +00025db0: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +00025dc0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00025dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00025df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00025e40: 204f 7574 7075 743d 3e48 6173 6846 6f72   Output=>HashFor
│ │ │ │ +00025e50: 6d20 636f 6e74 6169 6e73 0a64 6966 6665  m contains.diffe
│ │ │ │ +00025e60: 7265 6e74 2069 6e66 6f72 6d61 7469 6f6e  rent information
│ │ │ │ +00025e70: 2064 6570 656e 6469 6e67 206f 6e20 7468   depending on th
│ │ │ │ +00025e80: 6520 6d65 7468 6f64 2077 6974 6820 7768  e method with wh
│ │ │ │ +00025e90: 6963 6820 6974 2069 7320 7573 6564 2e0a  ich it is used..
│ │ │ │ +00025ea0: 4164 6469 7469 6f6e 616c 6c79 2069 6620  Additionally if 
│ │ │ │ +00025eb0: 7468 6520 6f70 7469 6f6e 202a 6e6f 7465  the option *note
│ │ │ │ +00025ec0: 2049 6e70 7574 4973 536d 6f6f 7468 3a20   InputIsSmooth: 
│ │ │ │ +00025ed0: 496e 7075 7449 7353 6d6f 6f74 682c 2069  InputIsSmooth, i
│ │ │ │ +00025ee0: 7320 7573 6564 2074 6865 6e20 7468 650a  s used then the.
│ │ │ │ +00025ef0: 6861 7368 2074 6162 6c65 2072 6574 7572  hash table retur
│ │ │ │ +00025f00: 6e65 6420 6279 2074 6865 206d 6574 686f  ned by the metho
│ │ │ │ +00025f10: 6473 2045 756c 6572 2061 6e64 2043 534d  ds Euler and CSM
│ │ │ │ +00025f20: 2077 696c 6c20 6265 2074 6865 2073 616d   will be the sam
│ │ │ │ +00025f30: 6520 6173 2074 6861 740a 7265 7475 726e  e as that.return
│ │ │ │ +00025f40: 6564 2062 7920 4368 6572 6e2e 2057 6865  ed by Chern. Whe
│ │ │ │ +00025f50: 6e20 7573 696e 6720 7468 6520 2a6e 6f74  n using the *not
│ │ │ │ +00025f60: 6520 4353 4d3a 2043 534d 2c20 2063 6f6d  e CSM: CSM,  com
│ │ │ │ +00025f70: 6d61 6e64 2069 6e20 7468 6520 6465 6661  mand in the defa
│ │ │ │ +00025f80: 756c 740a 636f 6e66 6967 7572 6174 696f  ult.configuratio
│ │ │ │ +00025f90: 6e73 2028 7468 6174 2069 7320 2a6e 6f74  ns (that is *not
│ │ │ │ +00025fa0: 6520 4d65 7468 6f64 3a20 4d65 7468 6f64  e Method: Method
│ │ │ │ +00025fb0: 2c3d 3e49 6e63 6c75 7369 6f6e 4578 636c  ,=>InclusionExcl
│ │ │ │ +00025fc0: 7573 696f 6e2c 202a 6e6f 7465 0a43 6f6d  usion, *note.Com
│ │ │ │ +00025fd0: 704d 6574 686f 643a 2043 6f6d 704d 6574  pMethod: CompMet
│ │ │ │ +00025fe0: 686f 642c 3d3e 5072 6f6a 6563 7469 7665  hod,=>Projective
│ │ │ │ +00025ff0: 4465 6772 6565 2920 7468 6572 6520 6973  Degree) there is
│ │ │ │ +00026000: 2074 6865 2061 6464 6974 696f 6e61 6c20   the additional 
│ │ │ │ +00026010: 6f70 7469 6f6e 2074 6f0a 7365 7420 4f75  option to.set Ou
│ │ │ │ +00026020: 7470 7574 3d3e 4861 7368 466f 726d 584c  tput=>HashFormXL
│ │ │ │ +00026030: 2e20 5468 6973 2072 6574 7572 6e73 2061  . This returns a
│ │ │ │ +00026040: 6c6c 2074 6865 2075 7375 616c 2069 6e66  ll the usual inf
│ │ │ │ +00026050: 6f72 6d61 7469 6f6e 2074 6861 740a 4f75  ormation that.Ou
│ │ │ │ +00026060: 7470 7574 3d3e 4861 7368 466f 726d 2077  tput=>HashForm w
│ │ │ │ +00026070: 6f75 6c64 2066 6f72 2074 6869 7320 636f  ould for this co
│ │ │ │ +00026080: 6e66 6967 7572 6174 696f 6e20 7769 7468  nfiguration with
│ │ │ │ +00026090: 2074 6865 2061 6464 6974 696f 6e20 6f66   the addition of
│ │ │ │ +000260a0: 2074 6865 0a70 726f 6a65 6374 6976 6520   the.projective 
│ │ │ │ +000260b0: 6465 6772 6565 7320 616e 6420 5365 6772  degrees and Segr
│ │ │ │ +000260c0: 6520 636c 6173 7365 7320 6f66 2073 696e  e classes of sin
│ │ │ │ +000260d0: 6775 6c61 7269 7479 2073 7562 7363 6865  gularity subsche
│ │ │ │ +000260e0: 6d65 7320 6765 6e65 7261 7465 6420 6279  mes generated by
│ │ │ │ +000260f0: 2074 6865 0a68 7970 6572 7375 7266 6163   the.hypersurfac
│ │ │ │ +00026100: 6573 2063 6f6e 7369 6465 7265 6420 696e  es considered in
│ │ │ │ +00026110: 2074 6865 2069 6e63 6c75 7369 6f6e 2f65   the inclusion/e
│ │ │ │ +00026120: 7863 6c75 7369 6f6e 2070 726f 6365 6475  xclusion procedu
│ │ │ │ +00026130: 7265 2c20 7468 6174 2069 7320 696e 0a66  re, that is in.f
│ │ │ │ +00026140: 696e 6469 6e67 2074 6865 2043 534d 2063  inding the CSM c
│ │ │ │ +00026150: 6c61 7373 206f 6620 616c 6c20 6879 7065  lass of all hype
│ │ │ │ +00026160: 7273 7572 6661 6365 7320 6765 6e65 7261  rsurfaces genera
│ │ │ │ +00026170: 7465 6420 6279 2074 616b 696e 6720 6120  ted by taking a 
│ │ │ │ +00026180: 7072 6f64 7563 7420 6f66 0a73 6f6d 6520  product of.some 
│ │ │ │ +00026190: 7375 6273 6574 7320 6f66 2067 656e 6572  subsets of gener
│ │ │ │ +000261a0: 6174 6f72 7320 6f66 2074 6865 2069 6e70  ators of the inp
│ │ │ │ +000261b0: 7574 2069 6465 616c 2e20 4e6f 7465 2074  ut ideal. Note t
│ │ │ │ +000261c0: 6861 742c 2073 696e 6365 2074 6865 2043  hat, since the C
│ │ │ │ +000261d0: 534d 2063 6c61 7373 0a6f 6620 6120 7375  SM class.of a su
│ │ │ │ +000261e0: 6273 6368 656d 6520 6571 7561 6c73 2074  bscheme equals t
│ │ │ │ +000261f0: 6865 2043 534d 2063 6c61 7373 206f 6620  he CSM class of 
│ │ │ │ +00026200: 6974 7320 7265 6475 6365 6420 7363 6865  its reduced sche
│ │ │ │ +00026210: 6d65 2c20 6f72 2065 7175 6976 616c 656e  me, or equivalen
│ │ │ │ +00026220: 746c 7920 666f 720a 7573 2074 6865 2043  tly for.us the C
│ │ │ │ +00026230: 534d 2063 6c61 7373 2063 6f72 7265 7370  SM class corresp
│ │ │ │ +00026240: 6f6e 6469 6e67 2074 6f20 616e 2069 6465  onding to an ide
│ │ │ │ +00026250: 616c 2049 2065 7175 616c 7320 7468 6520  al I equals the 
│ │ │ │ +00026260: 4353 4d20 636c 6173 7320 6f66 2074 6865  CSM class of the
│ │ │ │ +00026270: 0a72 6164 6963 616c 206f 6620 492c 2074  .radical of I, t
│ │ │ │ +00026280: 6865 6e20 696e 7465 726e 616c 6c79 2077  hen internally w
│ │ │ │ +00026290: 6520 616c 7761 7973 2077 6f72 6b20 7769  e always work wi
│ │ │ │ +000262a0: 7468 2072 6164 6963 616c 2069 6465 616c  th radical ideal
│ │ │ │ +000262b0: 7320 2866 6f72 0a65 6666 6963 6965 6e63  s (for.efficienc
│ │ │ │ +000262c0: 7920 7265 6173 6f6e 7329 2e20 4865 6e63  y reasons). Henc
│ │ │ │ +000262d0: 6520 7468 6520 7072 6f6a 6563 7469 7665  e the projective
│ │ │ │ +000262e0: 2064 6567 7265 6573 2061 6e64 2053 6567   degrees and Seg
│ │ │ │ +000262f0: 7265 2063 6c61 7373 6573 2063 6f6d 7075  re classes compu
│ │ │ │ +00026300: 7465 640a 696e 7465 726e 616c 6c79 2077  ted.internally w
│ │ │ │ +00026310: 696c 6c20 6265 2074 686f 7365 206f 6620  ill be those of 
│ │ │ │ +00026320: 7468 6520 7261 6469 6361 6c20 6f66 2061  the radical of a
│ │ │ │ +00026330: 6e20 6964 6561 6c20 6465 6669 6e65 6420  n ideal defined 
│ │ │ │ +00026340: 6279 2061 2070 6f6c 796e 6f6d 6961 6c0a  by a polynomial.
│ │ │ │ +00026350: 7768 6963 6820 6973 2061 2070 726f 6475  which is a produ
│ │ │ │ +00026360: 6374 206f 6620 736f 6d65 2073 7562 7365  ct of some subse
│ │ │ │ +00026370: 7420 6f66 2074 6865 2067 656e 6572 6174  t of the generat
│ │ │ │ +00026380: 6f72 732e 2057 6520 696c 6c75 7374 7261  ors. We illustra
│ │ │ │ +00026390: 7465 2074 6869 7320 7769 7468 2061 6e0a  te this with an.
│ │ │ │ +000263a0: 6578 616d 706c 6520 6265 6c6f 772e 0a0a  example below...
│ │ │ │ +000263b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +000263c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000263d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000263e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000263f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00026400: 7c69 3134 203a 2063 736d 584c 6861 7368  |i14 : csmXLhash
│ │ │ │ +00026410: 3d43 534d 2841 2c49 2c4f 7574 7075 743d  =CSM(A,I,Output=
│ │ │ │ +00026420: 3e48 6173 6846 6f72 6d58 4c29 2020 2020  >HashFormXL)    
│ │ │ │ +00026430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026440: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00026450: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00026460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026490: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000264a0: 7c6f 3134 203d 204d 7574 6162 6c65 4861  |o14 = MutableHa
│ │ │ │ +000264b0: 7368 5461 626c 657b 2e2e 2e31 302e 2e2e  shTable{...10...
│ │ │ │ +000264c0: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │  000264d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000264e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000264f0: 2020 2020 7c0a 7c6f 3134 203d 204d 7574      |.|o14 = Mut
│ │ │ │ -00026500: 6162 6c65 4861 7368 5461 626c 657b 2e2e  ableHashTable{..
│ │ │ │ -00026510: 2e31 302e 2e2e 7d20 2020 2020 2020 2020  .10...}         
│ │ │ │ +000264e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000264f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00026500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026540: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00026550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026530: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00026540: 7c6f 3134 203a 204d 7574 6162 6c65 4861  |o14 : MutableHa
│ │ │ │ +00026550: 7368 5461 626c 6520 2020 2020 2020 2020  shTable         
│ │ │ │  00026560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026590: 2020 2020 7c0a 7c6f 3134 203a 204d 7574      |.|o14 : Mut
│ │ │ │ -000265a0: 6162 6c65 4861 7368 5461 626c 6520 2020  ableHashTable   
│ │ │ │ -000265b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000265c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000265d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000265e0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -000265f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026630: 2d2d 2d2d 2b0a 7c69 3135 203a 2070 6565  ----+.|i15 : pee
│ │ │ │ -00026640: 6b20 6373 6d58 4c68 6173 6820 2020 2020  k csmXLhash     
│ │ │ │ +00026580: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00026590: 2b2d 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000265d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000265e0: 7c69 3135 203a 2070 6565 6b20 6373 6d58  |i15 : peek csmX
│ │ │ │ +000265f0: 4c68 6173 6820 2020 2020 2020 2020 2020  Lhash           
│ │ │ │ +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 7c0a                |.
│ │ │ │ +00026630: 7c20 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026680: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00026690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000266a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026670: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00026680: 7c6f 3135 203d 204d 7574 6162 6c65 4861  |o15 = MutableHa
│ │ │ │ +00026690: 7368 5461 626c 657b 4728 4a61 636f 6269  shTable{G(Jacobi
│ │ │ │ +000266a0: 616e 297b 307d 203d 3e20 3020 2020 2020  an){0} => 0     
│ │ │ │  000266b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000266c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000266d0: 2020 2020 7c0a 7c6f 3135 203d 204d 7574      |.|o15 = Mut
│ │ │ │ -000266e0: 6162 6c65 4861 7368 5461 626c 657b 4728  ableHashTable{G(
│ │ │ │ -000266f0: 4a61 636f 6269 616e 297b 307d 203d 3e20  Jacobian){0} => 
│ │ │ │ -00026700: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ -00026710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026720: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00026730: 2020 2020 2020 2020 2020 2020 2020 5365                Se
│ │ │ │ -00026740: 6772 6528 4a61 636f 6269 616e 297b 307d  gre(Jacobian){0}
│ │ │ │ -00026750: 203d 3e20 3020 2020 2020 2020 2020 2020   => 0           
│ │ │ │ -00026760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026770: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00026780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000267a0: 2020 2020 2020 2020 2020 2036 2020 2020             6    
│ │ │ │ -000267b0: 2020 2035 2020 2020 2020 2034 2020 2020     5       4    
│ │ │ │ -000267c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000267d0: 2020 2020 2020 2020 2020 2020 2020 5365                Se
│ │ │ │ -000267e0: 6772 6528 4a61 636f 6269 616e 297b 302c  gre(Jacobian){0,
│ │ │ │ -000267f0: 2031 7d20 3d3e 2033 3930 6820 202d 2033   1} => 390h  - 3
│ │ │ │ -00026800: 3836 6820 202b 2031 3530 6820 202d 2020  86h  + 150h  -  
│ │ │ │ -00026810: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000266c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000266d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000266e0: 2020 2020 2020 2020 5365 6772 6528 4a61          Segre(Ja
│ │ │ │ +000266f0: 636f 6269 616e 297b 307d 203d 3e20 3020  cobian){0} => 0 
│ │ │ │ +00026700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026710: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00026720: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00026730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026750: 2020 2020 2036 2020 2020 2020 2035 2020       6       5  
│ │ │ │ +00026760: 2020 2020 2034 2020 2020 2020 2020 7c0a       4        |.
│ │ │ │ +00026770: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00026780: 2020 2020 2020 2020 5365 6772 6528 4a61          Segre(Ja
│ │ │ │ +00026790: 636f 6269 616e 297b 302c 2031 7d20 3d3e  cobian){0, 1} =>
│ │ │ │ +000267a0: 2033 3930 6820 202d 2033 3836 6820 202b   390h  - 386h  +
│ │ │ │ +000267b0: 2031 3530 6820 202d 2020 2020 2020 7c0a   150h  -      |.
│ │ │ │ +000267c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000267d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000267e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000267f0: 2020 2020 2031 2020 2020 2020 2031 2020       1       1  
│ │ │ │ +00026800: 2020 2020 2031 2020 2020 2020 2020 7c0a       1        |.
│ │ │ │ +00026810: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00026820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026840: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ -00026850: 2020 2031 2020 2020 2020 2031 2020 2020     1       1    
│ │ │ │ -00026860: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00026870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026890: 2020 2020 2020 2020 2020 3620 2020 2020            6     
│ │ │ │ -000268a0: 2035 2020 2020 2033 2020 2020 2032 2020   5     3     2  
│ │ │ │ -000268b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000268c0: 2020 2020 2020 2020 2020 2020 2020 5365                Se
│ │ │ │ -000268d0: 6772 6528 4a61 636f 6269 616e 297b 317d  gre(Jacobian){1}
│ │ │ │ -000268e0: 203d 3e20 2d20 3136 3068 2020 2b20 3332   => - 160h  + 32
│ │ │ │ -000268f0: 6820 202d 2034 6820 202b 2032 6820 2020  h  - 4h  + 2h   
│ │ │ │ -00026900: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00026840: 2020 2020 3620 2020 2020 2035 2020 2020      6      5    
│ │ │ │ +00026850: 2033 2020 2020 2032 2020 2020 2020 7c0a   3     2      |.
│ │ │ │ +00026860: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00026870: 2020 2020 2020 2020 5365 6772 6528 4a61          Segre(Ja
│ │ │ │ +00026880: 636f 6269 616e 297b 317d 203d 3e20 2d20  cobian){1} => - 
│ │ │ │ +00026890: 3136 3068 2020 2b20 3332 6820 202d 2034  160h  + 32h  - 4
│ │ │ │ +000268a0: 6820 202b 2032 6820 2020 2020 2020 7c0a  h  + 2h       |.
│ │ │ │ +000268b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000268c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000268d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000268e0: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ +000268f0: 2031 2020 2020 2031 2020 2020 2020 7c0a   1     1      |.
│ │ │ │ +00026900: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00026910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026930: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ -00026940: 2031 2020 2020 2031 2020 2020 2031 2020   1     1     1  
│ │ │ │ -00026950: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00026960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026980: 2020 2020 2020 3620 2020 2020 2035 2020        6      5  
│ │ │ │ -00026990: 2020 2020 3420 2020 2020 2033 2020 2020      4      3    
│ │ │ │ -000269a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000269b0: 2020 2020 2020 2020 2020 2020 2020 4728                G(
│ │ │ │ -000269c0: 4a61 636f 6269 616e 297b 302c 2031 7d20  Jacobian){0, 1} 
│ │ │ │ -000269d0: 3d3e 2031 3068 2020 2b20 3130 6820 202b  => 10h  + 10h  +
│ │ │ │ -000269e0: 2031 3068 2020 2b20 3130 6820 202b 2020   10h  + 10h  +  
│ │ │ │ -000269f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00026930: 3620 2020 2020 2035 2020 2020 2020 3420  6      5      4 
│ │ │ │ +00026940: 2020 2020 2033 2020 2020 2020 2020 7c0a       3        |.
│ │ │ │ +00026950: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00026960: 2020 2020 2020 2020 4728 4a61 636f 6269          G(Jacobi
│ │ │ │ +00026970: 616e 297b 302c 2031 7d20 3d3e 2031 3068  an){0, 1} => 10h
│ │ │ │ +00026980: 2020 2b20 3130 6820 202b 2031 3068 2020    + 10h  + 10h  
│ │ │ │ +00026990: 2b20 3130 6820 202b 2020 2020 2020 7c0a  + 10h  +      |.
│ │ │ │ +000269a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000269b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000269c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000269d0: 3120 2020 2020 2031 2020 2020 2020 3120  1      1      1 
│ │ │ │ +000269e0: 2020 2020 2031 2020 2020 2020 2020 7c0a       1        |.
│ │ │ │ +000269f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00026a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026a20: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ -00026a30: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ -00026a40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00026a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026a70: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -00026a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026a90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00026aa0: 2020 2020 2020 2020 2020 2020 2020 4728                G(
│ │ │ │ -00026ab0: 4a61 636f 6269 616e 297b 317d 203d 3e20  Jacobian){1} => 
│ │ │ │ -00026ac0: 3268 2020 2b20 3268 2020 2b20 3120 2020  2h  + 2h  + 1   
│ │ │ │ -00026ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026ae0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00026a10: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +00026a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026a30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00026a40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00026a50: 2020 2020 2020 2020 4728 4a61 636f 6269          G(Jacobi
│ │ │ │ +00026a60: 616e 297b 317d 203d 3e20 3268 2020 2b20  an){1} => 2h  + 
│ │ │ │ +00026a70: 3268 2020 2b20 3120 2020 2020 2020 2020  2h  + 1         
│ │ │ │ +00026a80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00026a90: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00026aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026ab0: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ +00026ac0: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │ +00026ad0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00026ae0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00026af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026b10: 2020 3120 2020 2020 3120 2020 2020 2020    1     1       
│ │ │ │ -00026b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026b30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00026b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026b50: 2020 2020 2020 2020 2020 3620 2020 2020            6     
│ │ │ │ -00026b60: 2035 2020 2020 2020 3420 2020 2020 2033   5      4      3
│ │ │ │ -00026b70: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -00026b80: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00026b90: 2020 2020 2020 2020 2020 2020 2020 7b30                {0
│ │ │ │ -00026ba0: 2c20 317d 203d 3e20 3268 2020 2b20 3233  , 1} => 2h  + 23
│ │ │ │ -00026bb0: 6820 202b 2033 3268 2020 2b20 3333 6820  h  + 32h  + 33h 
│ │ │ │ -00026bc0: 202b 2031 3868 2020 2b20 3568 2020 2020   + 18h  + 5h    
│ │ │ │ -00026bd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00026b00: 2020 2020 3620 2020 2020 2035 2020 2020      6      5    
│ │ │ │ +00026b10: 2020 3420 2020 2020 2033 2020 2020 2020    4      3      
│ │ │ │ +00026b20: 3220 2020 2020 2020 2020 2020 2020 7c0a  2             |.
│ │ │ │ +00026b30: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00026b40: 2020 2020 2020 2020 7b30 2c20 317d 203d          {0, 1} =
│ │ │ │ +00026b50: 3e20 3268 2020 2b20 3233 6820 202b 2033  > 2h  + 23h  + 3
│ │ │ │ +00026b60: 3268 2020 2b20 3333 6820 202b 2031 3868  2h  + 33h  + 18h
│ │ │ │ +00026b70: 2020 2b20 3568 2020 2020 2020 2020 7c0a    + 5h        |.
│ │ │ │ +00026b80: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00026b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026ba0: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ +00026bb0: 2020 3120 2020 2020 2031 2020 2020 2020    1      1      
│ │ │ │ +00026bc0: 3120 2020 2020 3120 2020 2020 2020 7c0a  1     1       |.
│ │ │ │ +00026bd0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00026be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026bf0: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ -00026c00: 2031 2020 2020 2020 3120 2020 2020 2031   1      1      1
│ │ │ │ -00026c10: 2020 2020 2020 3120 2020 2020 3120 2020        1     1   
│ │ │ │ -00026c20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00026c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026c40: 2020 2020 2020 2020 3620 2020 2020 2035          6      5
│ │ │ │ -00026c50: 2020 2020 2020 3420 2020 2020 2033 2020        4      3  
│ │ │ │ -00026c60: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -00026c70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00026c80: 2020 2020 2020 2020 2020 2020 2020 4353                CS
│ │ │ │ -00026c90: 4d20 3d3e 2031 3068 2020 2b20 3132 6820  M => 10h  + 12h 
│ │ │ │ -00026ca0: 202b 2032 3268 2020 2b20 3136 6820 202b   + 22h  + 16h  +
│ │ │ │ -00026cb0: 2036 6820 2020 2020 2020 2020 2020 2020   6h             
│ │ │ │ -00026cc0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00026bf0: 2020 3620 2020 2020 2035 2020 2020 2020    6      5      
│ │ │ │ +00026c00: 3420 2020 2020 2033 2020 2020 2032 2020  4      3     2  
│ │ │ │ +00026c10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00026c20: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00026c30: 2020 2020 2020 2020 4353 4d20 3d3e 2031          CSM => 1
│ │ │ │ +00026c40: 3068 2020 2b20 3132 6820 202b 2032 3268  0h  + 12h  + 22h
│ │ │ │ +00026c50: 2020 2b20 3136 6820 202b 2036 6820 2020    + 16h  + 6h   
│ │ │ │ +00026c60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00026c70: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00026c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026c90: 2020 3120 2020 2020 2031 2020 2020 2020    1      1      
│ │ │ │ +00026ca0: 3120 2020 2020 2031 2020 2020 2031 2020  1      1     1  
│ │ │ │ +00026cb0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00026cc0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00026cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026ce0: 2020 2020 2020 2020 3120 2020 2020 2031          1      1
│ │ │ │ -00026cf0: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ -00026d00: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │ -00026d10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00026d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026d30: 2020 2020 2020 2036 2020 2020 2020 3520         6      5 
│ │ │ │ -00026d40: 2020 2020 2034 2020 2020 2020 3320 2020       4      3   
│ │ │ │ -00026d50: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -00026d60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00026d70: 2020 2020 2020 2020 2020 2020 2020 7b30                {0
│ │ │ │ -00026d80: 7d20 3d3e 2036 6820 202b 2031 3868 2020  } => 6h  + 18h  
│ │ │ │ -00026d90: 2b20 3236 6820 202b 2032 3268 2020 2b20  + 26h  + 22h  + 
│ │ │ │ -00026da0: 3130 6820 202b 2032 6820 2020 2020 2020  10h  + 2h       
│ │ │ │ -00026db0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00026ce0: 2036 2020 2020 2020 3520 2020 2020 2034   6      5      4
│ │ │ │ +00026cf0: 2020 2020 2020 3320 2020 2020 2032 2020        3      2  
│ │ │ │ +00026d00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00026d10: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00026d20: 2020 2020 2020 2020 7b30 7d20 3d3e 2036          {0} => 6
│ │ │ │ +00026d30: 6820 202b 2031 3868 2020 2b20 3236 6820  h  + 18h  + 26h 
│ │ │ │ +00026d40: 202b 2032 3268 2020 2b20 3130 6820 202b   + 22h  + 10h  +
│ │ │ │ +00026d50: 2032 6820 2020 2020 2020 2020 2020 7c0a   2h           |.
│ │ │ │ +00026d60: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00026d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026d80: 2031 2020 2020 2020 3120 2020 2020 2031   1      1      1
│ │ │ │ +00026d90: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ +00026da0: 2020 2031 2020 2020 2020 2020 2020 7c0a     1          |.
│ │ │ │ +00026db0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00026dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026dd0: 2020 2020 2020 2031 2020 2020 2020 3120         1      1 
│ │ │ │ -00026de0: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ -00026df0: 2020 2031 2020 2020 2031 2020 2020 2020     1     1      
│ │ │ │ -00026e00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00026e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026e20: 2020 2020 2020 2036 2020 2020 2020 3520         6      5 
│ │ │ │ -00026e30: 2020 2020 2034 2020 2020 2020 3320 2020       4      3   
│ │ │ │ -00026e40: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -00026e50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00026e60: 2020 2020 2020 2020 2020 2020 2020 7b31                {1
│ │ │ │ -00026e70: 7d20 3d3e 2036 6820 202b 2031 3768 2020  } => 6h  + 17h  
│ │ │ │ -00026e80: 2b20 3238 6820 202b 2032 3768 2020 2b20  + 28h  + 27h  + 
│ │ │ │ -00026e90: 3134 6820 202b 2033 6820 2020 2020 2020  14h  + 3h       
│ │ │ │ -00026ea0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00026eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026ec0: 2020 2020 2020 2031 2020 2020 2020 3120         1      1 
│ │ │ │ -00026ed0: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ -00026ee0: 2020 2031 2020 2020 2031 2020 2020 2020     1     1      
│ │ │ │ -00026ef0: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ -00026f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026f40: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020  ----|.|         
│ │ │ │ -00026f50: 2020 2020 7d20 2020 2020 2020 2020 2020      }           
│ │ │ │ +00026dd0: 2036 2020 2020 2020 3520 2020 2020 2034   6      5      4
│ │ │ │ +00026de0: 2020 2020 2020 3320 2020 2020 2032 2020        3      2  
│ │ │ │ +00026df0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00026e00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00026e10: 2020 2020 2020 2020 7b31 7d20 3d3e 2036          {1} => 6
│ │ │ │ +00026e20: 6820 202b 2031 3768 2020 2b20 3238 6820  h  + 17h  + 28h 
│ │ │ │ +00026e30: 202b 2032 3768 2020 2b20 3134 6820 202b   + 27h  + 14h  +
│ │ │ │ +00026e40: 2033 6820 2020 2020 2020 2020 2020 7c0a   3h           |.
│ │ │ │ +00026e50: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00026e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026e70: 2031 2020 2020 2020 3120 2020 2020 2031   1      1      1
│ │ │ │ +00026e80: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ +00026e90: 2020 2031 2020 2020 2020 2020 2020 7c0a     1          |.
│ │ │ │ +00026ea0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00026eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00026ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00026ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00026ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00026ef0: 7c20 2020 2020 2020 2020 2020 2020 7d20  |             } 
│ │ │ │ +00026f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026f30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00026f40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00026f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026f90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00026f80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00026f90: 7c20 2020 3320 2020 2020 3220 2020 2020  |   3     2     
│ │ │ │  00026fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026fe0: 2020 2020 7c0a 7c20 2020 3320 2020 2020      |.|   3     
│ │ │ │ -00026ff0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00026fd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00026fe0: 7c34 3268 2020 2b20 3868 2020 2020 2020  |42h  + 8h      
│ │ │ │ +00026ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027030: 2020 2020 7c0a 7c34 3268 2020 2b20 3868      |.|42h  + 8h
│ │ │ │ +00027020: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00027030: 7c20 2020 3120 2020 2020 3120 2020 2020  |   1     1     
│ │ │ │  00027040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027080: 2020 2020 7c0a 7c20 2020 3120 2020 2020      |.|   1     
│ │ │ │ -00027090: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +00027070: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00027080: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00027090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000270a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000270b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000270d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000270c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000270d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000270e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000270f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027110: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00027120: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00027130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027140: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00027160: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00027170: 7c20 2032 2020 2020 2020 2020 2020 2020  |  2            
│ │ │ │  00027180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000271a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000271b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000271c0: 2020 2020 7c0a 7c20 2032 2020 2020 2020      |.|  2      
│ │ │ │ +000271b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000271c0: 7c38 6820 202b 2034 6820 202b 2031 2020  |8h  + 4h  + 1  
│ │ │ │  000271d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000271e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000271f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027210: 2020 2020 7c0a 7c38 6820 202b 2034 6820      |.|8h  + 4h 
│ │ │ │ -00027220: 202b 2031 2020 2020 2020 2020 2020 2020   + 1            
│ │ │ │ +00027200: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00027210: 7c20 2031 2020 2020 2031 2020 2020 2020  |  1     1      
│ │ │ │ +00027220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027230: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00027250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027260: 2020 2020 7c0a 7c20 2031 2020 2020 2031      |.|  1     1
│ │ │ │ -00027270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000272a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000272b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -000272c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000272d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000272e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000272f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027300: 2d2d 2d2d 2b0a 7c69 3136 203a 204b 3d69  ----+.|i16 : K=i
│ │ │ │ -00027310: 6465 616c 2049 5f30 2a49 5f31 3b20 2020  deal I_0*I_1;   
│ │ │ │ +00027250: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00027260: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00027270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000272a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000272b0: 7c69 3136 203a 204b 3d69 6465 616c 2049  |i16 : K=ideal I
│ │ │ │ +000272c0: 5f30 2a49 5f31 3b20 2020 2020 2020 2020  _0*I_1;         
│ │ │ │ +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 7c0a                |.
│ │ │ │ +00027300: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00027310: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00027330: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00027360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027340: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00027350: 7c6f 3136 203a 2049 6465 616c 206f 6620  |o16 : Ideal of 
│ │ │ │ +00027360: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │  00027370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000273a0: 2020 2020 7c0a 7c6f 3136 203a 2049 6465      |.|o16 : Ide
│ │ │ │ -000273b0: 616c 206f 6620 5220 2020 2020 2020 2020  al of R         
│ │ │ │ -000273c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000273d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000273e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000273f0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00027400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027440: 2d2d 2d2d 2b0a 7c69 3137 203a 2043 534d  ----+.|i17 : CSM
│ │ │ │ -00027450: 2841 2c72 6164 6963 616c 204b 293d 3d43  (A,radical K)==C
│ │ │ │ -00027460: 534d 2841 2c4b 2920 2020 2020 2020 2020  SM(A,K)         
│ │ │ │ +00027390: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000273a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +000273b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000273c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000273d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000273e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000273f0: 7c69 3137 203a 2043 534d 2841 2c72 6164  |i17 : CSM(A,rad
│ │ │ │ +00027400: 6963 616c 204b 293d 3d43 534d 2841 2c4b  ical K)==CSM(A,K
│ │ │ │ +00027410: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +00027420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027430: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00027440: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00027450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027490: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00027480: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00027490: 7c6f 3137 203d 2074 7275 6520 2020 2020  |o17 = true     
│ │ │ │  000274a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000274b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000274c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000274d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000274e0: 2020 2020 7c0a 7c6f 3137 203d 2074 7275      |.|o17 = tru
│ │ │ │ -000274f0: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ -00027500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027530: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00027540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027580: 2d2d 2d2d 2b0a 7c69 3138 203a 204a 3d69  ----+.|i18 : J=i
│ │ │ │ -00027590: 6465 616c 206a 6163 6f62 6961 6e20 7261  deal jacobian ra
│ │ │ │ -000275a0: 6469 6361 6c20 4b3b 2020 2020 2020 2020  dical K;        
│ │ │ │ +000274d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000274e0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +000274f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00027530: 7c69 3138 203a 204a 3d69 6465 616c 206a  |i18 : J=ideal j
│ │ │ │ +00027540: 6163 6f62 6961 6e20 7261 6469 6361 6c20  acobian radical 
│ │ │ │ +00027550: 4b3b 2020 2020 2020 2020 2020 2020 2020  K;              
│ │ │ │ +00027560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027570: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00027580: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00027590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000275a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000275b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000275c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000275d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000275e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000275c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000275d0: 7c6f 3138 203a 2049 6465 616c 206f 6620  |o18 : Ideal of 
│ │ │ │ +000275e0: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │  000275f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027620: 2020 2020 7c0a 7c6f 3138 203a 2049 6465      |.|o18 : Ide
│ │ │ │ -00027630: 616c 206f 6620 5220 2020 2020 2020 2020  al of R         
│ │ │ │ -00027640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027670: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00027680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000276a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000276b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000276c0: 2d2d 2d2d 2b0a 7c69 3139 203a 2073 6567  ----+.|i19 : seg
│ │ │ │ -000276d0: 4a3d 5365 6772 6528 412c 4a2c 4f75 7470  J=Segre(A,J,Outp
│ │ │ │ -000276e0: 7574 3d3e 4861 7368 466f 726d 2920 2020  ut=>HashForm)   
│ │ │ │ +00027610: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00027620: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00027630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00027670: 7c69 3139 203a 2073 6567 4a3d 5365 6772  |i19 : segJ=Segr
│ │ │ │ +00027680: 6528 412c 4a2c 4f75 7470 7574 3d3e 4861  e(A,J,Output=>Ha
│ │ │ │ +00027690: 7368 466f 726d 2920 2020 2020 2020 2020  shForm)         
│ │ │ │ +000276a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000276b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000276c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000276d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000276e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000276f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027710: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00027720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027700: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00027710: 7c6f 3139 203d 204d 7574 6162 6c65 4861  |o19 = MutableHa
│ │ │ │ +00027720: 7368 5461 626c 657b 2e2e 2e34 2e2e 2e7d  shTable{...4...}
│ │ │ │  00027730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027760: 2020 2020 7c0a 7c6f 3139 203d 204d 7574      |.|o19 = Mut
│ │ │ │ -00027770: 6162 6c65 4861 7368 5461 626c 657b 2e2e  ableHashTable{..
│ │ │ │ -00027780: 2e34 2e2e 2e7d 2020 2020 2020 2020 2020  .4...}          
│ │ │ │ +00027750: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00027760: 7c20 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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000277a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000277b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000277c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000277a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000277b0: 7c6f 3139 203a 204d 7574 6162 6c65 4861  |o19 : MutableHa
│ │ │ │ +000277c0: 7368 5461 626c 6520 2020 2020 2020 2020  shTable         
│ │ │ │  000277d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000277e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000277f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027800: 2020 2020 7c0a 7c6f 3139 203a 204d 7574      |.|o19 : Mut
│ │ │ │ -00027810: 6162 6c65 4861 7368 5461 626c 6520 2020  ableHashTable   
│ │ │ │ -00027820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027850: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00027860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000278a0: 2d2d 2d2d 2b0a 7c69 3230 203a 2063 736d  ----+.|i20 : csm
│ │ │ │ -000278b0: 584c 6861 7368 2328 2247 284a 6163 6f62  XLhash#("G(Jacob
│ │ │ │ -000278c0: 6961 6e29 227c 746f 5374 7269 6e67 287b  ian)"|toString({
│ │ │ │ -000278d0: 302c 317d 2929 3d3d 7365 674a 2322 4722  0,1}))==segJ#"G"
│ │ │ │ -000278e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000278f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000277f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00027800: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00027810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00027850: 7c69 3230 203a 2063 736d 584c 6861 7368  |i20 : csmXLhash
│ │ │ │ +00027860: 2328 2247 284a 6163 6f62 6961 6e29 227c  #("G(Jacobian)"|
│ │ │ │ +00027870: 746f 5374 7269 6e67 287b 302c 317d 2929  toString({0,1}))
│ │ │ │ +00027880: 3d3d 7365 674a 2322 4722 2020 2020 2020  ==segJ#"G"      
│ │ │ │ +00027890: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000278a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000278b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000278c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000278d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000278e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000278f0: 7c6f 3230 203d 2074 7275 6520 2020 2020  |o20 = true     
│ │ │ │  00027900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027940: 2020 2020 7c0a 7c6f 3230 203d 2074 7275      |.|o20 = tru
│ │ │ │ -00027950: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ -00027960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027990: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -000279a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000279b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000279c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000279d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000279e0: 2d2d 2d2d 2b0a 7c69 3231 203a 2063 736d  ----+.|i21 : csm
│ │ │ │ -000279f0: 584c 6861 7368 2328 2253 6567 7265 284a  XLhash#("Segre(J
│ │ │ │ -00027a00: 6163 6f62 6961 6e29 227c 746f 5374 7269  acobian)"|toStri
│ │ │ │ -00027a10: 6e67 287b 302c 317d 2929 3d3d 7365 674a  ng({0,1}))==segJ
│ │ │ │ -00027a20: 2322 5365 6772 6522 2020 2020 2020 2020  #"Segre"        
│ │ │ │ -00027a30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00027930: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00027940: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00027950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00027990: 7c69 3231 203a 2063 736d 584c 6861 7368  |i21 : csmXLhash
│ │ │ │ +000279a0: 2328 2253 6567 7265 284a 6163 6f62 6961  #("Segre(Jacobia
│ │ │ │ +000279b0: 6e29 227c 746f 5374 7269 6e67 287b 302c  n)"|toString({0,
│ │ │ │ +000279c0: 317d 2929 3d3d 7365 674a 2322 5365 6772  1}))==segJ#"Segr
│ │ │ │ +000279d0: 6522 2020 2020 2020 2020 2020 2020 7c0a  e"            |.
│ │ │ │ +000279e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000279f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027a20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00027a30: 7c6f 3231 203d 2074 7275 6520 2020 2020  |o21 = true     
│ │ │ │  00027a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027a80: 2020 2020 7c0a 7c6f 3231 203d 2074 7275      |.|o21 = tru
│ │ │ │ -00027a90: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ -00027aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027ad0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00027ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027b20: 2d2d 2d2d 2b0a 0a46 756e 6374 696f 6e73  ----+..Functions
│ │ │ │ -00027b30: 2077 6974 6820 6f70 7469 6f6e 616c 2061   with optional a
│ │ │ │ -00027b40: 7267 756d 656e 7420 6e61 6d65 6420 4f75  rgument named Ou
│ │ │ │ -00027b50: 7470 7574 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d  tput:.==========
│ │ │ │ -00027b60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00027b70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00027b80: 3d3d 3d3d 0a0a 2020 2a20 2243 6865 726e  ====..  * "Chern
│ │ │ │ -00027b90: 282e 2e2e 2c4f 7574 7075 743d 3e2e 2e2e  (...,Output=>...
│ │ │ │ -00027ba0: 2922 202d 2d20 7365 6520 2a6e 6f74 6520  )" -- see *note 
│ │ │ │ -00027bb0: 4368 6572 6e3a 2043 6865 726e 2c20 2d2d  Chern: Chern, --
│ │ │ │ -00027bc0: 2054 6865 2043 6865 726e 2063 6c61 7373   The Chern class
│ │ │ │ -00027bd0: 0a20 202a 2022 4353 4d28 2e2e 2e2c 4f75  .  * "CSM(...,Ou
│ │ │ │ -00027be0: 7470 7574 3d3e 2e2e 2e29 2220 2d2d 2073  tput=>...)" -- s
│ │ │ │ -00027bf0: 6565 202a 6e6f 7465 2043 534d 3a20 4353  ee *note CSM: CS
│ │ │ │ -00027c00: 4d2c 202d 2d20 5468 650a 2020 2020 4368  M, -- The.    Ch
│ │ │ │ -00027c10: 6572 6e2d 5363 6877 6172 747a 2d4d 6163  ern-Schwartz-Mac
│ │ │ │ -00027c20: 5068 6572 736f 6e20 636c 6173 730a 2020  Pherson class.  
│ │ │ │ -00027c30: 2a20 2245 756c 6572 282e 2e2e 2c4f 7574  * "Euler(...,Out
│ │ │ │ -00027c40: 7075 743d 3e2e 2e2e 2922 202d 2d20 7365  put=>...)" -- se
│ │ │ │ -00027c50: 6520 2a6e 6f74 6520 4575 6c65 723a 2045  e *note Euler: E
│ │ │ │ -00027c60: 756c 6572 2c20 2d2d 2054 6865 2045 756c  uler, -- The Eul
│ │ │ │ -00027c70: 6572 0a20 2020 2043 6861 7261 6374 6572  er.    Character
│ │ │ │ -00027c80: 6973 7469 630a 2020 2a20 2253 6567 7265  istic.  * "Segre
│ │ │ │ -00027c90: 282e 2e2e 2c4f 7574 7075 743d 3e2e 2e2e  (...,Output=>...
│ │ │ │ -00027ca0: 2922 202d 2d20 7365 6520 2a6e 6f74 6520  )" -- see *note 
│ │ │ │ -00027cb0: 5365 6772 653a 2053 6567 7265 2c20 2d2d  Segre: Segre, --
│ │ │ │ -00027cc0: 2054 6865 2053 6567 7265 2063 6c61 7373   The Segre class
│ │ │ │ -00027cd0: 206f 6620 610a 2020 2020 7375 6273 6368   of a.    subsch
│ │ │ │ -00027ce0: 656d 650a 0a46 6f72 2074 6865 2070 726f  eme..For the pro
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│ │ │ │ +00027a80: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
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│ │ │ │ +00027b10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00027b20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +00027b30: 2020 2a20 2243 6865 726e 282e 2e2e 2c4f    * "Chern(...,O
│ │ │ │ +00027b40: 7574 7075 743d 3e2e 2e2e 2922 202d 2d20  utput=>...)" -- 
│ │ │ │ +00027b50: 7365 6520 2a6e 6f74 6520 4368 6572 6e3a  see *note Chern:
│ │ │ │ +00027b60: 2043 6865 726e 2c20 2d2d 2054 6865 2043   Chern, -- The C
│ │ │ │ +00027b70: 6865 726e 2063 6c61 7373 0a20 202a 2022  hern class.  * "
│ │ │ │ +00027b80: 4353 4d28 2e2e 2e2c 4f75 7470 7574 3d3e  CSM(...,Output=>
│ │ │ │ +00027b90: 2e2e 2e29 2220 2d2d 2073 6565 202a 6e6f  ...)" -- see *no
│ │ │ │ +00027ba0: 7465 2043 534d 3a20 4353 4d2c 202d 2d20  te CSM: CSM, -- 
│ │ │ │ +00027bb0: 5468 650a 2020 2020 4368 6572 6e2d 5363  The.    Chern-Sc
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│ │ │ │ +00027bf0: 2e2e 2922 202d 2d20 7365 6520 2a6e 6f74  ..)" -- see *not
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│ │ │ │ +00027c20: 2043 6861 7261 6374 6572 6973 7469 630a   Characteristic.
│ │ │ │ +00027c30: 2020 2a20 2253 6567 7265 282e 2e2e 2c4f    * "Segre(...,O
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│ │ │ │ +00027e30: 204f 7574 7075 742c 2055 703a 2054 6f70   Output, Up: Top
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│ │ │ │ +00027e50: 616c 676f 7269 7468 6d0a 2a2a 2a2a 2a2a  algorithm.******
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│ │ │ │ +00027f00: 610a 6765 6e65 7261 6c20 6368 6f69 6365  a.general choice
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│ │ │ │ +00027f70: 6c64 7320 7468 6520 636f 7272 6563 7420  lds the correct 
│ │ │ │ +00027f80: 636c 6173 732c 2069 2e65 2e2c 2074 6865  class, i.e., the
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│ │ │ │ +00027fa0: 7320 6361 6c63 756c 6174 6564 2077 6974  s calculated wit
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│ │ │ │ +00027fc0: 2048 6f77 6576 6572 2c20 7369 6e63 6520   However, since 
│ │ │ │ +00027fd0: 7468 6520 696d 706c 656d 656e 7461 7469  the implementati
│ │ │ │ +00027fe0: 6f6e 2077 6f72 6b73 206f 7665 720a 6120  on works over.a 
│ │ │ │ +00027ff0: 6469 7363 7265 7465 2070 726f 6261 6269  discrete probabi
│ │ │ │ +00028000: 6c69 7479 2073 7061 6365 2074 6865 7265  lity space there
│ │ │ │ +00028010: 2069 7320 6120 7665 7279 2073 6d61 6c6c   is a very small
│ │ │ │ +00028020: 2c20 6275 7420 6e6f 6e2d 7a65 726f 2c20  , but non-zero, 
│ │ │ │ +00028030: 7072 6f62 6162 696c 6974 790a 6f66 206e  probability.of n
│ │ │ │ +00028040: 6f74 2063 6f6d 7075 7469 6e67 2074 6865  ot computing the
│ │ │ │ +00028050: 2063 6f72 7265 6374 2063 6c61 7373 2e20   correct class. 
│ │ │ │ +00028060: 536b 6570 7469 6361 6c20 7573 6572 7320  Skeptical users 
│ │ │ │ +00028070: 7368 6f75 6c64 2072 6570 6561 7420 6361  should repeat ca
│ │ │ │ +00028080: 6c63 756c 6174 696f 6e73 0a73 6576 6572  lculations.sever
│ │ │ │ +00028090: 616c 2074 696d 6573 2074 6f20 696e 6372  al times to incr
│ │ │ │ +000280a0: 6561 7365 2074 6865 2070 726f 6261 6269  ease the probabi
│ │ │ │ +000280b0: 6c69 7479 206f 6620 636f 6d70 7574 696e  lity of computin
│ │ │ │ +000280c0: 6720 7468 6520 636f 7272 6563 7420 636c  g the correct cl
│ │ │ │ +000280d0: 6173 732e 0a0a 496e 2074 6865 2063 6173  ass...In the cas
│ │ │ │ +000280e0: 6520 6f66 2074 6865 2073 796d 626f 6c69  e of the symboli
│ │ │ │ +000280f0: 6320 696d 706c 656d 656e 7461 7469 6f6e  c implementation
│ │ │ │ +00028100: 206f 6620 7468 6520 5072 6f6a 6563 7469   of the Projecti
│ │ │ │ +00028110: 7665 4465 6772 6565 206d 6574 686f 640a  veDegree method.
│ │ │ │ +00028120: 7072 6163 7469 6361 6c20 6578 7065 7269  practical experi
│ │ │ │ +00028130: 656e 6365 2061 6e64 2061 6c67 6f72 6974  ence and algorit
│ │ │ │ +00028140: 686d 2074 6573 7469 6e67 2069 6e64 6963  hm testing indic
│ │ │ │ +00028150: 6174 6520 7468 6174 2061 2066 696e 6974  ate that a finit
│ │ │ │ +00028160: 6520 6669 656c 6420 7769 7468 0a6f 7665  e field with.ove
│ │ │ │ +00028170: 7220 3235 3030 3020 656c 656d 656e 7473  r 25000 elements
│ │ │ │ +00028180: 2069 7320 6d6f 7265 2074 6861 6e20 7375   is more than su
│ │ │ │ +00028190: 6666 6963 6965 6e74 2074 6f20 6578 7065  fficient to expe
│ │ │ │ +000281a0: 6374 2061 2063 6f72 7265 6374 2072 6573  ct a correct res
│ │ │ │ +000281b0: 756c 7420 7769 7468 0a68 6967 6820 7072  ult with.high pr
│ │ │ │ +000281c0: 6f62 6162 696c 6974 792c 2069 2e65 2e20  obability, i.e. 
│ │ │ │ +000281d0: 7573 696e 6720 7468 6520 6669 6e69 7465  using the finite
│ │ │ │ +000281e0: 2066 6965 6c64 206b 6b3d 5a5a 2f32 3530   field kk=ZZ/250
│ │ │ │ +000281f0: 3733 2074 6865 2065 7870 6572 696d 656e  73 the experimen
│ │ │ │ +00028200: 7461 6c0a 6368 616e 6365 206f 6620 6661  tal.chance of fa
│ │ │ │ +00028210: 696c 7572 6520 7769 7468 2074 6865 2050  ilure with the P
│ │ │ │ +00028220: 726f 6a65 6374 6976 6544 6567 7265 6520  rojectiveDegree 
│ │ │ │ +00028230: 616c 676f 7269 7468 6d20 6f6e 2061 2076  algorithm on a v
│ │ │ │ +00028240: 6172 6965 7479 206f 6620 6578 616d 706c  ariety of exampl
│ │ │ │ +00028250: 6573 0a77 6173 206c 6573 7320 7468 616e  es.was less than
│ │ │ │ +00028260: 2031 2f32 3030 302e 2055 7369 6e67 2074   1/2000. Using t
│ │ │ │ +00028270: 6865 2066 696e 6974 6520 6669 656c 6420  he finite field 
│ │ │ │ +00028280: 6b6b 3d5a 5a2f 3332 3734 3920 7265 7375  kk=ZZ/32749 resu
│ │ │ │ +00028290: 6c74 6564 2069 6e20 6e6f 0a66 6169 6c75  lted in no.failu
│ │ │ │ +000282a0: 7265 7320 696e 206f 7665 7220 3130 3030  res in over 1000
│ │ │ │ +000282b0: 3020 6174 7465 6d70 7473 206f 6620 7365  0 attempts of se
│ │ │ │ +000282c0: 7665 7261 6c20 6469 6666 6572 656e 7420  veral different 
│ │ │ │ +000282d0: 6578 616d 706c 6573 2e0a 0a57 6520 696c  examples...We il
│ │ │ │ +000282e0: 6c75 7374 7261 7465 2074 6865 2070 726f  lustrate the pro
│ │ │ │ +000282f0: 6261 6269 6c69 7374 6963 2062 6568 6176  babilistic behav
│ │ │ │ +00028300: 696f 7572 2077 6974 6820 616e 2065 7861  iour with an exa
│ │ │ │ +00028310: 6d70 6c65 2077 6865 7265 2074 6865 2063  mple where the c
│ │ │ │ +00028320: 686f 7365 6e0a 7261 6e64 6f6d 2073 6565  hosen.random see
│ │ │ │ +00028330: 6420 6c65 6164 7320 746f 2061 2077 726f  d leads to a wro
│ │ │ │ +00028340: 6e67 2072 6573 756c 7420 696e 2074 6865  ng result in the
│ │ │ │ +00028350: 2066 6972 7374 2063 616c 6375 6c61 7469   first calculati
│ │ │ │ +00028360: 6f6e 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  on...+----------
│ │ │ │ +00028370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028390: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 7365  ------+.|i1 : se
│ │ │ │ +000283a0: 7452 616e 646f 6d53 6565 6420 3132 313b  tRandomSeed 121;
│ │ │ │ +000283b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000283c0: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ +000283d0: 7365 7474 696e 6720 7261 6e64 6f6d 2073  setting random s
│ │ │ │ +000283e0: 6565 6420 746f 2031 3231 2020 2020 2020  eed to 121      
│ │ │ │ +000283f0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00028400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00028430: 0a7c 6932 203a 2052 203d 2051 515b 782c  .|i2 : R = QQ[x,
│ │ │ │ +00028440: 792c 7a2c 775d 2020 2020 2020 2020 2020  y,z,w]          
│ │ │ │ +00028450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028460: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00028470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028490: 2020 2020 207c 0a7c 6f32 203d 2052 2020       |.|o2 = R  
│ │ │ │  000284a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000284b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000284c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000284b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000284c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  000284d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000284e0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -000284f0: 203d 2052 2020 2020 2020 2020 2020 2020   = R            
│ │ │ │ -00028500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028510: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00028520: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00028530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028550: 207c 0a7c 6f32 203a 2050 6f6c 796e 6f6d   |.|o2 : Polynom
│ │ │ │ -00028560: 6961 6c52 696e 6720 2020 2020 2020 2020  ialRing         
│ │ │ │ -00028570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028580: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00028590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000285a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000285b0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2049  -------+.|i3 : I
│ │ │ │ -000285c0: 203d 206d 696e 6f72 7328 322c 6d61 7472   = minors(2,matr
│ │ │ │ -000285d0: 6978 7b7b 782c 792c 7a7d 2c7b 792c 7a2c  ix{{x,y,z},{y,z,
│ │ │ │ -000285e0: 777d 7d29 2020 2020 2020 7c0a 7c20 2020  w}})      |.|   
│ │ │ │ -000285f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028610: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00028620: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +000284e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000284f0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00028500: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e   : PolynomialRin
│ │ │ │ +00028510: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ +00028520: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00028530: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00028540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028560: 2d2b 0a7c 6933 203a 2049 203d 206d 696e  -+.|i3 : I = min
│ │ │ │ +00028570: 6f72 7328 322c 6d61 7472 6978 7b7b 782c  ors(2,matrix{{x,
│ │ │ │ +00028580: 792c 7a7d 2c7b 792c 7a2c 777d 7d29 2020  y,z},{y,z,w}})  
│ │ │ │ +00028590: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000285a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000285b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000285c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000285d0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +000285e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000285f0: 2020 3220 2020 2020 2020 7c0a 7c6f 3320    2       |.|o3 
│ │ │ │ +00028600: 3d20 6964 6561 6c20 282d 2079 2020 2b20  = ideal (- y  + 
│ │ │ │ +00028610: 782a 7a2c 202d 2079 2a7a 202b 2078 2a77  x*z, - y*z + x*w
│ │ │ │ +00028620: 2c20 2d20 7a20 202b 2079 2a77 297c 0a7c  , - z  + y*w)|.|
│ │ │ │  00028630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028640: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ -00028650: 7c0a 7c6f 3320 3d20 6964 6561 6c20 282d  |.|o3 = ideal (-
│ │ │ │ -00028660: 2079 2020 2b20 782a 7a2c 202d 2079 2a7a   y  + x*z, - y*z
│ │ │ │ -00028670: 202b 2078 2a77 2c20 2d20 7a20 202b 2079   + x*w, - z  + y
│ │ │ │ -00028680: 2a77 297c 0a7c 2020 2020 2020 2020 2020  *w)|.|          
│ │ │ │ -00028690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000286a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000286b0: 2020 2020 2020 7c0a 7c6f 3320 3a20 4964        |.|o3 : Id
│ │ │ │ -000286c0: 6561 6c20 6f66 2052 2020 2020 2020 2020  eal of R        
│ │ │ │ -000286d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000286e0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -000286f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00028720: 3420 3a20 4368 6572 6e20 2849 2c43 6f6d  4 : Chern (I,Com
│ │ │ │ -00028730: 704d 6574 686f 643d 3e50 6e52 6573 6964  pMethod=>PnResid
│ │ │ │ -00028740: 7561 6c29 2020 2020 2020 2020 2020 207c  ual)           |
│ │ │ │ -00028750: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00028760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028660: 7c0a 7c6f 3320 3a20 4964 6561 6c20 6f66  |.|o3 : Ideal of
│ │ │ │ +00028670: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ +00028680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028690: 2020 207c 0a2b 2d2d 2d2d 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 2b0a 7c69 3420 3a20 4368  ------+.|i4 : Ch
│ │ │ │ +000286d0: 6572 6e20 2849 2c43 6f6d 704d 6574 686f  ern (I,CompMetho
│ │ │ │ +000286e0: 643d 3e50 6e52 6573 6964 7561 6c29 2020  d=>PnResidual)  
│ │ │ │ +000286f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00028700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028720: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00028730: 2020 2020 2020 3320 2020 2020 3220 2020        3     2   
│ │ │ │ +00028740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028750: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028760: 0a7c 6f34 203d 2032 4820 202b 2033 4820  .|o4 = 2H  + 3H 
│ │ │ │  00028770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028780: 2020 7c0a 7c20 2020 2020 2020 3320 2020    |.|       3   
│ │ │ │ -00028790: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00028780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028790: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000287a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000287b0: 2020 2020 207c 0a7c 6f34 203d 2032 4820       |.|o4 = 2H 
│ │ │ │ -000287c0: 202b 2033 4820 2020 2020 2020 2020 2020   + 3H           
│ │ │ │ -000287d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000287e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000287f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028810: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00028820: 2020 205a 5a5b 485d 2020 2020 2020 2020     ZZ[H]        
│ │ │ │ -00028830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028840: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00028850: 7c6f 3420 3a20 2d2d 2d2d 2d20 2020 2020  |o4 : -----     
│ │ │ │ -00028860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000287b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000287c0: 2020 2020 207c 0a7c 2020 2020 205a 5a5b       |.|     ZZ[
│ │ │ │ +000287d0: 485d 2020 2020 2020 2020 2020 2020 2020  H]              
│ │ │ │ +000287e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000287f0: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20          |.|o4 : 
│ │ │ │ +00028800: 2d2d 2d2d 2d20 2020 2020 2020 2020 2020  -----           
│ │ │ │ +00028810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028820: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00028830: 2020 2020 2020 3420 2020 2020 2020 2020        4         
│ │ │ │ +00028840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028850: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00028860: 7c20 2020 2020 2020 4820 2020 2020 2020  |       H       
│ │ │ │  00028870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028880: 207c 0a7c 2020 2020 2020 2020 3420 2020   |.|        4   
│ │ │ │ -00028890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000288a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000288b0: 2020 2020 7c0a 7c20 2020 2020 2020 4820      |.|       H 
│ │ │ │ -000288c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000288d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000288e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -000288f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028910: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520  ----------+.|i5 
│ │ │ │ -00028920: 3a20 4368 6572 6e20 2849 2c43 6f6d 704d  : Chern (I,CompM
│ │ │ │ -00028930: 6574 686f 643d 3e50 6e52 6573 6964 7561  ethod=>PnResidua
│ │ │ │ -00028940: 6c29 2020 2020 2020 2020 2020 207c 0a7c  l)           |.|
│ │ │ │ -00028950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028890: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +000288a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000288b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000288c0: 2d2d 2d2d 2b0a 7c69 3520 3a20 4368 6572  ----+.|i5 : Cher
│ │ │ │ +000288d0: 6e20 2849 2c43 6f6d 704d 6574 686f 643d  n (I,CompMethod=
│ │ │ │ +000288e0: 3e50 6e52 6573 6964 7561 6c29 2020 2020  >PnResidual)    
│ │ │ │ +000288f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00028900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028920: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00028930: 2020 2020 3320 2020 2020 3220 2020 2020      3     2     
│ │ │ │ +00028940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028950: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00028960: 6f35 203d 2032 4820 202b 2033 4820 2020  o5 = 2H  + 3H   
│ │ │ │  00028970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028980: 7c0a 7c20 2020 2020 2020 3320 2020 2020  |.|       3     
│ │ │ │ -00028990: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00028980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028990: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000289a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000289b0: 2020 207c 0a7c 6f35 203d 2032 4820 202b     |.|o5 = 2H  +
│ │ │ │ -000289c0: 2033 4820 2020 2020 2020 2020 2020 2020   3H             
│ │ │ │ +000289b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000289c0: 2020 207c 0a7c 2020 2020 205a 5a5b 485d     |.|     ZZ[H]
│ │ │ │  000289d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000289e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000289f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028a10: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00028a20: 205a 5a5b 485d 2020 2020 2020 2020 2020   ZZ[H]          
│ │ │ │ -00028a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028a40: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00028a50: 3520 3a20 2d2d 2d2d 2d20 2020 2020 2020  5 : -----       
│ │ │ │ -00028a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028a70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028a80: 0a7c 2020 2020 2020 2020 3420 2020 2020  .|        4     
│ │ │ │ -00028a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028ab0: 2020 7c0a 7c20 2020 2020 2020 4820 2020    |.|       H   
│ │ │ │ -00028ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028ae0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00028af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028b10: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │ -00028b20: 4368 6572 6e20 2849 2c43 6f6d 704d 6574  Chern (I,CompMet
│ │ │ │ -00028b30: 686f 643d 3e50 6e52 6573 6964 7561 6c29  hod=>PnResidual)
│ │ │ │ -00028b40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00028b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028b70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00028b80: 7c20 2020 2020 2020 3320 2020 2020 3220  |       3     2 
│ │ │ │ -00028b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000289e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000289f0: 2020 2020 2020 7c0a 7c6f 3520 3a20 2d2d        |.|o5 : --
│ │ │ │ +00028a00: 2d2d 2d20 2020 2020 2020 2020 2020 2020  ---             
│ │ │ │ +00028a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028a20: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00028a30: 2020 2020 3420 2020 2020 2020 2020 2020      4           
│ │ │ │ +00028a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028a50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00028a60: 2020 2020 2020 4820 2020 2020 2020 2020        H         
│ │ │ │ +00028a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028a80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028a90: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00028aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028ac0: 2d2d 2b0a 7c69 3620 3a20 4368 6572 6e20  --+.|i6 : Chern 
│ │ │ │ +00028ad0: 2849 2c43 6f6d 704d 6574 686f 643d 3e50  (I,CompMethod=>P
│ │ │ │ +00028ae0: 6e52 6573 6964 7561 6c29 2020 2020 2020  nResidual)      
│ │ │ │ +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 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00028b30: 2020 3320 2020 2020 3220 2020 2020 2020    3     2       
│ │ │ │ +00028b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028b50: 2020 2020 2020 2020 2020 207c 0a7c 6f36             |.|o6
│ │ │ │ +00028b60: 203d 2032 4820 202b 2033 4820 2020 2020   = 2H  + 3H     
│ │ │ │ +00028b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028b80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00028b90: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00028ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028bb0: 207c 0a7c 6f36 203d 2032 4820 202b 2033   |.|o6 = 2H  + 3
│ │ │ │ -00028bc0: 4820 2020 2020 2020 2020 2020 2020 2020  H               
│ │ │ │ +00028bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028bc0: 207c 0a7c 2020 2020 205a 5a5b 485d 2020   |.|     ZZ[H]  
│ │ │ │  00028bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028be0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00028bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028c10: 2020 2020 2020 207c 0a7c 2020 2020 205a         |.|     Z
│ │ │ │ -00028c20: 5a5b 485d 2020 2020 2020 2020 2020 2020  Z[H]            
│ │ │ │ -00028c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028c40: 2020 2020 2020 2020 2020 7c0a 7c6f 3620            |.|o6 
│ │ │ │ -00028c50: 3a20 2d2d 2d2d 2d20 2020 2020 2020 2020  : -----         
│ │ │ │ -00028c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028c70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00028c80: 2020 2020 2020 2020 3420 2020 2020 2020          4       
│ │ │ │ -00028c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028cb0: 7c0a 7c20 2020 2020 2020 4820 2020 2020  |.|       H     
│ │ │ │ -00028cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028ce0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -00028cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028d10: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 4368  ------+.|i7 : Ch
│ │ │ │ -00028d20: 6572 6e28 492c 436f 6d70 4d65 7468 6f64  ern(I,CompMethod
│ │ │ │ -00028d30: 3d3e 5072 6f6a 6563 7469 7665 4465 6772  =>ProjectiveDegr
│ │ │ │ -00028d40: 6565 2920 2020 2020 207c 0a7c 2020 2020  ee)      |.|    
│ │ │ │ -00028d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028d70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00028d80: 2020 2020 2020 3320 2020 2020 3220 2020        3     2   
│ │ │ │ -00028d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028da0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028db0: 0a7c 6f37 203d 2032 6820 202b 2033 6820  .|o7 = 2h  + 3h 
│ │ │ │ -00028dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028bf0: 2020 2020 7c0a 7c6f 3620 3a20 2d2d 2d2d      |.|o6 : ----
│ │ │ │ +00028c00: 2d20 2020 2020 2020 2020 2020 2020 2020  -               
│ │ │ │ +00028c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028c20: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00028c30: 2020 3420 2020 2020 2020 2020 2020 2020    4             
│ │ │ │ +00028c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028c50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00028c60: 2020 2020 4820 2020 2020 2020 2020 2020      H           
│ │ │ │ +00028c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028c80: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +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: 2b0a 7c69 3720 3a20 4368 6572 6e28 492c  +.|i7 : Chern(I,
│ │ │ │ +00028cd0: 436f 6d70 4d65 7468 6f64 3d3e 5072 6f6a  CompMethod=>Proj
│ │ │ │ +00028ce0: 6563 7469 7665 4465 6772 6565 2920 2020  ectiveDegree)   
│ │ │ │ +00028cf0: 2020 207c 0a7c 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 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00028d30: 3320 2020 2020 3220 2020 2020 2020 2020  3     2         
│ │ │ │ +00028d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028d50: 2020 2020 2020 2020 207c 0a7c 6f37 203d           |.|o7 =
│ │ │ │ +00028d60: 2032 6820 202b 2033 6820 2020 2020 2020   2h  + 3h       
│ │ │ │ +00028d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028d80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00028d90: 2020 2020 2020 3120 2020 2020 3120 2020        1     1   
│ │ │ │ +00028da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028db0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028dc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00028dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028de0: 2020 7c0a 7c20 2020 2020 2020 3120 2020    |.|       1   
│ │ │ │ -00028df0: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │ +00028de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028df0: 2020 7c0a 7c20 2020 2020 5a5a 5b68 205d    |.|     ZZ[h ]
│ │ │ │  00028e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028e10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00028e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028e40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00028e50: 5a5a 5b68 205d 2020 2020 2020 2020 2020  ZZ[h ]          
│ │ │ │ -00028e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028e70: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00028e80: 2020 2020 2020 2031 2020 2020 2020 2020         1        
│ │ │ │ -00028e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028ea0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00028eb0: 7c6f 3720 3a20 2d2d 2d2d 2d2d 2020 2020  |o7 : ------    
│ │ │ │ -00028ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028e20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00028e30: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ +00028e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028e50: 2020 2020 2020 2020 7c0a 7c6f 3720 3a20          |.|o7 : 
│ │ │ │ +00028e60: 2d2d 2d2d 2d2d 2020 2020 2020 2020 2020  ------          
│ │ │ │ +00028e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028e80: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00028e90: 2020 2020 2020 3420 2020 2020 2020 2020        4         
│ │ │ │ +00028ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028eb0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00028ec0: 7c20 2020 2020 2020 6820 2020 2020 2020  |       h       
│ │ │ │  00028ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028ee0: 207c 0a7c 2020 2020 2020 2020 3420 2020   |.|        4   
│ │ │ │ -00028ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028ef0: 207c 0a7c 2020 2020 2020 2020 3120 2020   |.|        1   
│ │ │ │  00028f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028f10: 2020 2020 7c0a 7c20 2020 2020 2020 6820      |.|       h 
│ │ │ │ -00028f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028f40: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00028f50: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │ -00028f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028f70: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00028f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028f20: 2020 2020 7c0a 2b2d 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 2d2b 0a2d 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 2d2d  ----------------
│ │ │ │ -00028fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a2d  -------------+.-
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│ │ │ │ -00028fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00029000: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
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│ │ │ │ -00029030: 6369 626c 652d 7061 7468 2f6d 6163 6175  cible-path/macau
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│ │ │ │ -000290b0: 3a20 5365 6772 652c 204e 6578 743a 2054  : Segre, Next: T
│ │ │ │ -000290c0: 6f72 6963 4368 6f77 5269 6e67 2c20 5072  oricChowRing, Pr
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│ │ │ │ -00029100: 6865 2053 6567 7265 2063 6c61 7373 206f  he Segre class o
│ │ │ │ -00029110: 6620 6120 7375 6273 6368 656d 650a 2a2a  f a subscheme.**
│ │ │ │ -00029120: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00029130: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00029140: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765  *****..  * Usage
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│ │ │ │ -00029180: 6772 6528 582c 4a29 0a20 2020 2020 2020  gre(X,J).       
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│ │ │ │ -000291a0: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ -000291b0: 202a 2049 2c20 616e 202a 6e6f 7465 2069   * I, an *note i
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│ │ │ │ -00029210: 6d69 616c 2072 696e 6720 6f76 6572 2061  mial ring over a
│ │ │ │ -00029220: 2066 6965 6c64 2064 6566 696e 696e 6720   field defining 
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│ │ │ │ -00029240: 6d65 2056 206f 660a 2020 2020 2020 2020  me V of.        
│ │ │ │ -00029250: 5c50 505e 7b6e 5f31 7d78 2e2e 2e78 5c50  \PP^{n_1}x...x\P
│ │ │ │ -00029260: 505e 7b6e 5f6d 7d0a 2020 2020 2020 2a20  P^{n_m}.      * 
│ │ │ │ -00029270: 412c 2061 202a 6e6f 7465 2071 756f 7469  A, a *note quoti
│ │ │ │ -00029280: 656e 7420 7269 6e67 3a20 284d 6163 6175  ent ring: (Macau
│ │ │ │ -00029290: 6c61 7932 446f 6329 5175 6f74 6965 6e74  lay2Doc)Quotient
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│ │ │ │ -000292d0: 2e2e 2c68 5f6d 5e7b 6e5f 6d2b 317d 2920  ..,h_m^{n_m+1}) 
│ │ │ │ -000292e0: 7175 6f74 6965 6e74 2072 696e 670a 2020  quotient ring.  
│ │ │ │ -000292f0: 2020 2020 2020 7265 7072 6573 656e 7469        representi
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│ │ │ │ -00029340: 2020 2020 2020 6265 2062 7569 6c74 2075        be built u
│ │ │ │ -00029350: 7369 6e67 2074 6865 202a 6e6f 7465 2043  sing the *note C
│ │ │ │ -00029360: 686f 7752 696e 673a 2043 686f 7752 696e  howRing: ChowRin
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│ │ │ │ -00029380: 202a 204a 2c20 616e 202a 6e6f 7465 2069   * J, an *note i
│ │ │ │ -00029390: 6465 616c 3a20 284d 6163 6175 6c61 7932  deal: (Macaulay2
│ │ │ │ -000293a0: 446f 6329 4964 6561 6c2c 2c20 696e 2074  Doc)Ideal,, in t
│ │ │ │ -000293b0: 6865 2067 7261 6465 6420 706f 6c79 6e6f  he graded polyno
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│ │ │ │ -000293d0: 2020 7768 6963 6820 6973 2063 6f6f 7264    which is coord
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│ │ │ │ -00029430: 3a0a 2020 2020 2020 2020 284e 6f72 6d61  :.        (Norma
│ │ │ │ -00029440: 6c54 6f72 6963 5661 7269 6574 6965 7329  lToricVarieties)
│ │ │ │ -00029450: 4e6f 726d 616c 546f 7269 6356 6172 6965  NormalToricVarie
│ │ │ │ -00029460: 7479 2c2c 2077 6869 6368 2069 7320 7468  ty,, which is th
│ │ │ │ -00029470: 6520 616d 6269 656e 7420 7370 6163 650a  e ambient space.
│ │ │ │ -00029480: 2020 2020 2020 2020 7768 6963 6820 636f          which co
│ │ │ │ -00029490: 6e74 6169 6e73 2056 284a 290a 2020 2020  ntains V(J).    
│ │ │ │ -000294a0: 2020 2a20 4368 2c20 6120 2a6e 6f74 6520    * Ch, a *note 
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│ │ │ │ -00029530: 2020 2020 2020 5374 616e 6c65 792d 5265        Stanley-Re
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│ │ │ │ -000295c0: 2020 546f 7269 6343 686f 7752 696e 673a    ToricChowRing:
│ │ │ │ -000295d0: 2054 6f72 6963 4368 6f77 5269 6e67 2c20   ToricChowRing, 
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│ │ │ │ -00029ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00029be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029bf0: 2020 2020 207c 0a7c 202d 2d20 7365 7474       |.| -- sett
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│ │ │ │ -00029c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00028fa0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f  --------..The so
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│ │ │ │ +000290e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
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│ │ │ │ +00029100: 2020 2020 2053 6567 7265 2049 0a20 2020       Segre I.   
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│ │ │ │ +000293c0: 2a6e 6f74 6520 6e6f 726d 616c 2074 6f72  *note normal tor
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│ │ │ │ +000293e0: 2020 2020 284e 6f72 6d61 6c54 6f72 6963      (NormalToric
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│ │ │ │ +00029430: 2020 7768 6963 6820 636f 6e74 6169 6e73    which contains
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│ │ │ │ +000294b0: 2058 2c20 4368 3d28 7269 6e67 204a 292f   X, Ch=(ring J)/
│ │ │ │ +000294c0: 2853 522b 4c52 2920 7768 6572 6520 5352  (SR+LR) where SR
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│ │ │ │ +000294e0: 5374 616e 6c65 792d 5265 6973 6e65 7220  Stanley-Reisner 
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│ │ │ │ +00029530: 6e73 2069 6465 616c 2c20 7468 6973 2072  ns ideal, this r
│ │ │ │ +00029540: 696e 6720 7368 6f75 6c64 2062 6520 6275  ing should be bu
│ │ │ │ +00029550: 696c 7420 7573 696e 6720 7468 6520 2a6e  ilt using the *n
│ │ │ │ +00029560: 6f74 650a 2020 2020 2020 2020 546f 7269  ote.        Tori
│ │ │ │ +00029570: 6343 686f 7752 696e 673a 2054 6f72 6963  cChowRing: Toric
│ │ │ │ +00029580: 4368 6f77 5269 6e67 2c20 636f 6d6d 616e  ChowRing, comman
│ │ │ │ +00029590: 640a 2020 2a20 2a6e 6f74 6520 4f70 7469  d.  * *note Opti
│ │ │ │ +000295a0: 6f6e 616c 2069 6e70 7574 733a 2028 4d61  onal inputs: (Ma
│ │ │ │ +000295b0: 6361 756c 6179 3244 6f63 2975 7369 6e67  caulay2Doc)using
│ │ │ │ +000295c0: 2066 756e 6374 696f 6e73 2077 6974 6820   functions with 
│ │ │ │ +000295d0: 6f70 7469 6f6e 616c 2069 6e70 7574 732c  optional inputs,
│ │ │ │ +000295e0: 3a0a 2020 2020 2020 2a20 436f 6d70 4d65  :.      * CompMe
│ │ │ │ +000295f0: 7468 6f64 2028 6d69 7373 696e 6720 646f  thod (missing do
│ │ │ │ +00029600: 6375 6d65 6e74 6174 696f 6e29 203d 3e20  cumentation) => 
│ │ │ │ +00029610: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
│ │ │ │ +00029620: 7565 0a20 2020 2020 2020 2050 726f 6a65  ue.        Proje
│ │ │ │ +00029630: 6374 6976 6544 6567 7265 652c 2050 726f  ctiveDegree, Pro
│ │ │ │ +00029640: 6a65 6374 6976 6544 6567 7265 652c 2074  jectiveDegree, t
│ │ │ │ +00029650: 6869 7320 616c 676f 7269 7468 6d20 6d61  his algorithm ma
│ │ │ │ +00029660: 7920 6265 2075 7365 6420 666f 720a 2020  y be used for.  
│ │ │ │ +00029670: 2020 2020 2020 7375 6273 6368 656d 6573        subschemes
│ │ │ │ +00029680: 206f 6620 616e 7920 6170 706c 6963 6162   of any applicab
│ │ │ │ +00029690: 6c65 2074 6f72 6963 2076 6172 6965 7479  le toric variety
│ │ │ │ +000296a0: 2028 7468 6973 206d 6179 2062 6520 6368   (this may be ch
│ │ │ │ +000296b0: 6563 6b65 6420 7573 696e 670a 2020 2020  ecked using.    
│ │ │ │ +000296c0: 2020 2020 7468 6520 2a6e 6f74 6520 4368      the *note Ch
│ │ │ │ +000296d0: 6563 6b54 6f72 6963 5661 7269 6574 7956  eckToricVarietyV
│ │ │ │ +000296e0: 616c 6964 3a20 4368 6563 6b54 6f72 6963  alid: CheckToric
│ │ │ │ +000296f0: 5661 7269 6574 7956 616c 6964 2c20 636f  VarietyValid, co
│ │ │ │ +00029700: 6d6d 616e 6429 0a20 2020 2020 202a 2043  mmand).      * C
│ │ │ │ +00029710: 6f6d 704d 6574 686f 6420 286d 6973 7369  ompMethod (missi
│ │ │ │ +00029720: 6e67 2064 6f63 756d 656e 7461 7469 6f6e  ng documentation
│ │ │ │ +00029730: 2920 3d3e 202e 2e2e 2c20 6465 6661 756c  ) => ..., defaul
│ │ │ │ +00029740: 7420 7661 6c75 650a 2020 2020 2020 2020  t value.        
│ │ │ │ +00029750: 5072 6f6a 6563 7469 7665 4465 6772 6565  ProjectiveDegree
│ │ │ │ +00029760: 2c20 506e 5265 7369 6475 616c 2c20 7468  , PnResidual, th
│ │ │ │ +00029770: 6973 2061 6c67 6f72 6974 686d 206d 6179  is algorithm may
│ │ │ │ +00029780: 2062 6520 7573 6564 2066 6f72 2073 7562   be used for sub
│ │ │ │ +00029790: 7363 6865 6d65 730a 2020 2020 2020 2020  schemes.        
│ │ │ │ +000297a0: 6f66 205c 5050 5e6e 206f 6e6c 790a 2020  of \PP^n only.  
│ │ │ │ +000297b0: 2020 2020 2a20 4f75 7470 7574 203d 3e20      * Output => 
│ │ │ │ +000297c0: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
│ │ │ │ +000297d0: 7565 2043 686f 7752 696e 6745 6c65 6d65  ue ChowRingEleme
│ │ │ │ +000297e0: 6e74 2c20 4368 6f77 5269 6e67 456c 656d  nt, ChowRingElem
│ │ │ │ +000297f0: 656e 742c 2072 6574 7572 6e73 0a20 2020  ent, returns.   
│ │ │ │ +00029800: 2020 2020 2061 2052 696e 6745 6c65 6d65       a RingEleme
│ │ │ │ +00029810: 6e74 2069 6e20 7468 6520 4368 6f77 2072  nt in the Chow r
│ │ │ │ +00029820: 696e 6720 6f66 2074 6865 2061 7070 726f  ing of the appro
│ │ │ │ +00029830: 7072 6961 7465 2061 6d62 6965 6e74 2073  priate ambient s
│ │ │ │ +00029840: 7061 6365 0a20 2020 2020 202a 204f 7574  pace.      * Out
│ │ │ │ +00029850: 7075 7420 3d3e 202e 2e2e 2c20 6465 6661  put => ..., defa
│ │ │ │ +00029860: 756c 7420 7661 6c75 6520 4368 6f77 5269  ult value ChowRi
│ │ │ │ +00029870: 6e67 456c 656d 656e 742c 2048 6173 6846  ngElement, HashF
│ │ │ │ +00029880: 6f72 6d2c 2048 6173 6846 6f72 6d0a 2020  orm, HashForm.  
│ │ │ │ +00029890: 2020 2020 2020 7265 7475 726e 7320 6120        returns a 
│ │ │ │ +000298a0: 4d75 7461 626c 6548 6173 6854 6162 6c65  MutableHashTable
│ │ │ │ +000298b0: 2063 6f6e 7461 696e 696e 6720 7468 6520   containing the 
│ │ │ │ +000298c0: 666f 6c6c 6f77 696e 6720 6b65 7973 3a20  following keys: 
│ │ │ │ +000298d0: 2247 2220 2874 6865 0a20 2020 2020 2020  "G" (the.       
│ │ │ │ +000298e0: 2070 6f6c 796e 6f6d 6961 6c20 7769 7468   polynomial with
│ │ │ │ +000298f0: 2063 6f65 6666 6963 6965 6e74 7320 6f66   coefficients of
│ │ │ │ +00029900: 2074 6865 2068 7970 6572 706c 616e 6520   the hyperplane 
│ │ │ │ +00029910: 636c 6173 7365 7320 7265 7072 6573 656e  classes represen
│ │ │ │ +00029920: 7469 6e67 2074 6865 0a20 2020 2020 2020  ting the.       
│ │ │ │ +00029930: 2070 726f 6a65 6374 6976 6520 6465 6772   projective degr
│ │ │ │ +00029940: 6565 7329 2c20 2247 6c69 7374 2220 2874  ees), "Glist" (t
│ │ │ │ +00029950: 6865 206c 6973 7420 666f 726d 206f 6620  he list form of 
│ │ │ │ +00029960: 2247 2229 202c 2022 5365 6772 6522 2028  "G") , "Segre" (
│ │ │ │ +00029970: 7468 650a 2020 2020 2020 2020 746f 7461  the.        tota
│ │ │ │ +00029980: 6c20 5365 6772 6520 636c 6173 7320 6f66  l Segre class of
│ │ │ │ +00029990: 2074 6865 2069 6e70 7574 292c 2253 6567   the input),"Seg
│ │ │ │ +000299a0: 7265 4c69 7374 2220 2874 6865 206c 6973  reList" (the lis
│ │ │ │ +000299b0: 7420 666f 726d 206f 6620 2253 6567 7265  t form of "Segre
│ │ │ │ +000299c0: 2229 0a20 202a 204f 7574 7075 7473 3a0a  ").  * Outputs:.
│ │ │ │ +000299d0: 2020 2020 2020 2a20 6120 2a6e 6f74 6520        * a *note 
│ │ │ │ +000299e0: 7269 6e67 2065 6c65 6d65 6e74 3a20 284d  ring element: (M
│ │ │ │ +000299f0: 6163 6175 6c61 7932 446f 6329 5269 6e67  acaulay2Doc)Ring
│ │ │ │ +00029a00: 456c 656d 656e 742c 2c20 7468 6520 7075  Element,, the pu
│ │ │ │ +00029a10: 7368 666f 7277 6172 6420 6f66 0a20 2020  shforward of.   
│ │ │ │ +00029a20: 2020 2020 2074 6865 2074 6f74 616c 2053       the total S
│ │ │ │ +00029a30: 6567 7265 2063 6c61 7373 206f 6620 7468  egre class of th
│ │ │ │ +00029a40: 6520 7363 6865 6d65 2056 2064 6566 696e  e scheme V defin
│ │ │ │ +00029a50: 6564 2062 7920 7468 6520 696e 7075 7420  ed by the input 
│ │ │ │ +00029a60: 6964 6561 6c20 746f 2074 6865 0a20 2020  ideal to the.   
│ │ │ │ +00029a70: 2020 2020 2061 7070 726f 7072 6961 7465       appropriate
│ │ │ │ +00029a80: 2043 686f 7720 7269 6e67 0a0a 4465 7363   Chow ring..Desc
│ │ │ │ +00029a90: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ +00029aa0: 3d3d 3d0a 0a46 6f72 2061 2073 7562 7363  ===..For a subsc
│ │ │ │ +00029ab0: 6865 6d65 2056 206f 6620 616e 2061 7070  heme V of an app
│ │ │ │ +00029ac0: 6c69 6361 626c 6520 746f 7269 6320 7661  licable toric va
│ │ │ │ +00029ad0: 7269 6574 7920 5820 7468 6973 2063 6f6d  riety X this com
│ │ │ │ +00029ae0: 6d61 6e64 2063 6f6d 7075 7465 7320 7468  mand computes th
│ │ │ │ +00029af0: 650a 7075 7368 2d66 6f72 7761 7264 206f  e.push-forward o
│ │ │ │ +00029b00: 6620 7468 6520 746f 7461 6c20 5365 6772  f the total Segr
│ │ │ │ +00029b10: 6520 636c 6173 7320 7328 562c 5829 206f  e class s(V,X) o
│ │ │ │ +00029b20: 6620 5620 696e 2058 2074 6f20 7468 6520  f V in X to the 
│ │ │ │ +00029b30: 4368 6f77 2072 696e 6720 6f66 2058 2e0a  Chow ring of X..
│ │ │ │ +00029b40: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00029b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00029b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00029b70: 0a7c 6931 203a 2073 6574 5261 6e64 6f6d  .|i1 : setRandom
│ │ │ │ +00029b80: 5365 6564 2037 323b 2020 2020 2020 2020  Seed 72;        
│ │ │ │ +00029b90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029ba0: 0a7c 202d 2d20 7365 7474 696e 6720 7261  .| -- setting ra
│ │ │ │ +00029bb0: 6e64 6f6d 2073 6565 6420 746f 2037 3220  ndom seed to 72 
│ │ │ │ +00029bc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029bd0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00029be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00029bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00029c00: 0a7c 6932 203a 2052 203d 205a 5a2f 3332  .|i2 : R = ZZ/32
│ │ │ │ +00029c10: 3734 395b 772c 792c 7a5d 2020 2020 2020  749[w,y,z]      
│ │ │ │ +00029c20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029c30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00029c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029c50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029c60: 0a7c 6f32 203d 2052 2020 2020 2020 2020  .|o2 = R        
│ │ │ │  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                  
│ │ │ │ +00029c80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029c90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00029ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029cb0: 2020 2020 207c 0a7c 6f32 203d 2052 2020       |.|o2 = R  
│ │ │ │ -00029cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029ce0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00029cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029d10: 2020 2020 207c 0a7c 6f32 203a 2050 6f6c       |.|o2 : Pol
│ │ │ │ -00029d20: 796e 6f6d 6961 6c52 696e 6720 2020 2020  ynomialRing     
│ │ │ │ -00029d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029d40: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00029d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029d70: 2d2d 2d2d 2d2b 0a7c 6933 203a 2053 6567  -----+.|i3 : Seg
│ │ │ │ -00029d80: 7265 2869 6465 616c 2877 2a79 292c 436f  re(ideal(w*y),Co
│ │ │ │ -00029d90: 6d70 4d65 7468 6f64 3d3e 506e 5265 7369  mpMethod=>PnResi
│ │ │ │ -00029da0: 6475 616c 297c 0a7c 2020 2020 2020 2020  dual)|.|        
│ │ │ │ -00029db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029dd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00029de0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00029cb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029cc0: 0a7c 6f32 203a 2050 6f6c 796e 6f6d 6961  .|o2 : Polynomia
│ │ │ │ +00029cd0: 6c52 696e 6720 2020 2020 2020 2020 2020  lRing           
│ │ │ │ +00029ce0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029cf0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00029d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00029d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00029d20: 0a7c 6933 203a 2053 6567 7265 2869 6465  .|i3 : Segre(ide
│ │ │ │ +00029d30: 616c 2877 2a79 292c 436f 6d70 4d65 7468  al(w*y),CompMeth
│ │ │ │ +00029d40: 6f64 3d3e 506e 5265 7369 6475 616c 297c  od=>PnResidual)|
│ │ │ │ +00029d50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00029d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029d70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029d80: 0a7c 2020 2020 2020 2020 2032 2020 2020  .|         2    
│ │ │ │ +00029d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029da0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029db0: 0a7c 6f33 203d 202d 2034 4820 202b 2032  .|o3 = - 4H  + 2
│ │ │ │ +00029dc0: 4820 2020 2020 2020 2020 2020 2020 2020  H               
│ │ │ │ +00029dd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029de0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00029df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029e00: 2020 2020 207c 0a7c 6f33 203d 202d 2034       |.|o3 = - 4
│ │ │ │ -00029e10: 4820 202b 2032 4820 2020 2020 2020 2020  H  + 2H         
│ │ │ │ +00029e00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029e10: 0a7c 2020 2020 205a 5a5b 485d 2020 2020  .|     ZZ[H]    
│ │ │ │  00029e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029e30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00029e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029e30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029e40: 0a7c 6f33 203a 202d 2d2d 2d2d 2020 2020  .|o3 : -----    
│ │ │ │  00029e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029e60: 2020 2020 207c 0a7c 2020 2020 205a 5a5b       |.|     ZZ[
│ │ │ │ -00029e70: 485d 2020 2020 2020 2020 2020 2020 2020  H]              
│ │ │ │ +00029e60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029e70: 0a7c 2020 2020 2020 2020 3320 2020 2020  .|        3     
│ │ │ │  00029e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029e90: 2020 2020 207c 0a7c 6f33 203a 202d 2d2d       |.|o3 : ---
│ │ │ │ -00029ea0: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │ +00029e90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029ea0: 0a7c 2020 2020 2020 2048 2020 2020 2020  .|       H      
│ │ │ │  00029eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029ec0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00029ed0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ -00029ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029ef0: 2020 2020 207c 0a7c 2020 2020 2020 2048       |.|       H
│ │ │ │ -00029f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029f20: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00029f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029f50: 2d2d 2d2d 2d2b 0a7c 6934 203a 2041 3d43  -----+.|i4 : A=C
│ │ │ │ -00029f60: 686f 7752 696e 6728 5229 2020 2020 2020  howRing(R)      
│ │ │ │ +00029ec0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029ed0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00029ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00029ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00029f00: 0a7c 6934 203a 2041 3d43 686f 7752 696e  .|i4 : A=ChowRin
│ │ │ │ +00029f10: 6728 5229 2020 2020 2020 2020 2020 2020  g(R)            
│ │ │ │ +00029f20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029f30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00029f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029f50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029f60: 0a7c 6f34 203d 2041 2020 2020 2020 2020  .|o4 = A        
│ │ │ │  00029f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029f80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00029f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029f80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029f90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00029fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029fb0: 2020 2020 207c 0a7c 6f34 203d 2041 2020       |.|o4 = A  
│ │ │ │ -00029fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029fe0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00029ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a010: 2020 2020 207c 0a7c 6f34 203a 2051 756f       |.|o4 : Quo
│ │ │ │ -0002a020: 7469 656e 7452 696e 6720 2020 2020 2020  tientRing       
│ │ │ │ -0002a030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a040: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -0002a050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a070: 2d2d 2d2d 2d2b 0a7c 6935 203a 2053 6567  -----+.|i5 : Seg
│ │ │ │ -0002a080: 7265 2841 2c69 6465 616c 2877 5e32 2a79  re(A,ideal(w^2*y
│ │ │ │ -0002a090: 2c77 2a79 5e32 2929 2020 2020 2020 2020  ,w*y^2))        
│ │ │ │ -0002a0a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0002a0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a0d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0002a0e0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -0002a0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a100: 2020 2020 207c 0a7c 6f35 203d 202d 2033       |.|o5 = - 3
│ │ │ │ -0002a110: 6820 202b 2032 6820 2020 2020 2020 2020  h  + 2h         
│ │ │ │ +00029fb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029fc0: 0a7c 6f34 203a 2051 756f 7469 656e 7452  .|o4 : QuotientR
│ │ │ │ +00029fd0: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ +00029fe0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00029ff0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0002a000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0002a020: 0a7c 6935 203a 2053 6567 7265 2841 2c69  .|i5 : Segre(A,i
│ │ │ │ +0002a030: 6465 616c 2877 5e32 2a79 2c77 2a79 5e32  deal(w^2*y,w*y^2
│ │ │ │ +0002a040: 2929 2020 2020 2020 2020 2020 2020 207c  ))             |
│ │ │ │ +0002a050: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002a060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a070: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002a080: 0a7c 2020 2020 2020 2020 2032 2020 2020  .|         2    
│ │ │ │ +0002a090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a0a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002a0b0: 0a7c 6f35 203d 202d 2033 6820 202b 2032  .|o5 = - 3h  + 2
│ │ │ │ +0002a0c0: 6820 2020 2020 2020 2020 2020 2020 2020  h               
│ │ │ │ +0002a0d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002a0e0: 0a7c 2020 2020 2020 2020 2031 2020 2020  .|         1    
│ │ │ │ +0002a0f0: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ +0002a100: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002a110: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002a120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a130: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0002a140: 2031 2020 2020 2031 2020 2020 2020 2020   1     1        
│ │ │ │ +0002a130: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002a140: 0a7c 6f35 203a 2041 2020 2020 2020 2020  .|o5 : A        
│ │ │ │  0002a150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a160: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0002a170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a190: 2020 2020 207c 0a7c 6f35 203a 2041 2020       |.|o5 : A  
│ │ │ │ -0002a1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a1c0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -0002a1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a1f0: 2d2d 2d2d 2d2b 0a0a 4e6f 7720 636f 6e73  -----+..Now cons
│ │ │ │ -0002a200: 6964 6572 2061 6e20 6578 616d 706c 6520  ider an example 
│ │ │ │ -0002a210: 696e 205c 5050 5e32 205c 7469 6d65 7320  in \PP^2 \times 
│ │ │ │ -0002a220: 5c50 505e 322c 2069 6620 7765 2069 6e70  \PP^2, if we inp
│ │ │ │ -0002a230: 7574 2074 6865 2043 686f 7720 7269 6e67  ut the Chow ring
│ │ │ │ -0002a240: 2041 2074 6865 0a6f 7574 7075 7420 7769   A the.output wi
│ │ │ │ -0002a250: 6c6c 2062 6520 7265 7475 726e 6564 2069  ll be returned i
│ │ │ │ -0002a260: 6e20 7468 6520 7361 6d65 2072 696e 672e  n the same ring.
│ │ │ │ -0002a270: 2054 6f20 656e 7375 7265 2070 726f 7065   To ensure prope
│ │ │ │ -0002a280: 7220 6675 6e63 7469 6f6e 206f 6620 7468  r function of th
│ │ │ │ -0002a290: 650a 6d65 7468 6f64 7320 7765 2062 7569  e.methods we bui
│ │ │ │ -0002a2a0: 6c64 2074 6865 2043 686f 7720 7269 6e67  ld the Chow ring
│ │ │ │ -0002a2b0: 2075 7369 6e67 2074 6865 202a 6e6f 7465   using the *note
│ │ │ │ -0002a2c0: 2043 686f 7752 696e 673a 2043 686f 7752   ChowRing: ChowR
│ │ │ │ -0002a2d0: 696e 672c 2063 6f6d 6d61 6e64 2e20 5765  ing, command. We
│ │ │ │ -0002a2e0: 0a6d 6179 2061 6c73 6f20 7265 7475 726e  .may also return
│ │ │ │ -0002a2f0: 2061 204d 7574 6162 6c65 4861 7368 5461   a MutableHashTa
│ │ │ │ -0002a300: 626c 652e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  ble...+---------
│ │ │ │ -0002a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a350: 2d2d 2d2d 2b0a 7c69 3620 3a20 523d 4d75  ----+.|i6 : R=Mu
│ │ │ │ -0002a360: 6c74 6950 726f 6a43 6f6f 7264 5269 6e67  ltiProjCoordRing
│ │ │ │ -0002a370: 287b 322c 327d 2920 2020 2020 2020 2020  ({2,2})         
│ │ │ │ +0002a160: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002a170: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0002a180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0002a1a0: 0a0a 4e6f 7720 636f 6e73 6964 6572 2061  ..Now consider a
│ │ │ │ +0002a1b0: 6e20 6578 616d 706c 6520 696e 205c 5050  n example in \PP
│ │ │ │ +0002a1c0: 5e32 205c 7469 6d65 7320 5c50 505e 322c  ^2 \times \PP^2,
│ │ │ │ +0002a1d0: 2069 6620 7765 2069 6e70 7574 2074 6865   if we input the
│ │ │ │ +0002a1e0: 2043 686f 7720 7269 6e67 2041 2074 6865   Chow ring A the
│ │ │ │ +0002a1f0: 0a6f 7574 7075 7420 7769 6c6c 2062 6520  .output will be 
│ │ │ │ +0002a200: 7265 7475 726e 6564 2069 6e20 7468 6520  returned in the 
│ │ │ │ +0002a210: 7361 6d65 2072 696e 672e 2054 6f20 656e  same ring. To en
│ │ │ │ +0002a220: 7375 7265 2070 726f 7065 7220 6675 6e63  sure proper func
│ │ │ │ +0002a230: 7469 6f6e 206f 6620 7468 650a 6d65 7468  tion of the.meth
│ │ │ │ +0002a240: 6f64 7320 7765 2062 7569 6c64 2074 6865  ods we build the
│ │ │ │ +0002a250: 2043 686f 7720 7269 6e67 2075 7369 6e67   Chow ring using
│ │ │ │ +0002a260: 2074 6865 202a 6e6f 7465 2043 686f 7752   the *note ChowR
│ │ │ │ +0002a270: 696e 673a 2043 686f 7752 696e 672c 2063  ing: ChowRing, c
│ │ │ │ +0002a280: 6f6d 6d61 6e64 2e20 5765 0a6d 6179 2061  ommand. We.may a
│ │ │ │ +0002a290: 6c73 6f20 7265 7475 726e 2061 204d 7574  lso return a Mut
│ │ │ │ +0002a2a0: 6162 6c65 4861 7368 5461 626c 652e 0a0a  ableHashTable...
│ │ │ │ +0002a2b0: 2b2d 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 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0002a300: 7c69 3620 3a20 523d 4d75 6c74 6950 726f  |i6 : R=MultiPro
│ │ │ │ +0002a310: 6a43 6f6f 7264 5269 6e67 287b 322c 327d  jCoordRing({2,2}
│ │ │ │ +0002a320: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +0002a330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a340: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002a350: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002a360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a3a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002a390: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002a3a0: 7c6f 3620 3d20 5220 2020 2020 2020 2020  |o6 = R         
│ │ │ │  0002a3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a3f0: 2020 2020 7c0a 7c6f 3620 3d20 5220 2020      |.|o6 = R   
│ │ │ │ +0002a3e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002a3f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0002a400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a440: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002a450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a430: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002a440: 7c6f 3620 3a20 506f 6c79 6e6f 6d69 616c  |o6 : Polynomial
│ │ │ │ +0002a450: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │  0002a460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a490: 2020 2020 7c0a 7c6f 3620 3a20 506f 6c79      |.|o6 : Poly
│ │ │ │ -0002a4a0: 6e6f 6d69 616c 5269 6e67 2020 2020 2020  nomialRing      
│ │ │ │ -0002a4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a4e0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -0002a4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a530: 2d2d 2d2d 2b0a 7c69 3720 3a20 723d 6765  ----+.|i7 : r=ge
│ │ │ │ -0002a540: 6e73 2052 2020 2020 2020 2020 2020 2020  ns R            
│ │ │ │ +0002a480: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002a490: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0002a4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0002a4e0: 7c69 3720 3a20 723d 6765 6e73 2052 2020  |i7 : r=gens R  
│ │ │ │ +0002a4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a520: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002a530: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002a540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a580: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002a590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a570: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002a580: 7c6f 3720 3d20 7b78 202c 2078 202c 2078  |o7 = {x , x , x
│ │ │ │ +0002a590: 202c 2078 202c 2078 202c 2078 207d 2020   , x , x , x }  
│ │ │ │  0002a5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a5d0: 2020 2020 7c0a 7c6f 3720 3d20 7b78 202c      |.|o7 = {x ,
│ │ │ │ -0002a5e0: 2078 202c 2078 202c 2078 202c 2078 202c   x , x , x , x ,
│ │ │ │ -0002a5f0: 2078 207d 2020 2020 2020 2020 2020 2020   x }            
│ │ │ │ +0002a5c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002a5d0: 7c20 2020 2020 2020 3020 2020 3120 2020  |       0   1   
│ │ │ │ +0002a5e0: 3220 2020 3320 2020 3420 2020 3520 2020  2   3   4   5   
│ │ │ │ +0002a5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a620: 2020 2020 7c0a 7c20 2020 2020 2020 3020      |.|       0 
│ │ │ │ -0002a630: 2020 3120 2020 3220 2020 3320 2020 3420    1   2   3   4 
│ │ │ │ -0002a640: 2020 3520 2020 2020 2020 2020 2020 2020    5             
│ │ │ │ +0002a610: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002a620: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002a630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a670: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002a660: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002a670: 7c6f 3720 3a20 4c69 7374 2020 2020 2020  |o7 : List      
│ │ │ │  0002a680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a6c0: 2020 2020 7c0a 7c6f 3720 3a20 4c69 7374      |.|o7 : List
│ │ │ │ -0002a6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a710: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -0002a720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a760: 2d2d 2d2d 2b0a 7c69 3820 3a20 413d 4368  ----+.|i8 : A=Ch
│ │ │ │ -0002a770: 6f77 5269 6e67 2852 2920 2020 2020 2020  owRing(R)       
│ │ │ │ +0002a6b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002a6c0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0002a6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0002a710: 7c69 3820 3a20 413d 4368 6f77 5269 6e67  |i8 : A=ChowRing
│ │ │ │ +0002a720: 2852 2920 2020 2020 2020 2020 2020 2020  (R)             
│ │ │ │ +0002a730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a750: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002a760: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002a770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a7b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002a7a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002a7b0: 7c6f 3820 3d20 4120 2020 2020 2020 2020  |o8 = A         
│ │ │ │  0002a7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a800: 2020 2020 7c0a 7c6f 3820 3d20 4120 2020      |.|o8 = A   
│ │ │ │ +0002a7f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002a800: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0002a810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a850: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002a860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a840: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002a850: 7c6f 3820 3a20 5175 6f74 6965 6e74 5269  |o8 : QuotientRi
│ │ │ │ +0002a860: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │  0002a870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a8a0: 2020 2020 7c0a 7c6f 3820 3a20 5175 6f74      |.|o8 : Quot
│ │ │ │ -0002a8b0: 6965 6e74 5269 6e67 2020 2020 2020 2020  ientRing        
│ │ │ │ -0002a8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a8f0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -0002a900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a940: 2d2d 2d2d 2b0a 7c69 3920 3a20 493d 6964  ----+.|i9 : I=id
│ │ │ │ -0002a950: 6561 6c28 725f 305e 322a 725f 332d 725f  eal(r_0^2*r_3-r_
│ │ │ │ -0002a960: 342a 725f 312a 725f 322c 725f 325e 322a  4*r_1*r_2,r_2^2*
│ │ │ │ -0002a970: 725f 3529 2020 2020 2020 2020 2020 2020  r_5)            
│ │ │ │ -0002a980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a990: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002a9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a890: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002a8a0: 2b2d 2d2d 2d2d 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: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0002a8f0: 7c69 3920 3a20 493d 6964 6561 6c28 725f  |i9 : I=ideal(r_
│ │ │ │ +0002a900: 305e 322a 725f 332d 725f 342a 725f 312a  0^2*r_3-r_4*r_1*
│ │ │ │ +0002a910: 725f 322c 725f 325e 322a 725f 3529 2020  r_2,r_2^2*r_5)  
│ │ │ │ +0002a920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a930: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002a940: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002a950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a980: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002a990: 7c20 2020 2020 2020 2020 2020 2020 3220  |             2 
│ │ │ │ +0002a9a0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │  0002a9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a9e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002a9f0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -0002aa00: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +0002a9d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002a9e0: 7c6f 3920 3d20 6964 6561 6c20 2878 2078  |o9 = ideal (x x
│ │ │ │ +0002a9f0: 2020 2d20 7820 7820 7820 2c20 7820 7820    - x x x , x x 
│ │ │ │ +0002aa00: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  0002aa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aa30: 2020 2020 7c0a 7c6f 3920 3d20 6964 6561      |.|o9 = idea
│ │ │ │ -0002aa40: 6c20 2878 2078 2020 2d20 7820 7820 7820  l (x x  - x x x 
│ │ │ │ -0002aa50: 2c20 7820 7820 2920 2020 2020 2020 2020  , x x )         
│ │ │ │ +0002aa20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002aa30: 7c20 2020 2020 2020 2020 2020 2020 3020  |             0 
│ │ │ │ +0002aa40: 3320 2020 2031 2032 2034 2020 2032 2035  3    1 2 4   2 5
│ │ │ │ +0002aa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aa60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aa80: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002aa90: 2020 2020 3020 3320 2020 2031 2032 2034      0 3    1 2 4
│ │ │ │ -0002aaa0: 2020 2032 2035 2020 2020 2020 2020 2020     2 5          
│ │ │ │ +0002aa70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002aa80: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002aa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002aaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aad0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002aac0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002aad0: 7c6f 3920 3a20 4964 6561 6c20 6f66 2052  |o9 : Ideal of R
│ │ │ │  0002aae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ab00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ab10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ab20: 2020 2020 7c0a 7c6f 3920 3a20 4964 6561      |.|o9 : Idea
│ │ │ │ -0002ab30: 6c20 6f66 2052 2020 2020 2020 2020 2020  l of R          
│ │ │ │ -0002ab40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ab50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ab60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ab70: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -0002ab80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ab90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002aba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002abb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002abc0: 2d2d 2d2d 2b0a 7c69 3130 203a 2053 6567  ----+.|i10 : Seg
│ │ │ │ -0002abd0: 7265 2049 2020 2020 2020 2020 2020 2020  re I            
│ │ │ │ +0002ab10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002ab20: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0002ab30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002ab40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002ab50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002ab60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0002ab70: 7c69 3130 203a 2053 6567 7265 2049 2020  |i10 : Segre I  
│ │ │ │ +0002ab80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ab90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002aba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002abb0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002abc0: 7c20 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ac00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ac10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002ac20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ac30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ac40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ac50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ac60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002ac70: 3220 3220 2020 2020 2032 2020 2020 2020  2 2      2      
│ │ │ │ -0002ac80: 2020 2020 3220 2020 2020 3220 2020 2020      2     2     
│ │ │ │ -0002ac90: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -0002aca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002acb0: 2020 2020 7c0a 7c6f 3130 203d 2037 3268      |.|o10 = 72h
│ │ │ │ -0002acc0: 2068 2020 2d20 3234 6820 6820 202d 2031   h  - 24h h  - 1
│ │ │ │ -0002acd0: 3268 2068 2020 2b20 3468 2020 2b20 3468  2h h  + 4h  + 4h
│ │ │ │ -0002ace0: 2068 2020 2b20 6820 2020 2020 2020 2020   h  + h         
│ │ │ │ -0002acf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ad00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002ad10: 3120 3220 2020 2020 2031 2032 2020 2020  1 2      1 2    
│ │ │ │ -0002ad20: 2020 3120 3220 2020 2020 3120 2020 2020    1 2     1     
│ │ │ │ -0002ad30: 3120 3220 2020 2032 2020 2020 2020 2020  1 2    2        
│ │ │ │ -0002ad40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ad50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002ad60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ac00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002ac10: 7c20 2020 2020 2020 2020 3220 3220 2020  |         2 2   
│ │ │ │ +0002ac20: 2020 2032 2020 2020 2020 2020 2020 3220     2          2 
│ │ │ │ +0002ac30: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0002ac40: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0002ac50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002ac60: 7c6f 3130 203d 2037 3268 2068 2020 2d20  |o10 = 72h h  - 
│ │ │ │ +0002ac70: 3234 6820 6820 202d 2031 3268 2068 2020  24h h  - 12h h  
│ │ │ │ +0002ac80: 2b20 3468 2020 2b20 3468 2068 2020 2b20  + 4h  + 4h h  + 
│ │ │ │ +0002ac90: 6820 2020 2020 2020 2020 2020 2020 2020  h               
│ │ │ │ +0002aca0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002acb0: 7c20 2020 2020 2020 2020 3120 3220 2020  |         1 2   
│ │ │ │ +0002acc0: 2020 2031 2032 2020 2020 2020 3120 3220     1 2      1 2 
│ │ │ │ +0002acd0: 2020 2020 3120 2020 2020 3120 3220 2020      1     1 2   
│ │ │ │ +0002ace0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0002acf0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002ad00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002ad10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ad20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ad30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ad40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002ad50: 7c20 2020 2020 205a 5a5b 6820 2e2e 6820  |      ZZ[h ..h 
│ │ │ │ +0002ad60: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │  0002ad70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ad80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ad90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ada0: 2020 2020 7c0a 7c20 2020 2020 205a 5a5b      |.|      ZZ[
│ │ │ │ -0002adb0: 6820 2e2e 6820 5d20 2020 2020 2020 2020  h ..h ]         
│ │ │ │ +0002ad90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002ada0: 7c20 2020 2020 2020 2020 2031 2020 2032  |          1   2
│ │ │ │ +0002adb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002adc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002add0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ade0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002adf0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002ae00: 2031 2020 2032 2020 2020 2020 2020 2020   1   2          
│ │ │ │ +0002ade0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002adf0: 7c6f 3130 203a 202d 2d2d 2d2d 2d2d 2d2d  |o10 : ---------
│ │ │ │ +0002ae00: 2d20 2020 2020 2020 2020 2020 2020 2020  -               
│ │ │ │  0002ae10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ae20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ae30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ae40: 2020 2020 7c0a 7c6f 3130 203a 202d 2d2d      |.|o10 : ---
│ │ │ │ -0002ae50: 2d2d 2d2d 2d2d 2d20 2020 2020 2020 2020  -------         
│ │ │ │ +0002ae30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002ae40: 7c20 2020 2020 2020 2020 3320 2020 3320  |         3   3 
│ │ │ │ +0002ae50: 2020 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 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002aea0: 3320 2020 3320 2020 2020 2020 2020 2020  3   3           
│ │ │ │ +0002ae80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002ae90: 7c20 2020 2020 2020 2868 202c 2068 2029  |       (h , h )
│ │ │ │ +0002aea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aee0: 2020 2020 7c0a 7c20 2020 2020 2020 2868      |.|       (h
│ │ │ │ -0002aef0: 202c 2068 2029 2020 2020 2020 2020 2020   , h )          
│ │ │ │ +0002aed0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002aee0: 7c20 2020 2020 2020 2020 3120 2020 3220  |         1   2 
│ │ │ │ +0002aef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002af00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002af10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002af20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002af30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002af40: 3120 2020 3220 2020 2020 2020 2020 2020  1   2           
│ │ │ │ -0002af50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002af60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002af70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002af80: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -0002af90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002afa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002afb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002afc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002afd0: 2d2d 2d2d 2b0a 7c69 3131 203a 2073 313d  ----+.|i11 : s1=
│ │ │ │ -0002afe0: 5365 6772 6528 412c 4929 2020 2020 2020  Segre(A,I)      
│ │ │ │ +0002af20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002af30: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0002af40: 2d2d 2d2d 2d2d 2d2d 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 2b0a  --------------+.
│ │ │ │ +0002af80: 7c69 3131 203a 2073 313d 5365 6772 6528  |i11 : s1=Segre(
│ │ │ │ +0002af90: 412c 4929 2020 2020 2020 2020 2020 2020  A,I)            
│ │ │ │ +0002afa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002afb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002afc0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002afd0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002afe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b020: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002b030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b070: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002b080: 3220 3220 2020 2020 2032 2020 2020 2020  2 2      2      
│ │ │ │ -0002b090: 2020 2020 3220 2020 2020 3220 2020 2020      2     2     
│ │ │ │ -0002b0a0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -0002b0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b0c0: 2020 2020 7c0a 7c6f 3131 203d 2037 3268      |.|o11 = 72h
│ │ │ │ -0002b0d0: 2068 2020 2d20 3234 6820 6820 202d 2031   h  - 24h h  - 1
│ │ │ │ -0002b0e0: 3268 2068 2020 2b20 3468 2020 2b20 3468  2h h  + 4h  + 4h
│ │ │ │ -0002b0f0: 2068 2020 2b20 6820 2020 2020 2020 2020   h  + h         
│ │ │ │ -0002b100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b110: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002b120: 3120 3220 2020 2020 2031 2032 2020 2020  1 2      1 2    
│ │ │ │ -0002b130: 2020 3120 3220 2020 2020 3120 2020 2020    1 2     1     
│ │ │ │ -0002b140: 3120 3220 2020 2032 2020 2020 2020 2020  1 2    2        
│ │ │ │ -0002b150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b160: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002b010: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b020: 7c20 2020 2020 2020 2020 3220 3220 2020  |         2 2   
│ │ │ │ +0002b030: 2020 2032 2020 2020 2020 2020 2020 3220     2          2 
│ │ │ │ +0002b040: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0002b050: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0002b060: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b070: 7c6f 3131 203d 2037 3268 2068 2020 2d20  |o11 = 72h h  - 
│ │ │ │ +0002b080: 3234 6820 6820 202d 2031 3268 2068 2020  24h h  - 12h h  
│ │ │ │ +0002b090: 2b20 3468 2020 2b20 3468 2068 2020 2b20  + 4h  + 4h h  + 
│ │ │ │ +0002b0a0: 6820 2020 2020 2020 2020 2020 2020 2020  h               
│ │ │ │ +0002b0b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b0c0: 7c20 2020 2020 2020 2020 3120 3220 2020  |         1 2   
│ │ │ │ +0002b0d0: 2020 2031 2032 2020 2020 2020 3120 3220     1 2      1 2 
│ │ │ │ +0002b0e0: 2020 2020 3120 2020 2020 3120 3220 2020      1     1 2   
│ │ │ │ +0002b0f0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0002b100: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b110: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002b120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b150: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b160: 7c6f 3131 203a 2041 2020 2020 2020 2020  |o11 : A        
│ │ │ │  0002b170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b1b0: 2020 2020 7c0a 7c6f 3131 203a 2041 2020      |.|o11 : A  
│ │ │ │ -0002b1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b200: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -0002b210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b250: 2d2d 2d2d 2b0a 7c69 3132 203a 2053 6567  ----+.|i12 : Seg
│ │ │ │ -0002b260: 4861 7368 3d53 6567 7265 2841 2c49 2c4f  Hash=Segre(A,I,O
│ │ │ │ -0002b270: 7574 7075 743d 3e48 6173 6846 6f72 6d29  utput=>HashForm)
│ │ │ │ +0002b1a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b1b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0002b1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002b1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002b1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002b1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0002b200: 7c69 3132 203a 2053 6567 4861 7368 3d53  |i12 : SegHash=S
│ │ │ │ +0002b210: 6567 7265 2841 2c49 2c4f 7574 7075 743d  egre(A,I,Output=
│ │ │ │ +0002b220: 3e48 6173 6846 6f72 6d29 2020 2020 2020  >HashForm)      
│ │ │ │ +0002b230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b240: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b250: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002b260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b2a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002b2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b290: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b2a0: 7c6f 3132 203d 204d 7574 6162 6c65 4861  |o12 = MutableHa
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│ │ │ │  0002b2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b2f0: 2020 2020 7c0a 7c6f 3132 203d 204d 7574      |.|o12 = Mut
│ │ │ │ -0002b300: 6162 6c65 4861 7368 5461 626c 657b 2e2e  ableHashTable{..
│ │ │ │ -0002b310: 2e34 2e2e 2e7d 2020 2020 2020 2020 2020  .4...}          
│ │ │ │ +0002b2e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b2f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002b300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b340: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002b350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b330: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b340: 7c6f 3132 203a 204d 7574 6162 6c65 4861  |o12 : MutableHa
│ │ │ │ +0002b350: 7368 5461 626c 6520 2020 2020 2020 2020  shTable         
│ │ │ │  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 7c0a 7c6f 3132 203a 204d 7574      |.|o12 : Mut
│ │ │ │ -0002b3a0: 6162 6c65 4861 7368 5461 626c 6520 2020  ableHashTable   
│ │ │ │ -0002b3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b3e0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -0002b3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b430: 2d2d 2d2d 2b0a 7c69 3133 203a 2070 6565  ----+.|i13 : pee
│ │ │ │ -0002b440: 6b20 5365 6748 6173 6820 2020 2020 2020  k SegHash       
│ │ │ │ +0002b380: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b390: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0002b3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002b3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002b3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002b3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0002b3e0: 7c69 3133 203a 2070 6565 6b20 5365 6748  |i13 : peek SegH
│ │ │ │ +0002b3f0: 6173 6820 2020 2020 2020 2020 2020 2020  ash             
│ │ │ │ +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 7c0a                |.
│ │ │ │ +0002b430: 7c20 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b480: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002b490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b470: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b480: 7c6f 3133 203d 204d 7574 6162 6c65 4861  |o13 = MutableHa
│ │ │ │ +0002b490: 7368 5461 626c 657b 4720 3d3e 2032 6820  shTable{G => 2h 
│ │ │ │ +0002b4a0: 202b 2068 2020 2b20 3120 2020 2020 2020   + h  + 1       
│ │ │ │  0002b4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b4d0: 2020 2020 7c0a 7c6f 3133 203d 204d 7574      |.|o13 = Mut
│ │ │ │ -0002b4e0: 6162 6c65 4861 7368 5461 626c 657b 4720  ableHashTable{G 
│ │ │ │ -0002b4f0: 3d3e 2032 6820 202b 2068 2020 2b20 3120  => 2h  + h  + 1 
│ │ │ │ +0002b4c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b4d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002b4e0: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +0002b4f0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  0002b500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b520: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002b530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b540: 2020 2020 2031 2020 2020 3220 2020 2020       1    2     
│ │ │ │ -0002b550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b570: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002b580: 2020 2020 2020 2020 2020 2020 2020 476c                Gl
│ │ │ │ -0002b590: 6973 7420 3d3e 207b 312c 2032 6820 202b  ist => {1, 2h  +
│ │ │ │ -0002b5a0: 2068 202c 2030 2c20 302c 2030 7d20 2020   h , 0, 0, 0}   
│ │ │ │ -0002b5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b5c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002b510: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b520: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002b530: 2020 2020 2020 2020 476c 6973 7420 3d3e          Glist =>
│ │ │ │ +0002b540: 207b 312c 2032 6820 202b 2068 202c 2030   {1, 2h  + h , 0
│ │ │ │ +0002b550: 2c20 302c 2030 7d20 2020 2020 2020 2020  , 0, 0}         
│ │ │ │ +0002b560: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b570: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002b580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b590: 2020 2020 2020 2031 2020 2020 3220 2020         1    2   
│ │ │ │ +0002b5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b5b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b5c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0002b5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b5e0: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ -0002b5f0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -0002b600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b610: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002b620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b640: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -0002b650: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -0002b660: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002b670: 2020 2020 2020 2020 2020 2020 2020 5365                Se
│ │ │ │ -0002b680: 6772 654c 6973 7420 3d3e 207b 302c 2030  greList => {0, 0
│ │ │ │ -0002b690: 2c20 3468 2020 2b20 3468 2068 2020 2b20  , 4h  + 4h h  + 
│ │ │ │ -0002b6a0: 6820 2c20 2d20 2020 2020 2020 2020 2020  h , -           
│ │ │ │ -0002b6b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002b5e0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +0002b5f0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +0002b600: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b610: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002b620: 2020 2020 2020 2020 5365 6772 654c 6973          SegreLis
│ │ │ │ +0002b630: 7420 3d3e 207b 302c 2030 2c20 3468 2020  t => {0, 0, 4h  
│ │ │ │ +0002b640: 2b20 3468 2068 2020 2b20 6820 2c20 2d20  + 4h h  + h , - 
│ │ │ │ +0002b650: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b660: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002b670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b680: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ +0002b690: 2020 2020 3120 3220 2020 2032 2020 2020      1 2    2    
│ │ │ │ +0002b6a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b6b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0002b6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b6e0: 2020 2020 3120 2020 2020 3120 3220 2020      1     1 2   
│ │ │ │ -0002b6f0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -0002b700: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002b710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b720: 2020 2020 2020 2020 2020 3220 3220 2020            2 2   
│ │ │ │ -0002b730: 2020 2032 2020 2020 2020 2020 2020 3220     2          2 
│ │ │ │ -0002b740: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -0002b750: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002b760: 2020 2020 2020 2020 2020 2020 2020 5365                Se
│ │ │ │ -0002b770: 6772 6520 3d3e 2037 3268 2068 2020 2d20  gre => 72h h  - 
│ │ │ │ -0002b780: 3234 6820 6820 202d 2031 3268 2068 2020  24h h  - 12h h  
│ │ │ │ -0002b790: 2b20 3468 2020 2020 2020 2020 2020 2020  + 4h            
│ │ │ │ -0002b7a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002b7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b7c0: 2020 2020 2020 2020 2020 3120 3220 2020            1 2   
│ │ │ │ -0002b7d0: 2020 2031 2032 2020 2020 2020 3120 3220     1 2      1 2 
│ │ │ │ -0002b7e0: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │ -0002b7f0: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ -0002b800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b840: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020  ----|.|         
│ │ │ │ -0002b850: 2020 2020 2020 2020 2020 2020 2020 207d                 }
│ │ │ │ +0002b6d0: 2020 2020 3220 3220 2020 2020 2032 2020      2 2      2  
│ │ │ │ +0002b6e0: 2020 2020 2020 2020 3220 2020 2020 3220          2     2 
│ │ │ │ +0002b6f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b700: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002b710: 2020 2020 2020 2020 5365 6772 6520 3d3e          Segre =>
│ │ │ │ +0002b720: 2037 3268 2068 2020 2d20 3234 6820 6820   72h h  - 24h h 
│ │ │ │ +0002b730: 202d 2031 3268 2068 2020 2b20 3468 2020   - 12h h  + 4h  
│ │ │ │ +0002b740: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b750: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002b760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b770: 2020 2020 3120 3220 2020 2020 2031 2032      1 2      1 2
│ │ │ │ +0002b780: 2020 2020 2020 3120 3220 2020 2020 3120        1 2     1 
│ │ │ │ +0002b790: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b7a0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +0002b7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002b7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002b7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002b7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +0002b7f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002b800: 2020 2020 2020 2020 207d 2020 2020 2020           }      
│ │ │ │ +0002b810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b830: 2020 2020 2020 2020 2020 2020 2020 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 2020 2020 2020                  
│ │ │ │ -0002b890: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002b880: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b890: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0002b8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b8e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002b8d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b8e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0002b8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b930: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002b940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b920: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b930: 7c20 2020 3220 2020 2020 2020 2020 2032  |   2          2
│ │ │ │ +0002b940: 2020 2020 2032 2032 2020 2020 2020 2020       2 2        
│ │ │ │  0002b950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b980: 2020 2020 7c0a 7c20 2020 3220 2020 2020      |.|   2     
│ │ │ │ -0002b990: 2020 2020 2032 2020 2020 2032 2032 2020       2     2 2  
│ │ │ │ +0002b970: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b980: 7c32 3468 2068 2020 2d20 3132 6820 6820  |24h h  - 12h h 
│ │ │ │ +0002b990: 2c20 3732 6820 6820 7d20 2020 2020 2020  , 72h h }       
│ │ │ │  0002b9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b9d0: 2020 2020 7c0a 7c32 3468 2068 2020 2d20      |.|24h h  - 
│ │ │ │ -0002b9e0: 3132 6820 6820 2c20 3732 6820 6820 7d20  12h h , 72h h } 
│ │ │ │ +0002b9c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b9d0: 7c20 2020 3120 3220 2020 2020 2031 2032  |   1 2      1 2
│ │ │ │ +0002b9e0: 2020 2020 2031 2032 2020 2020 2020 2020       1 2        
│ │ │ │  0002b9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ba00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ba10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ba20: 2020 2020 7c0a 7c20 2020 3120 3220 2020      |.|   1 2   
│ │ │ │ -0002ba30: 2020 2031 2032 2020 2020 2031 2032 2020     1 2     1 2  
│ │ │ │ +0002ba10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002ba20: 7c20 2020 2020 2020 2020 2020 3220 2020  |           2   
│ │ │ │ +0002ba30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ba40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ba50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ba60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ba70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002ba80: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0002ba60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002ba70: 7c2b 2034 6820 6820 202b 2068 2020 2020  |+ 4h h  + h    
│ │ │ │ +0002ba80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ba90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002baa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bac0: 2020 2020 7c0a 7c2b 2034 6820 6820 202b      |.|+ 4h h  +
│ │ │ │ -0002bad0: 2068 2020 2020 2020 2020 2020 2020 2020   h              
│ │ │ │ +0002bab0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002bac0: 7c20 2020 2031 2032 2020 2020 3220 2020  |    1 2    2   
│ │ │ │ +0002bad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 7c0a 7c20 2020 2031 2032 2020      |.|    1 2  
│ │ │ │ -0002bb20: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -0002bb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bb60: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -0002bb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bbb0: 2d2d 2d2d 2b0a 7c69 3134 203a 2073 313d  ----+.|i14 : s1=
│ │ │ │ -0002bbc0: 3d53 6567 4861 7368 2322 5365 6772 6522  =SegHash#"Segre"
│ │ │ │ +0002bb00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002bb10: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0002bb20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002bb30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002bb40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002bb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0002bb60: 7c69 3134 203a 2073 313d 3d53 6567 4861  |i14 : s1==SegHa
│ │ │ │ +0002bb70: 7368 2322 5365 6772 6522 2020 2020 2020  sh#"Segre"      
│ │ │ │ +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 7c0a                |.
│ │ │ │ +0002bbb0: 7c20 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bc00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002bbf0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002bc00: 7c6f 3134 203d 2074 7275 6520 2020 2020  |o14 = true     
│ │ │ │  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 7c0a 7c6f 3134 203d 2074 7275      |.|o14 = tru
│ │ │ │ -0002bc60: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ -0002bc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bca0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -0002bcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bcf0: 2d2d 2d2d 2b0a 0a49 6e20 7468 6520 6361  ----+..In the ca
│ │ │ │ -0002bd00: 7365 2077 6865 7265 2074 6865 2061 6d62  se where the amb
│ │ │ │ -0002bd10: 6965 6e74 2073 7061 6365 2069 7320 6120  ient space is a 
│ │ │ │ -0002bd20: 746f 7269 6320 7661 7269 6574 7920 7768  toric variety wh
│ │ │ │ -0002bd30: 6963 6820 6973 206e 6f74 2061 2070 726f  ich is not a pro
│ │ │ │ -0002bd40: 6475 6374 0a6f 6620 7072 6f6a 6563 7469  duct.of projecti
│ │ │ │ -0002bd50: 7665 2073 7061 6365 7320 7765 206d 7573  ve spaces we mus
│ │ │ │ -0002bd60: 7420 6c6f 6164 2074 6865 204e 6f72 6d61  t load the Norma
│ │ │ │ -0002bd70: 6c54 6f72 6963 5661 7269 6574 6965 7320  lToricVarieties 
│ │ │ │ -0002bd80: 7061 636b 6167 6520 616e 6420 6d75 7374  package and must
│ │ │ │ -0002bd90: 0a61 6c73 6f20 696e 7075 7420 7468 6520  .also input the 
│ │ │ │ -0002bda0: 746f 7269 6320 7661 7269 6574 792e 2049  toric variety. I
│ │ │ │ -0002bdb0: 6620 7468 6520 746f 7269 6320 7661 7269  f the toric vari
│ │ │ │ -0002bdc0: 6574 7920 6973 2061 2070 726f 6475 6374  ety is a product
│ │ │ │ -0002bdd0: 206f 6620 7072 6f6a 6563 7469 7665 0a73   of projective.s
│ │ │ │ -0002bde0: 7061 6365 2069 7420 6973 2072 6563 6f6d  pace it is recom
│ │ │ │ -0002bdf0: 6d65 6e64 6564 2074 6f20 7573 6520 7468  mended to use th
│ │ │ │ -0002be00: 6520 666f 726d 2061 626f 7665 2072 6174  e form above rat
│ │ │ │ -0002be10: 6865 7220 7468 616e 2069 6e70 7574 7469  her than inputti
│ │ │ │ -0002be20: 6e67 2074 6865 2074 6f72 6963 0a76 6172  ng the toric.var
│ │ │ │ -0002be30: 6965 7479 2066 6f72 2065 6666 6963 6965  iety for efficie
│ │ │ │ -0002be40: 6e63 7920 7265 6173 6f6e 732e 0a0a 2b2d  ncy reasons...+-
│ │ │ │ -0002be50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002be60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002be70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002be80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002be90: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3520 3a20  -------+.|i15 : 
│ │ │ │ -0002bea0: 6e65 6564 7350 6163 6b61 6765 2022 4e6f  needsPackage "No
│ │ │ │ -0002beb0: 726d 616c 546f 7269 6356 6172 6965 7469  rmalToricVarieti
│ │ │ │ -0002bec0: 6573 2220 2020 2020 2020 2020 2020 2020  es"             
│ │ │ │ -0002bed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bee0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0002bef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bc40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002bc50: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0002bc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002bc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002bc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002bc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0002bca0: 0a49 6e20 7468 6520 6361 7365 2077 6865  .In the case whe
│ │ │ │ +0002bcb0: 7265 2074 6865 2061 6d62 6965 6e74 2073  re the ambient s
│ │ │ │ +0002bcc0: 7061 6365 2069 7320 6120 746f 7269 6320  pace is a toric 
│ │ │ │ +0002bcd0: 7661 7269 6574 7920 7768 6963 6820 6973  variety which is
│ │ │ │ +0002bce0: 206e 6f74 2061 2070 726f 6475 6374 0a6f   not a product.o
│ │ │ │ +0002bcf0: 6620 7072 6f6a 6563 7469 7665 2073 7061  f projective spa
│ │ │ │ +0002bd00: 6365 7320 7765 206d 7573 7420 6c6f 6164  ces we must load
│ │ │ │ +0002bd10: 2074 6865 204e 6f72 6d61 6c54 6f72 6963   the NormalToric
│ │ │ │ +0002bd20: 5661 7269 6574 6965 7320 7061 636b 6167  Varieties packag
│ │ │ │ +0002bd30: 6520 616e 6420 6d75 7374 0a61 6c73 6f20  e and must.also 
│ │ │ │ +0002bd40: 696e 7075 7420 7468 6520 746f 7269 6320  input the toric 
│ │ │ │ +0002bd50: 7661 7269 6574 792e 2049 6620 7468 6520  variety. If the 
│ │ │ │ +0002bd60: 746f 7269 6320 7661 7269 6574 7920 6973  toric variety is
│ │ │ │ +0002bd70: 2061 2070 726f 6475 6374 206f 6620 7072   a product of pr
│ │ │ │ +0002bd80: 6f6a 6563 7469 7665 0a73 7061 6365 2069  ojective.space i
│ │ │ │ +0002bd90: 7420 6973 2072 6563 6f6d 6d65 6e64 6564  t is recommended
│ │ │ │ +0002bda0: 2074 6f20 7573 6520 7468 6520 666f 726d   to use the form
│ │ │ │ +0002bdb0: 2061 626f 7665 2072 6174 6865 7220 7468   above rather th
│ │ │ │ +0002bdc0: 616e 2069 6e70 7574 7469 6e67 2074 6865  an inputting the
│ │ │ │ +0002bdd0: 2074 6f72 6963 0a76 6172 6965 7479 2066   toric.variety f
│ │ │ │ +0002bde0: 6f72 2065 6666 6963 6965 6e63 7920 7265  or efficiency re
│ │ │ │ +0002bdf0: 6173 6f6e 732e 0a0a 2b2d 2d2d 2d2d 2d2d  asons...+-------
│ │ │ │ +0002be00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002be10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002be20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002be30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002be40: 2d2b 0a7c 6931 3520 3a20 6e65 6564 7350  -+.|i15 : needsP
│ │ │ │ +0002be50: 6163 6b61 6765 2022 4e6f 726d 616c 546f  ackage "NormalTo
│ │ │ │ +0002be60: 7269 6356 6172 6965 7469 6573 2220 2020  ricVarieties"   
│ │ │ │ +0002be70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002be80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0002be90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002beb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bed0: 2020 2020 2020 207c 0a7c 6f31 3520 3d20         |.|o15 = 
│ │ │ │ +0002bee0: 4e6f 726d 616c 546f 7269 6356 6172 6965  NormalToricVarie
│ │ │ │ +0002bef0: 7469 6573 2020 2020 2020 2020 2020 2020  ties            
│ │ │ │  0002bf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bf20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0002bf30: 6f31 3520 3d20 4e6f 726d 616c 546f 7269  o15 = NormalTori
│ │ │ │ -0002bf40: 6356 6172 6965 7469 6573 2020 2020 2020  cVarieties      
│ │ │ │ +0002bf20: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002bf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bf40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bf70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002bf60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002bf70: 6f31 3520 3a20 5061 636b 6167 6520 2020  o15 : Package   
│ │ │ │  0002bf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bfc0: 2020 207c 0a7c 6f31 3520 3a20 5061 636b     |.|o15 : Pack
│ │ │ │ -0002bfd0: 6167 6520 2020 2020 2020 2020 2020 2020  age             
│ │ │ │ -0002bfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c000: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002c010: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -0002c020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c050: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3620  ---------+.|i16 
│ │ │ │ -0002c060: 3a20 5268 6f20 3d20 7b7b 312c 302c 307d  : Rho = {{1,0,0}
│ │ │ │ -0002c070: 2c7b 302c 312c 307d 2c7b 302c 302c 317d  ,{0,1,0},{0,0,1}
│ │ │ │ -0002c080: 2c7b 2d31 2c2d 312c 307d 2c7b 302c 302c  ,{-1,-1,0},{0,0,
│ │ │ │ -0002c090: 2d31 7d7d 2020 2020 2020 2020 2020 2020  -1}}            
│ │ │ │ -0002c0a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002c0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c0e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002c0f0: 0a7c 6f31 3620 3d20 7b7b 312c 2030 2c20  .|o16 = {{1, 0, 
│ │ │ │ -0002c100: 307d 2c20 7b30 2c20 312c 2030 7d2c 207b  0}, {0, 1, 0}, {
│ │ │ │ -0002c110: 302c 2030 2c20 317d 2c20 7b2d 312c 202d  0, 0, 1}, {-1, -
│ │ │ │ -0002c120: 312c 2030 7d2c 207b 302c 2030 2c20 2d31  1, 0}, {0, 0, -1
│ │ │ │ -0002c130: 7d7d 2020 2020 2020 2020 7c0a 7c20 2020  }}        |.|   
│ │ │ │ +0002bfb0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002bfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002bfd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002bfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002bff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002c000: 2d2d 2d2b 0a7c 6931 3620 3a20 5268 6f20  ---+.|i16 : Rho 
│ │ │ │ +0002c010: 3d20 7b7b 312c 302c 307d 2c7b 302c 312c  = {{1,0,0},{0,1,
│ │ │ │ +0002c020: 307d 2c7b 302c 302c 317d 2c7b 2d31 2c2d  0},{0,0,1},{-1,-
│ │ │ │ +0002c030: 312c 307d 2c7b 302c 302c 2d31 7d7d 2020  1,0},{0,0,-1}}  
│ │ │ │ +0002c040: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002c050: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002c060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c090: 2020 2020 2020 2020 207c 0a7c 6f31 3620           |.|o16 
│ │ │ │ +0002c0a0: 3d20 7b7b 312c 2030 2c20 307d 2c20 7b30  = {{1, 0, 0}, {0
│ │ │ │ +0002c0b0: 2c20 312c 2030 7d2c 207b 302c 2030 2c20  , 1, 0}, {0, 0, 
│ │ │ │ +0002c0c0: 317d 2c20 7b2d 312c 202d 312c 2030 7d2c  1}, {-1, -1, 0},
│ │ │ │ +0002c0d0: 207b 302c 2030 2c20 2d31 7d7d 2020 2020   {0, 0, -1}}    
│ │ │ │ +0002c0e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002c0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c120: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c130: 0a7c 6f31 3620 3a20 4c69 7374 2020 2020  .|o16 : List    
│ │ │ │  0002c140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c180: 2020 2020 207c 0a7c 6f31 3620 3a20 4c69       |.|o16 : Li
│ │ │ │ -0002c190: 7374 2020 2020 2020 2020 2020 2020 2020  st              
│ │ │ │ -0002c1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c1d0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -0002c1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -0002c220: 3720 3a20 5369 676d 6120 3d20 7b7b 302c  7 : Sigma = {{0,
│ │ │ │ -0002c230: 312c 327d 2c7b 312c 322c 337d 2c7b 302c  1,2},{1,2,3},{0,
│ │ │ │ -0002c240: 322c 337d 2c7b 302c 312c 347d 2c7b 312c  2,3},{0,1,4},{1,
│ │ │ │ -0002c250: 332c 347d 2c7b 302c 332c 347d 7d20 2020  3,4},{0,3,4}}   
│ │ │ │ -0002c260: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0002c270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c2b0: 207c 0a7c 6f31 3720 3d20 7b7b 302c 2031   |.|o17 = {{0, 1
│ │ │ │ -0002c2c0: 2c20 327d 2c20 7b31 2c20 322c 2033 7d2c  , 2}, {1, 2, 3},
│ │ │ │ -0002c2d0: 207b 302c 2032 2c20 337d 2c20 7b30 2c20   {0, 2, 3}, {0, 
│ │ │ │ -0002c2e0: 312c 2034 7d2c 207b 312c 2033 2c20 347d  1, 4}, {1, 3, 4}
│ │ │ │ -0002c2f0: 2c20 7b30 2c20 332c 2034 7d7d 7c0a 7c20  , {0, 3, 4}}|.| 
│ │ │ │ +0002c170: 2020 2020 2020 2020 2020 7c0a 2b2d 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 2d2d  ----------------
│ │ │ │ +0002c1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002c1c0: 2d2d 2d2d 2d2b 0a7c 6931 3720 3a20 5369  -----+.|i17 : Si
│ │ │ │ +0002c1d0: 676d 6120 3d20 7b7b 302c 312c 327d 2c7b  gma = {{0,1,2},{
│ │ │ │ +0002c1e0: 312c 322c 337d 2c7b 302c 322c 337d 2c7b  1,2,3},{0,2,3},{
│ │ │ │ +0002c1f0: 302c 312c 347d 2c7b 312c 332c 347d 2c7b  0,1,4},{1,3,4},{
│ │ │ │ +0002c200: 302c 332c 347d 7d20 2020 2020 2020 2020  0,3,4}}         
│ │ │ │ +0002c210: 7c0a 7c20 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 2020                  
│ │ │ │ +0002c250: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0002c260: 3720 3d20 7b7b 302c 2031 2c20 327d 2c20  7 = {{0, 1, 2}, 
│ │ │ │ +0002c270: 7b31 2c20 322c 2033 7d2c 207b 302c 2032  {1, 2, 3}, {0, 2
│ │ │ │ +0002c280: 2c20 337d 2c20 7b30 2c20 312c 2034 7d2c  , 3}, {0, 1, 4},
│ │ │ │ +0002c290: 207b 312c 2033 2c20 347d 2c20 7b30 2c20   {1, 3, 4}, {0, 
│ │ │ │ +0002c2a0: 332c 2034 7d7d 7c0a 7c20 2020 2020 2020  3, 4}}|.|       
│ │ │ │ +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 2020                  
│ │ │ │ +0002c2f0: 207c 0a7c 6f31 3720 3a20 4c69 7374 2020   |.|o17 : List  
│ │ │ │  0002c300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c340: 2020 2020 2020 207c 0a7c 6f31 3720 3a20         |.|o17 : 
│ │ │ │ -0002c350: 4c69 7374 2020 2020 2020 2020 2020 2020  List            
│ │ │ │ -0002c360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c390: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -0002c3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -0002c3e0: 6931 3820 3a20 5820 3d20 6e6f 726d 616c  i18 : X = normal
│ │ │ │ -0002c3f0: 546f 7269 6356 6172 6965 7479 2852 686f  ToricVariety(Rho
│ │ │ │ -0002c400: 2c53 6967 6d61 2c43 6f65 6666 6963 6965  ,Sigma,Coefficie
│ │ │ │ -0002c410: 6e74 5269 6e67 203d 3e5a 5a2f 3332 3734  ntRing =>ZZ/3274
│ │ │ │ -0002c420: 3929 2020 2020 2020 7c0a 7c20 2020 2020  9)      |.|     
│ │ │ │ +0002c330: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002c380: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3820 3a20  -------+.|i18 : 
│ │ │ │ +0002c390: 5820 3d20 6e6f 726d 616c 546f 7269 6356  X = normalToricV
│ │ │ │ +0002c3a0: 6172 6965 7479 2852 686f 2c53 6967 6d61  ariety(Rho,Sigma
│ │ │ │ +0002c3b0: 2c43 6f65 6666 6963 6965 6e74 5269 6e67  ,CoefficientRing
│ │ │ │ +0002c3c0: 203d 3e5a 5a2f 3332 3734 3929 2020 2020   =>ZZ/32749)    
│ │ │ │ +0002c3d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002c3e0: 2020 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 207c 0a7c               |.|
│ │ │ │ +0002c420: 6f31 3820 3d20 5820 2020 2020 2020 2020  o18 = X         
│ │ │ │  0002c430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c470: 2020 207c 0a7c 6f31 3820 3d20 5820 2020     |.|o18 = X   
│ │ │ │ +0002c460: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002c470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c4b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002c4c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002c4b0: 2020 207c 0a7c 6f31 3820 3a20 4e6f 726d     |.|o18 : Norm
│ │ │ │ +0002c4c0: 616c 546f 7269 6356 6172 6965 7479 2020  alToricVariety  
│ │ │ │  0002c4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c500: 2020 2020 2020 2020 207c 0a7c 6f31 3820           |.|o18 
│ │ │ │ -0002c510: 3a20 4e6f 726d 616c 546f 7269 6356 6172  : NormalToricVar
│ │ │ │ -0002c520: 6965 7479 2020 2020 2020 2020 2020 2020  iety            
│ │ │ │ -0002c530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c550: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -0002c560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0002c5a0: 0a7c 6931 3920 3a20 4368 6563 6b54 6f72  .|i19 : CheckTor
│ │ │ │ -0002c5b0: 6963 5661 7269 6574 7956 616c 6964 2858  icVarietyValid(X
│ │ │ │ -0002c5c0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -0002c5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c5e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002c4f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002c500: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0002c510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002c520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002c530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002c540: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3920  ---------+.|i19 
│ │ │ │ +0002c550: 3a20 4368 6563 6b54 6f72 6963 5661 7269  : CheckToricVari
│ │ │ │ +0002c560: 6574 7956 616c 6964 2858 2920 2020 2020  etyValid(X)     
│ │ │ │ +0002c570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c590: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002c5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c5d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c5e0: 0a7c 6f31 3920 3d20 7472 7565 2020 2020  .|o19 = true    
│ │ │ │  0002c5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c630: 2020 2020 207c 0a7c 6f31 3920 3d20 7472       |.|o19 = tr
│ │ │ │ -0002c640: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ -0002c650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c680: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -0002c690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ -0002c6d0: 3020 3a20 523d 7269 6e67 2858 2920 2020  0 : R=ring(X)   
│ │ │ │ +0002c620: 2020 2020 2020 2020 2020 7c0a 2b2d 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 2d2d 2d2d  ----------------
│ │ │ │ +0002c660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002c670: 2d2d 2d2d 2d2b 0a7c 6932 3020 3a20 523d  -----+.|i20 : R=
│ │ │ │ +0002c680: 7269 6e67 2858 2920 2020 2020 2020 2020  ring(X)         
│ │ │ │ +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: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002c6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c710: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002c700: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0002c710: 3020 3d20 5220 2020 2020 2020 2020 2020  0 = R           
│ │ │ │  0002c720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c760: 207c 0a7c 6f32 3020 3d20 5220 2020 2020   |.|o20 = R     
│ │ │ │ +0002c750: 2020 2020 2020 7c0a 7c20 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 2020 2020 2020 2020                  
│ │ │ │  0002c790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c7a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0002c7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c7a0: 207c 0a7c 6f32 3020 3a20 506f 6c79 6e6f   |.|o20 : Polyno
│ │ │ │ +0002c7b0: 6d69 616c 5269 6e67 2020 2020 2020 2020  mialRing        
│ │ │ │  0002c7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c7f0: 2020 2020 2020 207c 0a7c 6f32 3020 3a20         |.|o20 : 
│ │ │ │ -0002c800: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020  PolynomialRing  
│ │ │ │ -0002c810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c840: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -0002c850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -0002c890: 6932 3120 3a20 493d 6964 6561 6c28 525f  i21 : I=ideal(R_
│ │ │ │ -0002c8a0: 305e 342a 525f 312c 525f 302a 525f 332a  0^4*R_1,R_0*R_3*
│ │ │ │ -0002c8b0: 525f 342a 525f 322d 525f 325e 322a 525f  R_4*R_2-R_2^2*R_
│ │ │ │ -0002c8c0: 305e 3229 2020 2020 2020 2020 2020 2020  0^2)            
│ │ │ │ -0002c8d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0002c8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c7e0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0002c7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002c800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002c810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002c820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002c830: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3120 3a20  -------+.|i21 : 
│ │ │ │ +0002c840: 493d 6964 6561 6c28 525f 305e 342a 525f  I=ideal(R_0^4*R_
│ │ │ │ +0002c850: 312c 525f 302a 525f 332a 525f 342a 525f  1,R_0*R_3*R_4*R_
│ │ │ │ +0002c860: 322d 525f 325e 322a 525f 305e 3229 2020  2-R_2^2*R_0^2)  
│ │ │ │ +0002c870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c880: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0002c890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c8c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002c8d0: 2020 2020 2020 2020 2020 2020 2020 3420                4 
│ │ │ │ +0002c8e0: 2020 2020 2020 3220 3220 2020 2020 2020        2 2       
│ │ │ │  0002c8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c920: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0002c930: 2020 2020 3420 2020 2020 2020 3220 3220      4       2 2 
│ │ │ │ +0002c910: 2020 2020 2020 2020 7c0a 7c6f 3231 203d          |.|o21 =
│ │ │ │ +0002c920: 2069 6465 616c 2028 7820 7820 2c20 2d20   ideal (x x , - 
│ │ │ │ +0002c930: 7820 7820 202b 2078 2078 2078 2078 2029  x x  + x x x x )
│ │ │ │  0002c940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c960: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002c970: 7c6f 3231 203d 2069 6465 616c 2028 7820  |o21 = ideal (x 
│ │ │ │ -0002c980: 7820 2c20 2d20 7820 7820 202b 2078 2078  x , - x x  + x x
│ │ │ │ -0002c990: 2078 2078 2029 2020 2020 2020 2020 2020   x x )          
│ │ │ │ -0002c9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c9b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0002c9c0: 2020 2020 2020 2020 2020 3020 3120 2020            0 1   
│ │ │ │ -0002c9d0: 2020 3020 3220 2020 2030 2032 2033 2034    0 2    0 2 3 4
│ │ │ │ +0002c960: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0002c970: 2020 2020 3020 3120 2020 2020 3020 3220      0 1     0 2 
│ │ │ │ +0002c980: 2020 2030 2032 2033 2034 2020 2020 2020     0 2 3 4      
│ │ │ │ +0002c990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c9a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002c9b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002c9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ca00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002c9f0: 2020 2020 2020 2020 207c 0a7c 6f32 3120           |.|o21 
│ │ │ │ +0002ca00: 3a20 4964 6561 6c20 6f66 2052 2020 2020  : Ideal of R    
│ │ │ │  0002ca10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ca20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ca30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ca40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002ca50: 0a7c 6f32 3120 3a20 4964 6561 6c20 6f66  .|o21 : Ideal of
│ │ │ │ -0002ca60: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ -0002ca70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ca80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ca90: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -0002caa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002cab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002cac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002cad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002cae0: 2d2d 2d2d 2d2b 0a7c 6932 3220 3a20 5365  -----+.|i22 : Se
│ │ │ │ -0002caf0: 6772 6528 582c 4929 2020 2020 2020 2020  gre(X,I)        
│ │ │ │ +0002ca40: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0002ca50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002ca60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002ca70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002ca80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0002ca90: 0a7c 6932 3220 3a20 5365 6772 6528 582c  .|i22 : Segre(X,
│ │ │ │ +0002caa0: 4929 2020 2020 2020 2020 2020 2020 2020  I)              
│ │ │ │ +0002cab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cad0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002cae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002caf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cb30: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002cb20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002cb30: 2020 2032 2020 2020 2020 2032 2020 2020     2       2    
│ │ │ │  0002cb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cb70: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002cb80: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ -0002cb90: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0002cb70: 7c0a 7c6f 3232 203d 202d 2037 3278 2078  |.|o22 = - 72x x
│ │ │ │ +0002cb80: 2020 2b20 3378 2020 2b20 3878 2078 2020    + 3x  + 8x x  
│ │ │ │ +0002cb90: 2b20 7820 2020 2020 2020 2020 2020 2020  + x             
│ │ │ │  0002cba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cbc0: 2020 2020 2020 7c0a 7c6f 3232 203d 202d        |.|o22 = -
│ │ │ │ -0002cbd0: 2037 3278 2078 2020 2b20 3378 2020 2b20   72x x  + 3x  + 
│ │ │ │ -0002cbe0: 3878 2078 2020 2b20 7820 2020 2020 2020  8x x  + x       
│ │ │ │ +0002cbb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002cbc0: 2020 2020 2020 2020 2033 2034 2020 2020           3 4    
│ │ │ │ +0002cbd0: 2033 2020 2020 2033 2034 2020 2020 3320   3     3 4    3 
│ │ │ │ +0002cbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cc10: 207c 0a7c 2020 2020 2020 2020 2020 2033   |.|           3
│ │ │ │ -0002cc20: 2034 2020 2020 2033 2020 2020 2033 2034   4     3     3 4
│ │ │ │ -0002cc30: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │ +0002cc00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002cc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cc50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0002cc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cc50: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002cc60: 2020 2020 2020 2020 2020 5a5a 5b78 202e            ZZ[x .
│ │ │ │ +0002cc70: 2e78 205d 2020 2020 2020 2020 2020 2020  .x ]            
│ │ │ │  0002cc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cca0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0002ccb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ccc0: 5a5a 5b78 202e 2e78 205d 2020 2020 2020  ZZ[x ..x ]      
│ │ │ │ +0002cc90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0002cca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ccb0: 2020 2020 2020 2020 2030 2020 2034 2020           0   4  
│ │ │ │ +0002ccc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ccd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ccf0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0002cd00: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -0002cd10: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ +0002cce0: 2020 2020 2020 207c 0a7c 6f32 3220 3a20         |.|o22 : 
│ │ │ │ +0002ccf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002cd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002cd10: 2d2d 2d2d 2d2d 2d2d 2d20 2020 2020 2020  ---------       
│ │ │ │  0002cd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cd30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0002cd40: 6f32 3220 3a20 2d2d 2d2d 2d2d 2d2d 2d2d  o22 : ----------
│ │ │ │ -0002cd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002cd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d20  --------------- 
│ │ │ │ -0002cd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cd80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0002cd90: 2028 7820 7820 2c20 7820 7820 7820 2c20   (x x , x x x , 
│ │ │ │ -0002cda0: 7820 202d 2078 202c 2078 2020 2d20 7820  x  - x , x  - x 
│ │ │ │ -0002cdb0: 2c20 7820 202d 2078 2029 2020 2020 2020  , x  - x )      
│ │ │ │ -0002cdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cdd0: 2020 207c 0a7c 2020 2020 2020 2020 3220     |.|        2 
│ │ │ │ -0002cde0: 3420 2020 3020 3120 3320 2020 3020 2020  4   0 1 3   0   
│ │ │ │ -0002cdf0: 2033 2020 2031 2020 2020 3320 2020 3220   3   1    3   2 
│ │ │ │ -0002ce00: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ -0002ce10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002ce20: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -0002ce30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0002d4d0: 2022 5365 6772 6528 4964 6561 6c2c 5379   "Segre(Ideal,Sy
│ │ │ │ -0002d4e0: 6d62 6f6c 2922 0a20 202a 2022 5365 6772  mbol)".  * "Segr
│ │ │ │ -0002d4f0: 6528 5175 6f74 6965 6e74 5269 6e67 2c49  e(QuotientRing,I
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│ │ │ │ -0002d540: 5365 6772 653a 2053 6567 7265 2c20 6973  Segre: Segre, is
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│ │ │ │ -0002d6b0: 6e67 2c20 5072 6576 3a20 5365 6772 652c  ng, Prev: Segre,
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│ │ │ │ -0002d710: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002d720: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002d730: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002d740: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20  *************.. 
│ │ │ │ -0002d750: 202a 2055 7361 6765 3a20 0a20 2020 2020   * Usage: .     
│ │ │ │ -0002d760: 2020 2054 6f72 6963 4368 6f77 5269 6e67     ToricChowRing
│ │ │ │ -0002d770: 2058 0a20 202a 2049 6e70 7574 733a 0a20   X.  * Inputs:. 
│ │ │ │ -0002d780: 2020 2020 202a 2052 2c20 6120 2a6e 6f74       * R, a *not
│ │ │ │ -0002d790: 6520 6e6f 726d 616c 2074 6f72 6963 2076  e normal toric v
│ │ │ │ -0002d7a0: 6172 6965 7479 3a0a 2020 2020 2020 2020  ariety:.        
│ │ │ │ -0002d7b0: 284e 6f72 6d61 6c54 6f72 6963 5661 7269  (NormalToricVari
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│ │ │ │ -0002d7d0: 6356 6172 6965 7479 2c2c 2041 206e 6f72  cVariety,, A nor
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│ │ │ │ -0002d7f0: 790a 2020 2a20 4f75 7470 7574 733a 0a20  y.  * Outputs:. 
│ │ │ │ -0002d800: 2020 2020 202a 2061 202a 6e6f 7465 2071       * a *note q
│ │ │ │ -0002d810: 756f 7469 656e 7420 7269 6e67 3a20 284d  uotient ring: (M
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│ │ │ │ -0002d850: 3d3d 3d3d 0a0a 4c65 7420 5820 6265 2061  ====..Let X be a
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│ │ │ │ -0002d930: 6e65 6172 0a72 656c 6174 696f 6e73 2061  near.relations a
│ │ │ │ -0002d940: 6d6f 756e 7420 7468 6520 7261 7973 2e20  mount the rays. 
│ │ │ │ -0002d950: 4974 2069 7320 6e65 6564 6564 2066 6f72  It is needed for
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│ │ │ │ -0002d970: 6d65 7468 6f64 7320 2a6e 6f74 6520 5365  methods *note Se
│ │ │ │ -0002d980: 6772 653a 0a53 6567 7265 2c2c 202a 6e6f  gre:.Segre,, *no
│ │ │ │ -0002d990: 7465 2043 6865 726e 3a20 4368 6572 6e2c  te Chern: Chern,
│ │ │ │ -0002d9a0: 2061 6e64 202a 6e6f 7465 2043 534d 3a20   and *note CSM: 
│ │ │ │ -0002d9b0: 4353 4d2c 2069 6e20 7468 6520 6361 7365  CSM, in the case
│ │ │ │ -0002d9c0: 7320 7768 6572 6520 6120 746f 7269 630a  s where a toric.
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│ │ │ │ -0002d9e0: 696e 7075 7420 746f 2065 6e73 7572 6520  input to ensure 
│ │ │ │ -0002d9f0: 7468 6174 2074 6865 7365 206d 6574 686f  that these metho
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│ │ │ │ -0002da90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0002d210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0002d3b0: 2061 6e64 206d 6179 2067 6976 6520 6120   and may give a 
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│ │ │ │ +0002d3f0: 792e 2052 6561 6420 6d6f 7265 2075 6e64  y. Read more und
│ │ │ │ +0002d400: 6572 202a 6e6f 7465 0a70 726f 6261 6269  er *note.probabi
│ │ │ │ +0002d410: 6c69 7374 6963 2061 6c67 6f72 6974 686d  listic algorithm
│ │ │ │ +0002d420: 3a20 7072 6f62 6162 696c 6973 7469 6320  : probabilistic 
│ │ │ │ +0002d430: 616c 676f 7269 7468 6d2c 2e0a 0a57 6179  algorithm,...Way
│ │ │ │ +0002d440: 7320 746f 2075 7365 2053 6567 7265 3a0a  s to use Segre:.
│ │ │ │ +0002d450: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0002d460: 3d3d 0a0a 2020 2a20 2253 6567 7265 2849  ==..  * "Segre(I
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│ │ │ │ +0002d480: 6528 4964 6561 6c2c 5379 6d62 6f6c 2922  e(Ideal,Symbol)"
│ │ │ │ +0002d490: 0a20 202a 2022 5365 6772 6528 5175 6f74  .  * "Segre(Quot
│ │ │ │ +0002d4a0: 6965 6e74 5269 6e67 2c49 6465 616c 2922  ientRing,Ideal)"
│ │ │ │ +0002d4b0: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
│ │ │ │ +0002d4c0: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
│ │ │ │ +0002d4d0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
│ │ │ │ +0002d4e0: 6563 7420 2a6e 6f74 6520 5365 6772 653a  ect *note Segre:
│ │ │ │ +0002d4f0: 2053 6567 7265 2c20 6973 2061 202a 6e6f   Segre, is a *no
│ │ │ │ +0002d500: 7465 206d 6574 686f 6420 6675 6e63 7469  te method functi
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│ │ │ │ +0002d520: 0a28 4d61 6361 756c 6179 3244 6f63 294d  .(Macaulay2Doc)M
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│ │ │ │ +0002d570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002d580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002d590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
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│ │ │ │ +0002d680: 6720 2d2d 2043 6f6d 7075 7465 7320 7468  g -- Computes th
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│ │ │ │ +0002d6a0: 206e 6f72 6d61 6c20 746f 7269 6320 7661   normal toric va
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│ │ │ │ +0002d6c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0002d6d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0002d6e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0002d6f0: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361  *******..  * Usa
│ │ │ │ +0002d700: 6765 3a20 0a20 2020 2020 2020 2054 6f72  ge: .        Tor
│ │ │ │ +0002d710: 6963 4368 6f77 5269 6e67 2058 0a20 202a  icChowRing X.  *
│ │ │ │ +0002d720: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ +0002d730: 2052 2c20 6120 2a6e 6f74 6520 6e6f 726d   R, a *note norm
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│ │ │ │ +0002d750: 3a0a 2020 2020 2020 2020 284e 6f72 6d61  :.        (Norma
│ │ │ │ +0002d760: 6c54 6f72 6963 5661 7269 6574 6965 7329  lToricVarieties)
│ │ │ │ +0002d770: 4e6f 726d 616c 546f 7269 6356 6172 6965  NormalToricVarie
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│ │ │ │ +0002d790: 7269 6320 7661 7269 6574 790a 2020 2a20  ric variety.  * 
│ │ │ │ +0002d7a0: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ +0002d7b0: 2061 202a 6e6f 7465 2071 756f 7469 656e   a *note quotien
│ │ │ │ +0002d7c0: 7420 7269 6e67 3a20 284d 6163 6175 6c61  t ring: (Macaula
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│ │ │ │ +0002d800: 4c65 7420 5820 6265 2061 2074 6f72 6963  Let X be a toric
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│ │ │ │ +0002d830: 696e 6720 2843 6f78 2072 696e 6729 2052  ing (Cox ring) R
│ │ │ │ +0002d840: 2e20 5468 6973 206d 6574 686f 640a 636f  . This method.co
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│ │ │ │ +0002d860: 7269 6e67 2020 4368 6f77 2072 696e 6720  ring  Chow ring 
│ │ │ │ +0002d870: 4368 3d52 2f28 5352 2b4c 5229 206f 6620  Ch=R/(SR+LR) of 
│ │ │ │ +0002d880: 583b 2068 6572 6520 5352 2069 7320 7468  X; here SR is th
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│ │ │ │ +0002d8a0: 7220 6964 6561 6c20 6f66 2074 6865 2063  r ideal of the c
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│ │ │ │ +0002d8c0: 2061 6e64 204c 5220 6973 2074 6865 2069   and LR is the i
│ │ │ │ +0002d8d0: 6465 616c 206f 6620 6c69 6e65 6172 0a72  deal of linear.r
│ │ │ │ +0002d8e0: 656c 6174 696f 6e73 2061 6d6f 756e 7420  elations amount 
│ │ │ │ +0002d8f0: 7468 6520 7261 7973 2e20 4974 2069 7320  the rays. It is 
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│ │ │ │ +0002d910: 2069 6e74 6f20 7468 6520 6d65 7468 6f64   into the method
│ │ │ │ +0002d920: 7320 2a6e 6f74 6520 5365 6772 653a 0a53  s *note Segre:.S
│ │ │ │ +0002d930: 6567 7265 2c2c 202a 6e6f 7465 2043 6865  egre,, *note Che
│ │ │ │ +0002d940: 726e 3a20 4368 6572 6e2c 2061 6e64 202a  rn: Chern, and *
│ │ │ │ +0002d950: 6e6f 7465 2043 534d 3a20 4353 4d2c 2069  note CSM: CSM, i
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│ │ │ │ +0002d990: 746f 2065 6e73 7572 6520 7468 6174 2074  to ensure that t
│ │ │ │ +0002d9a0: 6865 7365 206d 6574 686f 6473 2072 6574  hese methods ret
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│ │ │ │ +0002d9d0: 2067 6976 6520 616e 2065 7861 6d70 6c65   give an example
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│ │ │ │ +0002da10: 206f 6620 7468 650a 4368 6f77 2072 696e   of the.Chow rin
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│ │ │ │ +0002da40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002da50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0002da70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0002dad0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
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│ │ │ │  0002db10: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0002db30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002db20: 2020 2020 7c0a 7c6f 3120 3d20 4e6f 726d      |.|o1 = Norm
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│ │ │ │  0002db50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0002dd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dd50: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ -0002dd60: 3d20 7b7b 312c 2030 2c20 307d 2c20 7b30  = {{1, 0, 0}, {0
│ │ │ │ -0002dd70: 2c20 312c 2030 7d2c 207b 302c 2030 2c20  , 1, 0}, {0, 0, 
│ │ │ │ -0002dd80: 317d 2c20 7b2d 312c 202d 312c 2030 7d2c  1}, {-1, -1, 0},
│ │ │ │ -0002dd90: 207b 302c 2030 2c20 2d31 7d7d 2020 2020   {0, 0, -1}}    
│ │ │ │ -0002dda0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002dc10: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0002dc20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002dc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002dc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002dc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002dc60: 2d2d 2d2d 2b0a 7c69 3220 3a20 5268 6f20  ----+.|i2 : Rho 
│ │ │ │ +0002dc70: 3d20 7b7b 312c 302c 307d 2c7b 302c 312c  = {{1,0,0},{0,1,
│ │ │ │ +0002dc80: 307d 2c7b 302c 302c 317d 2c7b 2d31 2c2d  0},{0,0,1},{-1,-
│ │ │ │ +0002dc90: 312c 307d 2c7b 302c 302c 2d31 7d7d 2020  1,0},{0,0,-1}}  
│ │ │ │ +0002dca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dcb0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002dcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dd00: 2020 2020 7c0a 7c6f 3220 3d20 7b7b 312c      |.|o2 = {{1,
│ │ │ │ +0002dd10: 2030 2c20 307d 2c20 7b30 2c20 312c 2030   0, 0}, {0, 1, 0
│ │ │ │ +0002dd20: 7d2c 207b 302c 2030 2c20 317d 2c20 7b2d  }, {0, 0, 1}, {-
│ │ │ │ +0002dd30: 312c 202d 312c 2030 7d2c 207b 302c 2030  1, -1, 0}, {0, 0
│ │ │ │ +0002dd40: 2c20 2d31 7d7d 2020 2020 2020 2020 2020  , -1}}          
│ │ │ │ +0002dd50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002dd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dda0: 2020 2020 7c0a 7c6f 3220 3a20 4c69 7374      |.|o2 : List
│ │ │ │  0002ddb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ddc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ddd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ddf0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ -0002de00: 3a20 4c69 7374 2020 2020 2020 2020 2020  : List          
│ │ │ │ -0002de10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002de20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002de30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002de40: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -0002de50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002de60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002de70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002de80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002de90: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320  ----------+.|i3 
│ │ │ │ -0002dea0: 3a20 5369 676d 6120 3d20 7b7b 302c 312c  : Sigma = {{0,1,
│ │ │ │ -0002deb0: 327d 2c7b 312c 322c 337d 2c7b 302c 322c  2},{1,2,3},{0,2,
│ │ │ │ -0002dec0: 337d 2c7b 302c 312c 347d 2c7b 312c 332c  3},{0,1,4},{1,3,
│ │ │ │ -0002ded0: 347d 2c7b 302c 332c 347d 7d20 2020 2020  4},{0,3,4}}     
│ │ │ │ -0002dee0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002def0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002df00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002df10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002df20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002df30: 2020 2020 2020 2020 2020 7c0a 7c6f 3320            |.|o3 
│ │ │ │ -0002df40: 3d20 7b7b 302c 2031 2c20 327d 2c20 7b31  = {{0, 1, 2}, {1
│ │ │ │ -0002df50: 2c20 322c 2033 7d2c 207b 302c 2032 2c20  , 2, 3}, {0, 2, 
│ │ │ │ -0002df60: 337d 2c20 7b30 2c20 312c 2034 7d2c 207b  3}, {0, 1, 4}, {
│ │ │ │ -0002df70: 312c 2033 2c20 347d 2c20 7b30 2c20 332c  1, 3, 4}, {0, 3,
│ │ │ │ -0002df80: 2034 7d7d 2020 2020 2020 7c0a 7c20 2020   4}}      |.|   
│ │ │ │ +0002ddf0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0002de00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002de10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002de20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002de30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002de40: 2d2d 2d2d 2b0a 7c69 3320 3a20 5369 676d  ----+.|i3 : Sigm
│ │ │ │ +0002de50: 6120 3d20 7b7b 302c 312c 327d 2c7b 312c  a = {{0,1,2},{1,
│ │ │ │ +0002de60: 322c 337d 2c7b 302c 322c 337d 2c7b 302c  2,3},{0,2,3},{0,
│ │ │ │ +0002de70: 312c 347d 2c7b 312c 332c 347d 2c7b 302c  1,4},{1,3,4},{0,
│ │ │ │ +0002de80: 332c 347d 7d20 2020 2020 2020 2020 2020  3,4}}           
│ │ │ │ +0002de90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002dea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002deb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ded0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dee0: 2020 2020 7c0a 7c6f 3320 3d20 7b7b 302c      |.|o3 = {{0,
│ │ │ │ +0002def0: 2031 2c20 327d 2c20 7b31 2c20 322c 2033   1, 2}, {1, 2, 3
│ │ │ │ +0002df00: 7d2c 207b 302c 2032 2c20 337d 2c20 7b30  }, {0, 2, 3}, {0
│ │ │ │ +0002df10: 2c20 312c 2034 7d2c 207b 312c 2033 2c20  , 1, 4}, {1, 3, 
│ │ │ │ +0002df20: 347d 2c20 7b30 2c20 332c 2034 7d7d 2020  4}, {0, 3, 4}}  
│ │ │ │ +0002df30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002df40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002df50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002df60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002df70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002df80: 2020 2020 7c0a 7c6f 3320 3a20 4c69 7374      |.|o3 : List
│ │ │ │  0002df90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dfd0: 2020 2020 2020 2020 2020 7c0a 7c6f 3320            |.|o3 
│ │ │ │ -0002dfe0: 3a20 4c69 7374 2020 2020 2020 2020 2020  : List          
│ │ │ │ -0002dff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e020: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -0002e030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e070: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420  ----------+.|i4 
│ │ │ │ -0002e080: 3a20 5820 3d20 6e6f 726d 616c 546f 7269  : X = normalTori
│ │ │ │ -0002e090: 6356 6172 6965 7479 2852 686f 2c53 6967  cVariety(Rho,Sig
│ │ │ │ -0002e0a0: 6d61 2c43 6f65 6666 6963 6965 6e74 5269  ma,CoefficientRi
│ │ │ │ -0002e0b0: 6e67 203d 3e5a 5a2f 3332 3734 3929 2020  ng =>ZZ/32749)  
│ │ │ │ -0002e0c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002dfd0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0002dfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002dff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e020: 2d2d 2d2d 2b0a 7c69 3420 3a20 5820 3d20  ----+.|i4 : X = 
│ │ │ │ +0002e030: 6e6f 726d 616c 546f 7269 6356 6172 6965  normalToricVarie
│ │ │ │ +0002e040: 7479 2852 686f 2c53 6967 6d61 2c43 6f65  ty(Rho,Sigma,Coe
│ │ │ │ +0002e050: 6666 6963 6965 6e74 5269 6e67 203d 3e5a  fficientRing =>Z
│ │ │ │ +0002e060: 5a2f 3332 3734 3929 2020 2020 2020 2020  Z/32749)        
│ │ │ │ +0002e070: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002e080: 2020 2020 2020 2020 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 7c6f 3420 3d20 5820 2020      |.|o4 = X   
│ │ │ │  0002e0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e110: 2020 2020 2020 2020 2020 7c0a 7c6f 3420            |.|o4 
│ │ │ │ -0002e120: 3d20 5820 2020 2020 2020 2020 2020 2020  = X             
│ │ │ │ +0002e110: 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020                  
│ │ │ │  0002e150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e160: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002e170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e160: 2020 2020 7c0a 7c6f 3420 3a20 4e6f 726d      |.|o4 : Norm
│ │ │ │ +0002e170: 616c 546f 7269 6356 6172 6965 7479 2020  alToricVariety  
│ │ │ │  0002e180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e1b0: 2020 2020 2020 2020 2020 7c0a 7c6f 3420            |.|o4 
│ │ │ │ -0002e1c0: 3a20 4e6f 726d 616c 546f 7269 6356 6172  : NormalToricVar
│ │ │ │ -0002e1d0: 6965 7479 2020 2020 2020 2020 2020 2020  iety            
│ │ │ │ -0002e1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e200: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -0002e210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e250: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520  ----------+.|i5 
│ │ │ │ -0002e260: 3a20 523d 7269 6e67 2058 2020 2020 2020  : R=ring X      
│ │ │ │ +0002e1b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0002e1c0: 2d2d 2d2d 2d2d 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 2d2d 2b0a 7c69 3520 3a20 523d 7269  ----+.|i5 : R=ri
│ │ │ │ +0002e210: 6e67 2058 2020 2020 2020 2020 2020 2020  ng X            
│ │ │ │ +0002e220: 2020 2020 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 7c0a 7c20 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 2020 2020 2020 2020 2020                  
│ │ │ │  0002e290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e2a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002e2a0: 2020 2020 7c0a 7c6f 3520 3d20 5220 2020      |.|o5 = R   
│ │ │ │  0002e2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e2f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ -0002e300: 3d20 5220 2020 2020 2020 2020 2020 2020  = R             
│ │ │ │ +0002e2f0: 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020                  
│ │ │ │  0002e330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e340: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002e350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e340: 2020 2020 7c0a 7c6f 3520 3a20 506f 6c79      |.|o5 : Poly
│ │ │ │ +0002e350: 6e6f 6d69 616c 5269 6e67 2020 2020 2020  nomialRing      
│ │ │ │  0002e360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e390: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ -0002e3a0: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67  : PolynomialRing
│ │ │ │ -0002e3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e3e0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -0002e3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e430: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620  ----------+.|i6 
│ │ │ │ -0002e440: 3a20 4368 3d54 6f72 6963 4368 6f77 5269  : Ch=ToricChowRi
│ │ │ │ -0002e450: 6e67 2858 2920 2020 2020 2020 2020 2020  ng(X)           
│ │ │ │ +0002e390: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0002e3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e3e0: 2d2d 2d2d 2b0a 7c69 3620 3a20 4368 3d54  ----+.|i6 : Ch=T
│ │ │ │ +0002e3f0: 6f72 6963 4368 6f77 5269 6e67 2858 2920  oricChowRing(X) 
│ │ │ │ +0002e400: 2020 2020 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 7c0a 7c20 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 2020 2020 2020 2020 2020                  
│ │ │ │  0002e470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e480: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002e480: 2020 2020 7c0a 7c6f 3620 3d20 4368 2020      |.|o6 = Ch  
│ │ │ │  0002e490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e4d0: 2020 2020 2020 2020 2020 7c0a 7c6f 3620            |.|o6 
│ │ │ │ -0002e4e0: 3d20 4368 2020 2020 2020 2020 2020 2020  = Ch            
│ │ │ │ +0002e4d0: 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020                  
│ │ │ │  0002e510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e520: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002e530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e520: 2020 2020 7c0a 7c6f 3620 3a20 5175 6f74      |.|o6 : Quot
│ │ │ │ +0002e530: 6965 6e74 5269 6e67 2020 2020 2020 2020  ientRing        
│ │ │ │  0002e540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e570: 2020 2020 2020 2020 2020 7c0a 7c6f 3620            |.|o6 
│ │ │ │ -0002e580: 3a20 5175 6f74 6965 6e74 5269 6e67 2020  : QuotientRing  
│ │ │ │ -0002e590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e5c0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -0002e5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e610: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720  ----------+.|i7 
│ │ │ │ -0002e620: 3a20 6465 7363 7269 6265 2043 6820 2020  : describe Ch   
│ │ │ │ +0002e570: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0002e580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e5c0: 2d2d 2d2d 2b0a 7c69 3720 3a20 6465 7363  ----+.|i7 : desc
│ │ │ │ +0002e5d0: 7269 6265 2043 6820 2020 2020 2020 2020  ribe Ch         
│ │ │ │ +0002e5e0: 2020 2020 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 7c0a 7c20 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 2020 2020 2020 2020 2020                  
│ │ │ │  0002e650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e660: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002e670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e660: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002e670: 2020 2020 2020 2020 2020 2020 5a5a 5b78              ZZ[x
│ │ │ │ +0002e680: 202e 2e78 205d 2020 2020 2020 2020 2020   ..x ]          
│ │ │ │  0002e690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e6b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002e6b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0002e6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e6d0: 2020 5a5a 5b78 202e 2e78 205d 2020 2020    ZZ[x ..x ]    
│ │ │ │ +0002e6d0: 3020 2020 3420 2020 2020 2020 2020 2020  0   4           
│ │ │ │  0002e6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e700: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002e710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e720: 2020 2020 2020 3020 2020 3420 2020 2020        0   4     
│ │ │ │ -0002e730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e700: 2020 2020 7c0a 7c6f 3720 3d20 2d2d 2d2d      |.|o7 = ----
│ │ │ │ +0002e710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e730: 2d2d 2d2d 2d20 2020 2020 2020 2020 2020  -----           
│ │ │ │  0002e740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e750: 2020 2020 2020 2020 2020 7c0a 7c6f 3720            |.|o7 
│ │ │ │ -0002e760: 3d20 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  = --------------
│ │ │ │ -0002e770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d20 2020 2020  -----------     
│ │ │ │ +0002e750: 2020 2020 7c0a 7c20 2020 2020 2878 2078      |.|     (x x
│ │ │ │ +0002e760: 202c 2078 2078 2078 202c 2078 2020 2d20   , x x x , x  - 
│ │ │ │ +0002e770: 7820 2c20 7820 202d 2078 202c 2078 2020  x , x  - x , x  
│ │ │ │ +0002e780: 2d20 7820 2920 2020 2020 2020 2020 2020  - x )           
│ │ │ │  0002e790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e7a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002e7b0: 2020 2878 2078 202c 2078 2078 2078 202c    (x x , x x x ,
│ │ │ │ -0002e7c0: 2078 2020 2d20 7820 2c20 7820 202d 2078   x  - x , x  - x
│ │ │ │ -0002e7d0: 202c 2078 2020 2d20 7820 2920 2020 2020   , x  - x )     
│ │ │ │ +0002e7a0: 2020 2020 7c0a 7c20 2020 2020 2020 3220      |.|       2 
│ │ │ │ +0002e7b0: 3420 2020 3020 3120 3320 2020 3020 2020  4   0 1 3   0   
│ │ │ │ +0002e7c0: 2033 2020 2031 2020 2020 3320 2020 3220   3   1    3   2 
│ │ │ │ +0002e7d0: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │  0002e7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e7f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002e800: 2020 2020 3220 3420 2020 3020 3120 3320      2 4   0 1 3 
│ │ │ │ -0002e810: 2020 3020 2020 2033 2020 2031 2020 2020    0    3   1    
│ │ │ │ -0002e820: 3320 2020 3220 2020 2034 2020 2020 2020  3   2    4      
│ │ │ │ -0002e830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e840: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -0002e850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e890: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820  ----------+.|i8 
│ │ │ │ -0002e8a0: 3a20 723d 6765 6e73 2052 2020 2020 2020  : r=gens R      
│ │ │ │ +0002e7f0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0002e800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e840: 2d2d 2d2d 2b0a 7c69 3820 3a20 723d 6765  ----+.|i8 : r=ge
│ │ │ │ +0002e850: 6e73 2052 2020 2020 2020 2020 2020 2020  ns R            
│ │ │ │ +0002e860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e890: 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020                  
│ │ │ │  0002e8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e8e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002e8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e8e0: 2020 2020 7c0a 7c6f 3820 3d20 7b78 202c      |.|o8 = {x ,
│ │ │ │ +0002e8f0: 2078 202c 2078 202c 2078 202c 2078 207d   x , x , x , x }
│ │ │ │  0002e900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e930: 2020 2020 2020 2020 2020 7c0a 7c6f 3820            |.|o8 
│ │ │ │ -0002e940: 3d20 7b78 202c 2078 202c 2078 202c 2078  = {x , x , x , x
│ │ │ │ -0002e950: 202c 2078 207d 2020 2020 2020 2020 2020   , x }          
│ │ │ │ +0002e930: 2020 2020 7c0a 7c20 2020 2020 2020 3020      |.|       0 
│ │ │ │ +0002e940: 2020 3120 2020 3220 2020 3320 2020 3420    1   2   3   4 
│ │ │ │ +0002e950: 2020 2020 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 7c0a 7c20 2020            |.|   
│ │ │ │ -0002e990: 2020 2020 3020 2020 3120 2020 3220 2020      0   1   2   
│ │ │ │ -0002e9a0: 3320 2020 3420 2020 2020 2020 2020 2020  3   4           
│ │ │ │ +0002e980: 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020                  
│ │ │ │  0002e9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e9d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002e9d0: 2020 2020 7c0a 7c6f 3820 3a20 4c69 7374      |.|o8 : List
│ │ │ │  0002e9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ea00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ea10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ea20: 2020 2020 2020 2020 2020 7c0a 7c6f 3820            |.|o8 
│ │ │ │ -0002ea30: 3a20 4c69 7374 2020 2020 2020 2020 2020  : List          
│ │ │ │ -0002ea40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ea50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ea60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ea70: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -0002ea80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ea90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002eaa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002eab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002eac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920  ----------+.|i9 
│ │ │ │ -0002ead0: 3a20 493d 6964 6561 6c28 7261 6e64 6f6d  : I=ideal(random
│ │ │ │ -0002eae0: 287b 312c 307d 2c52 2929 2020 2020 2020  ({1,0},R))      
│ │ │ │ +0002ea20: 2020 2020 7c0a 2b2d 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 2d2d  ----------------
│ │ │ │ +0002ea70: 2d2d 2d2d 2b0a 7c69 3920 3a20 493d 6964  ----+.|i9 : I=id
│ │ │ │ +0002ea80: 6561 6c28 7261 6e64 6f6d 287b 312c 307d  eal(random({1,0}
│ │ │ │ +0002ea90: 2c52 2929 2020 2020 2020 2020 2020 2020  ,R))            
│ │ │ │ +0002eaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002eab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002eac0: 2020 2020 7c0a 7c20 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 2020                  
│ │ │ │ -0002eb10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002eb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002eb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002eb10: 2020 2020 7c0a 7c6f 3920 3d20 6964 6561      |.|o9 = idea
│ │ │ │ +0002eb20: 6c28 3130 3778 2020 2b20 3433 3736 7820  l(107x  + 4376x 
│ │ │ │ +0002eb30: 202d 2036 3331 3678 2029 2020 2020 2020   - 6316x )      
│ │ │ │  0002eb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002eb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002eb60: 2020 2020 2020 2020 2020 7c0a 7c6f 3920            |.|o9 
│ │ │ │ -0002eb70: 3d20 6964 6561 6c28 3130 3778 2020 2b20  = ideal(107x  + 
│ │ │ │ -0002eb80: 3433 3736 7820 202d 2036 3331 3678 2029  4376x  - 6316x )
│ │ │ │ +0002eb60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002eb70: 2020 2020 2020 3020 2020 2020 2020 2031        0        1
│ │ │ │ +0002eb80: 2020 2020 2020 2020 3320 2020 2020 2020          3       
│ │ │ │  0002eb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002eba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ebb0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002ebc0: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ -0002ebd0: 2020 2020 2031 2020 2020 2020 2020 3320       1        3 
│ │ │ │ +0002ebb0: 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020                  
│ │ │ │  0002ebf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ec00: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002ec10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ec00: 2020 2020 7c0a 7c6f 3920 3a20 4964 6561      |.|o9 : Idea
│ │ │ │ +0002ec10: 6c20 6f66 2052 2020 2020 2020 2020 2020  l of R          
│ │ │ │  0002ec20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ec30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ec40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ec50: 2020 2020 2020 2020 2020 7c0a 7c6f 3920            |.|o9 
│ │ │ │ -0002ec60: 3a20 4964 6561 6c20 6f66 2052 2020 2020  : Ideal of R    
│ │ │ │ -0002ec70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ec80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ec90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002eca0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -0002ecb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ecc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ecd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ece0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ecf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130  ----------+.|i10
│ │ │ │ -0002ed00: 203a 204b 3d69 6465 616c 2872 616e 646f   : K=ideal(rando
│ │ │ │ -0002ed10: 6d28 7b31 2c31 7d2c 5229 2920 2020 2020  m({1,1},R))     
│ │ │ │ +0002ec50: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0002ec60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002ec70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002ec80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002ec90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002eca0: 2d2d 2d2d 2b0a 7c69 3130 203a 204b 3d69  ----+.|i10 : K=i
│ │ │ │ +0002ecb0: 6465 616c 2872 616e 646f 6d28 7b31 2c31  deal(random({1,1
│ │ │ │ +0002ecc0: 7d2c 5229 2920 2020 2020 2020 2020 2020  },R))           
│ │ │ │ +0002ecd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ece0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ecf0: 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020                  
│ │ │ │  0002ed30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ed40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002ed50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ed60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ed70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ed80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ed90: 2020 2020 2020 2020 2020 7c0a 7c6f 3130            |.|o10
│ │ │ │ -0002eda0: 203d 2069 6465 616c 2833 3138 3778 2078   = ideal(3187x x
│ │ │ │ -0002edb0: 2020 2d20 3630 3533 7820 7820 202d 2031    - 6053x x  - 1
│ │ │ │ -0002edc0: 3630 3930 7820 7820 202b 2033 3738 3378  6090x x  + 3783x
│ │ │ │ -0002edd0: 2078 2020 2b20 3835 3730 7820 7820 202b   x  + 8570x x  +
│ │ │ │ -0002ede0: 2038 3434 3478 2078 2029 7c0a 7c20 2020   8444x x )|.|   
│ │ │ │ -0002edf0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -0002ee00: 3220 2020 2020 2020 2031 2032 2020 2020  2        1 2    
│ │ │ │ -0002ee10: 2020 2020 2032 2033 2020 2020 2020 2020       2 3        
│ │ │ │ -0002ee20: 3020 3420 2020 2020 2020 2031 2034 2020  0 4        1 4  
│ │ │ │ -0002ee30: 2020 2020 2020 3320 3420 7c0a 7c20 2020        3 4 |.|   
│ │ │ │ -0002ee40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ed40: 2020 2020 7c0a 7c6f 3130 203d 2069 6465      |.|o10 = ide
│ │ │ │ +0002ed50: 616c 2833 3138 3778 2078 2020 2d20 3630  al(3187x x  - 60
│ │ │ │ +0002ed60: 3533 7820 7820 202d 2031 3630 3930 7820  53x x  - 16090x 
│ │ │ │ +0002ed70: 7820 202b 2033 3738 3378 2078 2020 2b20  x  + 3783x x  + 
│ │ │ │ +0002ed80: 3835 3730 7820 7820 202b 2038 3434 3478  8570x x  + 8444x
│ │ │ │ +0002ed90: 2078 2029 7c0a 7c20 2020 2020 2020 2020   x )|.|         
│ │ │ │ +0002eda0: 2020 2020 2020 2020 3020 3220 2020 2020          0 2     
│ │ │ │ +0002edb0: 2020 2031 2032 2020 2020 2020 2020 2032     1 2         2
│ │ │ │ +0002edc0: 2033 2020 2020 2020 2020 3020 3420 2020   3        0 4   
│ │ │ │ +0002edd0: 2020 2020 2031 2034 2020 2020 2020 2020       1 4        
│ │ │ │ +0002ede0: 3320 3420 7c0a 7c20 2020 2020 2020 2020  3 4 |.|         
│ │ │ │ +0002edf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ee00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ee10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ee20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ee30: 2020 2020 7c0a 7c6f 3130 203a 2049 6465      |.|o10 : Ide
│ │ │ │ +0002ee40: 616c 206f 6620 5220 2020 2020 2020 2020  al of R         
│ │ │ │  0002ee50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ee60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ee70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ee80: 2020 2020 2020 2020 2020 7c0a 7c6f 3130            |.|o10
│ │ │ │ -0002ee90: 203a 2049 6465 616c 206f 6620 5220 2020   : Ideal of R   
│ │ │ │ -0002eea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002eeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002eec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002eed0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -0002eee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002eef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ef00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ef10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ef20: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131  ----------+.|i11
│ │ │ │ -0002ef30: 203a 2063 3d43 6865 726e 2843 682c 582c   : c=Chern(Ch,X,
│ │ │ │ -0002ef40: 4929 2020 2020 2020 2020 2020 2020 2020  I)              
│ │ │ │ +0002ee80: 2020 2020 7c0a 2b2d 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 2d2d  ----------------
│ │ │ │ +0002eed0: 2d2d 2d2d 2b0a 7c69 3131 203a 2063 3d43  ----+.|i11 : c=C
│ │ │ │ +0002eee0: 6865 726e 2843 682c 582c 4929 2020 2020  hern(Ch,X,I)    
│ │ │ │ +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 2020                  
│ │ │ │ +0002ef20: 2020 2020 7c0a 7c20 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 2020                  
│ │ │ │ -0002ef70: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002ef80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ef70: 2020 2020 7c0a 7c20 2020 2020 2020 2032      |.|        2
│ │ │ │ +0002ef80: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │  0002ef90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002efa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002efb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002efc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002efd0: 2020 2020 2032 2020 2020 2020 2032 2020       2       2  
│ │ │ │ -0002efe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002efc0: 2020 2020 7c0a 7c6f 3131 203d 2034 7820      |.|o11 = 4x 
│ │ │ │ +0002efd0: 7820 202b 2032 7820 202b 2032 7820 7820  x  + 2x  + 2x x 
│ │ │ │ +0002efe0: 202b 2078 2020 2020 2020 2020 2020 2020   + x            
│ │ │ │  0002eff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f010: 2020 2020 2020 2020 2020 7c0a 7c6f 3131            |.|o11
│ │ │ │ -0002f020: 203d 2034 7820 7820 202b 2032 7820 202b   = 4x x  + 2x  +
│ │ │ │ -0002f030: 2032 7820 7820 202b 2078 2020 2020 2020   2x x  + x      
│ │ │ │ +0002f010: 2020 2020 7c0a 7c20 2020 2020 2020 2033      |.|        3
│ │ │ │ +0002f020: 2034 2020 2020 2033 2020 2020 2033 2034   4     3     3 4
│ │ │ │ +0002f030: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │  0002f040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f060: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002f070: 2020 2020 2033 2034 2020 2020 2033 2020       3 4     3  
│ │ │ │ -0002f080: 2020 2033 2034 2020 2020 3320 2020 2020     3 4    3     
│ │ │ │ +0002f060: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002f070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f0b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002f0b0: 2020 2020 7c0a 7c6f 3131 203a 2043 6820      |.|o11 : Ch 
│ │ │ │  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 2020                  
│ │ │ │ -0002f100: 2020 2020 2020 2020 2020 7c0a 7c6f 3131            |.|o11
│ │ │ │ -0002f110: 203a 2043 6820 2020 2020 2020 2020 2020   : Ch           
│ │ │ │ -0002f120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f150: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -0002f160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3132  ----------+.|i12
│ │ │ │ -0002f1b0: 203a 2073 3d53 6567 7265 2843 682c 582c   : s=Segre(Ch,X,
│ │ │ │ -0002f1c0: 4b29 2020 2020 2020 2020 2020 2020 2020  K)              
│ │ │ │ +0002f100: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0002f110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f150: 2d2d 2d2d 2b0a 7c69 3132 203a 2073 3d53  ----+.|i12 : s=S
│ │ │ │ +0002f160: 6567 7265 2843 682c 582c 4b29 2020 2020  egre(Ch,X,K)    
│ │ │ │ +0002f170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f1a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002f1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f1f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002f200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f1f0: 2020 2020 7c0a 7c20 2020 2020 2020 2032      |.|        2
│ │ │ │ +0002f200: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  0002f210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f240: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002f250: 2020 2020 2032 2020 2020 2020 3220 2020       2      2   
│ │ │ │ -0002f260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f240: 2020 2020 7c0a 7c6f 3132 203d 2033 7820      |.|o12 = 3x 
│ │ │ │ +0002f250: 7820 202d 2078 2020 2d20 3278 2078 2020  x  - x  - 2x x  
│ │ │ │ +0002f260: 2b20 7820 202b 2078 2020 2020 2020 2020  + x  + x        
│ │ │ │  0002f270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f290: 2020 2020 2020 2020 2020 7c0a 7c6f 3132            |.|o12
│ │ │ │ -0002f2a0: 203d 2033 7820 7820 202d 2078 2020 2d20   = 3x x  - x  - 
│ │ │ │ -0002f2b0: 3278 2078 2020 2b20 7820 202b 2078 2020  2x x  + x  + x  
│ │ │ │ +0002f290: 2020 2020 7c0a 7c20 2020 2020 2020 2033      |.|        3
│ │ │ │ +0002f2a0: 2034 2020 2020 3320 2020 2020 3320 3420   4    3     3 4 
│ │ │ │ +0002f2b0: 2020 2033 2020 2020 3420 2020 2020 2020     3    4       
│ │ │ │  0002f2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f2e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002f2f0: 2020 2020 2033 2034 2020 2020 3320 2020       3 4    3   
│ │ │ │ -0002f300: 2020 3320 3420 2020 2033 2020 2020 3420    3 4    3    4 
│ │ │ │ +0002f2e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002f2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f330: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002f330: 2020 2020 7c0a 7c6f 3132 203a 2043 6820      |.|o12 : Ch 
│ │ │ │  0002f340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f380: 2020 2020 2020 2020 2020 7c0a 7c6f 3132            |.|o12
│ │ │ │ -0002f390: 203a 2043 6820 2020 2020 2020 2020 2020   : Ch           
│ │ │ │ -0002f3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f3d0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -0002f3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f420: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3133  ----------+.|i13
│ │ │ │ -0002f430: 203a 2073 2d63 2020 2020 2020 2020 2020   : s-c          
│ │ │ │ +0002f380: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0002f390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f3d0: 2d2d 2d2d 2b0a 7c69 3133 203a 2073 2d63  ----+.|i13 : s-c
│ │ │ │ +0002f3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f420: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002f430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f470: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002f480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f470: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002f480: 3220 2020 2020 2020 3220 2020 2020 2020  2       2       
│ │ │ │  0002f490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f4c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002f4d0: 2020 2020 2020 3220 2020 2020 2020 3220        2       2 
│ │ │ │ -0002f4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f4c0: 2020 2020 7c0a 7c6f 3133 203d 202d 2078      |.|o13 = - x
│ │ │ │ +0002f4d0: 2078 2020 2d20 3378 2020 2d20 3478 2078   x  - 3x  - 4x x
│ │ │ │ +0002f4e0: 2020 2b20 7820 2020 2020 2020 2020 2020    + x           
│ │ │ │  0002f4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f510: 2020 2020 2020 2020 2020 7c0a 7c6f 3133            |.|o13
│ │ │ │ -0002f520: 203d 202d 2078 2078 2020 2d20 3378 2020   = - x x  - 3x  
│ │ │ │ -0002f530: 2d20 3478 2078 2020 2b20 7820 2020 2020  - 4x x  + x     
│ │ │ │ +0002f510: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002f520: 3320 3420 2020 2020 3320 2020 2020 3320  3 4     3     3 
│ │ │ │ +0002f530: 3420 2020 2034 2020 2020 2020 2020 2020  4    4          
│ │ │ │  0002f540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f560: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002f570: 2020 2020 2020 3320 3420 2020 2020 3320        3 4     3 
│ │ │ │ -0002f580: 2020 2020 3320 3420 2020 2034 2020 2020      3 4    4    
│ │ │ │ +0002f560: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002f570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f5b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002f5b0: 2020 2020 7c0a 7c6f 3133 203a 2043 6820      |.|o13 : Ch 
│ │ │ │  0002f5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f600: 2020 2020 2020 2020 2020 7c0a 7c6f 3133            |.|o13
│ │ │ │ -0002f610: 203a 2043 6820 2020 2020 2020 2020 2020   : Ch           
│ │ │ │ -0002f620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f650: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -0002f660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134  ----------+.|i14
│ │ │ │ -0002f6b0: 203a 2073 2a63 2020 2020 2020 2020 2020   : s*c          
│ │ │ │ +0002f600: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0002f610: 2d2d 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 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f650: 2d2d 2d2d 2b0a 7c69 3134 203a 2073 2a63  ----+.|i14 : s*c
│ │ │ │ +0002f660: 2020 2020 2020 2020 2020 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 2020                  
│ │ │ │ +0002f6a0: 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020                  
│ │ │ │  0002f6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f6f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002f700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f6f0: 2020 2020 7c0a 7c20 2020 2020 2020 2032      |.|        2
│ │ │ │ +0002f700: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  0002f710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f740: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002f750: 2020 2020 2032 2020 2020 2020 3220 2020       2      2   
│ │ │ │ +0002f740: 2020 2020 7c0a 7c6f 3134 203d 2032 7820      |.|o14 = 2x 
│ │ │ │ +0002f750: 7820 202b 2078 2020 2b20 7820 7820 2020  x  + x  + x x   
│ │ │ │  0002f760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f790: 2020 2020 2020 2020 2020 7c0a 7c6f 3134            |.|o14
│ │ │ │ -0002f7a0: 203d 2032 7820 7820 202b 2078 2020 2b20   = 2x x  + x  + 
│ │ │ │ -0002f7b0: 7820 7820 2020 2020 2020 2020 2020 2020  x x             
│ │ │ │ +0002f790: 2020 2020 7c0a 7c20 2020 2020 2020 2033      |.|        3
│ │ │ │ +0002f7a0: 2034 2020 2020 3320 2020 2033 2034 2020   4    3    3 4  
│ │ │ │ +0002f7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f7e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002f7f0: 2020 2020 2033 2034 2020 2020 3320 2020       3 4    3   
│ │ │ │ -0002f800: 2033 2034 2020 2020 2020 2020 2020 2020   3 4            
│ │ │ │ +0002f7e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002f7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f830: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002f830: 2020 2020 7c0a 7c6f 3134 203a 2043 6820      |.|o14 : Ch 
│ │ │ │  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 7c0a 7c6f 3134            |.|o14
│ │ │ │ -0002f890: 203a 2043 6820 2020 2020 2020 2020 2020   : Ch           
│ │ │ │ -0002f8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f8d0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -0002f8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f920: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a46 6f72  ----------+..For
│ │ │ │ -0002f930: 2074 6865 2070 726f 6772 616d 6d65 720a   the programmer.
│ │ │ │ -0002f940: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0002f950: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a  ==..The object *
│ │ │ │ -0002f960: 6e6f 7465 2054 6f72 6963 4368 6f77 5269  note ToricChowRi
│ │ │ │ -0002f970: 6e67 3a20 546f 7269 6343 686f 7752 696e  ng: ToricChowRin
│ │ │ │ -0002f980: 672c 2069 7320 6120 2a6e 6f74 6520 6d65  g, is a *note me
│ │ │ │ -0002f990: 7468 6f64 2066 756e 6374 696f 6e3a 0a28  thod function:.(
│ │ │ │ -0002f9a0: 4d61 6361 756c 6179 3244 6f63 294d 6574  Macaulay2Doc)Met
│ │ │ │ -0002f9b0: 686f 6446 756e 6374 696f 6e2c 2e0a 0a2d  hodFunction,...-
│ │ │ │ -0002f9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002fa00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
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│ │ │ │ +0002f8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0002fc40: 686d 7f31 3633 3239 310a 4e6f 6465 3a20  hm.163291.Node: 
│ │ │ │ +0002fc50: 5365 6772 657f 3136 3739 3836 0a4e 6f64  Segre.167986.Nod
│ │ │ │ +0002fc60: 653a 2054 6f72 6963 4368 6f77 5269 6e67  e: ToricChowRing
│ │ │ │ +0002fc70: 7f31 3835 3839 330a 1f0a 456e 6420 5461  .185893...End Ta
│ │ │ │ +0002fc80: 6720 5461 626c 650a                      g Table.
│ │ ├── ./usr/share/info/Chordal.info.gz
│ │ │ ├── Chordal.info
│ │ │ │ @@ -3949,30 +3949,30 @@
│ │ │ │  0000f6c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0000f6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f710: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │  0000f720: 203d 2045 6c69 6d54 7265 657b 6120 3d3e   = ElimTree{a =>
│ │ │ │ -0000f730: 2062 2020 207d 2020 2020 2020 2020 2020   b   }          
│ │ │ │ +0000f730: 2063 7d20 2020 2020 2020 2020 2020 2020   c}             
│ │ │ │  0000f740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f760: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0000f770: 2020 2020 2020 2020 2020 2020 6220 3d3e              b =>
│ │ │ │  0000f780: 2063 2020 2020 2020 2020 2020 2020 2020   c              
│ │ │ │  0000f790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f7b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0000f7c0: 2020 2020 2020 2020 2020 2020 6320 3d3e              c =>
│ │ │ │  0000f7d0: 2064 2020 2020 2020 2020 2020 2020 2020   d              
│ │ │ │  0000f7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f800: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0000f810: 2020 2020 2020 2020 2020 2020 6420 3d3e              d =>
│ │ │ │ -0000f820: 206e 756c 6c20 2020 2020 2020 2020 2020   null           
│ │ │ │ +0000f820: 2062 2020 2020 2020 2020 2020 2020 2020   b              
│ │ │ │  0000f830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f850: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0000f860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4033,31 +4033,31 @@
│ │ │ │  0000fc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fc10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0000fc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fc60: 2020 2020 2020 2020 2020 207c 0a7c 6f37             |.|o7
│ │ │ │ -0000fc70: 203d 2043 686f 7264 616c 4e65 747b 2064   = ChordalNet{ d
│ │ │ │ -0000fc80: 203d 3e20 7b20 2c20 647d 2020 2020 7d20   => { , d}    } 
│ │ │ │ +0000fc70: 203d 2043 686f 7264 616c 4e65 747b 2061   = ChordalNet{ a
│ │ │ │ +0000fc80: 203d 3e20 7b61 2c20 207d 2020 2020 7d20   => {a,  }    } 
│ │ │ │  0000fc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fcb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0000fcc0: 2020 2020 2020 2020 2020 2020 2020 2062                 b
│ │ │ │ -0000fcd0: 203d 3e20 7b62 2c20 202c 2062 7d20 2020   => {b,  , b}   
│ │ │ │ +0000fcc0: 2020 2020 2020 2020 2020 2020 2020 2063                 c
│ │ │ │ +0000fcd0: 203d 3e20 7b20 2c20 637d 2020 2020 2020   => { , c}      
│ │ │ │  0000fce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fd00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0000fd10: 2020 2020 2020 2020 2020 2020 2020 2061                 a
│ │ │ │ -0000fd20: 203d 3e20 7b61 2c20 207d 2020 2020 2020   => {a,  }      
│ │ │ │ +0000fd10: 2020 2020 2020 2020 2020 2020 2020 2064                 d
│ │ │ │ +0000fd20: 203d 3e20 7b20 2c20 647d 2020 2020 2020   => { , d}      
│ │ │ │  0000fd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fd50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0000fd60: 2020 2020 2020 2020 2020 2020 2020 2063                 c
│ │ │ │ -0000fd70: 203d 3e20 7b20 2c20 637d 2020 2020 2020   => { , c}      
│ │ │ │ +0000fd60: 2020 2020 2020 2020 2020 2020 2020 2062                 b
│ │ │ │ +0000fd70: 203d 3e20 7b62 2c20 202c 2062 7d20 2020   => {b,  , b}   
│ │ │ │  0000fd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fda0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0000fdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fde0: 2020 2020 2020 2020 2020 2020 2020 2020
│ │ ├── ./usr/share/info/CohomCalg.info.gz
│ │ │ ├── CohomCalg.info
│ │ │ │ @@ -1042,15 +1042,15 @@
│ │ │ │  00004110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  00004130: 7c69 3230 203a 2065 6c61 7073 6564 5469  |i20 : elapsedTi
│ │ │ │  00004140: 6d65 2068 7665 6373 203d 2063 6f68 6f6d  me hvecs = cohom
│ │ │ │  00004150: 4361 6c67 2858 2c20 4432 2920 2020 2020  Calg(X, D2)     
│ │ │ │  00004160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004170: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00004180: 7c20 2d2d 2032 2e36 3734 3038 7320 656c  | -- 2.67408s el
│ │ │ │ +00004180: 7c20 2d2d 2033 2e31 3131 3834 7320 656c  | -- 3.11184s el
│ │ │ │  00004190: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │  000041a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000041b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000041c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  000041d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000041e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000041f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -1677,15 +1677,15 @@
│ │ │ │  000068c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000068d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  000068e0: 7c69 3233 203a 2065 6c61 7073 6564 5469  |i23 : elapsedTi
│ │ │ │  000068f0: 6d65 2063 6f68 6f6d 7665 6331 203d 2063  me cohomvec1 = c
│ │ │ │  00006900: 6f68 6f6d 4361 6c67 2858 5f33 202b 2058  ohomCalg(X_3 + X
│ │ │ │  00006910: 5f37 202b 2058 5f38 2920 2020 2020 2020  _7 + X_8)       
│ │ │ │  00006920: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00006930: 7c20 2d2d 202e 3330 3239 3135 7320 656c  | -- .302915s el
│ │ │ │ +00006930: 7c20 2d2d 202e 3531 3436 3638 7320 656c  | -- .514668s el
│ │ │ │  00006940: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │  00006950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006970: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00006980: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00006990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000069a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -1712,15 +1712,15 @@
│ │ │ │  00006af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00006b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  00006b10: 7c69 3234 203a 2065 6c61 7073 6564 5469  |i24 : elapsedTi
│ │ │ │  00006b20: 6d65 2063 6f68 6f6d 7665 6332 203d 2066  me cohomvec2 = f
│ │ │ │  00006b30: 6f72 206a 2066 726f 6d20 3020 746f 2064  or j from 0 to d
│ │ │ │  00006b40: 696d 2058 206c 6973 7420 7261 6e6b 2048  im X list rank H
│ │ │ │  00006b50: 485e 6a28 582c 2020 2020 2020 2020 7c0a  H^j(X,        |.
│ │ │ │ -00006b60: 7c20 2d2d 2031 302e 3937 3932 7320 656c  | -- 10.9792s el
│ │ │ │ +00006b60: 7c20 2d2d 2039 2e32 3134 3233 7320 656c  | -- 9.21423s 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 3334 3231 3236 7320 656c  | -- .342126s el
│ │ │ │ +000070b0: 7c20 2d2d 202e 3533 3034 3434 7320 656c  | -- .530444s 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 3531 3733 3436 7320 656c  | -- .517346s el
│ │ │ │ +000072e0: 7c20 2d2d 202e 3439 3235 3437 7320 656c  | -- .492547s 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 3531 3733 3733 7320 656c  | -- .517373s el
│ │ │ │ +00007330: 7c20 2d2d 202e 3439 3235 3734 7320 656c  | -- .492574s 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
│ │ │ │ @@ -4343,18 +4343,18 @@
│ │ │ │  00010f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010f80: 2d2d 2d2b 0a7c 6937 203a 2074 696d 6520  ---+.|i7 : time 
│ │ │ │  00010f90: 4720 3d20 4569 7365 6e62 7564 5368 616d  G = EisenbudSham
│ │ │ │  00010fa0: 6173 6828 6666 2c46 2c6c 656e 2920 2020  ash(ff,F,len)   
│ │ │ │  00010fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010fd0: 2020 207c 0a7c 202d 2d20 7573 6564 2036     |.| -- used 6
│ │ │ │ -00010fe0: 2e36 3632 3932 7320 2863 7075 293b 2034  .66292s (cpu); 4
│ │ │ │ -00010ff0: 2e38 3937 3937 7320 2874 6872 6561 6429  .89797s (thread)
│ │ │ │ -00011000: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ +00010fd0: 2020 207c 0a7c 202d 2d20 7573 6564 2037     |.| -- used 7
│ │ │ │ +00010fe0: 2e36 3437 3273 2028 6370 7529 3b20 352e  .6472s (cpu); 5.
│ │ │ │ +00010ff0: 3737 3834 3473 2028 7468 7265 6164 293b  77844s (thread);
│ │ │ │ +00011000: 2030 7320 2867 6329 2020 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                  
│ │ │ │  00011070: 2020 207c 0a7c 2020 2020 202f 2020 2020     |.|     /    
│ │ │ │ @@ -4884,17 +4884,17 @@
│ │ │ │  00013130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013150: 2d2b 0a7c 6932 3020 3a20 4646 203d 2074  -+.|i20 : FF = t
│ │ │ │  00013160: 696d 6520 5368 616d 6173 6828 5231 2c46  ime Shamash(R1,F
│ │ │ │  00013170: 2c34 2920 2020 2020 2020 2020 2020 2020  ,4)             
│ │ │ │  00013180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013190: 2020 207c 0a7c 202d 2d20 7573 6564 2030     |.| -- used 0
│ │ │ │ -000131a0: 2e31 3639 3335 3673 2028 6370 7529 3b20  .169356s (cpu); 
│ │ │ │ -000131b0: 302e 3039 3630 3434 3573 2028 7468 7265  0.0960445s (thre
│ │ │ │ -000131c0: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │ +000131a0: 2e32 3035 3334 3873 2028 6370 7529 3b20  .205348s (cpu); 
│ │ │ │ +000131b0: 302e 3131 3530 3534 7320 2874 6872 6561  0.115054s (threa
│ │ │ │ +000131c0: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │  000131d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000131e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000131f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013210: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00013220: 2020 3120 2020 2020 2020 3620 2020 2020    1       6     
│ │ │ │  00013230: 2020 3138 2020 2020 2020 2033 3820 2020    18       38   
│ │ │ │ @@ -4925,17 +4925,17 @@
│ │ │ │  000133c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000133d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000133e0: 2d2d 2d2d 2d2b 0a7c 6932 3120 3a20 4747  -----+.|i21 : GG
│ │ │ │  000133f0: 203d 2074 696d 6520 4569 7365 6e62 7564   = time Eisenbud
│ │ │ │  00013400: 5368 616d 6173 6828 6666 2c46 2c34 2920  Shamash(ff,F,4) 
│ │ │ │  00013410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013420: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -00013430: 6564 2030 2e39 3337 3435 3973 2028 6370  ed 0.937459s (cp
│ │ │ │ -00013440: 7529 3b20 302e 3732 3039 3132 7320 2874  u); 0.720912s (t
│ │ │ │ -00013450: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ +00013430: 6564 2031 2e32 3038 3334 7320 2863 7075  ed 1.20834s (cpu
│ │ │ │ +00013440: 293b 2030 2e39 3531 3438 3673 2028 7468  ); 0.951486s (th
│ │ │ │ +00013450: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │  00013460: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00013470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000134a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000134b0: 2020 2020 2f20 525c 3120 2020 2020 2f20      / R\1     / 
│ │ │ │  000134c0: 525c 3620 2020 2020 2f20 525c 3138 2020  R\6     / R\18  
│ │ │ │ @@ -4982,16 +4982,16 @@
│ │ │ │  00013750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  00013780: 0a7c 6932 3220 3a20 4747 203d 2074 696d  .|i22 : GG = tim
│ │ │ │  00013790: 6520 4569 7365 6e62 7564 5368 616d 6173  e EisenbudShamas
│ │ │ │  000137a0: 6828 5231 2c46 5b32 5d2c 3429 2020 2020  h(R1,F[2],4)    
│ │ │ │  000137b0: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ -000137c0: 6420 302e 3933 3032 3673 2028 6370 7529  d 0.93026s (cpu)
│ │ │ │ -000137d0: 3b20 302e 3730 3231 3332 7320 2874 6872  ; 0.702132s (thr
│ │ │ │ +000137c0: 6420 312e 3038 3334 3573 2028 6370 7529  d 1.08345s (cpu)
│ │ │ │ +000137d0: 3b20 302e 3834 3238 3634 7320 2874 6872  ; 0.842864s (thr
│ │ │ │  000137e0: 6561 6429 3b20 3073 2028 6763 297c 0a7c  ead); 0s (gc)|.|
│ │ │ │  000137f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013820: 2020 2020 7c0a 7c20 2020 2020 2020 2031      |.|        1
│ │ │ │  00013830: 2020 2020 2020 2036 2020 2020 2020 2031         6       1
│ │ │ │  00013840: 3820 2020 2020 2020 3338 2020 2020 2020  8       38      
│ │ │ │ @@ -28230,45 +28230,45 @@
│ │ │ │  0006e450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006e460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006e470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  0006e480: 6932 203a 2073 756d 5477 6f4d 6f6e 6f6d  i2 : sumTwoMonom
│ │ │ │  0006e490: 6961 6c73 2832 2c33 2920 2020 2020 2020  ials(2,3)       
│ │ │ │  0006e4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e4b0: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -0006e4c0: 6564 2030 2e33 3730 3233 3973 2028 6370  ed 0.370239s (cp
│ │ │ │ -0006e4d0: 7529 3b20 302e 3331 3937 3339 7320 2874  u); 0.319739s (t
│ │ │ │ -0006e4e0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ +0006e4c0: 6564 2030 2e35 3833 3473 2028 6370 7529  ed 0.5834s (cpu)
│ │ │ │ +0006e4d0: 3b20 302e 3431 3131 3135 7320 2874 6872  ; 0.411115s (thr
│ │ │ │ +0006e4e0: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │  0006e4f0: 207c 0a7c 3220 2020 2020 2020 2020 2020   |.|2           
│ │ │ │  0006e500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e520: 2020 2020 2020 2020 2020 207c 0a7c 5461             |.|Ta
│ │ │ │  0006e530: 6c6c 797b 7b7b 322c 2032 7d2c 207b 312c  lly{{{2, 2}, {1,
│ │ │ │  0006e540: 2032 7d7d 203d 3e20 337d 2020 2020 2020   2}} => 3}      
│ │ │ │  0006e550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e560: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0006e570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e590: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0006e5a0: 0a7c 202d 2d20 7573 6564 2030 2e32 3130  .| -- used 0.210
│ │ │ │ -0006e5b0: 3130 3473 2028 6370 7529 3b20 302e 3133  104s (cpu); 0.13
│ │ │ │ -0006e5c0: 3831 3539 7320 2874 6872 6561 6429 3b20  8159s (thread); 
│ │ │ │ +0006e5a0: 0a7c 202d 2d20 7573 6564 2030 2e33 3235  .| -- used 0.325
│ │ │ │ +0006e5b0: 3639 3873 2028 6370 7529 3b20 302e 3138  698s (cpu); 0.18
│ │ │ │ +0006e5c0: 3930 3531 7320 2874 6872 6561 6429 3b20  9051s (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 207c 0a7c 5461 6c6c 797b 7b7b 322c     |.|Tally{{{2,
│ │ │ │  0006e620: 2032 7d2c 207b 312c 2032 7d7d 203d 3e20   2}, {1, 2}} => 
│ │ │ │  0006e630: 317d 2020 2020 2020 2020 2020 2020 2020  1}              
│ │ │ │  0006e640: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0006e650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e680: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -0006e690: 6564 2033 2e36 3937 652d 3036 7320 2863  ed 3.697e-06s (c
│ │ │ │ -0006e6a0: 7075 293b 2033 2e33 3236 652d 3036 7320  pu); 3.326e-06s 
│ │ │ │ +0006e690: 6564 2033 2e34 3137 652d 3036 7320 2863  ed 3.417e-06s (c
│ │ │ │ +0006e6a0: 7075 293b 2033 2e32 3132 652d 3036 7320  pu); 3.212e-06s 
│ │ │ │  0006e6b0: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │  0006e6c0: 297c 0a7c 3420 2020 2020 2020 2020 2020  )|.|4           
│ │ │ │  0006e6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e6f0: 2020 2020 2020 2020 2020 207c 0a7c 5461             |.|Ta
│ │ │ │  0006e700: 6c6c 797b 7d20 2020 2020 2020 2020 2020  lly{}           
│ │ │ │  0006e710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -28812,34 +28812,34 @@
│ │ │ │  000708b0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  000708c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000708d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000708e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │  000708f0: 203a 2074 776f 4d6f 6e6f 6d69 616c 7328   : twoMonomials(
│ │ │ │  00070900: 322c 3329 2020 2020 2020 2020 2020 2020  2,3)            
│ │ │ │  00070910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070920: 2020 207c 0a7c 202d 2d20 7573 6564 2030     |.| -- used 0
│ │ │ │ -00070930: 2e38 3032 3531 3773 2028 6370 7529 3b20  .802517s (cpu); 
│ │ │ │ -00070940: 302e 3538 3533 3538 7320 2874 6872 6561  0.585358s (threa
│ │ │ │ -00070950: 6429 3b20 3073 2028 6763 297c 0a7c 3220  d); 0s (gc)|.|2 
│ │ │ │ +00070920: 2020 207c 0a7c 202d 2d20 7573 6564 2031     |.| -- used 1
│ │ │ │ +00070930: 2e32 3139 3637 7320 2863 7075 293b 2030  .21967s (cpu); 0
│ │ │ │ +00070940: 2e37 3339 3334 3173 2028 7468 7265 6164  .739341s (thread
│ │ │ │ +00070950: 293b 2030 7320 2867 6329 207c 0a7c 3220  ); 0s (gc) |.|2 
│ │ │ │  00070960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070990: 2020 207c 0a7c 5461 6c6c 797b 7b7b 312c     |.|Tally{{{1,
│ │ │ │  000709a0: 2031 7d7d 203d 3e20 3220 2020 2020 2020   1}} => 2       
│ │ │ │  000709b0: 207d 2020 2020 2020 2020 2020 2020 2020   }              
│ │ │ │  000709c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000709d0: 2020 2020 7b7b 322c 2032 7d2c 207b 312c      {{2, 2}, {1,
│ │ │ │  000709e0: 2032 7d7d 203d 3e20 3420 2020 2020 2020   2}} => 4       
│ │ │ │  000709f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070a00: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00070a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070a30: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -00070a40: 2d20 7573 6564 2030 2e34 3031 3330 3373  - used 0.401303s
│ │ │ │ -00070a50: 2028 6370 7529 3b20 302e 3333 3530 3535   (cpu); 0.335055
│ │ │ │ +00070a40: 2d20 7573 6564 2030 2e35 3837 3931 3873  - used 0.587918s
│ │ │ │ +00070a50: 2028 6370 7529 3b20 302e 3432 3731 3938   (cpu); 0.427198
│ │ │ │  00070a60: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │  00070a70: 6763 297c 0a7c 3320 2020 2020 2020 2020  gc)|.|3         
│ │ │ │  00070a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070aa0: 2020 2020 2020 2020 2020 207c 0a7c 5461             |.|Ta
│ │ │ │  00070ab0: 6c6c 797b 7b7b 322c 2032 7d2c 207b 312c  lly{{{2, 2}, {1,
│ │ │ │  00070ac0: 2032 7d7d 203d 3e20 327d 2020 2020 2020   2}} => 2}      
│ │ │ │ @@ -28848,17 +28848,17 @@
│ │ │ │  00070af0: 2033 7d2c 207b 322c 2033 7d7d 203d 3e20   3}, {2, 3}} => 
│ │ │ │  00070b00: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │  00070b10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00070b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070b50: 2020 207c 0a7c 202d 2d20 7573 6564 2030     |.| -- used 0
│ │ │ │ -00070b60: 2e32 3032 3232 3173 2028 6370 7529 3b20  .202221s (cpu); 
│ │ │ │ -00070b70: 302e 3133 3830 3435 7320 2874 6872 6561  0.138045s (threa
│ │ │ │ -00070b80: 6429 3b20 3073 2028 6763 297c 0a7c 3420  d); 0s (gc)|.|4 
│ │ │ │ +00070b60: 2e32 3139 3234 7320 2863 7075 293b 2030  .21924s (cpu); 0
│ │ │ │ +00070b70: 2e31 3337 3738 3773 2028 7468 7265 6164  .137787s (thread
│ │ │ │ +00070b80: 293b 2030 7320 2867 6329 207c 0a7c 3420  ); 0s (gc) |.|4 
│ │ │ │  00070b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00070bc0: 2020 207c 0a7c 5461 6c6c 797b 7b7b 322c     |.|Tally{{{2,
│ │ │ │  00070bd0: 2032 7d2c 207b 312c 2032 7d7d 203d 3e20   2}, {1, 2}} => 
│ │ │ │  00070be0: 317d 2020 2020 2020 2020 2020 2020 2020  1}              
│ │ │ │  00070bf0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ ├── ./usr/share/info/ConnectionMatrices.info.gz
│ │ │ ├── ConnectionMatrices.info
│ │ │ │ @@ -2415,30 +2415,30 @@
│ │ │ │  000096e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000096f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  00009700: 7c69 3920 3a20 656c 6170 7365 6454 696d  |i9 : elapsedTim
│ │ │ │  00009710: 6520 4120 3d20 636f 6e6e 6563 7469 6f6e  e A = connection
│ │ │ │  00009720: 4d61 7472 6963 6573 2049 3b20 2020 2020  Matrices I;     
│ │ │ │  00009730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009740: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00009750: 7c20 2d2d 2032 2e38 3130 3738 7320 656c  | -- 2.81078s el
│ │ │ │ +00009750: 7c20 2d2d 2032 2e33 3632 3435 7320 656c  | -- 2.36245s el
│ │ │ │  00009760: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │  00009770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009790: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  000097a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000097b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000097c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000097d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000097e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  000097f0: 7c69 3130 203a 2065 6c61 7073 6564 5469  |i10 : elapsedTi
│ │ │ │  00009800: 6d65 2061 7373 6572 7420 6973 496e 7465  me assert isInte
│ │ │ │  00009810: 6772 6162 6c65 2041 2020 2020 2020 2020  grable A        
│ │ │ │  00009820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009830: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00009840: 7c20 2d2d 2035 2e39 3430 3133 7320 656c  | -- 5.94013s el
│ │ │ │ +00009840: 7c20 2d2d 2033 2e39 3533 3334 7320 656c  | -- 3.95334s el
│ │ │ │  00009850: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │  00009860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009880: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00009890: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000098a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000098b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -4559,16 +4559,16 @@
│ │ │ │  00011ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011d00: 2b0a 7c69 3134 203a 2065 6c61 7073 6564  +.|i14 : elapsed
│ │ │ │  00011d10: 5469 6d65 2067 203d 2067 6175 6765 4d61  Time g = gaugeMa
│ │ │ │  00011d20: 7472 6978 2849 2c20 4229 3b20 2020 2020  trix(I, B);     
│ │ │ │  00011d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011d50: 7c0a 7c20 2d2d 202e 3733 3532 3373 2065  |.| -- .73523s e
│ │ │ │ -00011d60: 6c61 7073 6564 2020 2020 2020 2020 2020  lapsed          
│ │ │ │ +00011d50: 7c0a 7c20 2d2d 202e 3439 3132 3131 7320  |.| -- .491211s 
│ │ │ │ +00011d60: 656c 6170 7365 6420 2020 2020 2020 2020  elapsed         
│ │ │ │  00011d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011da0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00011db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4589,30 +4589,30 @@
│ │ │ │  00011ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011ee0: 2b0a 7c69 3135 203a 2065 6c61 7073 6564  +.|i15 : elapsed
│ │ │ │  00011ef0: 5469 6d65 2041 3120 3d20 6761 7567 6554  Time A1 = gaugeT
│ │ │ │  00011f00: 7261 6e73 666f 726d 2867 2c20 4129 3b20  ransform(g, A); 
│ │ │ │  00011f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011f30: 7c0a 7c20 2d2d 2031 2e35 3730 3432 7320  |.| -- 1.57042s 
│ │ │ │ +00011f30: 7c0a 7c20 2d2d 2031 2e30 3734 3131 7320  |.| -- 1.07411s 
│ │ │ │  00011f40: 656c 6170 7365 6420 2020 2020 2020 2020  elapsed         
│ │ │ │  00011f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011f80: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00011f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011fd0: 2b0a 7c69 3136 203a 2065 6c61 7073 6564  +.|i16 : elapsed
│ │ │ │  00011fe0: 5469 6d65 2061 7373 6572 7420 6973 496e  Time assert isIn
│ │ │ │  00011ff0: 7465 6772 6162 6c65 2041 3120 2020 2020  tegrable A1     
│ │ │ │  00012000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012020: 7c0a 7c20 2d2d 202e 3830 3836 3135 7320  |.| -- .808615s 
│ │ │ │ +00012020: 7c0a 7c20 2d2d 202e 3738 3732 3332 7320  |.| -- .787232s 
│ │ │ │  00012030: 656c 6170 7365 6420 2020 2020 2020 2020  elapsed         
│ │ │ │  00012040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012070: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00012080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -5030,31 +5030,31 @@
│ │ │ │  00013a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013a70: 2d2d 2d2d 2d2b 0a7c 6931 3920 3a20 656c  -----+.|i19 : el
│ │ │ │  00013a80: 6170 7365 6454 696d 6520 4132 203d 2067  apsedTime A2 = g
│ │ │ │  00013a90: 6175 6765 5472 616e 7366 6f72 6d28 6368  augeTransform(ch
│ │ │ │  00013aa0: 616e 6765 4570 732c 2041 3129 3b20 2020  angeEps, A1);   
│ │ │ │  00013ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013ac0: 2020 2020 207c 0a7c 202d 2d20 2e34 3936       |.| -- .496
│ │ │ │ -00013ad0: 3137 3373 2065 6c61 7073 6564 2020 2020  173s elapsed    
│ │ │ │ +00013ac0: 2020 2020 207c 0a7c 202d 2d20 2e33 3137       |.| -- .317
│ │ │ │ +00013ad0: 3633 7320 656c 6170 7365 6420 2020 2020  63s elapsed     
│ │ │ │  00013ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013b10: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  00013b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013b60: 2d2d 2d2d 2d2b 0a7c 6932 3020 3a20 656c  -----+.|i20 : el
│ │ │ │  00013b70: 6170 7365 6454 696d 6520 6173 7365 7274  apsedTime assert
│ │ │ │  00013b80: 2069 7349 6e74 6567 7261 626c 6520 4132   isIntegrable A2
│ │ │ │  00013b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013bb0: 2020 2020 207c 0a7c 202d 2d20 2e38 3330       |.| -- .830
│ │ │ │ -00013bc0: 3437 3973 2065 6c61 7073 6564 2020 2020  479s elapsed    
│ │ │ │ +00013bb0: 2020 2020 207c 0a7c 202d 2d20 2e36 3039       |.| -- .609
│ │ │ │ +00013bc0: 3234 3373 2065 6c61 7073 6564 2020 2020  243s 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  ----------------
│ │ │ │ @@ -5440,31 +5440,31 @@
│ │ │ │  000153f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  00015410: 6937 203a 2065 6c61 7073 6564 5469 6d65  i7 : elapsedTime
│ │ │ │  00015420: 2041 203d 2063 6f6e 6e65 6374 696f 6e4d   A = connectionM
│ │ │ │  00015430: 6174 7269 6365 7320 493b 2020 2020 2020  atrices I;      
│ │ │ │  00015440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015450: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00015460: 202d 2d20 2e32 3637 3232 3673 2065 6c61   -- .267226s ela
│ │ │ │ +00015460: 202d 2d20 2e32 3033 3937 3273 2065 6c61   -- .203972s ela
│ │ │ │  00015470: 7073 6564 2020 2020 2020 2020 2020 2020  psed            
│ │ │ │  00015480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000154a0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │  000154b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000154c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000154d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000154e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000154f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  00015500: 6938 203a 2065 6c61 7073 6564 5469 6d65  i8 : elapsedTime
│ │ │ │  00015510: 2061 7373 6572 7420 6973 496e 7465 6772   assert isIntegr
│ │ │ │  00015520: 6162 6c65 2041 2020 2020 2020 2020 2020  able A          
│ │ │ │  00015530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015540: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00015550: 202d 2d20 2e32 3033 3838 3373 2065 6c61   -- .203883s ela
│ │ │ │ -00015560: 7073 6564 2020 2020 2020 2020 2020 2020  psed            
│ │ │ │ +00015550: 202d 2d20 2e31 3638 3936 7320 656c 6170   -- .16896s elap
│ │ │ │ +00015560: 7365 6420 2020 2020 2020 2020 2020 2020  sed             
│ │ │ │  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  ----------------
│ │ │ │  000155d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ ├── ./usr/share/info/Cremona.info.gz
│ │ │ ├── Cremona.info
│ │ │ │ @@ -147,16 +147,16 @@
│ │ │ │  00000920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00000930: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2074  -------+.|i2 : t
│ │ │ │  00000940: 696d 6520 7068 6920 3d20 746f 4d61 7020  ime phi = toMap 
│ │ │ │  00000950: 6d69 6e6f 7273 2833 2c6d 6174 7269 787b  minors(3,matrix{
│ │ │ │  00000960: 7b74 5f30 2e2e 745f 347d 2c7b 745f 312e  {t_0..t_4},{t_1.
│ │ │ │  00000970: 2e74 5f35 7d2c 7b74 5f32 2e2e 745f 367d  .t_5},{t_2..t_6}
│ │ │ │  00000980: 7d29 2020 2020 207c 0a7c 202d 2d20 7573  })     |.| -- us
│ │ │ │ -00000990: 6564 2030 2e30 3034 3330 3231 3573 2028  ed 0.00430215s (
│ │ │ │ -000009a0: 6370 7529 3b20 302e 3030 3432 3938 3432  cpu); 0.00429842
│ │ │ │ +00000990: 6564 2030 2e30 3035 3234 3131 3273 2028  ed 0.00524112s (
│ │ │ │ +000009a0: 6370 7529 3b20 302e 3030 3532 3339 3736  cpu); 0.00523976
│ │ │ │  000009b0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │  000009c0: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │  000009d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000009e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000009f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -322,16 +322,16 @@
│ │ │ │  00001410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00001420: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2074  -------+.|i3 : t
│ │ │ │  00001430: 696d 6520 4a20 3d20 6b65 726e 656c 2870  ime J = kernel(p
│ │ │ │  00001440: 6869 2c32 2920 2020 2020 2020 2020 2020  hi,2)           
│ │ │ │  00001450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001470: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -00001480: 6564 2030 2e31 3337 3233 3173 2028 6370  ed 0.137231s (cp
│ │ │ │ -00001490: 7529 3b20 302e 3036 3939 3637 3973 2028  u); 0.0699679s (
│ │ │ │ +00001480: 6564 2030 2e31 3439 3339 3973 2028 6370  ed 0.149399s (cp
│ │ │ │ +00001490: 7529 3b20 302e 3037 3330 3335 3573 2028  u); 0.0730355s (
│ │ │ │  000014a0: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │  000014b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000014c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000014d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000014e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000014f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -387,18 +387,18 @@
│ │ │ │  00001820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00001830: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2074  -------+.|i4 : t
│ │ │ │  00001840: 696d 6520 6465 6772 6565 4d61 7020 7068  ime degreeMap ph
│ │ │ │  00001850: 6920 2020 2020 2020 2020 2020 2020 2020  i               
│ │ │ │  00001860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001880: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -00001890: 6564 2030 2e30 3239 3434 7320 2863 7075  ed 0.02944s (cpu
│ │ │ │ -000018a0: 293b 2030 2e30 3239 3434 3435 7320 2874  ); 0.0294445s (t
│ │ │ │ -000018b0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ -000018c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00001890: 6564 2030 2e30 3333 3431 3736 7320 2863  ed 0.0334176s (c
│ │ │ │ +000018a0: 7075 293b 2030 2e30 3333 3432 3234 7320  pu); 0.0334224s 
│ │ │ │ +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                  
│ │ │ │ @@ -412,17 +412,17 @@
│ │ │ │  000019b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000019c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2074  -------+.|i5 : t
│ │ │ │  000019d0: 696d 6520 7072 6f6a 6563 7469 7665 4465  ime projectiveDe
│ │ │ │  000019e0: 6772 6565 7320 7068 6920 2020 2020 2020  grees phi       
│ │ │ │  000019f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001a10: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -00001a20: 6564 2030 2e36 3837 3536 7320 2863 7075  ed 0.68756s (cpu
│ │ │ │ -00001a30: 293b 2030 2e34 3837 3538 3673 2028 7468  ); 0.487586s (th
│ │ │ │ -00001a40: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │ +00001a20: 6564 2030 2e37 3136 3439 3673 2028 6370  ed 0.716496s (cp
│ │ │ │ +00001a30: 7529 3b20 302e 3534 3939 3936 7320 2874  u); 0.549996s (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                  
│ │ │ │  00001ab0: 2020 2020 2020 207c 0a7c 6f35 203d 207b         |.|o5 = {
│ │ │ │ @@ -447,16 +447,16 @@
│ │ │ │  00001be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00001bf0: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2074  -------+.|i6 : t
│ │ │ │  00001c00: 696d 6520 7072 6f6a 6563 7469 7665 4465  ime projectiveDe
│ │ │ │  00001c10: 6772 6565 7328 7068 692c 4e75 6d44 6567  grees(phi,NumDeg
│ │ │ │  00001c20: 7265 6573 3d3e 3029 2020 2020 2020 2020  rees=>0)        
│ │ │ │  00001c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001c40: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -00001c50: 6564 2030 2e30 3632 3332 3037 7320 2863  ed 0.0623207s (c
│ │ │ │ -00001c60: 7075 293b 2030 2e30 3632 3236 3533 7320  pu); 0.0622653s 
│ │ │ │ +00001c50: 6564 2030 2e30 3730 3435 3835 7320 2863  ed 0.0704585s (c
│ │ │ │ +00001c60: 7075 293b 2030 2e30 3730 3436 3731 7320  pu); 0.0704671s 
│ │ │ │  00001c70: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │  00001c80: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  00001c90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00001ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -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 692c 2020 2020 2020 2020 2020 2020  phi,            
│ │ │ │  00001e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001e70: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -00001e80: 6564 2030 2e30 3032 3134 3836 7320 2863  ed 0.0021486s (c
│ │ │ │ +00001e80: 6564 2030 2e30 3032 3535 3439 7320 2863  ed 0.0025549s (c
│ │ │ │  00001e90: 7075 293b 2020 2020 2020 2020 2020 2020  pu);            
│ │ │ │  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 446f 6d69 6e61  -------|.|Domina
│ │ │ │  00002380: 6e74 3d3e 4a29 2020 2020 2020 2020 2020  nt=>J)          
│ │ │ │  00002390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000023a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000023b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000023c0: 2020 2020 2020 207c 0a7c 2030 2e30 3032         |.| 0.002
│ │ │ │ -000023d0: 3134 3933 3973 2028 7468 7265 6164 293b  14939s (thread);
│ │ │ │ +000023d0: 3535 3839 3873 2028 7468 7265 6164 293b  55898s (thread);
│ │ │ │  000023e0: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │  000023f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002410: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00002420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -832,16 +832,16 @@
│ │ │ │  000033f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003400: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2074  -------+.|i8 : t
│ │ │ │  00003410: 696d 6520 7073 6920 3d20 696e 7665 7273  ime psi = invers
│ │ │ │  00003420: 654d 6170 2070 6869 2020 2020 2020 2020  eMap phi        
│ │ │ │  00003430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003450: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -00003460: 6564 2030 2e34 3734 3235 3273 2028 6370  ed 0.474252s (cp
│ │ │ │ -00003470: 7529 3b20 302e 3339 3435 3334 7320 2874  u); 0.394534s (t
│ │ │ │ +00003460: 6564 2030 2e34 3232 3932 3573 2028 6370  ed 0.422925s (cp
│ │ │ │ +00003470: 7529 3b20 302e 3432 3239 3331 7320 2874  u); 0.422931s (t
│ │ │ │  00003480: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │  00003490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000034a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000034b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000034c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000034d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000034e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -1117,18 +1117,18 @@
│ │ │ │  000045c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000045d0: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2074  -------+.|i9 : t
│ │ │ │  000045e0: 696d 6520 6973 496e 7665 7273 654d 6170  ime isInverseMap
│ │ │ │  000045f0: 2870 6869 2c70 7369 2920 2020 2020 2020  (phi,psi)       
│ │ │ │  00004600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004620: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -00004630: 6564 2030 2e30 3039 3331 3630 3373 2028  ed 0.00931603s (
│ │ │ │ -00004640: 6370 7529 3b20 302e 3030 3933 3138 3535  cpu); 0.00931855
│ │ │ │ -00004650: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ -00004660: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │ +00004630: 6564 2030 2e30 3130 3936 3437 7320 2863  ed 0.0109647s (c
│ │ │ │ +00004640: 7075 293b 2030 2e30 3130 3936 3631 7320  pu); 0.0109661s 
│ │ │ │ +00004650: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ +00004660: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  00004670: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00004680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000046a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000046b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000046c0: 2020 2020 2020 207c 0a7c 6f39 203d 2074         |.|o9 = t
│ │ │ │  000046d0: 7275 6520 2020 2020 2020 2020 2020 2020  rue             
│ │ │ │ @@ -1142,16 +1142,16 @@
│ │ │ │  00004750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004760: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20  -------+.|i10 : 
│ │ │ │  00004770: 7469 6d65 2064 6567 7265 654d 6170 2070  time degreeMap p
│ │ │ │  00004780: 7369 2020 2020 2020 2020 2020 2020 2020  si              
│ │ │ │  00004790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000047a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000047b0: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -000047c0: 6564 2030 2e34 3538 3439 3373 2028 6370  ed 0.458493s (cp
│ │ │ │ -000047d0: 7529 3b20 302e 3239 3432 3539 7320 2874  u); 0.294259s (t
│ │ │ │ +000047c0: 6564 2030 2e34 3836 3733 3173 2028 6370  ed 0.486731s (cp
│ │ │ │ +000047d0: 7529 3b20 302e 3236 3234 3339 7320 2874  u); 0.262439s (t
│ │ │ │  000047e0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │  000047f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004800: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00004810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -1167,16 +1167,16 @@
│ │ │ │  000048e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000048f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20  -------+.|i11 : 
│ │ │ │  00004900: 7469 6d65 2070 726f 6a65 6374 6976 6544  time projectiveD
│ │ │ │  00004910: 6567 7265 6573 2070 7369 2020 2020 2020  egrees psi      
│ │ │ │  00004920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004940: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -00004950: 6564 2035 2e32 3930 3034 7320 2863 7075  ed 5.29004s (cpu
│ │ │ │ -00004960: 293b 2034 2e36 3339 3638 7320 2874 6872  ); 4.63968s (thr
│ │ │ │ +00004950: 6564 2035 2e34 3433 3731 7320 2863 7075  ed 5.44371s (cpu
│ │ │ │ +00004960: 293b 2035 2e30 3234 3932 7320 2874 6872  ); 5.02492s (thr
│ │ │ │  00004970: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │  00004980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004990: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000049a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000049b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000049c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000049d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -1214,17 +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: 3230 3233 3873 2028 6370 7529 3b20 302e  20238s (cpu); 0.
│ │ │ │ -00004c50: 3030 3232 3033 3133 7320 2874 6872 6561  00220313s (threa
│ │ │ │ -00004c60: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +00004c40: 3631 3173 2028 6370 7529 3b20 302e 3030  611s (cpu); 0.00
│ │ │ │ +00004c50: 3236 3135 3431 7320 2874 6872 6561 6429  261541s (thread)
│ │ │ │ +00004c60: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 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 3537  .| -- used 0.157
│ │ │ │ -00005dc0: 3331 7320 2863 7075 293b 2030 2e30 3834  31s (cpu); 0.084
│ │ │ │ -00005dd0: 3938 3639 7320 2874 6872 6561 6429 3b20  9869s (thread); 
│ │ │ │ -00005de0: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │ +00005db0: 0a7c 202d 2d20 7573 6564 2030 2e31 3734  .| -- used 0.174
│ │ │ │ +00005dc0: 3732 7320 2863 7075 293b 2030 2e30 3930  72s (cpu); 0.090
│ │ │ │ +00005dd0: 3830 3273 2028 7468 7265 6164 293b 2030  802s (thread); 0
│ │ │ │ +00005de0: 7320 2867 6329 2020 2020 2020 2020 2020  s (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 3132  .| -- used 0.512
│ │ │ │ -00008700: 3035 3973 2028 6370 7529 3b20 302e 3432  059s (cpu); 0.42
│ │ │ │ -00008710: 3638 3638 7320 2874 6872 6561 6429 3b20  6868s (thread); 
│ │ │ │ +000086f0: 0a7c 202d 2d20 7573 6564 2030 2e34 3430  .| -- used 0.440
│ │ │ │ +00008700: 3031 3873 2028 6370 7529 3b20 302e 3434  018s (cpu); 0.44
│ │ │ │ +00008710: 3030 3231 7320 2874 6872 6561 6429 3b20  0021s (thread); 
│ │ │ │  00008720: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │  00008730: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00008740: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00008750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008780: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ @@ -2708,17 +2708,17 @@
│ │ │ │  0000a930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000a950: 0a7c 6931 3520 3a20 7469 6d65 2064 6567  .|i15 : time deg
│ │ │ │  0000a960: 7265 6573 2070 6869 5e28 2d31 2920 2020  rees phi^(-1)   
│ │ │ │  0000a970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a990: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000a9a0: 0a7c 202d 2d20 7573 6564 2030 2e33 3437  .| -- used 0.347
│ │ │ │ -0000a9b0: 3734 3973 2028 6370 7529 3b20 302e 3237  749s (cpu); 0.27
│ │ │ │ -0000a9c0: 3431 3431 7320 2874 6872 6561 6429 3b20  4141s (thread); 
│ │ │ │ +0000a9a0: 0a7c 202d 2d20 7573 6564 2030 2e34 3735  .| -- used 0.475
│ │ │ │ +0000a9b0: 3337 3373 2028 6370 7529 3b20 302e 3334  373s (cpu); 0.34
│ │ │ │ +0000a9c0: 3031 3532 7320 2874 6872 6561 6429 3b20  0152s (thread); 
│ │ │ │  0000a9d0: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │  0000a9e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000a9f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000aa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000aa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000aa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000aa30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ @@ -2743,17 +2743,17 @@
│ │ │ │  0000ab60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000ab70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000ab80: 0a7c 6931 3620 3a20 7469 6d65 2064 6567  .|i16 : time deg
│ │ │ │  0000ab90: 7265 6573 2070 6869 2020 2020 2020 2020  rees phi        
│ │ │ │  0000aba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000abb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000abc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000abd0: 0a7c 202d 2d20 7573 6564 2030 2e30 3138  .| -- used 0.018
│ │ │ │ -0000abe0: 3031 3133 7320 2863 7075 293b 2030 2e30  0113s (cpu); 0.0
│ │ │ │ -0000abf0: 3137 3639 3933 7320 2874 6872 6561 6429  176993s (thread)
│ │ │ │ +0000abd0: 0a7c 202d 2d20 7573 6564 2030 2e30 3739  .| -- used 0.079
│ │ │ │ +0000abe0: 3339 3532 7320 2863 7075 293b 2030 2e30  3952s (cpu); 0.0
│ │ │ │ +0000abf0: 3236 3739 3431 7320 2874 6872 6561 6429  267941s (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: 3230 3731 3873 2028 6370 7529 3b20 302e  20718s (cpu); 0.
│ │ │ │ -0000ae20: 3030 3332 3037 3434 7320 2874 6872 6561  00320744s (threa
│ │ │ │ +0000ae10: 3732 3633 3473 2028 6370 7529 3b20 302e  72634s (cpu); 0.
│ │ │ │ +0000ae20: 3030 3337 3331 3737 7320 2874 6872 6561  00373177s (threa
│ │ │ │  0000ae30: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │  0000ae40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000ae50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000ae60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ae70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ae80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ae90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ @@ -2843,18 +2843,18 @@
│ │ │ │  0000b1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000b1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000b1c0: 0a7c 6931 3820 3a20 7469 6d65 2064 6573  .|i18 : time des
│ │ │ │  0000b1d0: 6372 6962 6520 7068 695e 282d 3129 2020  cribe phi^(-1)  
│ │ │ │  0000b1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b200: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000b210: 0a7c 202d 2d20 7573 6564 2030 2e30 3039  .| -- used 0.009
│ │ │ │ -0000b220: 3937 3939 3773 2028 6370 7529 3b20 302e  97997s (cpu); 0.
│ │ │ │ -0000b230: 3030 3939 3830 3773 2028 7468 7265 6164  0099807s (thread
│ │ │ │ -0000b240: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ +0000b210: 0a7c 202d 2d20 7573 6564 2030 2e30 3130  .| -- used 0.010
│ │ │ │ +0000b220: 3932 3032 7320 2863 7075 293b 2030 2e30  9202s (cpu); 0.0
│ │ │ │ +0000b230: 3130 3932 3633 7320 2874 6872 6561 6429  109263s (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: 3530 3537 3173 2028 6370 7529 3b20 302e  50571s (cpu); 0.
│ │ │ │ -0000b730: 3030 3935 3036 3539 7320 2874 6872 6561  00950659s (threa
│ │ │ │ -0000b740: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +0000b710: 0a7c 202d 2d20 7573 6564 2030 2e30 3131  .| -- used 0.011
│ │ │ │ +0000b720: 3533 3133 7320 2863 7075 293b 2030 2e30  5313s (cpu); 0.0
│ │ │ │ +0000b730: 3131 3533 3639 7320 2874 6872 6561 6429  115369s (thread)
│ │ │ │ +0000b740: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │  0000b750: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000b760: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000b770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b7a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000b7b0: 0a7c 6f32 3020 3a20 4d75 6c74 6968 6f6d  .|o20 : Multihom
│ │ │ │ @@ -2958,18 +2958,18 @@
│ │ │ │  0000b8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000b8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000b8f0: 0a7c 6932 3120 3a20 7469 6d65 2064 6567  .|i21 : time deg
│ │ │ │  0000b900: 7265 6573 2066 2020 2020 2020 2020 2020  rees f          
│ │ │ │  0000b910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b930: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000b940: 0a7c 202d 2d20 7573 6564 2031 2e33 3333  .| -- used 1.333
│ │ │ │ -0000b950: 3237 7320 2863 7075 293b 2030 2e39 3535  27s (cpu); 0.955
│ │ │ │ -0000b960: 3132 7320 2874 6872 6561 6429 3b20 3073  12s (thread); 0s
│ │ │ │ -0000b970: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │ +0000b940: 0a7c 202d 2d20 7573 6564 2031 2e32 3037  .| -- used 1.207
│ │ │ │ +0000b950: 3331 7320 2863 7075 293b 2030 2e39 3837  31s (cpu); 0.987
│ │ │ │ +0000b960: 3330 3573 2028 7468 7265 6164 293b 2030  305s (thread); 0
│ │ │ │ +0000b970: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │  0000b980: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000b990: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000b9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b9d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000b9e0: 0a7c 6f32 3120 3d20 7b39 3034 2c20 3530  .|o21 = {904, 50
│ │ │ │ @@ -2993,18 +2993,18 @@
│ │ │ │  0000bb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000bb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000bb20: 0a7c 6932 3220 3a20 7469 6d65 2064 6567  .|i22 : time deg
│ │ │ │  0000bb30: 7265 6520 6620 2020 2020 2020 2020 2020  ree f           
│ │ │ │  0000bb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bb60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000bb70: 0a7c 202d 2d20 7573 6564 2031 2e36 3235  .| -- used 1.625
│ │ │ │ -0000bb80: 652d 3035 7320 2863 7075 293b 2031 2e35  e-05s (cpu); 1.5
│ │ │ │ -0000bb90: 3933 652d 3035 7320 2874 6872 6561 6429  93e-05s (thread)
│ │ │ │ -0000bba0: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ +0000bb70: 0a7c 202d 2d20 7573 6564 2032 2e31 3037  .| -- used 2.107
│ │ │ │ +0000bb80: 3465 2d30 3573 2028 6370 7529 3b20 312e  4e-05s (cpu); 1.
│ │ │ │ +0000bb90: 3935 3537 652d 3035 7320 2874 6872 6561  9557e-05s (threa
│ │ │ │ +0000bba0: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │  0000bbb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000bbc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000bbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bc00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000bc10: 0a7c 6f32 3220 3d20 3120 2020 2020 2020  .|o22 = 1       
│ │ │ │ @@ -3019,16 +3019,16 @@
│ │ │ │  0000bca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000bcb0: 0a7c 6932 3320 3a20 7469 6d65 2064 6573  .|i23 : time des
│ │ │ │  0000bcc0: 6372 6962 6520 6620 2020 2020 2020 2020  cribe f         
│ │ │ │  0000bcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bcf0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000bd00: 0a7c 202d 2d20 7573 6564 2030 2e30 3031  .| -- used 0.001
│ │ │ │ -0000bd10: 3631 3436 3573 2028 6370 7529 3b20 302e  61465s (cpu); 0.
│ │ │ │ -0000bd20: 3030 3136 3135 3535 7320 2874 6872 6561  00161555s (threa
│ │ │ │ +0000bd10: 3635 3139 3973 2028 6370 7529 3b20 302e  65199s (cpu); 0.
│ │ │ │ +0000bd20: 3030 3136 3537 3438 7320 2874 6872 6561  00165748s (threa
│ │ │ │  0000bd30: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │  0000bd40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000bd50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000bd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bd90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ @@ -4676,16 +4676,16 @@
│ │ │ │  00012430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012440: 2b0a 7c69 3420 3a20 7469 6d65 2070 7369  +.|i4 : time psi
│ │ │ │  00012450: 203d 2061 6273 7472 6163 7452 6174 696f   = abstractRatio
│ │ │ │  00012460: 6e61 6c4d 6170 2850 342c 5035 2c66 2920  nalMap(P4,P5,f) 
│ │ │ │  00012470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012490: 7c0a 7c20 2d2d 2075 7365 6420 302e 3030  |.| -- used 0.00
│ │ │ │ -000124a0: 3034 3130 3031 3973 2028 6370 7529 3b20  0410019s (cpu); 
│ │ │ │ -000124b0: 302e 3030 3034 3036 3239 3273 2028 7468  0.000406292s (th
│ │ │ │ +000124a0: 3034 3634 3131 3973 2028 6370 7529 3b20  0464119s (cpu); 
│ │ │ │ +000124b0: 302e 3030 3034 3538 3336 3973 2028 7468  0.000458369s (th
│ │ │ │  000124c0: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │  000124d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000124e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000124f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4746,18 +4746,18 @@
│ │ │ │  00012890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000128a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000128b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  000128c0: 6935 203a 2074 696d 6520 7072 6f6a 6563  i5 : time projec
│ │ │ │  000128d0: 7469 7665 4465 6772 6565 7328 7073 692c  tiveDegrees(psi,
│ │ │ │  000128e0: 3329 2020 2020 2020 2020 2020 2020 2020  3)              
│ │ │ │  000128f0: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -00012900: 2d20 7573 6564 2030 2e32 3938 3835 3373  - used 0.298853s
│ │ │ │ -00012910: 2028 6370 7529 3b20 302e 3138 3532 3773   (cpu); 0.18527s
│ │ │ │ -00012920: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ -00012930: 6329 2020 2020 2020 207c 0a7c 2020 2020  c)       |.|    
│ │ │ │ +00012900: 2d20 7573 6564 2030 2e33 3538 3632 3673  - used 0.358626s
│ │ │ │ +00012910: 2028 6370 7529 3b20 302e 3230 3136 3134   (cpu); 0.201614
│ │ │ │ +00012920: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ +00012930: 6763 2920 2020 2020 207c 0a7c 2020 2020  gc)      |.|    
│ │ │ │  00012940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012970: 2020 2020 2020 207c 0a7c 6f35 203d 2032         |.|o5 = 2
│ │ │ │  00012980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000129a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4765,17 +4765,17 @@
│ │ │ │  000129c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000129d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000129e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000129f0: 2d2d 2d2b 0a7c 6936 203a 2074 696d 6520  ---+.|i6 : time 
│ │ │ │  00012a00: 7261 7469 6f6e 616c 4d61 7020 7073 6920  rationalMap psi 
│ │ │ │  00012a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012a30: 207c 0a7c 202d 2d20 7573 6564 2030 2e35   |.| -- used 0.5
│ │ │ │ -00012a40: 3034 3032 3673 2028 6370 7529 3b20 302e  04026s (cpu); 0.
│ │ │ │ -00012a50: 3336 3634 3939 7320 2874 6872 6561 6429  366499s (thread)
│ │ │ │ +00012a30: 207c 0a7c 202d 2d20 7573 6564 2030 2e34   |.| -- used 0.4
│ │ │ │ +00012a40: 3638 3135 3273 2028 6370 7529 3b20 302e  68152s (cpu); 0.
│ │ │ │ +00012a50: 3338 3436 3339 7320 2874 6872 6561 6429  384639s (thread)
│ │ │ │  00012a60: 3b20 3073 2028 6763 2920 2020 2020 207c  ; 0s (gc)      |
│ │ │ │  00012a70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00012a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012aa0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00012ab0: 6f36 203d 202d 2d20 7261 7469 6f6e 616c  o6 = -- rational
│ │ │ │  00012ac0: 206d 6170 202d 2d20 2020 2020 2020 2020   map --         
│ │ │ │ @@ -5189,16 +5189,16 @@
│ │ │ │  00014440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014450: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134 203a  --------+.|i14 :
│ │ │ │  00014460: 2074 696d 6520 5420 3d20 6162 7374 7261   time T = abstra
│ │ │ │  00014470: 6374 5261 7469 6f6e 616c 4d61 7028 492c  ctRationalMap(I,
│ │ │ │  00014480: 224f 4144 5022 2920 2020 2020 2020 2020  "OADP")         
│ │ │ │  00014490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000144a0: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ -000144b0: 6420 302e 3134 3931 3835 7320 2863 7075  d 0.149185s (cpu
│ │ │ │ -000144c0: 293b 2030 2e30 3736 3932 3939 7320 2874  ); 0.0769299s (t
│ │ │ │ +000144b0: 6420 302e 3136 3338 3439 7320 2863 7075  d 0.163849s (cpu
│ │ │ │ +000144c0: 293b 2030 2e30 3735 3230 3831 7320 2874  ); 0.0752081s (t
│ │ │ │  000144d0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 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                  
│ │ │ │ @@ -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 342e 3037 3736 3473 2028 6370  sed 4.07764s (cp
│ │ │ │ -00014980: 7529 3b20 322e 3132 3032 3273 2028 7468  u); 2.12022s (th
│ │ │ │ +00014970: 7365 6420 342e 3433 3234 3773 2028 6370  sed 4.43247s (cp
│ │ │ │ +00014980: 7529 3b20 322e 3234 3339 3173 2028 7468  u); 2.24391s (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 322e 3835 3634 652d 3035 7320  sed 2.8564e-05s 
│ │ │ │ -00014b20: 2863 7075 293b 2032 2e38 3237 3365 2d30  (cpu); 2.8273e-0
│ │ │ │ +00014b10: 7365 6420 322e 3634 3032 652d 3035 7320  sed 2.6402e-05s 
│ │ │ │ +00014b20: 2863 7075 293b 2032 2e35 3239 3365 2d30  (cpu); 2.5293e-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 --   
│ │ │ │ @@ -5344,17 +5344,17 @@
│ │ │ │  00014df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014e20: 2d2b 0a7c 6931 3720 3a20 7469 6d65 2070  -+.|i17 : time p
│ │ │ │  00014e30: 726f 6a65 6374 6976 6544 6567 7265 6573  rojectiveDegrees
│ │ │ │  00014e40: 2854 322c 3229 2020 2020 2020 2020 2020  (T2,2)          
│ │ │ │  00014e50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00014e60: 7c20 2d2d 2075 7365 6420 362e 3635 3930  | -- used 6.6590
│ │ │ │ -00014e70: 3173 2028 6370 7529 3b20 332e 3435 3639  1s (cpu); 3.4569
│ │ │ │ -00014e80: 3973 2028 7468 7265 6164 293b 2030 7320  9s (thread); 0s 
│ │ │ │ +00014e60: 7c20 2d2d 2075 7365 6420 372e 3031 3135  | -- used 7.0115
│ │ │ │ +00014e70: 3973 2028 6370 7529 3b20 332e 3534 3834  9s (cpu); 3.5484
│ │ │ │ +00014e80: 3373 2028 7468 7265 6164 293b 2030 7320  3s (thread); 0s 
│ │ │ │  00014e90: 2867 6329 2020 2020 2020 207c 0a7c 2020  (gc)       |.|  
│ │ │ │  00014ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014ed0: 2020 2020 2020 2020 7c0a 7c6f 3137 203d          |.|o17 =
│ │ │ │  00014ee0: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │  00014ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -5430,18 +5430,18 @@
│ │ │ │  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 2035 2e33 3833  .| -- used 5.383
│ │ │ │ -000153d0: 3637 7320 2863 7075 293b 2032 2e39 3231  67s (cpu); 2.921
│ │ │ │ -000153e0: 3139 7320 2874 6872 6561 6429 3b20 3073  19s (thread); 0s
│ │ │ │ -000153f0: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │ +000153c0: 0a7c 202d 2d20 7573 6564 2035 2e38 3233  .| -- used 5.823
│ │ │ │ +000153d0: 3136 7320 2863 7075 293b 2032 2e39 3131  16s (cpu); 2.911
│ │ │ │ +000153e0: 3373 2028 7468 7265 6164 293b 2030 7320  3s (thread); 0s 
│ │ │ │ +000153f0: 2867 6329 2020 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 
│ │ │ │  00015460: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │ @@ -6678,17 +6678,17 @@
│ │ │ │  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 3237 3230 3137  -- used 0.272017
│ │ │ │ -0001a1d0: 7320 2863 7075 293b 2030 2e32 3036 3936  s (cpu); 0.20696
│ │ │ │ -0001a1e0: 3973 2028 7468 7265 6164 293b 2030 7320  9s (thread); 0s 
│ │ │ │ +0001a1c0: 2d2d 2075 7365 6420 302e 3238 3735 3833  -- used 0.287583
│ │ │ │ +0001a1d0: 7320 2863 7075 293b 2030 2e32 3236 3137  s (cpu); 0.22617
│ │ │ │ +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
│ │ │ │ @@ -8043,16 +8043,16 @@
│ │ │ │  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 3232 3739 3335  -- used 0.227935
│ │ │ │ -0001f720: 7320 2863 7075 293b 2030 2e31 3632 3333  s (cpu); 0.16233
│ │ │ │ +0001f710: 2d2d 2075 7365 6420 302e 3233 3630 3673  -- used 0.23606s
│ │ │ │ +0001f720: 2028 6370 7529 3b20 302e 3137 3836 3332   (cpu); 0.178632
│ │ │ │  0001f730: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │  0001f740: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │  0001f750: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  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                  
│ │ │ │ @@ -10405,17 +10405,17 @@
│ │ │ │  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 2e33 3134 3637 7320 2863 7075 293b   2.31467s (cpu);
│ │ │ │ -00028ac0: 2031 2e37 3837 3336 7320 2874 6872 6561   1.78736s (threa
│ │ │ │ -00028ad0: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +00028ab0: 2032 2e31 3036 3734 7320 2863 7075 293b   2.10674s (cpu);
│ │ │ │ +00028ac0: 2031 2e37 3938 3473 2028 7468 7265 6164   1.7984s (thread
│ │ │ │ +00028ad0: 293b 2030 7320 2867 6329 2020 2020 2020  ); 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                  
│ │ │ │  00028b40: 2020 2020 207c 0a7c 6f38 203d 202d 2d20       |.|o8 = -- 
│ │ │ │ @@ -11710,17 +11710,17 @@
│ │ │ │  0002dbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dbe0: 2020 2020 207c 0a7c 4365 7274 6966 793a       |.|Certify:
│ │ │ │  0002dbf0: 206f 7574 7075 7420 6365 7274 6966 6965   output certifie
│ │ │ │  0002dc00: 6421 2020 2020 2020 2020 2020 2020 2020  d!              
│ │ │ │  0002dc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dc30: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -0002dc40: 2033 2e38 3833 3333 7320 2863 7075 293b   3.88333s (cpu);
│ │ │ │ -0002dc50: 2033 2e31 3136 3538 7320 2874 6872 6561   3.11658s (threa
│ │ │ │ -0002dc60: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +0002dc40: 2032 2e39 3031 3534 7320 2863 7075 293b   2.90154s (cpu);
│ │ │ │ +0002dc50: 2032 2e35 3373 2028 7468 7265 6164 293b   2.53s (thread);
│ │ │ │ +0002dc60: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │  0002dc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dc80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002dc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dcd0: 2020 2020 207c 0a7c 6f31 3020 3d20 2d2d       |.|o10 = --
│ │ │ │ @@ -13366,16 +13366,16 @@
│ │ │ │  00034350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034380: 2b0a 7c69 3320 3a20 7469 6d65 2043 6865  +.|i3 : time Che
│ │ │ │  00034390: 726e 5363 6877 6172 747a 4d61 6350 6865  rnSchwartzMacPhe
│ │ │ │  000343a0: 7273 6f6e 2043 2020 2020 2020 2020 2020  rson C          
│ │ │ │  000343b0: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ -000343c0: 7573 6564 2032 2e32 3538 3632 7320 2863  used 2.25862s (c
│ │ │ │ -000343d0: 7075 293b 2031 2e31 3833 3232 7320 2874  pu); 1.18322s (t
│ │ │ │ +000343c0: 7573 6564 2032 2e33 3639 3734 7320 2863  used 2.36974s (c
│ │ │ │ +000343d0: 7075 293b 2031 2e32 3535 3032 7320 2874  pu); 1.25502s (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              
│ │ │ │ @@ -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 2e36 3137 3834 7320 2863 7075 293b   1.61784s (cpu);
│ │ │ │ -00034680: 2031 2e31 3430 3331 7320 2874 6872 6561   1.14031s (threa
│ │ │ │ -00034690: 6429 3b20 3073 2028 6763 2920 2020 7c0a  d); 0s (gc)   |.
│ │ │ │ +00034670: 2031 2e34 3333 3039 7320 2863 7075 293b   1.43309s (cpu);
│ │ │ │ +00034680: 2030 2e39 3836 3034 3973 2028 7468 7265   0.986049s (thre
│ │ │ │ +00034690: 6164 293b 2030 7320 2867 6329 2020 7c0a  ad); 0s (gc)  |.
│ │ │ │  000346a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000346b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000346c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000346d0: 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 2e34 3339 3935 3273  - used 0.439952s
│ │ │ │ -000353a0: 2028 6370 7529 3b20 302e 3235 3839 3332   (cpu); 0.258932
│ │ │ │ +00035390: 2d20 7573 6564 2030 2e33 3530 3834 3673  - used 0.350846s
│ │ │ │ +000353a0: 2028 6370 7529 3b20 302e 3139 3437 3931   (cpu); 0.194791
│ │ │ │  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 3332 3334 3773  - used 0.132347s
│ │ │ │ -00035760: 2028 6370 7529 3b20 302e 3034 3439 3738   (cpu); 0.044978
│ │ │ │ -00035770: 3973 2028 7468 7265 6164 293b 2030 7320  9s (thread); 0s 
│ │ │ │ +00035750: 2d20 7573 6564 2030 2e31 3332 3433 3173  - used 0.132431s
│ │ │ │ +00035760: 2028 6370 7529 3b20 302e 3033 3837 3935   (cpu); 0.038795
│ │ │ │ +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: 3435 3336 3936 7320 2863 7075 293b 2030  453696s (cpu); 0
│ │ │ │ -0003fd70: 2e30 3435 3337 3037 7320 2874 6872 6561  .0453707s (threa
│ │ │ │ +0003fd60: 3535 3335 3337 7320 2863 7075 293b 2030  553537s (cpu); 0
│ │ │ │ +0003fd70: 2e30 3535 3038 3836 7320 2874 6872 6561  .0550886s (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                  
│ │ │ │ @@ -17511,16 +17511,16 @@
│ │ │ │  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 2e32   |.| -- used 1.2
│ │ │ │ -000446d0: 3438 3839 7320 2863 7075 293b 2030 2e37  4889s (cpu); 0.7
│ │ │ │ -000446e0: 3036 3430 3173 2028 7468 7265 6164 293b  06401s (thread);
│ │ │ │ +000446d0: 3236 3139 7320 2863 7075 293b 2030 2e37  2619s (cpu); 0.7
│ │ │ │ +000446e0: 3136 3037 3873 2028 7468 7265 6164 293b  16078s (thread);
│ │ │ │  000446f0: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │  00044700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044710: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00044720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -18325,17 +18325,17 @@
│ │ │ │  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: 3236 3537 3436 7320 2863 7075 293b 2030  265746s (cpu); 0
│ │ │ │ -000479c0: 2e31 3532 3938 7320 2874 6872 6561 6429  .15298s (thread)
│ │ │ │ -000479d0: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ +000479b0: 3331 3336 3438 7320 2863 7075 293b 2030  313648s (cpu); 0
│ │ │ │ +000479c0: 2e31 3731 3939 3173 2028 7468 7265 6164  .171991s (thread
│ │ │ │ +000479d0: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │  000479e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000479f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00047a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047a40: 2020 7c0a 7c6f 3220 3d20 3130 2020 2020    |.|o2 = 10    
│ │ │ │ @@ -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 3135 3135 3873 2028 6370 7529 3b20  0115158s (cpu); 
│ │ │ │ -00047ba0: 302e 3031 3039 3731 3573 2028 7468 7265  0.0109715s (thre
│ │ │ │ +00047b90: 3037 3436 3230 3373 2028 6370 7529 3b20  0746203s (cpu); 
│ │ │ │ +00047ba0: 302e 3032 3036 3339 3173 2028 7468 7265  0.0206391s (thre
│ │ │ │  00047bb0: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │  00047bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047bd0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00047be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -19033,17 +19033,17 @@
│ │ │ │  0004a580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004a590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  0004a5a0: 3420 3a20 7469 6d65 2066 6f72 6365 496d  4 : time forceIm
│ │ │ │  0004a5b0: 6167 6528 5068 692c 6964 6561 6c20 305f  age(Phi,ideal 0_
│ │ │ │  0004a5c0: 2874 6172 6765 7420 5068 6929 2920 2020  (target Phi))   
│ │ │ │  0004a5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a5e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0004a5f0: 2d2d 2075 7365 6420 302e 3030 3036 3037  -- used 0.000607
│ │ │ │ -0004a600: 3231 3973 2028 6370 7529 3b20 302e 3030  219s (cpu); 0.00
│ │ │ │ -0004a610: 3036 3031 3933 3973 2028 7468 7265 6164  0601939s (thread
│ │ │ │ +0004a5f0: 2d2d 2075 7365 6420 302e 3030 3038 3539  -- used 0.000859
│ │ │ │ +0004a600: 3734 3173 2028 6370 7529 3b20 302e 3030  741s (cpu); 0.00
│ │ │ │ +0004a610: 3038 3532 3931 3673 2028 7468 7265 6164  0852916s (thread
│ │ │ │  0004a620: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │  0004a630: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  0004a640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004a650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004a660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004a670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004a680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ @@ -19645,18 +19645,18 @@
│ │ │ │  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 3138 3831 3738 7320 2863  ed 0.0188178s (c
│ │ │ │ -0004cc40: 7075 293b 2030 2e30 3138 3433 3534 7320  pu); 0.0184354s 
│ │ │ │ -0004cc50: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ -0004cc60: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +0004cc30: 6564 2030 2e30 3831 3238 3873 2028 6370  ed 0.081288s (cp
│ │ │ │ +0004cc40: 7529 3b20 302e 3033 3033 3631 3473 2028  u); 0.0303614s (
│ │ │ │ +0004cc50: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │ +0004cc60: 2020 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  ----------------
│ │ │ │  0004ccc0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2070  -------+.|i4 : p
│ │ │ │  0004ccd0: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ @@ -20942,17 +20942,17 @@
│ │ │ │  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 3137 3136 3573 2028 6370 7529  0.0317165s (cpu)
│ │ │ │ -00051d50: 3b20 302e 3033 3132 3632 3473 2028 7468  ; 0.0312624s (th
│ │ │ │ -00051d60: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │ +00051d40: 302e 3035 3336 3837 3773 2028 6370 7529  0.0536877s (cpu)
│ │ │ │ +00051d50: 3b20 302e 3033 3738 3034 7320 2874 6872  ; 0.037804s (thr
│ │ │ │ +00051d60: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │  00051d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051d80: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  00051d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051dd0: 2d2d 2d2d 2b0a 7c69 3130 203a 2067 5f30  ----+.|i10 : g_0
│ │ │ │ @@ -21662,18 +21662,18 @@
│ │ │ │  000549d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000549e0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │  000549f0: 7469 6d65 2069 6465 616c 2070 6869 2020  time ideal phi  
│ │ │ │  00054a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054a30: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ -00054a40: 7365 6420 302e 3030 3335 3539 3973 2028  sed 0.0035599s (
│ │ │ │ -00054a50: 6370 7529 3b20 302e 3030 3335 3535 3232  cpu); 0.00355522
│ │ │ │ -00054a60: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ -00054a70: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │ +00054a40: 7365 6420 302e 3030 3338 3532 3333 7320  sed 0.00385233s 
│ │ │ │ +00054a50: 2863 7075 293b 2030 2e30 3033 3834 3931  (cpu); 0.0038491
│ │ │ │ +00054a60: 3673 2028 7468 7265 6164 293b 2030 7320  6s (thread); 0s 
│ │ │ │ +00054a70: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │  00054a80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00054a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054ad0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00054ae0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ @@ -22297,18 +22297,18 @@
│ │ │ │  00057180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057190: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │  000571a0: 7469 6d65 2069 6465 616c 2070 6869 2720  time ideal phi' 
│ │ │ │  000571b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000571c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000571d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000571e0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ -000571f0: 7365 6420 302e 3039 3330 3639 3173 2028  sed 0.0930691s (
│ │ │ │ -00057200: 6370 7529 3b20 302e 3039 3330 3438 3873  cpu); 0.0930488s
│ │ │ │ -00057210: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ -00057220: 6329 2020 2020 2020 2020 2020 2020 2020  c)              
│ │ │ │ +000571f0: 7365 6420 302e 3130 3338 3737 7320 2863  sed 0.103877s (c
│ │ │ │ +00057200: 7075 293b 2030 2e31 3033 3837 3973 2028  pu); 0.103879s (
│ │ │ │ +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 3536 3936 3973 2028 6370 7529   0.056969s (cpu)
│ │ │ │ -000611f0: 3b20 302e 3035 3639 3638 3773 2028 7468  ; 0.0569687s (th
│ │ │ │ +000611e0: 2030 2e30 3634 3234 3273 2028 6370 7529   0.064242s (cpu)
│ │ │ │ +000611f0: 3b20 302e 3036 3431 3239 3173 2028 7468  ; 0.0641291s (th
│ │ │ │  00061200: 7265 6164 293b 2030 7320 2867 6329 2020  read); 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 2e31 3834 3233 3273 2028  used 0.184232s (
│ │ │ │ -0006cd60: 6370 7529 3b20 302e 3132 3131 3573 2028  cpu); 0.12115s (
│ │ │ │ +0006cd50: 7573 6564 2030 2e31 3937 3834 7320 2863  used 0.19784s (c
│ │ │ │ +0006cd60: 7075 293b 2030 2e31 3133 3237 3673 2028  pu); 0.113276s (
│ │ │ │  0006cd70: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │  0006cd80: 2020 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 2e33   |.| -- used 0.3
│ │ │ │ -0006f820: 3731 3432 3873 2028 6370 7529 3b20 302e  71428s (cpu); 0.
│ │ │ │ -0006f830: 3232 3436 3339 7320 2874 6872 6561 6429  224639s (thread)
│ │ │ │ +0006f820: 3530 3038 3873 2028 6370 7529 3b20 302e  50088s (cpu); 0.
│ │ │ │ +0006f830: 3230 3834 3139 7320 2874 6872 6561 6429  208419s (thread)
│ │ │ │  0006f840: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │  0006f850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f860: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0006f870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -29536,16 +29536,16 @@
│ │ │ │  000735f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00073600: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │  00073610: 7469 6d65 2069 7342 6972 6174 696f 6e61  time isBirationa
│ │ │ │  00073620: 6c20 7068 6920 2020 2020 2020 2020 2020  l phi           
│ │ │ │  00073630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00073640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00073650: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ -00073660: 7365 6420 302e 3031 3933 3230 3173 2028  sed 0.0193201s (
│ │ │ │ -00073670: 6370 7529 3b20 302e 3031 3933 3230 3673  cpu); 0.0193206s
│ │ │ │ +00073660: 7365 6420 302e 3032 3232 3836 3173 2028  sed 0.0222861s (
│ │ │ │ +00073670: 6370 7529 3b20 302e 3032 3232 3835 3873  cpu); 0.0222858s
│ │ │ │  00073680: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │  00073690: 6329 2020 2020 2020 2020 2020 2020 2020  c)              
│ │ │ │  000736a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  000736b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000736c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000736d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000736e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -29566,16 +29566,16 @@
│ │ │ │  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 3336 3932 3573 2028  sed 0.0136925s (
│ │ │ │ -00073850: 6370 7529 3b20 302e 3031 3332 3931 3573  cpu); 0.0132915s
│ │ │ │ +00073840: 7365 6420 302e 3032 3635 3334 3973 2028  sed 0.0265349s (
│ │ │ │ +00073850: 6370 7529 3b20 302e 3031 3437 3431 3273  cpu); 0.0147412s
│ │ │ │  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                  
│ │ │ │ @@ -29739,17 +29739,17 @@
│ │ │ │  000742a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000742b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  000742c0: 0a7c 4365 7274 6966 793a 206f 7574 7075  .|Certify: outpu
│ │ │ │  000742d0: 7420 6365 7274 6966 6965 6421 2020 2020  t certified!    
│ │ │ │  000742e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000742f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074300: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00074310: 0a7c 202d 2d20 7573 6564 2032 2e35 3833  .| -- used 2.583
│ │ │ │ -00074320: 3435 7320 2863 7075 293b 2032 2e30 3131  45s (cpu); 2.011
│ │ │ │ -00074330: 3334 7320 2874 6872 6561 6429 3b20 3073  34s (thread); 0s
│ │ │ │ +00074310: 0a7c 202d 2d20 7573 6564 2032 2e35 3633  .| -- used 2.563
│ │ │ │ +00074320: 3636 7320 2863 7075 293b 2032 2e32 3339  66s (cpu); 2.239
│ │ │ │ +00074330: 3538 7320 2874 6872 6561 6429 3b20 3073  58s (thread); 0s
│ │ │ │  00074340: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │  00074350: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00074360: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00074370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00074390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000743a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ @@ -30174,17 +30174,17 @@
│ │ │ │  00075dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00075de0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00075df0: 0a7c 4365 7274 6966 793a 206f 7574 7075  .|Certify: outpu
│ │ │ │  00075e00: 7420 6365 7274 6966 6965 6421 2020 2020  t certified!    
│ │ │ │  00075e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00075e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00075e30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00075e40: 0a7c 202d 2d20 7573 6564 2033 2e38 3838  .| -- used 3.888
│ │ │ │ -00075e50: 3434 7320 2863 7075 293b 2032 2e35 3438  44s (cpu); 2.548
│ │ │ │ -00075e60: 3832 7320 2874 6872 6561 6429 3b20 3073  82s (thread); 0s
│ │ │ │ +00075e40: 0a7c 202d 2d20 7573 6564 2033 2e38 3131  .| -- used 3.811
│ │ │ │ +00075e50: 3531 7320 2863 7075 293b 2032 2e37 3431  51s (cpu); 2.741
│ │ │ │ +00075e60: 3433 7320 2874 6872 6561 6429 3b20 3073  43s (thread); 0s
│ │ │ │  00075e70: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │  00075e80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00075e90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00075ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00075eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00075ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00075ed0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ @@ -31483,18 +31483,18 @@
│ │ │ │  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 3734  | -- used 0.0174
│ │ │ │ -0007b020: 3434 3673 2028 6370 7529 3b20 302e 3031  446s (cpu); 0.01
│ │ │ │ -0007b030: 3734 3431 3173 2028 7468 7265 6164 293b  74411s (thread);
│ │ │ │ -0007b040: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │ +0007b010: 7c20 2d2d 2075 7365 6420 302e 3032 3132  | -- used 0.0212
│ │ │ │ +0007b020: 3934 7320 2863 7075 293b 2030 2e30 3231  94s (cpu); 0.021
│ │ │ │ +0007b030: 3239 3339 7320 2874 6872 6561 6429 3b20  2939s (thread); 
│ │ │ │ +0007b040: 3073 2028 6763 2920 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                |.
│ │ │ │  0007b0b0: 7c6f 3220 3d20 6964 6561 6c20 2829 2020  |o2 = ideal ()  
│ │ │ │ @@ -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 3931 3136  | -- used 0.9116
│ │ │ │ -0007b2a0: 3835 7320 2863 7075 293b 2030 2e34 3439  85s (cpu); 0.449
│ │ │ │ -0007b2b0: 3333 3573 2028 7468 7265 6164 293b 2030  335s (thread); 0
│ │ │ │ -0007b2c0: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │ +0007b290: 7c20 2d2d 2075 7365 6420 312e 3034 3139  | -- used 1.0419
│ │ │ │ +0007b2a0: 3473 2028 6370 7529 3b20 302e 3437 3631  4s (cpu); 0.4761
│ │ │ │ +0007b2b0: 3835 7320 2874 6872 6561 6429 3b20 3073  85s (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  |               
│ │ │ │ @@ -32424,18 +32424,18 @@
│ │ │ │  0007ea70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0007ea80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  0007ea90: 7c69 3320 3a20 7469 6d65 2070 6172 616d  |i3 : time param
│ │ │ │  0007eaa0: 6574 7269 7a65 204c 2020 2020 2020 2020  etrize L        
│ │ │ │  0007eab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007eac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007ead0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0007eae0: 7c20 2d2d 2075 7365 6420 302e 3030 3530  | -- used 0.0050
│ │ │ │ -0007eaf0: 3036 3138 7320 2863 7075 293b 2030 2e30  0618s (cpu); 0.0
│ │ │ │ -0007eb00: 3035 3030 3136 3873 2028 7468 7265 6164  0500168s (thread
│ │ │ │ -0007eb10: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ +0007eae0: 7c20 2d2d 2075 7365 6420 302e 3030 3537  | -- used 0.0057
│ │ │ │ +0007eaf0: 3739 3934 7320 2863 7075 293b 2030 2e30  7994s (cpu); 0.0
│ │ │ │ +0007eb00: 3035 3737 3635 7320 2874 6872 6561 6429  057765s (thread)
│ │ │ │ +0007eb10: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │  0007eb20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  0007eb30: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0007eb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007eb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007eb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0007eb70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  0007eb80: 7c6f 3320 3d20 2d2d 2072 6174 696f 6e61  |o3 = -- rationa
│ │ │ │ @@ -32934,18 +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 3534 3636  | -- used 0.5466
│ │ │ │ -00080ad0: 7320 2863 7075 293b 2030 2e33 3934 3339  s (cpu); 0.39439
│ │ │ │ -00080ae0: 3273 2028 7468 7265 6164 293b 2030 7320  2s (thread); 0s 
│ │ │ │ -00080af0: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │ +00080ac0: 7c20 2d2d 2075 7365 6420 302e 3533 3038  | -- used 0.5308
│ │ │ │ +00080ad0: 3936 7320 2863 7075 293b 2030 2e34 3237  96s (cpu); 0.427
│ │ │ │ +00080ae0: 3037 3673 2028 7468 7265 6164 293b 2030  076s (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
│ │ │ │ @@ -34395,17 +34395,17 @@
│ │ │ │  000865a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000865b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  000865c0: 6932 203a 2074 696d 6520 7020 3d20 706f  i2 : time p = po
│ │ │ │  000865d0: 696e 7420 736f 7572 6365 2066 2020 2020  int source f    
│ │ │ │  000865e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000865f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00086600: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00086610: 202d 2d20 7573 6564 2030 2e34 3633 3036   -- used 0.46306
│ │ │ │ -00086620: 3873 2028 6370 7529 3b20 302e 3230 3831  8s (cpu); 0.2081
│ │ │ │ -00086630: 3934 7320 2874 6872 6561 6429 3b20 3073  94s (thread); 0s
│ │ │ │ +00086610: 202d 2d20 7573 6564 2030 2e34 3730 3335   -- used 0.47035
│ │ │ │ +00086620: 3273 2028 6370 7529 3b20 302e 3232 3535  2s (cpu); 0.2255
│ │ │ │ +00086630: 3432 7320 2874 6872 6561 6429 3b20 3073  42s (thread); 0s
│ │ │ │  00086640: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │  00086650: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00086660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00086670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00086680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00086690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000866a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ @@ -34510,17 +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 3132 3436   -- used 0.21246
│ │ │ │ -00086d50: 3873 2028 6370 7529 3b20 302e 3133 3538  8s (cpu); 0.1358
│ │ │ │ -00086d60: 3132 7320 2874 6872 6561 6429 3b20 3073  12s (thread); 0s
│ │ │ │ +00086d40: 202d 2d20 7573 6564 2030 2e32 3139 3538   -- used 0.21958
│ │ │ │ +00086d50: 3173 2028 6370 7529 3b20 302e 3133 3634  1s (cpu); 0.1364
│ │ │ │ +00086d60: 3031 7320 2874 6872 6561 6429 3b20 3073  01s (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               |.|
│ │ │ │ @@ -34842,16 +34842,16 @@
│ │ │ │  00088190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000881a0: 2020 207c 0a7c 4365 7274 6966 793a 206f     |.|Certify: o
│ │ │ │  000881b0: 7574 7075 7420 6365 7274 6966 6965 6421  utput certified!
│ │ │ │  000881c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000881d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000881e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000881f0: 2020 207c 0a7c 202d 2d20 7573 6564 2030     |.| -- used 0
│ │ │ │ -00088200: 2e30 3135 3133 3736 7320 2863 7075 293b  .0151376s (cpu);
│ │ │ │ -00088210: 2030 2e30 3134 3830 3639 7320 2874 6872   0.0148069s (thr
│ │ │ │ +00088200: 2e30 3634 3132 3933 7320 2863 7075 293b  .0641293s (cpu);
│ │ │ │ +00088210: 2030 2e30 3233 3133 3839 7320 2874 6872   0.0231389s (thr
│ │ │ │  00088220: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │  00088230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00088240: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00088250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00088260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00088270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00088280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -34972,16 +34972,16 @@
│ │ │ │  000889b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000889c0: 2020 207c 0a7c 4365 7274 6966 793a 206f     |.|Certify: o
│ │ │ │  000889d0: 7574 7075 7420 6365 7274 6966 6965 6421  utput certified!
│ │ │ │  000889e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000889f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00088a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00088a10: 2020 207c 0a7c 202d 2d20 7573 6564 2030     |.| -- used 0
│ │ │ │ -00088a20: 2e30 3131 3634 3839 7320 2863 7075 293b  .0116489s (cpu);
│ │ │ │ -00088a30: 2030 2e30 3131 3336 3437 7320 2874 6872   0.0113647s (thr
│ │ │ │ +00088a20: 2e30 3739 3634 3537 7320 2863 7075 293b  .0796457s (cpu);
│ │ │ │ +00088a30: 2030 2e30 3230 3938 3735 7320 2874 6872   0.0209875s (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 2035     |.| -- used 5
│ │ │ │ -00089e70: 2e38 3539 652d 3035 7320 2863 7075 293b  .859e-05s (cpu);
│ │ │ │ -00089e80: 2035 2e33 3631 652d 3035 7320 2874 6872   5.361e-05s (thr
│ │ │ │ -00089e90: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │ +00089e60: 2020 207c 0a7c 202d 2d20 7573 6564 2036     |.| -- used 6
│ │ │ │ +00089e70: 2e31 3839 3565 2d30 3573 2028 6370 7529  .1895e-05s (cpu)
│ │ │ │ +00089e80: 3b20 352e 3435 3733 652d 3035 7320 2874  ; 5.4573e-05s (t
│ │ │ │ +00089e90: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │  00089ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00089eb0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00089ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00089ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00089ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00089ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00089f00: 2020 207c 0a7c 6f37 203d 207b 312c 2032     |.|o7 = {1, 2
│ │ │ │ @@ -35331,18 +35331,18 @@
│ │ │ │  0008a020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008a030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0008a040: 2d2d 2d2b 0a7c 6938 203a 2074 696d 6520  ---+.|i8 : time 
│ │ │ │  0008a050: 7072 6f6a 6563 7469 7665 4465 6772 6565  projectiveDegree
│ │ │ │  0008a060: 7328 7068 692c 4e75 6d44 6567 7265 6573  s(phi,NumDegrees
│ │ │ │  0008a070: 3d3e 3129 2020 2020 2020 2020 2020 2020  =>1)            
│ │ │ │  0008a080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0008a090: 2020 207c 0a7c 202d 2d20 7573 6564 2032     |.| -- used 2
│ │ │ │ -0008a0a0: 2e36 3130 3965 2d30 3573 2028 6370 7529  .6109e-05s (cpu)
│ │ │ │ -0008a0b0: 3b20 322e 3539 3339 652d 3035 7320 2874  ; 2.5939e-05s (t
│ │ │ │ -0008a0c0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ +0008a090: 2020 207c 0a7c 202d 2d20 7573 6564 2033     |.| -- used 3
│ │ │ │ +0008a0a0: 2e36 3837 3765 2d30 3573 2028 6370 7529  .6877e-05s (cpu)
│ │ │ │ +0008a0b0: 3b20 332e 3637 3465 2d30 3573 2028 7468  ; 3.674e-05s (th
│ │ │ │ +0008a0c0: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │  0008a0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008a0e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0008a0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008a100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008a110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008a120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0008a130: 2020 207c 0a7c 6f38 203d 207b 342c 2031     |.|o8 = {4, 1
│ │ │ │ @@ -37824,17 +37824,17 @@
│ │ │ │  00093bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00093c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00093c10: 2b0a 7c69 3420 3a20 7469 6d65 2070 6869  +.|i4 : time phi
│ │ │ │  00093c20: 2120 3b20 2020 2020 2020 2020 2020 2020  ! ;             
│ │ │ │  00093c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00093c60: 7c0a 7c20 2d2d 2075 7365 6420 302e 3035  |.| -- used 0.05
│ │ │ │ -00093c70: 3332 3432 3973 2028 6370 7529 3b20 302e  32429s (cpu); 0.
│ │ │ │ -00093c80: 3035 3238 3939 3273 2028 7468 7265 6164  0528992s (thread
│ │ │ │ +00093c60: 7c0a 7c20 2d2d 2075 7365 6420 302e 3037  |.| -- used 0.07
│ │ │ │ +00093c70: 3236 3532 3173 2028 6370 7529 3b20 302e  26521s (cpu); 0.
│ │ │ │ +00093c80: 3036 3039 3835 3873 2028 7468 7265 6164  0609858s (thread
│ │ │ │  00093c90: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │  00093ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093cb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00093cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00093cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -37999,17 +37999,17 @@
│ │ │ │  000946e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000946f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00094700: 2b0a 7c69 3920 3a20 7469 6d65 2070 6869  +.|i9 : time phi
│ │ │ │  00094710: 2120 3b20 2020 2020 2020 2020 2020 2020  ! ;             
│ │ │ │  00094720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00094740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00094750: 7c0a 7c20 2d2d 2075 7365 6420 302e 3033  |.| -- used 0.03
│ │ │ │ -00094760: 3630 3632 3773 2028 6370 7529 3b20 302e  60627s (cpu); 0.
│ │ │ │ -00094770: 3033 3537 3034 3973 2028 7468 7265 6164  0357049s (thread
│ │ │ │ +00094750: 7c0a 7c20 2d2d 2075 7365 6420 302e 3036  |.| -- used 0.06
│ │ │ │ +00094760: 3632 3437 3473 2028 6370 7529 3b20 302e  62474s (cpu); 0.
│ │ │ │ +00094770: 3034 3539 3330 3473 2028 7468 7265 6164  0459304s (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 3735  | -- used 0.1575
│ │ │ │ -0009c720: 3331 7320 2863 7075 293b 2030 2e31 3537  31s (cpu); 0.157
│ │ │ │ -0009c730: 3532 3773 2028 7468 7265 6164 293b 2030  527s (thread); 0
│ │ │ │ +0009c710: 7c20 2d2d 2075 7365 6420 302e 3136 3632  | -- used 0.1662
│ │ │ │ +0009c720: 3135 7320 2863 7075 293b 2030 2e31 3636  15s (cpu); 0.166
│ │ │ │ +0009c730: 3231 3473 2028 7468 7265 6164 293b 2030  214s (thread); 0
│ │ │ │  0009c740: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │  0009c750: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  0009c760: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0009c770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0009c7a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ @@ -42163,18 +42163,18 @@
│ │ │ │  000a4b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a4b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000a4b40: 2d2b 0a7c 6933 203a 2074 696d 6520 7068  -+.|i3 : time ph
│ │ │ │  000a4b50: 6920 3d20 7261 7469 6f6e 616c 4d61 7028  i = rationalMap(
│ │ │ │  000a4b60: 562c 332c 3229 2020 2020 2020 2020 2020  V,3,2)          
│ │ │ │  000a4b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000a4b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000a4b90: 207c 0a7c 202d 2d20 7573 6564 2030 2e30   |.| -- used 0.0
│ │ │ │ -000a4ba0: 3935 3332 3173 2028 6370 7529 3b20 302e  95321s (cpu); 0.
│ │ │ │ -000a4bb0: 3039 3533 3231 3573 2028 7468 7265 6164  0953215s (thread
│ │ │ │ -000a4bc0: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ +000a4b90: 207c 0a7c 202d 2d20 7573 6564 2030 2e31   |.| -- used 0.1
│ │ │ │ +000a4ba0: 3039 3933 3473 2028 6370 7529 3b20 302e  09934s (cpu); 0.
│ │ │ │ +000a4bb0: 3130 3939 3334 7320 2874 6872 6561 6429  109934s (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 3031 3438 3173 2028  sed 0.0301481s (
│ │ │ │ -000aab70: 6370 7529 3b20 302e 3033 3031 3433 3273  cpu); 0.0301432s
│ │ │ │ +000aab60: 7365 6420 302e 3033 3431 3730 3673 2028  sed 0.0341706s (
│ │ │ │ +000aab70: 6370 7529 3b20 302e 3033 3431 3535 3873  cpu); 0.0341558s
│ │ │ │  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 3234 3131 3673 2028 6370  sed 1.24116s (cp
│ │ │ │ -000aea90: 7529 3b20 302e 3730 3131 3939 7320 2874  u); 0.701199s (t
│ │ │ │ +000aea80: 7365 6420 312e 3432 3635 3373 2028 6370  sed 1.42653s (cp
│ │ │ │ +000aea90: 7529 3b20 302e 3634 3830 3739 7320 2874  u); 0.648079s (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                  
│ │ │ │ @@ -46244,17 +46244,17 @@
│ │ │ │  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 3836 3538 3235 7320 2863 7075 293b  0.865825s (cpu);
│ │ │ │ -000b4ab0: 2030 2e35 3139 3834 3273 2028 7468 7265   0.519842s (thre
│ │ │ │ -000b4ac0: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │ +000b4aa0: 302e 3738 3731 3536 7320 2863 7075 293b  0.787156s (cpu);
│ │ │ │ +000b4ab0: 2030 2e35 3336 3135 7320 2874 6872 6561   0.53615s (threa
│ │ │ │ +000b4ac0: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 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                  
│ │ │ │  000b4b30: 2020 2020 7c0a 7c20 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 3536 3530 3933 7320 2863 7075 293b  0.565093s (cpu);
│ │ │ │ -000b4e20: 2030 2e33 3639 3031 7320 2874 6872 6561   0.36901s (threa
│ │ │ │ +000b4e10: 302e 3632 3039 3934 7320 2863 7075 293b  0.620994s (cpu);
│ │ │ │ +000b4e20: 2030 2e33 3633 3431 7320 2874 6872 6561   0.36341s (threa
│ │ │ │  000b4e30: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 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 3132 3930 3973 2028 6370 7529  0.0212909s (cpu)
│ │ │ │ -000b51e0: 3b20 302e 3032 3038 3735 3573 2028 7468  ; 0.0208755s (th
│ │ │ │ +000b51d0: 302e 3036 3734 3237 3973 2028 6370 7529  0.0674279s (cpu)
│ │ │ │ +000b51e0: 3b20 302e 3033 3136 3434 3673 2028 7468  ; 0.0316446s (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 3737 3135 7320 2863 7075 293b  0.097715s (cpu);
│ │ │ │ -000b55a0: 2030 2e30 3937 3336 3539 7320 2874 6872   0.0973659s (thr
│ │ │ │ -000b55b0: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │ +000b5590: 302e 3132 3736 3734 7320 2863 7075 293b  0.127674s (cpu);
│ │ │ │ +000b55a0: 2030 2e31 3134 3637 3373 2028 7468 7265   0.114673s (thre
│ │ │ │ +000b55b0: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │  000b55c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b55d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  000b55e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b55f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b5620: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ @@ -46535,26 +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 203d 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: 202d 2d2d 2d2d 2d5b 7820 2e2e 7820 5d20   ------[x ..x ] 
│ │ │ │ -000b5d30: 2020 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: 2031 3030 3030 3320 2030 2020 2036 2020   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 3136 3330   -- used 0.21630
│ │ │ │ -000b5f10: 3373 2028 6370 7529 3b20 302e 3130 3237  3s (cpu); 0.1027
│ │ │ │ -000b5f20: 3673 2028 7468 7265 6164 293b 2030 7320  6s (thread); 0s 
│ │ │ │ -000b5f30: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │ +000b5f00: 202d 2d20 7573 6564 2030 2e30 3636 3232   -- used 0.06622
│ │ │ │ +000b5f10: 3931 7320 2863 7075 293b 2030 2e30 3636  91s (cpu); 0.066
│ │ │ │ +000b5f20: 3233 3236 7320 2874 6872 6561 6429 3b20  2326s (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 2e33 3338 3630   -- used 0.33860
│ │ │ │ -000b6be0: 3473 2028 6370 7529 3b20 302e 3232 3831  4s (cpu); 0.2281
│ │ │ │ -000b6bf0: 3634 7320 2874 6872 6561 6429 3b20 3073  64s (thread); 0s
│ │ │ │ +000b6bd0: 202d 2d20 7573 6564 2030 2e33 3732 3737   -- used 0.37277
│ │ │ │ +000b6be0: 3573 2028 6370 7529 3b20 302e 3235 3332  5s (cpu); 0.2532
│ │ │ │ +000b6bf0: 3538 7320 2874 6872 6561 6429 3b20 3073  58s (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               |.|
│ │ │ │ @@ -46935,18 +46935,18 @@
│ │ │ │  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 2e33 3936 3436   -- used 0.39646
│ │ │ │ -000b75e0: 7320 2863 7075 293b 2030 2e32 3932 3132  s (cpu); 0.29212
│ │ │ │ -000b75f0: 3973 2028 7468 7265 6164 293b 2030 7320  9s (thread); 0s 
│ │ │ │ -000b7600: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │ +000b75d0: 202d 2d20 7573 6564 2030 2e34 3233 3734   -- used 0.42374
│ │ │ │ +000b75e0: 3573 2028 6370 7529 3b20 302e 3239 3435  5s (cpu); 0.2945
│ │ │ │ +000b75f0: 3339 7320 2874 6872 6561 6429 3b20 3073  39s (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               |.|
│ │ │ │  000b7670: 2020 2020 2020 2020 2039 2020 2020 2020           9      
│ │ │ │ @@ -46995,17 +46995,17 @@
│ │ │ │  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 2e34 3137 3937   -- used 1.41797
│ │ │ │ -000b79a0: 7320 2863 7075 293b 2030 2e39 3030 3631  s (cpu); 0.90061
│ │ │ │ -000b79b0: 3273 2028 7468 7265 6164 293b 2030 7320  2s (thread); 0s 
│ │ │ │ +000b7990: 202d 2d20 7573 6564 2031 2e35 3437 3832   -- used 1.54782
│ │ │ │ +000b79a0: 7320 2863 7075 293b 2030 2e39 3336 3339  s (cpu); 0.93639
│ │ │ │ +000b79b0: 3573 2028 7468 7265 6164 293b 2030 7320  5s (thread); 0s 
│ │ │ │  000b79c0: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │  000b79d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000b79e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b79f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b7a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b7a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000b7a20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ @@ -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 3539 3337  | -- used 1.5937
│ │ │ │ -000b8940: 3673 2028 6370 7529 3b20 312e 3136 3539  6s (cpu); 1.1659
│ │ │ │ -000b8950: 3173 2028 7468 7265 6164 293b 2030 7320  1s (thread); 0s 
│ │ │ │ +000b8930: 7c20 2d2d 2075 7365 6420 312e 3530 3132  | -- used 1.5012
│ │ │ │ +000b8940: 3973 2028 6370 7529 3b20 312e 3136 3636  9s (cpu); 1.1666
│ │ │ │ +000b8950: 3873 2028 7468 7265 6164 293b 2030 7320  8s (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                |.
│ │ │ │ @@ -48068,914 +48068,914 @@
│ │ │ │  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
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│ │ │ │  000bbc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000bbcb0: 2863 7075 293b 2030 2e30 3935 3431 3034  (cpu); 0.0954104
│ │ │ │ -000bbcc0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
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│ │ │ │ +000bbcb0: 2863 7075 293b 2030 2e30 3839 3737 3673  (cpu); 0.089776s
│ │ │ │ +000bbcc0: 2028 7468 7265 6164 2020 2020 2020 2020   (thread        
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│ │ │ │  000bbcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000bbd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bbd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bbd30: 2020 2020 2020 2020 207c 0a7c 6f31 203d           |.|o1 =
│ │ │ │  000bbd40: 202d 2d20 7261 7469 6f6e 616c 206d 6170   -- rational map
│ │ │ │  000bbd50: 202d 2d20 2020 2020 2020 2020 2020 2020   --             
│ │ │ │  000bbd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bbd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bbd80: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bbd90: 2073 6f75 7263 653a 2050 726f 6a28 5151   source: Proj(QQ
│ │ │ │  000bbda0: 5b78 202c 2078 202c 2078 202c 2078 202c  [x , x , x , x ,
│ │ │ │ -000bbdb0: 2078 202c 2078 202c 2078 205d 2920 2020   x , x , x ])   
│ │ │ │ +000bbdb0: 2078 202c 2078 202c 2020 2020 2020 2020   x , x ,        
│ │ │ │  000bbdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bbdd0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bbde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000bbe00: 2020 3420 2020 3520 2020 3620 2020 2020    4   5   6     
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│ │ │ │  000bbe10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bbe20: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bbe30: 2074 6172 6765 743a 2073 7562 7661 7269   target: subvari
│ │ │ │  000bbe40: 6574 7920 6f66 2050 726f 6a28 5151 5b74  ety of Proj(QQ[t
│ │ │ │ -000bbe50: 202c 2074 202c 2074 202c 2074 202c 2074   , t , t , t , t
│ │ │ │ +000bbe50: 202c 2074 202c 2074 2020 2020 2020 2020   , t , t        
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│ │ │ │  000bbed0: 2020 2020 2020 2020 207b 2020 2020 2020           {      
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│ │ │ │  000bbf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000bbf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000bbf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bbf60: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bbf70: 2020 2020 2020 2020 2020 3374 2074 2020            3t t  
│ │ │ │  000bbf80: 2b20 3130 7420 7420 202b 2031 3074 2074  + 10t t  + 10t t
│ │ │ │ -000bbf90: 2020 2d20 3374 2074 2020 2d20 3130 7420    - 3t t  - 10t 
│ │ │ │ +000bbf90: 2020 2d20 3374 2074 2020 2020 2020 2020    - 3t t        
│ │ │ │  000bbfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bbfb0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bbfc0: 2020 2020 2020 2020 2020 2020 3220 3420              2 4 
│ │ │ │  000bbfd0: 2020 2020 2033 2034 2020 2020 2020 3020       3 4      0 
│ │ │ │ -000bbfe0: 3520 2020 2020 3120 3520 2020 2020 2032  5     1 5      2
│ │ │ │ +000bbfe0: 3520 2020 2020 3120 2020 2020 2020 2020  5     1         
│ │ │ │  000bbff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000bc0b0: 2020 2020 2020 2020 2020 3374 2074 2020            3t t  
│ │ │ │  000bc0c0: 2d20 3874 2074 2020 2d20 3574 2074 2020  - 8t t  - 5t t  
│ │ │ │ -000bc0d0: 2b20 3874 2074 2020 2d20 3874 2074 2020  + 8t t  - 8t t  
│ │ │ │ +000bc0d0: 2b20 3874 2074 2020 2020 2020 2020 2020  + 8t t          
│ │ │ │  000bc0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000bc120: 2020 2020 3220 3520 2020 2020 2020 2020      2 5         
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│ │ │ │  000bc1e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bc1f0: 2020 2020 2020 2020 2020 7420 7420 202d            t t  -
│ │ │ │  000bc200: 2074 2074 2020 2d20 3274 2074 2020 2d20   t t  - 2t t  - 
│ │ │ │ -000bc210: 3274 2074 2020 2b20 3274 2074 2020 2d20  2t t  + 2t t  - 
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│ │ │ │  000bc2d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bc2e0: 2064 6566 696e 696e 6720 666f 726d 733a   defining forms:
│ │ │ │  000bc2f0: 207b 2020 2020 2020 2020 2020 2020 2020   {              
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│ │ │ │  000bc390: 2020 2d20 3678 2078 2020 2d20 3278 2078    - 6x x  - 2x x
│ │ │ │ -000bc3a0: 2078 2020 2b20 3878 2078 2020 2b20 3578   x  + 8x x  + 5x
│ │ │ │ +000bc3a0: 2078 2020 2b20 3878 2020 2020 2020 2020   x  + 8x        
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│ │ │ │ -000bc3f0: 3120 3220 2020 2020 3120 3220 2020 2020  1 2     1 2     
│ │ │ │ +000bc3f0: 3120 3220 2020 2020 2020 2020 2020 2020  1 2             
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│ │ │ │ +000bc4e0: 2d20 3678 2078 2020 2020 2020 2020 2020  - 6x x          
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│ │ │ │  000bc510: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000bc570: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000bc5d0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
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│ │ │ │ +000bc620: 3130 7820 7820 202d 2020 2020 2020 2020  10x x  -        
│ │ │ │  000bc630: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000bc670: 2020 2030 2031 2020 2020 2020 3020 3220     0 1      0 2 
│ │ │ │ +000bc670: 2020 2030 2031 2020 2020 2020 2020 2020     0 1          
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│ │ │ │  000bc6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bc700: 2020 2020 2020 2033 2020 2020 2032 2020         3     2  
│ │ │ │  000bc710: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  000bc720: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000bc750: 2020 2d20 3132 7820 202d 2038 7820 7820    - 12x  - 8x x 
│ │ │ │ -000bc760: 202b 2037 7820 7820 202b 2036 7820 7820   + 7x x  + 6x x 
│ │ │ │ +000bc760: 202b 2037 7820 7820 2020 2020 2020 2020   + 7x x         
│ │ │ │  000bc770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bc780: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bc790: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000bc7b0: 2020 2020 2030 2032 2020 2020 2020 2020       0 2        
│ │ │ │  000bc7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000bc810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bc820: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
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│ │ │ │ -000bc8a0: 2d20 3132 7820 7820 202b 2031 3278 2020  - 12x x  + 12x  
│ │ │ │ +000bc8a0: 2d20 3132 7820 7820 2020 2020 2020 2020  - 12x x         
│ │ │ │  000bc8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bc8c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bc8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bc8e0: 2020 2020 2030 2020 2020 2020 3020 3120       0      0 1 
│ │ │ │ -000bc8f0: 2020 2020 2030 2031 2020 2020 2020 3120       0 1      1 
│ │ │ │ +000bc8f0: 2020 2020 2030 2031 2020 2020 2020 2020       0 1        
│ │ │ │  000bc900: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000bc920: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000bc970: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000bc9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bc9b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bc9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bc9d0: 2020 2d20 3678 2020 2b20 3230 7820 7820    - 6x  + 20x x 
│ │ │ │ -000bc9e0: 202d 2032 7820 7820 202d 2031 3278 2020   - 2x x  - 12x  
│ │ │ │ +000bc9e0: 202d 2032 7820 7820 2020 2020 2020 2020   - 2x x         
│ │ │ │  000bc9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bca00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bca10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bca20: 2020 2020 2020 3020 2020 2020 2030 2031        0      0 1
│ │ │ │ -000bca30: 2020 2020 2030 2031 2020 2020 2020 3120       0 1      1 
│ │ │ │ +000bca30: 2020 2020 2030 2031 2020 2020 2020 2020       0 1        
│ │ │ │  000bca40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bca50: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bca60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bca70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bca80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bca90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcaa0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bcab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcac0: 2020 2020 2020 2033 2020 2020 2032 2020         3     2  
│ │ │ │ -000bcad0: 2020 2020 2020 2020 3220 2020 2020 3220          2     2 
│ │ │ │ +000bcad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcaf0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bcb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcb10: 2020 2d20 3234 7820 202b 2038 7820 7820    - 24x  + 8x x 
│ │ │ │ -000bcb20: 202b 2031 3678 2078 2020 2d20 3478 2078   + 16x x  - 4x x
│ │ │ │ +000bcb20: 202b 2031 3678 2078 2020 2020 2020 2020   + 16x x        
│ │ │ │  000bcb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcb40: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bcb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcb60: 2020 2020 2020 2030 2020 2020 2030 2031         0     0 1
│ │ │ │ -000bcb70: 2020 2020 2020 3020 3120 2020 2020 3020        0 1     0 
│ │ │ │ +000bcb70: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │  000bcb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcb90: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bcba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcbe0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bcbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcc00: 2020 2020 3220 2020 2020 2020 2020 3220      2         2 
│ │ │ │ -000bcc10: 2020 2020 2033 2020 2020 2032 2020 2020       3     2    
│ │ │ │ +000bcc10: 2020 2020 2033 2020 2020 2020 2020 2020       3          
│ │ │ │  000bcc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcc30: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bcc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcc50: 2020 3878 2078 2020 2d20 3478 2078 2020    8x x  - 4x x  
│ │ │ │ -000bcc60: 2d20 3234 7820 202d 2037 7820 7820 202b  - 24x  - 7x x  +
│ │ │ │ +000bcc60: 2d20 3234 7820 202d 2020 2020 2020 2020  - 24x  -        
│ │ │ │  000bcc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcc80: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bcc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcca0: 2020 2020 3020 3120 2020 2020 3020 3120      0 1     0 1 
│ │ │ │ -000bccb0: 2020 2020 2031 2020 2020 2030 2032 2020       1     0 2  
│ │ │ │ +000bccb0: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │  000bccc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bccd0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bcce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bccf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcd20: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bcd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcd40: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │  000bcd50: 3320 2020 2020 2032 2020 2020 2020 2020  3      2        
│ │ │ │  000bcd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcd70: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bcd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcd90: 2020 2d20 3132 7820 7820 202b 2031 3278    - 12x x  + 12x
│ │ │ │ -000bcda0: 2020 2b20 3132 7820 7820 202d 2034 7820    + 12x x  - 4x 
│ │ │ │ +000bcda0: 2020 2b20 3132 7820 2020 2020 2020 2020    + 12x         
│ │ │ │  000bcdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcdc0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bcdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcde0: 2020 2020 2020 2030 2031 2020 2020 2020         0 1      
│ │ │ │ -000bcdf0: 3120 2020 2020 2030 2032 2020 2020 2030  1      0 2     0
│ │ │ │ +000bcdf0: 3120 2020 2020 2030 2020 2020 2020 2020  1      0        
│ │ │ │  000bce00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bce10: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bce20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bce30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bce40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bce50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bce60: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bce70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bce80: 2020 2020 3220 2020 2020 2020 2020 2032      2          2
│ │ │ │ -000bce90: 2020 2020 2020 3320 2020 2020 2032 2020        3      2  
│ │ │ │ +000bce90: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │  000bcea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bceb0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bcec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bced0: 2020 3878 2078 2020 2d20 3132 7820 7820    8x x  - 12x x 
│ │ │ │ -000bcee0: 202b 2032 3478 2020 2d20 3131 7820 7820   + 24x  - 11x x 
│ │ │ │ +000bcee0: 202b 2032 3478 2020 2020 2020 2020 2020   + 24x          
│ │ │ │  000bcef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcf00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bcf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcf20: 2020 2020 3020 3120 2020 2020 2030 2031      0 1      0 1
│ │ │ │ -000bcf30: 2020 2020 2020 3120 2020 2020 2030 2032        1      0 2
│ │ │ │ +000bcf30: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │  000bcf40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcf50: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bcf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcf70: 207d 2020 2020 2020 2020 2020 2020 2020   }              
│ │ │ │  000bcf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcfa0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bcfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bcff0: 2020 2020 2020 2020 207c 0a7c 6f31 203a           |.|o1 :
│ │ │ │  000bd000: 2052 6174 696f 6e61 6c4d 6170 2028 6375   RationalMap (cu
│ │ │ │  000bd010: 6269 6320 6269 7261 7469 6f6e 616c 206d  bic birational m
│ │ │ │ -000bd020: 6170 2066 726f 6d20 5050 5e36 2074 6f20  ap from PP^6 to 
│ │ │ │ +000bd020: 6170 2066 726f 6d20 2020 2020 2020 2020  ap from         
│ │ │ │  000bd030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd040: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d           |.|----
│ │ │ │  000bd050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000bd060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000bd070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000bd080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000bd090: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 6763 2920  ---------|.|gc) 
│ │ │ │ -000bd0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000bd090: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 293b 2030  ---------|.|); 0
│ │ │ │ +000bd0a0: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │  000bd0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd0e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bd0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd130: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bd140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000bd180: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000bd190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000bd180: 2020 2020 2020 2020 207c 0a7c 2078 205d           |.| x ]
│ │ │ │ +000bd190: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  000bd1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000bd1d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000bd1d0: 2020 2020 2020 2020 207c 0a7c 2020 3620           |.|  6 
│ │ │ │  000bd1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd220: 2020 2020 2020 2020 207c 0a7c 202c 2074           |.| , t
│ │ │ │  000bd230: 202c 2074 202c 2074 202c 2074 202c 2074   , t , t , t , t
│ │ │ │ -000bd240: 205d 2920 6465 6669 6e65 6420 6279 2020   ]) defined by  
│ │ │ │ -000bd250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000bd240: 202c 2074 202c 2074 205d 2920 6465 6669   , t , t ]) defi
│ │ │ │ +000bd250: 6e65 6420 6279 2020 2020 2020 2020 2020  ned by          
│ │ │ │  000bd260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000bd270: 2020 2020 2020 2020 207c 0a7c 3420 2020           |.|4   
│ │ │ │ -000bd280: 3520 2020 3620 2020 3720 2020 3820 2020  5   6   7   8   
│ │ │ │ -000bd290: 3920 2020 2020 2020 2020 2020 2020 2020  9               
│ │ │ │ +000bd270: 2020 2020 2020 2020 207c 0a7c 3220 2020           |.|2   
│ │ │ │ +000bd280: 3320 2020 3420 2020 3520 2020 3620 2020  3   4   5   6   
│ │ │ │ +000bd290: 3720 2020 3820 2020 3920 2020 2020 2020  7   8   9       
│ │ │ │  000bd2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd2c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bd2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd310: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bd320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000bd330: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +000bd330: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │  000bd340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000bd360: 2020 2020 2020 2020 207c 0a7c 7420 202b           |.|t  +
│ │ │ │ -000bd370: 2036 7420 7420 202b 2031 3374 2074 2020   6t t  + 13t t  
│ │ │ │ -000bd380: 2b20 3133 7420 202d 2031 3074 2074 2020  + 13t  - 10t t  
│ │ │ │ -000bd390: 2d20 3130 7420 7420 202b 2074 2074 2020  - 10t t  + t t  
│ │ │ │ -000bd3a0: 2b20 3130 7420 7420 202d 2032 7420 7420  + 10t t  - 2t t 
│ │ │ │ -000bd3b0: 202b 2020 2020 2020 207c 0a7c 2035 2020   +       |.| 5  
│ │ │ │ -000bd3c0: 2020 2033 2035 2020 2020 2020 3420 3520     3 5      4 5 
│ │ │ │ -000bd3d0: 2020 2020 2035 2020 2020 2020 3020 3620       5      0 6 
│ │ │ │ -000bd3e0: 2020 2020 2031 2036 2020 2020 3220 3620       1 6    2 6 
│ │ │ │ -000bd3f0: 2020 2020 2034 2036 2020 2020 2035 2036       4 6     5 6
│ │ │ │ -000bd400: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000bd360: 2020 2020 2020 2020 207c 0a7c 2020 2d20           |.|  - 
│ │ │ │ +000bd370: 3130 7420 7420 202b 2036 7420 7420 202b  10t t  + 6t t  +
│ │ │ │ +000bd380: 2031 3374 2074 2020 2b20 3133 7420 202d   13t t  + 13t  -
│ │ │ │ +000bd390: 2031 3074 2074 2020 2d20 3130 7420 7420   10t t  - 10t t 
│ │ │ │ +000bd3a0: 202b 2074 2074 2020 2b20 3130 7420 7420   + t t  + 10t t 
│ │ │ │ +000bd3b0: 202d 2032 7420 7420 207c 0a7c 3520 2020   - 2t t  |.|5   
│ │ │ │ +000bd3c0: 2020 2032 2035 2020 2020 2033 2035 2020     2 5     3 5  
│ │ │ │ +000bd3d0: 2020 2020 3420 3520 2020 2020 2035 2020      4 5      5  
│ │ │ │ +000bd3e0: 2020 2020 3020 3620 2020 2020 2031 2036      0 6      1 6
│ │ │ │ +000bd3f0: 2020 2020 3220 3620 2020 2020 2034 2036      2 6      4 6
│ │ │ │ +000bd400: 2020 2020 2035 2036 207c 0a7c 2020 2020       5 6 |.|    
│ │ │ │  000bd410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd450: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000bd460: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +000bd460: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  000bd470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000bd480: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -000bd490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000bd480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000bd490: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  000bd4a0: 2020 2020 2020 2020 207c 0a7c 2d20 3874           |.|- 8t
│ │ │ │ -000bd4b0: 2020 2b20 3874 2074 2020 2b20 3874 2074    + 8t t  + 8t t
│ │ │ │ -000bd4c0: 2020 2b20 7420 7420 202d 2038 7420 7420    + t t  - 8t t 
│ │ │ │ -000bd4d0: 202b 2074 2074 2020 2d20 3874 2020 2d20   + t t  - 8t  - 
│ │ │ │ -000bd4e0: 3274 2074 2020 2d20 7420 7420 202d 2032  2t t  - t t  - 2
│ │ │ │ -000bd4f0: 7420 2020 2020 2020 207c 0a7c 2020 2020  t        |.|    
│ │ │ │ -000bd500: 3520 2020 2020 3020 3620 2020 2020 3120  5     0 6     1 
│ │ │ │ -000bd510: 3620 2020 2032 2036 2020 2020 2034 2036  6    2 6     4 6
│ │ │ │ -000bd520: 2020 2020 3520 3620 2020 2020 3620 2020      5 6     6   
│ │ │ │ -000bd530: 2020 3020 3720 2020 2031 2037 2020 2020    0 7    1 7    
│ │ │ │ -000bd540: 2032 2020 2020 2020 207c 0a7c 2020 2020   2       |.|    
│ │ │ │ +000bd4b0: 2074 2020 2d20 3874 2020 2b20 3874 2074   t  - 8t  + 8t t
│ │ │ │ +000bd4c0: 2020 2b20 3874 2074 2020 2b20 7420 7420    + 8t t  + t t 
│ │ │ │ +000bd4d0: 202d 2038 7420 7420 202b 2074 2074 2020   - 8t t  + t t  
│ │ │ │ +000bd4e0: 2d20 3874 2020 2d20 3274 2074 2020 2d20  - 8t  - 2t t  - 
│ │ │ │ +000bd4f0: 7420 7420 202d 2032 747c 0a7c 2020 2020  t t  - 2t|.|    
│ │ │ │ +000bd500: 3420 3520 2020 2020 3520 2020 2020 3020  4 5     5     0 
│ │ │ │ +000bd510: 3620 2020 2020 3120 3620 2020 2032 2036  6     1 6    2 6
│ │ │ │ +000bd520: 2020 2020 2034 2036 2020 2020 3520 3620       4 6    5 6 
│ │ │ │ +000bd530: 2020 2020 3620 2020 2020 3020 3720 2020      6     0 7   
│ │ │ │ +000bd540: 2031 2037 2020 2020 207c 0a7c 2020 2020   1 7     |.|    
│ │ │ │  000bd550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd590: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000bd5a0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +000bd5a0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  000bd5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000bd5c0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -000bd5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000bd5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000bd5d0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │  000bd5e0: 2020 2020 2020 2020 207c 0a7c 3274 2074           |.|2t t
│ │ │ │ -000bd5f0: 2020 2d20 3274 2020 2b20 3274 2074 2020    - 2t  + 2t t  
│ │ │ │ -000bd600: 2b20 3274 2074 2020 2b20 7420 7420 202d  + 2t t  + t t  -
│ │ │ │ -000bd610: 2033 7420 7420 202d 2033 7420 202d 2074   3t t  - 3t  - t
│ │ │ │ -000bd620: 2074 2020 2d20 7420 7420 202b 2074 2074   t  - t t  + t t
│ │ │ │ -000bd630: 2020 2020 2020 2020 207c 0a7c 2020 3420           |.|  4 
│ │ │ │ -000bd640: 3520 2020 2020 3520 2020 2020 3020 3620  5     5     0 6 
│ │ │ │ -000bd650: 2020 2020 3120 3620 2020 2033 2036 2020      1 6    3 6  
│ │ │ │ -000bd660: 2020 2034 2036 2020 2020 2036 2020 2020     4 6     6    
│ │ │ │ -000bd670: 3020 3720 2020 2032 2037 2020 2020 3420  0 7    2 7    4 
│ │ │ │ -000bd680: 3720 2020 2020 2020 207c 0a7c 2020 2020  7        |.|    
│ │ │ │ +000bd5f0: 2020 2d20 3274 2074 2020 2d20 3274 2020    - 2t t  - 2t  
│ │ │ │ +000bd600: 2b20 3274 2074 2020 2b20 3274 2074 2020  + 2t t  + 2t t  
│ │ │ │ +000bd610: 2b20 7420 7420 202d 2033 7420 7420 202d  + t t  - 3t t  -
│ │ │ │ +000bd620: 2033 7420 202d 2074 2074 2020 2d20 7420   3t  - t t  - t 
│ │ │ │ +000bd630: 7420 202b 2074 2074 207c 0a7c 2020 3220  t  + t t |.|  2 
│ │ │ │ +000bd640: 3520 2020 2020 3420 3520 2020 2020 3520  5     4 5     5 
│ │ │ │ +000bd650: 2020 2020 3020 3620 2020 2020 3120 3620      0 6     1 6 
│ │ │ │ +000bd660: 2020 2033 2036 2020 2020 2034 2036 2020     3 6     4 6  
│ │ │ │ +000bd670: 2020 2036 2020 2020 3020 3720 2020 2032     6    0 7    2
│ │ │ │ +000bd680: 2037 2020 2020 3420 377c 0a7c 2020 2020   7    4 7|.|    
│ │ │ │  000bd690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd6d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bd6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000bd720: 2020 2020 2020 2020 207c 0a7c 2020 3220           |.|  2 
│ │ │ │ -000bd730: 2020 2020 2020 3220 2020 2033 2020 2020        2    3    
│ │ │ │ -000bd740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000bd750: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -000bd760: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +000bd720: 2020 2020 2020 2020 207c 0a7c 3220 2020           |.|2   
│ │ │ │ +000bd730: 2020 2020 2020 3220 2020 2020 2020 3220        2       2 
│ │ │ │ +000bd740: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ +000bd750: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +000bd760: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  000bd770: 2020 2020 2020 2020 207c 0a7c 2078 2020           |.| x  
│ │ │ │ -000bd780: 2d20 3678 2078 2020 2b20 7820 202b 2032  - 6x x  + x  + 2
│ │ │ │ -000bd790: 7820 7820 7820 202d 2031 3678 2078 2078  x x x  - 16x x x
│ │ │ │ -000bd7a0: 2020 2b20 3778 2078 2020 2b20 3878 2078    + 7x x  + 8x x
│ │ │ │ -000bd7b0: 2020 2d20 3578 2078 2078 2020 2d20 3278    - 5x x x  - 2x
│ │ │ │ -000bd7c0: 2078 2020 2020 2020 207c 0a7c 3020 3220   x       |.|0 2 
│ │ │ │ -000bd7d0: 2020 2020 3120 3220 2020 2032 2020 2020      1 2    2    
│ │ │ │ -000bd7e0: 2030 2032 2033 2020 2020 2020 3120 3220   0 2 3      1 2 
│ │ │ │ -000bd7f0: 3320 2020 2020 3220 3320 2020 2020 3220  3     2 3     2 
│ │ │ │ -000bd800: 3320 2020 2020 3020 3220 3420 2020 2020  3     0 2 4     
│ │ │ │ -000bd810: 3120 2020 2020 2020 207c 0a7c 2020 2020  1        |.|    
│ │ │ │ +000bd780: 2b20 3578 2078 2020 2d20 3678 2078 2020  + 5x x  - 6x x  
│ │ │ │ +000bd790: 2b20 7820 202b 2032 7820 7820 7820 202d  + x  + 2x x x  -
│ │ │ │ +000bd7a0: 2031 3678 2078 2078 2020 2b20 3778 2078   16x x x  + 7x x
│ │ │ │ +000bd7b0: 2020 2b20 3878 2078 2020 2d20 3578 2078    + 8x x  - 5x x
│ │ │ │ +000bd7c0: 2078 2020 2d20 3278 207c 0a7c 3120 3220   x  - 2x |.|1 2 
│ │ │ │ +000bd7d0: 2020 2020 3020 3220 2020 2020 3120 3220      0 2     1 2 
│ │ │ │ +000bd7e0: 2020 2032 2020 2020 2030 2032 2033 2020     2     0 2 3  
│ │ │ │ +000bd7f0: 2020 2020 3120 3220 3320 2020 2020 3220      1 2 3     2 
│ │ │ │ +000bd800: 3320 2020 2020 3220 3320 2020 2020 3020  3     2 3     0 
│ │ │ │ +000bd810: 3220 3420 2020 2020 317c 0a7c 2020 2020  2 4     1|.|    
│ │ │ │  000bd820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd860: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000bd870: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -000bd880: 2020 2020 3220 2020 2020 2032 2020 2020      2      2    
│ │ │ │ -000bd890: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -000bd8a0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -000bd8b0: 2020 2020 2020 2020 207c 0a7c 2b20 3778           |.|+ 7x
│ │ │ │ -000bd8c0: 2078 2078 2020 2b20 3678 2078 2020 2b20   x x  + 6x x  + 
│ │ │ │ -000bd8d0: 3278 2078 2020 2b20 7820 7820 202b 2031  2x x  + x x  + 1
│ │ │ │ -000bd8e0: 3278 2078 2020 2b20 3678 2078 2078 2020  2x x  + 6x x x  
│ │ │ │ -000bd8f0: 2d20 3678 2078 2020 2d20 3678 2078 2078  - 6x x  - 6x x x
│ │ │ │ -000bd900: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000bd910: 3020 3120 3220 2020 2020 3120 3220 2020  0 1 2     1 2   
│ │ │ │ -000bd920: 2020 3020 3220 2020 2031 2032 2020 2020    0 2    1 2    
│ │ │ │ -000bd930: 2020 3020 3320 2020 2020 3020 3120 3320    0 3     0 1 3 
│ │ │ │ -000bd940: 2020 2020 3120 3320 2020 2020 3020 3220      1 3     0 2 
│ │ │ │ -000bd950: 3320 2020 2020 2020 207c 0a7c 2020 2020  3        |.|    
│ │ │ │ +000bd870: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +000bd880: 2020 3220 2020 2020 2020 2020 3220 2020    2         2   
│ │ │ │ +000bd890: 2020 2032 2020 2020 2020 3220 2020 2020     2      2     
│ │ │ │ +000bd8a0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +000bd8b0: 2020 2020 2020 2020 207c 0a7c 2b20 3478           |.|+ 4x
│ │ │ │ +000bd8c0: 2078 2020 2b20 3778 2078 2078 2020 2b20   x  + 7x x x  + 
│ │ │ │ +000bd8d0: 3678 2078 2020 2b20 3278 2078 2020 2b20  6x x  + 2x x  + 
│ │ │ │ +000bd8e0: 7820 7820 202b 2031 3278 2078 2020 2b20  x x  + 12x x  + 
│ │ │ │ +000bd8f0: 3678 2078 2078 2020 2d20 3678 2078 2020  6x x x  - 6x x  
│ │ │ │ +000bd900: 2d20 3678 2078 2078 207c 0a7c 2020 2020  - 6x x x |.|    
│ │ │ │ +000bd910: 3020 3220 2020 2020 3020 3120 3220 2020  0 2     0 1 2   
│ │ │ │ +000bd920: 2020 3120 3220 2020 2020 3020 3220 2020    1 2     0 2   
│ │ │ │ +000bd930: 2031 2032 2020 2020 2020 3020 3320 2020   1 2      0 3   
│ │ │ │ +000bd940: 2020 3020 3120 3320 2020 2020 3120 3320    0 1 3     1 3 
│ │ │ │ +000bd950: 2020 2020 3020 3220 337c 0a7c 2020 2020      0 2 3|.|    
│ │ │ │  000bd960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bd9a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000bd9b0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -000bd9c0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +000bd9b0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +000bd9c0: 2020 2032 2020 2020 2020 2020 3220 2020     2        2   
│ │ │ │  000bd9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000bd9e0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -000bd9f0: 2020 2020 2020 2020 207c 0a7c 2d20 3132           |.|- 12
│ │ │ │ -000bda00: 7820 7820 7820 202d 2036 7820 7820 202d  x x x  - 6x x  -
│ │ │ │ -000bda10: 2031 3078 2078 2020 2b20 3678 2078 2078   10x x  + 6x x x
│ │ │ │ -000bda20: 2020 2b20 3138 7820 7820 7820 202b 2031    + 18x x x  + 1
│ │ │ │ -000bda30: 3078 2078 2078 2020 2b20 3478 2078 2020  0x x x  + 4x x  
│ │ │ │ -000bda40: 2d20 2020 2020 2020 207c 0a7c 2020 2020  -        |.|    
│ │ │ │ -000bda50: 2030 2031 2032 2020 2020 2031 2032 2020   0 1 2     1 2  
│ │ │ │ -000bda60: 2020 2020 3020 3320 2020 2020 3020 3120      0 3     0 1 
│ │ │ │ -000bda70: 3320 2020 2020 2030 2032 2033 2020 2020  3      0 2 3    
│ │ │ │ -000bda80: 2020 3120 3220 3320 2020 2020 3020 3320    1 2 3     0 3 
│ │ │ │ -000bda90: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000bd9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000bd9f0: 2020 2020 2020 3220 207c 0a7c 2031 3478        2  |.| 14x
│ │ │ │ +000bda00: 2078 2020 2d20 3132 7820 7820 7820 202d   x  - 12x x x  -
│ │ │ │ +000bda10: 2036 7820 7820 202d 2031 3078 2078 2020   6x x  - 10x x  
│ │ │ │ +000bda20: 2b20 3678 2078 2078 2020 2b20 3138 7820  + 6x x x  + 18x 
│ │ │ │ +000bda30: 7820 7820 202b 2031 3078 2078 2078 2020  x x  + 10x x x  
│ │ │ │ +000bda40: 2b20 3478 2078 2020 2d7c 0a7c 2020 2020  + 4x x  -|.|    
│ │ │ │ +000bda50: 3020 3220 2020 2020 2030 2031 2032 2020  0 2      0 1 2  
│ │ │ │ +000bda60: 2020 2031 2032 2020 2020 2020 3020 3320     1 2      0 3 
│ │ │ │ +000bda70: 2020 2020 3020 3120 3320 2020 2020 2030      0 1 3      0
│ │ │ │ +000bda80: 2032 2033 2020 2020 2020 3120 3220 3320   2 3      1 2 3 
│ │ │ │ +000bda90: 2020 2020 3020 3320 207c 0a7c 2020 2020      0 3  |.|    
│ │ │ │  000bdaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bdab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bdac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bdad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bdae0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000bdaf0: 2020 2032 2020 2020 2020 2020 2032 2020     2         2  
│ │ │ │ -000bdb00: 2020 2020 2032 2020 2020 2033 2020 2020       2     3    
│ │ │ │ -000bdb10: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +000bdaf0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +000bdb00: 2020 2020 2032 2020 2020 2020 2032 2020       2       2  
│ │ │ │ +000bdb10: 2020 2033 2020 2020 2032 2020 2020 2020     3     2      
│ │ │ │  000bdb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000bdb30: 2020 2020 2020 2020 207c 0a7c 7820 202b           |.|x  +
│ │ │ │ -000bdb40: 2038 7820 7820 202b 2033 7820 7820 202d   8x x  + 3x x  -
│ │ │ │ -000bdb50: 2036 7820 7820 202b 2032 7820 202b 2038   6x x  + 2x  + 8
│ │ │ │ -000bdb60: 7820 7820 202d 2038 7820 7820 7820 202b  x x  - 8x x x  +
│ │ │ │ -000bdb70: 2035 7820 7820 7820 202d 2031 3678 2078   5x x x  - 16x x
│ │ │ │ -000bdb80: 2078 2020 2020 2020 207c 0a7c 2032 2020   x       |.| 2  
│ │ │ │ -000bdb90: 2020 2031 2032 2020 2020 2030 2032 2020     1 2     0 2  
│ │ │ │ -000bdba0: 2020 2031 2032 2020 2020 2032 2020 2020     1 2     2    
│ │ │ │ -000bdbb0: 2030 2033 2020 2020 2030 2031 2033 2020   0 3     0 1 3  
│ │ │ │ -000bdbc0: 2020 2030 2032 2033 2020 2020 2020 3120     0 2 3      1 
│ │ │ │ -000bdbd0: 3220 2020 2020 2020 207c 0a7c 2020 2020  2        |.|    
│ │ │ │ +000bdb30: 2020 2020 2020 2020 207c 0a7c 202b 2036           |.| + 6
│ │ │ │ +000bdb40: 7820 7820 7820 202b 2038 7820 7820 202b  x x x  + 8x x  +
│ │ │ │ +000bdb50: 2033 7820 7820 202d 2036 7820 7820 202b   3x x  - 6x x  +
│ │ │ │ +000bdb60: 2032 7820 202b 2038 7820 7820 202d 2038   2x  + 8x x  - 8
│ │ │ │ +000bdb70: 7820 7820 7820 202b 2035 7820 7820 7820  x x x  + 5x x x 
│ │ │ │ +000bdb80: 202d 2031 3678 2078 207c 0a7c 2020 2020   - 16x x |.|    
│ │ │ │ +000bdb90: 2030 2031 2032 2020 2020 2031 2032 2020   0 1 2     1 2  
│ │ │ │ +000bdba0: 2020 2030 2032 2020 2020 2031 2032 2020     0 2     1 2  
│ │ │ │ +000bdbb0: 2020 2032 2020 2020 2030 2033 2020 2020     2     0 3    
│ │ │ │ +000bdbc0: 2030 2031 2033 2020 2020 2030 2032 2033   0 1 3     0 2 3
│ │ │ │ +000bdbd0: 2020 2020 2020 3120 327c 0a7c 2020 2020        1 2|.|    
│ │ │ │  000bdbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bdbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bdc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bdc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bdc20: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000bdc30: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -000bdc40: 2020 2020 3220 2020 2020 2020 3220 2020      2       2   
│ │ │ │ -000bdc50: 2033 2020 2020 2020 3220 2020 2020 2020   3      2       
│ │ │ │ -000bdc60: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -000bdc70: 2020 2020 2020 2020 207c 0a7c 2b20 3678           |.|+ 6x
│ │ │ │ -000bdc80: 2078 2078 2020 2d20 3678 2078 2020 2d20   x x  - 6x x  - 
│ │ │ │ -000bdc90: 3378 2078 2020 2b20 3478 2078 2020 2d20  3x x  + 4x x  - 
│ │ │ │ -000bdca0: 7820 202b 2031 3478 2078 2020 2b20 3234  x  + 14x x  + 24
│ │ │ │ -000bdcb0: 7820 7820 7820 202d 2033 3878 2078 2020  x x x  - 38x x  
│ │ │ │ -000bdcc0: 2d20 2020 2020 2020 207c 0a7c 2020 2020  -        |.|    
│ │ │ │ -000bdcd0: 3020 3120 3220 2020 2020 3120 3220 2020  0 1 2     1 2   
│ │ │ │ -000bdce0: 2020 3020 3220 2020 2020 3120 3220 2020    0 2     1 2   
│ │ │ │ -000bdcf0: 2032 2020 2020 2020 3020 3320 2020 2020   2      0 3     
│ │ │ │ -000bdd00: 2030 2031 2033 2020 2020 2020 3120 3320   0 1 3      1 3 
│ │ │ │ -000bdd10: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000bdc30: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ +000bdc40: 2020 3220 2020 2020 2020 2020 3220 2020    2         2   
│ │ │ │ +000bdc50: 2020 2020 3220 2020 2033 2020 2020 2020      2    3      
│ │ │ │ +000bdc60: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +000bdc70: 2020 2020 3220 2020 207c 0a7c 202b 2031      2    |.| + 1
│ │ │ │ +000bdc80: 3278 2020 2b20 3678 2078 2078 2020 2d20  2x  + 6x x x  - 
│ │ │ │ +000bdc90: 3678 2078 2020 2d20 3378 2078 2020 2b20  6x x  - 3x x  + 
│ │ │ │ +000bdca0: 3478 2078 2020 2d20 7820 202b 2031 3478  4x x  - x  + 14x
│ │ │ │ +000bdcb0: 2078 2020 2b20 3234 7820 7820 7820 202d   x  + 24x x x  -
│ │ │ │ +000bdcc0: 2033 3878 2078 2020 2d7c 0a7c 2020 2020   38x x  -|.|    
│ │ │ │ +000bdcd0: 2020 3120 2020 2020 3020 3120 3220 2020    1     0 1 2   
│ │ │ │ +000bdce0: 2020 3120 3220 2020 2020 3020 3220 2020    1 2     0 2   
│ │ │ │ +000bdcf0: 2020 3120 3220 2020 2032 2020 2020 2020    1 2    2      
│ │ │ │ +000bdd00: 3020 3320 2020 2020 2030 2031 2033 2020  0 3      0 1 3  
│ │ │ │ +000bdd10: 2020 2020 3120 3320 207c 0a7c 2020 2020      1 3  |.|    
│ │ │ │  000bdd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bdd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bdd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bdd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000bdd60: 2020 2020 2020 2020 207c 0a7c 2020 2032           |.|   2
│ │ │ │ -000bdd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000bdd80: 2032 2020 2020 2020 2020 3220 2020 2020   2        2     
│ │ │ │ -000bdd90: 2020 3220 2020 2020 2032 2020 2020 2020    2      2      
│ │ │ │ -000bdda0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -000bddb0: 2020 2020 2020 2020 207c 0a7c 2d20 7820           |.|- x 
│ │ │ │ -000bddc0: 7820 202d 2032 7820 7820 7820 202b 2038  x  - 2x x x  + 8
│ │ │ │ -000bddd0: 7820 7820 202d 2078 2078 2020 2d20 3478  x x  - x x  - 4x
│ │ │ │ -000bdde0: 2078 2020 2d20 3134 7820 7820 202b 2032   x  - 14x x  + 2
│ │ │ │ -000bddf0: 7820 7820 7820 202b 2033 3278 2078 2020  x x x  + 32x x  
│ │ │ │ -000bde00: 2b20 2020 2020 2020 207c 0a7c 2020 2030  +        |.|   0
│ │ │ │ -000bde10: 2032 2020 2020 2030 2031 2032 2020 2020   2     0 1 2    
│ │ │ │ -000bde20: 2031 2032 2020 2020 3020 3220 2020 2020   1 2    0 2     
│ │ │ │ -000bde30: 3120 3220 2020 2020 2030 2033 2020 2020  1 2      0 3    
│ │ │ │ -000bde40: 2030 2031 2033 2020 2020 2020 3120 3320   0 1 3      1 3 
│ │ │ │ -000bde50: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000bdd60: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000bdd70: 2020 3320 2020 2032 2020 2020 2020 2020    3    2        
│ │ │ │ +000bdd80: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +000bdd90: 2020 3220 2020 2020 2020 3220 2020 2020    2       2     
│ │ │ │ +000bdda0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +000bddb0: 2020 2020 3220 2020 207c 0a7c 202d 2031      2    |.| - 1
│ │ │ │ +000bddc0: 3278 2020 2d20 7820 7820 202d 2032 7820  2x  - x x  - 2x 
│ │ │ │ +000bddd0: 7820 7820 202b 2038 7820 7820 202d 2078  x x  + 8x x  - x
│ │ │ │ +000bdde0: 2078 2020 2d20 3478 2078 2020 2d20 3134   x  - 4x x  - 14
│ │ │ │ +000bddf0: 7820 7820 202b 2032 7820 7820 7820 202b  x x  + 2x x x  +
│ │ │ │ +000bde00: 2033 3278 2078 2020 2b7c 0a7c 2020 2020   32x x  +|.|    
│ │ │ │ +000bde10: 2020 3120 2020 2030 2032 2020 2020 2030    1    0 2     0
│ │ │ │ +000bde20: 2031 2032 2020 2020 2031 2032 2020 2020   1 2     1 2    
│ │ │ │ +000bde30: 3020 3220 2020 2020 3120 3220 2020 2020  0 2     1 2     
│ │ │ │ +000bde40: 2030 2033 2020 2020 2030 2031 2033 2020   0 3     0 1 3  
│ │ │ │ +000bde50: 2020 2020 3120 3320 207c 0a7c 2020 2020      1 3  |.|    
│ │ │ │  000bde60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bde70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bde80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bde90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000bdea0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000bdeb0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -000bdec0: 2020 2020 2020 3220 2020 2020 2020 3220        2       2 
│ │ │ │ -000bded0: 2020 2033 2020 2020 2032 2020 2020 2020     3     2      
│ │ │ │ -000bdee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000bdea0: 2020 2020 2020 2020 207c 0a7c 3220 2020           |.|2   
│ │ │ │ +000bdeb0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +000bdec0: 2020 2020 3220 2020 2020 2020 2020 3220      2         2 
│ │ │ │ +000bded0: 2020 2020 2020 3220 2020 2033 2020 2020        2    3    
│ │ │ │ +000bdee0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  000bdef0: 2020 2020 2020 2020 207c 0a7c 2020 2d20           |.|  - 
│ │ │ │ -000bdf00: 3478 2078 2078 2020 2b20 3878 2078 2020  4x x x  + 8x x  
│ │ │ │ -000bdf10: 2b20 3378 2078 2020 2d20 3478 2078 2020  + 3x x  - 4x x  
│ │ │ │ -000bdf20: 2b20 7820 202d 2038 7820 7820 202d 2033  + x  - 8x x  - 3
│ │ │ │ -000bdf30: 3278 2078 2078 2020 2b20 3878 2078 2078  2x x x  + 8x x x
│ │ │ │ -000bdf40: 2020 2020 2020 2020 207c 0a7c 3220 2020           |.|2   
│ │ │ │ -000bdf50: 2020 3020 3120 3220 2020 2020 3120 3220    0 1 2     1 2 
│ │ │ │ -000bdf60: 2020 2020 3020 3220 2020 2020 3120 3220      0 2     1 2 
│ │ │ │ -000bdf70: 2020 2032 2020 2020 2030 2033 2020 2020     2     0 3    
│ │ │ │ -000bdf80: 2020 3020 3120 3320 2020 2020 3020 3220    0 1 3     0 2 
│ │ │ │ -000bdf90: 3320 2020 2020 2020 207c 0a7c 2020 2020  3        |.|    
│ │ │ │ +000bdf00: 3478 2078 2020 2d20 3478 2078 2078 2020  4x x  - 4x x x  
│ │ │ │ +000bdf10: 2b20 3878 2078 2020 2b20 3378 2078 2020  + 8x x  + 3x x  
│ │ │ │ +000bdf20: 2d20 3478 2078 2020 2b20 7820 202d 2038  - 4x x  + x  - 8
│ │ │ │ +000bdf30: 7820 7820 202d 2033 3278 2078 2078 2020  x x  - 32x x x  
│ │ │ │ +000bdf40: 2b20 3878 2078 2078 207c 0a7c 3120 2020  + 8x x x |.|1   
│ │ │ │ +000bdf50: 2020 3020 3220 2020 2020 3020 3120 3220    0 2     0 1 2 
│ │ │ │ +000bdf60: 2020 2020 3120 3220 2020 2020 3020 3220      1 2     0 2 
│ │ │ │ +000bdf70: 2020 2020 3120 3220 2020 2032 2020 2020      1 2    2    
│ │ │ │ +000bdf80: 2030 2033 2020 2020 2020 3020 3120 3320   0 3      0 1 3 
│ │ │ │ +000bdf90: 2020 2020 3020 3220 337c 0a7c 2020 2020      0 2 3|.|    
│ │ │ │  000bdfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bdfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bdfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bdfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000bdfe0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000bdff0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -000be000: 2020 2020 3220 2020 2020 2020 3220 2020      2       2   
│ │ │ │ -000be010: 2020 3320 2020 2020 2032 2020 2020 2020    3      2      
│ │ │ │ -000be020: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -000be030: 2020 2020 2020 2020 207c 0a7c 2035 7820           |.| 5x 
│ │ │ │ -000be040: 7820 7820 202b 2032 3478 2078 2020 2b20  x x  + 24x x  + 
│ │ │ │ -000be050: 3478 2078 2020 2d20 3978 2078 2020 2b20  4x x  - 9x x  + 
│ │ │ │ -000be060: 3378 2020 2d20 3130 7820 7820 202b 2034  3x  - 10x x  + 4
│ │ │ │ -000be070: 7820 7820 7820 202b 2037 3078 2078 2020  x x x  + 70x x  
│ │ │ │ -000be080: 2d20 2020 2020 2020 207c 0a7c 2020 2030  -        |.|   0
│ │ │ │ -000be090: 2031 2032 2020 2020 2020 3120 3220 2020   1 2      1 2   
│ │ │ │ -000be0a0: 2020 3020 3220 2020 2020 3120 3220 2020    0 2     1 2   
│ │ │ │ -000be0b0: 2020 3220 2020 2020 2030 2033 2020 2020    2      0 3    
│ │ │ │ -000be0c0: 2030 2031 2033 2020 2020 2020 3120 3320   0 1 3      1 3 
│ │ │ │ -000be0d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000bdfe0: 2020 2020 2020 2020 207c 0a7c 2020 2032           |.|   2
│ │ │ │ +000bdff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000be000: 2020 3220 2020 2020 2020 2020 3220 2020    2         2   
│ │ │ │ +000be010: 2020 2020 3220 2020 2020 3320 2020 2020      2     3     
│ │ │ │ +000be020: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +000be030: 2020 2020 3220 2020 207c 0a7c 2037 7820      2    |.| 7x 
│ │ │ │ +000be040: 7820 202b 2035 7820 7820 7820 202b 2032  x  + 5x x x  + 2
│ │ │ │ +000be050: 3478 2078 2020 2b20 3478 2078 2020 2d20  4x x  + 4x x  - 
│ │ │ │ +000be060: 3978 2078 2020 2b20 3378 2020 2d20 3130  9x x  + 3x  - 10
│ │ │ │ +000be070: 7820 7820 202b 2034 7820 7820 7820 202b  x x  + 4x x x  +
│ │ │ │ +000be080: 2037 3078 2078 2020 2d7c 0a7c 2020 2030   70x x  -|.|   0
│ │ │ │ +000be090: 2032 2020 2020 2030 2031 2032 2020 2020   2     0 1 2    
│ │ │ │ +000be0a0: 2020 3120 3220 2020 2020 3020 3220 2020    1 2     0 2   
│ │ │ │ +000be0b0: 2020 3120 3220 2020 2020 3220 2020 2020    1 2     2     
│ │ │ │ +000be0c0: 2030 2033 2020 2020 2030 2031 2033 2020   0 3     0 1 3  
│ │ │ │ +000be0d0: 2020 2020 3120 3320 207c 0a7c 2020 2020      1 3  |.|    
│ │ │ │  000be0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be120: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000be130: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -000be140: 3220 2020 2020 2020 3220 2020 2020 2032  2       2      2
│ │ │ │ -000be150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000be160: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -000be170: 2020 2020 2020 2020 207c 0a7c 7820 7820           |.|x x 
│ │ │ │ -000be180: 202d 2031 3478 2078 2020 2b20 3878 2078   - 14x x  + 8x x
│ │ │ │ -000be190: 2020 2b20 3678 2078 2020 2b20 3132 7820    + 6x x  + 12x 
│ │ │ │ -000be1a0: 7820 202d 2034 7820 7820 7820 202d 2033  x  - 4x x x  - 3
│ │ │ │ -000be1b0: 3278 2078 2020 2b20 3130 7820 7820 7820  2x x  + 10x x x 
│ │ │ │ -000be1c0: 202b 2020 2020 2020 207c 0a7c 2031 2032   +       |.| 1 2
│ │ │ │ -000be1d0: 2020 2020 2020 3120 3220 2020 2020 3020        1 2     0 
│ │ │ │ -000be1e0: 3220 2020 2020 3120 3220 2020 2020 2030  2     1 2      0
│ │ │ │ -000be1f0: 2033 2020 2020 2030 2031 2033 2020 2020   3     0 1 3    
│ │ │ │ -000be200: 2020 3120 3320 2020 2020 2030 2032 2033    1 3      0 2 3
│ │ │ │ -000be210: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000be130: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +000be140: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +000be150: 3220 2020 2020 2032 2020 2020 2020 2020  2      2        
│ │ │ │ +000be160: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +000be170: 2020 2020 2020 2020 207c 0a7c 7820 202d           |.|x  -
│ │ │ │ +000be180: 2034 7820 7820 7820 202d 2031 3478 2078   4x x x  - 14x x
│ │ │ │ +000be190: 2020 2b20 3878 2078 2020 2b20 3678 2078    + 8x x  + 6x x
│ │ │ │ +000be1a0: 2020 2b20 3132 7820 7820 202d 2034 7820    + 12x x  - 4x 
│ │ │ │ +000be1b0: 7820 7820 202d 2033 3278 2078 2020 2b20  x x  - 32x x  + 
│ │ │ │ +000be1c0: 3130 7820 7820 7820 207c 0a7c 2032 2020  10x x x  |.| 2  
│ │ │ │ +000be1d0: 2020 2030 2031 2032 2020 2020 2020 3120     0 1 2      1 
│ │ │ │ +000be1e0: 3220 2020 2020 3020 3220 2020 2020 3120  2     0 2     1 
│ │ │ │ +000be1f0: 3220 2020 2020 2030 2033 2020 2020 2030  2      0 3     0
│ │ │ │ +000be200: 2031 2033 2020 2020 2020 3120 3320 2020   1 3      1 3   
│ │ │ │ +000be210: 2020 2030 2032 2033 207c 0a7c 2020 2020     0 2 3 |.|    
│ │ │ │  000be220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be260: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000be270: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -000be280: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ -000be290: 2032 2020 2020 2033 2020 2020 2032 2020   2     3     2  
│ │ │ │ -000be2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000be2b0: 2032 2020 2020 2020 207c 0a7c 202b 2031   2       |.| + 1
│ │ │ │ -000be2c0: 3778 2078 2078 2020 2d20 3234 7820 7820  7x x x  - 24x x 
│ │ │ │ -000be2d0: 202d 2031 3078 2078 2020 2b20 3131 7820   - 10x x  + 11x 
│ │ │ │ -000be2e0: 7820 202d 2033 7820 202d 2036 7820 7820  x  - 3x  - 6x x 
│ │ │ │ -000be2f0: 202b 2032 3878 2078 2078 2020 2d20 3730   + 28x x x  - 70
│ │ │ │ -000be300: 7820 2020 2020 2020 207c 0a7c 2020 2020  x        |.|    
│ │ │ │ -000be310: 2020 3020 3120 3220 2020 2020 2031 2032    0 1 2      1 2
│ │ │ │ -000be320: 2020 2020 2020 3020 3220 2020 2020 2031        0 2      1
│ │ │ │ -000be330: 2032 2020 2020 2032 2020 2020 2030 2033   2     2     0 3
│ │ │ │ -000be340: 2020 2020 2020 3020 3120 3320 2020 2020        0 1 3     
│ │ │ │ -000be350: 2031 2020 2020 2020 207c 0a7c 2020 2020   1       |.|    
│ │ │ │ +000be270: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +000be280: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +000be290: 3220 2020 2020 2020 2032 2020 2020 2033  2        2     3
│ │ │ │ +000be2a0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +000be2b0: 2020 2020 2020 2020 207c 0a7c 2d20 3131           |.|- 11
│ │ │ │ +000be2c0: 7820 7820 202b 2031 3778 2078 2078 2020  x x  + 17x x x  
│ │ │ │ +000be2d0: 2d20 3234 7820 7820 202d 2031 3078 2078  - 24x x  - 10x x
│ │ │ │ +000be2e0: 2020 2b20 3131 7820 7820 202d 2033 7820    + 11x x  - 3x 
│ │ │ │ +000be2f0: 202d 2036 7820 7820 202b 2032 3878 2078   - 6x x  + 28x x
│ │ │ │ +000be300: 2078 2020 2d20 3730 787c 0a7c 2020 2020   x  - 70x|.|    
│ │ │ │ +000be310: 2030 2032 2020 2020 2020 3020 3120 3220   0 2      0 1 2 
│ │ │ │ +000be320: 2020 2020 2031 2032 2020 2020 2020 3020       1 2      0 
│ │ │ │ +000be330: 3220 2020 2020 2031 2032 2020 2020 2032  2      1 2     2
│ │ │ │ +000be340: 2020 2020 2030 2033 2020 2020 2020 3020       0 3      0 
│ │ │ │ +000be350: 3120 3320 2020 2020 207c 0a7c 2020 2020  1 3      |.|    
│ │ │ │  000be360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be3a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000be3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000be3f0: 2020 2020 2020 2020 207c 0a7c 362d 6469           |.|6-di
│ │ │ │ -000be400: 6d65 6e73 696f 6e61 6c20 7375 6276 6172  mensional subvar
│ │ │ │ -000be410: 6965 7479 206f 6620 5050 5e39 2920 2020  iety of PP^9)   
│ │ │ │ -000be420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000be3f0: 2020 2020 2020 2020 207c 0a7c 5050 5e36           |.|PP^6
│ │ │ │ +000be400: 2074 6f20 362d 6469 6d65 6e73 696f 6e61   to 6-dimensiona
│ │ │ │ +000be410: 6c20 7375 6276 6172 6965 7479 206f 6620  l subvariety of 
│ │ │ │ +000be420: 5050 5e39 2920 2020 2020 2020 2020 2020  PP^9)           
│ │ │ │  000be430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be440: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d           |.|----
│ │ │ │  000be450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000be460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000be470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000be480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000be490: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020  ---------|.|    
│ │ │ │ -000be4a0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +000be4a0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  000be4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000be4d0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -000be4e0: 2020 2020 2020 2020 207c 0a7c 2031 3074           |.| 10t
│ │ │ │ -000be4f0: 2020 2b20 3474 2074 2020 2d20 3774 2074    + 4t t  - 7t t
│ │ │ │ -000be500: 2020 2b20 3474 2074 2020 2d20 3274 2074    + 4t t  - 2t t
│ │ │ │ -000be510: 2020 2d20 3874 2074 2020 2b20 3239 7420    - 8t t  + 29t 
│ │ │ │ -000be520: 7420 202d 2034 7420 202b 2031 3074 2074  t  - 4t  + 10t t
│ │ │ │ -000be530: 2020 2d20 3474 2020 207c 0a7c 2020 2020    - 4t   |.|    
│ │ │ │ -000be540: 3620 2020 2020 3020 3720 2020 2020 3120  6     0 7     1 
│ │ │ │ -000be550: 3720 2020 2020 3220 3720 2020 2020 3420  7     2 7     4 
│ │ │ │ -000be560: 3720 2020 2020 3520 3720 2020 2020 2036  7     5 7      6
│ │ │ │ -000be570: 2037 2020 2020 2037 2020 2020 2020 3120   7     7      1 
│ │ │ │ -000be580: 3820 2020 2020 2020 207c 0a7c 2020 2020  8        |.|    
│ │ │ │ +000be4d0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +000be4e0: 2020 2020 2020 2020 207c 0a7c 2b20 3130           |.|+ 10
│ │ │ │ +000be4f0: 7420 202b 2034 7420 7420 202d 2037 7420  t  + 4t t  - 7t 
│ │ │ │ +000be500: 7420 202b 2034 7420 7420 202d 2032 7420  t  + 4t t  - 2t 
│ │ │ │ +000be510: 7420 202d 2038 7420 7420 202b 2032 3974  t  - 8t t  + 29t
│ │ │ │ +000be520: 2074 2020 2d20 3474 2020 2b20 3130 7420   t  - 4t  + 10t 
│ │ │ │ +000be530: 7420 202d 2034 7420 207c 0a7c 2020 2020  t  - 4t  |.|    
│ │ │ │ +000be540: 2036 2020 2020 2030 2037 2020 2020 2031   6     0 7     1
│ │ │ │ +000be550: 2037 2020 2020 2032 2037 2020 2020 2034   7     2 7     4
│ │ │ │ +000be560: 2037 2020 2020 2035 2037 2020 2020 2020   7     5 7      
│ │ │ │ +000be570: 3620 3720 2020 2020 3720 2020 2020 2031  6 7     7      1
│ │ │ │ +000be580: 2038 2020 2020 2020 207c 0a7c 2020 2020   8       |.|    
│ │ │ │  000be590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be5d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000be5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000be600: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +000be600: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  000be610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000be620: 2020 2020 2020 2020 207c 0a7c 7420 202b           |.|t  +
│ │ │ │ -000be630: 2033 7420 7420 202d 2032 7420 7420 202d   3t t  - 2t t  -
│ │ │ │ -000be640: 2035 7420 7420 202d 2031 3374 2074 2020   5t t  - 13t t  
│ │ │ │ -000be650: 2d20 7420 202d 2033 7420 7420 202d 2038  - t  - 3t t  - 8
│ │ │ │ -000be660: 7420 7420 202b 2032 7420 7420 202b 2038  t t  + 2t t  + 8
│ │ │ │ -000be670: 7420 7420 202d 2020 207c 0a7c 2037 2020  t t  -   |.| 7  
│ │ │ │ -000be680: 2020 2033 2037 2020 2020 2034 2037 2020     3 7     4 7  
│ │ │ │ -000be690: 2020 2035 2037 2020 2020 2020 3620 3720     5 7      6 7 
│ │ │ │ -000be6a0: 2020 2037 2020 2020 2030 2038 2020 2020     7     0 8    
│ │ │ │ -000be6b0: 2031 2038 2020 2020 2033 2038 2020 2020   1 8     3 8    
│ │ │ │ -000be6c0: 2034 2038 2020 2020 207c 0a7c 2020 2020   4 8     |.|    
│ │ │ │ +000be620: 2020 2020 2020 2020 207c 0a7c 2074 2020           |.| t  
│ │ │ │ +000be630: 2b20 3374 2074 2020 2d20 3274 2074 2020  + 3t t  - 2t t  
│ │ │ │ +000be640: 2d20 3574 2074 2020 2d20 3133 7420 7420  - 5t t  - 13t t 
│ │ │ │ +000be650: 202d 2074 2020 2d20 3374 2074 2020 2d20   - t  - 3t t  - 
│ │ │ │ +000be660: 3874 2074 2020 2b20 3274 2074 2020 2b20  8t t  + 2t t  + 
│ │ │ │ +000be670: 3874 2074 2020 2d20 207c 0a7c 3220 3720  8t t  -  |.|2 7 
│ │ │ │ +000be680: 2020 2020 3320 3720 2020 2020 3420 3720      3 7     4 7 
│ │ │ │ +000be690: 2020 2020 3520 3720 2020 2020 2036 2037      5 7      6 7
│ │ │ │ +000be6a0: 2020 2020 3720 2020 2020 3020 3820 2020      7     0 8   
│ │ │ │ +000be6b0: 2020 3120 3820 2020 2020 3320 3820 2020    1 8     3 8   
│ │ │ │ +000be6c0: 2020 3420 3820 2020 207c 0a7c 2020 2020    4 8    |.|    
│ │ │ │  000be6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be710: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000be720: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +000be720: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  000be730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000be760: 2020 2020 2020 2020 207c 0a7c 2d20 3474           |.|- 4t
│ │ │ │ -000be770: 2074 2020 2b20 7420 202d 2032 7420 7420   t  + t  - 2t t 
│ │ │ │ -000be780: 202b 2074 2074 2020 2b20 3274 2074 2020   + t t  + 2t t  
│ │ │ │ -000be790: 2d20 3274 2074 2020 2d20 3374 2074 2020  - 2t t  - 3t t  
│ │ │ │ -000be7a0: 2b20 7420 7420 202d 2074 2074 2020 2d20  + t t  - t t  - 
│ │ │ │ -000be7b0: 7420 7420 202b 2020 207c 0a7c 2020 2020  t t  +   |.|    
│ │ │ │ -000be7c0: 3620 3720 2020 2037 2020 2020 2031 2038  6 7    7     1 8
│ │ │ │ -000be7d0: 2020 2020 3320 3820 2020 2020 3420 3820      3 8     4 8 
│ │ │ │ -000be7e0: 2020 2020 3520 3820 2020 2020 3620 3820      5 8     6 8 
│ │ │ │ -000be7f0: 2020 2037 2038 2020 2020 3020 3920 2020     7 8    0 9   
│ │ │ │ -000be800: 2032 2039 2020 2020 207c 0a7c 2020 2020   2 9     |.|    
│ │ │ │ +000be760: 2020 2020 2020 2020 207c 0a7c 202d 2034           |.| - 4
│ │ │ │ +000be770: 7420 7420 202b 2074 2020 2d20 3274 2074  t t  + t  - 2t t
│ │ │ │ +000be780: 2020 2b20 7420 7420 202b 2032 7420 7420    + t t  + 2t t 
│ │ │ │ +000be790: 202d 2032 7420 7420 202d 2033 7420 7420   - 2t t  - 3t t 
│ │ │ │ +000be7a0: 202b 2074 2074 2020 2d20 7420 7420 202d   + t t  - t t  -
│ │ │ │ +000be7b0: 2074 2074 2020 2b20 207c 0a7c 2020 2020   t t  +  |.|    
│ │ │ │ +000be7c0: 2036 2037 2020 2020 3720 2020 2020 3120   6 7    7     1 
│ │ │ │ +000be7d0: 3820 2020 2033 2038 2020 2020 2034 2038  8    3 8     4 8
│ │ │ │ +000be7e0: 2020 2020 2035 2038 2020 2020 2036 2038       5 8     6 8
│ │ │ │ +000be7f0: 2020 2020 3720 3820 2020 2030 2039 2020      7 8    0 9  
│ │ │ │ +000be800: 2020 3220 3920 2020 207c 0a7c 2020 2020    2 9    |.|    
│ │ │ │  000be810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be850: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000be860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be8a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000be8b0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -000be8c0: 2020 2020 2020 2032 2020 2020 2032 2020         2     2  
│ │ │ │ -000be8d0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -000be8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000be8f0: 2020 2020 2020 2020 207c 0a7c 2078 2020           |.| x  
│ │ │ │ -000be900: 2b20 3378 2078 2020 2b20 3278 2078 2078  + 3x x  + 2x x x
│ │ │ │ -000be910: 2020 2d20 7820 7820 202b 2036 7820 7820    - x x  + 6x x 
│ │ │ │ -000be920: 202d 2032 7820 7820 7820 202d 2034 7820   - 2x x x  - 4x 
│ │ │ │ -000be930: 7820 202d 2078 2078 2078 2020 2b20 3878  x  - x x x  + 8x
│ │ │ │ -000be940: 2078 2078 2020 2020 207c 0a7c 3220 3420   x x     |.|2 4 
│ │ │ │ -000be950: 2020 2020 3220 3420 2020 2020 3220 3320      2 4     2 3 
│ │ │ │ -000be960: 3420 2020 2032 2034 2020 2020 2030 2035  4    2 4     0 5
│ │ │ │ -000be970: 2020 2020 2030 2031 2035 2020 2020 2031       0 1 5     1
│ │ │ │ -000be980: 2035 2020 2020 3020 3220 3520 2020 2020   5    0 2 5     
│ │ │ │ -000be990: 3120 3220 3520 2020 207c 0a7c 2020 2020  1 2 5    |.|    
│ │ │ │ +000be8b0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +000be8c0: 2020 2020 2020 2020 3220 2020 2020 3220          2     2 
│ │ │ │ +000be8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000be8e0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +000be8f0: 2020 2020 2020 2020 207c 0a7c 7820 7820           |.|x x 
│ │ │ │ +000be900: 202b 2033 7820 7820 202b 2032 7820 7820   + 3x x  + 2x x 
│ │ │ │ +000be910: 7820 202d 2078 2078 2020 2b20 3678 2078  x  - x x  + 6x x
│ │ │ │ +000be920: 2020 2d20 3278 2078 2078 2020 2d20 3478    - 2x x x  - 4x
│ │ │ │ +000be930: 2078 2020 2d20 7820 7820 7820 202b 2038   x  - x x x  + 8
│ │ │ │ +000be940: 7820 7820 7820 2020 207c 0a7c 2032 2034  x x x    |.| 2 4
│ │ │ │ +000be950: 2020 2020 2032 2034 2020 2020 2032 2033       2 4     2 3
│ │ │ │ +000be960: 2034 2020 2020 3220 3420 2020 2020 3020   4    2 4     0 
│ │ │ │ +000be970: 3520 2020 2020 3020 3120 3520 2020 2020  5     0 1 5     
│ │ │ │ +000be980: 3120 3520 2020 2030 2032 2035 2020 2020  1 5    0 2 5    
│ │ │ │ +000be990: 2031 2032 2035 2020 207c 0a7c 2020 2020   1 2 5   |.|    
│ │ │ │  000be9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000be9e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000be9f0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -000bea00: 2020 2020 3220 2020 2020 2020 2032 2020      2        2  
│ │ │ │ -000bea10: 2020 2020 2032 2020 2020 2033 2020 2020       2     3    
│ │ │ │ -000bea20: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -000bea30: 2020 2020 2020 2020 207c 0a7c 2d20 3978           |.|- 9x
│ │ │ │ -000bea40: 2078 2078 2020 2d20 3278 2078 2020 2d20   x x  - 2x x  - 
│ │ │ │ -000bea50: 3278 2078 2020 2b20 3130 7820 7820 202b  2x x  + 10x x  +
│ │ │ │ -000bea60: 2032 7820 7820 202d 2034 7820 202d 2078   2x x  - 4x  - x
│ │ │ │ -000bea70: 2078 2020 2d20 3378 2078 2078 2020 2b20   x  - 3x x x  + 
│ │ │ │ -000bea80: 7820 7820 7820 2020 207c 0a7c 2020 2020  x x x    |.|    
│ │ │ │ -000bea90: 3120 3220 3320 2020 2020 3220 3320 2020  1 2 3     2 3   
│ │ │ │ -000beaa0: 2020 3020 3320 2020 2020 2031 2033 2020    0 3      1 3  
│ │ │ │ -000beab0: 2020 2032 2033 2020 2020 2033 2020 2020     2 3     3    
│ │ │ │ -000beac0: 3020 3420 2020 2020 3020 3120 3420 2020  0 4     0 1 4   
│ │ │ │ -000bead0: 2030 2032 2034 2020 207c 0a7c 2020 2020   0 2 4   |.|    
│ │ │ │ +000be9f0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +000bea00: 2020 2020 2032 2020 2020 2020 2020 3220       2        2 
│ │ │ │ +000bea10: 2020 2020 2020 3220 2020 2020 3320 2020        2     3   
│ │ │ │ +000bea20: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +000bea30: 2020 2020 2020 2020 207c 0a7c 202d 2039           |.| - 9
│ │ │ │ +000bea40: 7820 7820 7820 202d 2032 7820 7820 202d  x x x  - 2x x  -
│ │ │ │ +000bea50: 2032 7820 7820 202b 2031 3078 2078 2020   2x x  + 10x x  
│ │ │ │ +000bea60: 2b20 3278 2078 2020 2d20 3478 2020 2d20  + 2x x  - 4x  - 
│ │ │ │ +000bea70: 7820 7820 202d 2033 7820 7820 7820 202b  x x  - 3x x x  +
│ │ │ │ +000bea80: 2078 2078 2078 2020 207c 0a7c 2020 2020   x x x   |.|    
│ │ │ │ +000bea90: 2031 2032 2033 2020 2020 2032 2033 2020   1 2 3     2 3  
│ │ │ │ +000beaa0: 2020 2030 2033 2020 2020 2020 3120 3320     0 3      1 3 
│ │ │ │ +000beab0: 2020 2020 3220 3320 2020 2020 3320 2020      2 3     3   
│ │ │ │ +000beac0: 2030 2034 2020 2020 2030 2031 2034 2020   0 4     0 1 4  
│ │ │ │ +000bead0: 2020 3020 3220 3420 207c 0a7c 2020 2020    0 2 4  |.|    
│ │ │ │  000beae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000beaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000beb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000beb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000beb20: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000beb30: 3220 2020 2020 2032 2020 2020 2020 2020  2      2        
│ │ │ │ -000beb40: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +000beb30: 2032 2020 2020 2020 3220 2020 2020 2020   2      2       
│ │ │ │ +000beb40: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │  000beb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000beb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000beb70: 2020 2020 2020 2020 207c 0a7c 3478 2078           |.|4x x
│ │ │ │ -000beb80: 2020 2b20 3131 7820 7820 202b 2078 2078    + 11x x  + x x
│ │ │ │ -000beb90: 2078 2020 2d20 3278 2078 2020 2d20 3578   x  - 2x x  - 5x
│ │ │ │ -000beba0: 2078 2078 2020 2d20 3578 2078 2078 2020   x x  - 5x x x  
│ │ │ │ -000bebb0: 2d20 3678 2078 2078 2020 2d20 3278 2078  - 6x x x  - 2x x
│ │ │ │ -000bebc0: 2078 2020 2b20 2020 207c 0a7c 2020 3220   x  +    |.|  2 
│ │ │ │ -000bebd0: 3320 2020 2020 2030 2034 2020 2020 3020  3      0 4    0 
│ │ │ │ -000bebe0: 3120 3420 2020 2020 3120 3420 2020 2020  1 4     1 4     
│ │ │ │ -000bebf0: 3020 3220 3420 2020 2020 3120 3220 3420  0 2 4     1 2 4 
│ │ │ │ -000bec00: 2020 2020 3020 3320 3420 2020 2020 3120      0 3 4     1 
│ │ │ │ -000bec10: 3320 3420 2020 2020 207c 0a7c 2020 2020  3 4      |.|    
│ │ │ │ +000beb70: 2020 2020 2020 2020 207c 0a7c 2034 7820           |.| 4x 
│ │ │ │ +000beb80: 7820 202b 2031 3178 2078 2020 2b20 7820  x  + 11x x  + x 
│ │ │ │ +000beb90: 7820 7820 202d 2032 7820 7820 202d 2035  x x  - 2x x  - 5
│ │ │ │ +000beba0: 7820 7820 7820 202d 2035 7820 7820 7820  x x x  - 5x x x 
│ │ │ │ +000bebb0: 202d 2036 7820 7820 7820 202d 2032 7820   - 6x x x  - 2x 
│ │ │ │ +000bebc0: 7820 7820 202b 2020 207c 0a7c 2020 2032  x x  +   |.|   2
│ │ │ │ +000bebd0: 2033 2020 2020 2020 3020 3420 2020 2030   3      0 4    0
│ │ │ │ +000bebe0: 2031 2034 2020 2020 2031 2034 2020 2020   1 4     1 4    
│ │ │ │ +000bebf0: 2030 2032 2034 2020 2020 2031 2032 2034   0 2 4     1 2 4
│ │ │ │ +000bec00: 2020 2020 2030 2033 2034 2020 2020 2031       0 3 4     1
│ │ │ │ +000bec10: 2033 2034 2020 2020 207c 0a7c 2020 2020   3 4     |.|    
│ │ │ │  000bec20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bec30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bec40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bec50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bec60: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000bec70: 2020 3220 2020 2020 2020 2020 3220 2020    2         2   
│ │ │ │ -000bec80: 2020 2020 3220 2020 2020 2032 2020 2020      2      2    
│ │ │ │ +000bec70: 2020 2032 2020 2020 2020 2020 2032 2020     2         2  
│ │ │ │ +000bec80: 2020 2020 2032 2020 2020 2020 3220 2020       2      2   
│ │ │ │  000bec90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000beca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000becb0: 2032 2020 2020 2020 207c 0a7c 2020 2b20   2       |.|  + 
│ │ │ │ -000becc0: 3878 2078 2020 2b20 3878 2078 2020 2b20  8x x  + 8x x  + 
│ │ │ │ -000becd0: 3878 2078 2020 2d20 3130 7820 7820 202d  8x x  - 10x x  -
│ │ │ │ -000bece0: 2034 7820 7820 7820 202b 2036 7820 7820   4x x x  + 6x x 
│ │ │ │ -000becf0: 7820 202b 2032 7820 7820 7820 202b 2032  x  + 2x x x  + 2
│ │ │ │ -000bed00: 7820 7820 202b 2020 207c 0a7c 3320 2020  x x  +   |.|3   
│ │ │ │ -000bed10: 2020 3220 3320 2020 2020 3020 3320 2020    2 3     0 3   
│ │ │ │ -000bed20: 2020 3220 3320 2020 2020 2030 2034 2020    2 3      0 4  
│ │ │ │ -000bed30: 2020 2030 2031 2034 2020 2020 2030 2032     0 1 4     0 2
│ │ │ │ -000bed40: 2034 2020 2020 2031 2032 2034 2020 2020   4     1 2 4    
│ │ │ │ -000bed50: 2032 2034 2020 2020 207c 0a7c 2020 2020   2 4     |.|    
│ │ │ │ +000becb0: 2020 3220 2020 2020 207c 0a7c 7820 202b    2      |.|x  +
│ │ │ │ +000becc0: 2038 7820 7820 202b 2038 7820 7820 202b   8x x  + 8x x  +
│ │ │ │ +000becd0: 2038 7820 7820 202d 2031 3078 2078 2020   8x x  - 10x x  
│ │ │ │ +000bece0: 2d20 3478 2078 2078 2020 2b20 3678 2078  - 4x x x  + 6x x
│ │ │ │ +000becf0: 2078 2020 2b20 3278 2078 2078 2020 2b20   x  + 2x x x  + 
│ │ │ │ +000bed00: 3278 2078 2020 2b20 207c 0a7c 2033 2020  2x x  +  |.| 3  
│ │ │ │ +000bed10: 2020 2032 2033 2020 2020 2030 2033 2020     2 3     0 3  
│ │ │ │ +000bed20: 2020 2032 2033 2020 2020 2020 3020 3420     2 3      0 4 
│ │ │ │ +000bed30: 2020 2020 3020 3120 3420 2020 2020 3020      0 1 4     0 
│ │ │ │ +000bed40: 3220 3420 2020 2020 3120 3220 3420 2020  2 4     1 2 4   
│ │ │ │ +000bed50: 2020 3220 3420 2020 207c 0a7c 2020 2020    2 4    |.|    
│ │ │ │  000bed60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bed70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bed80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bed90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000beda0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bedb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000bedc0: 2020 2032 2020 2020 2020 2020 2020 3220     2          2 
│ │ │ │ -000bedd0: 2020 2020 2020 2032 2020 2020 2020 2032         2       2
│ │ │ │ -000bede0: 2020 2020 2020 3320 2020 2020 2032 2020        3      2  
│ │ │ │ -000bedf0: 2020 2020 2020 2020 207c 0a7c 3778 2078           |.|7x x
│ │ │ │ -000bee00: 2078 2020 2b20 3134 7820 7820 7820 202d   x  + 14x x x  -
│ │ │ │ -000bee10: 2035 7820 7820 202d 2031 3078 2078 2020   5x x  - 10x x  
│ │ │ │ -000bee20: 2b20 3338 7820 7820 202d 2038 7820 7820  + 38x x  - 8x x 
│ │ │ │ -000bee30: 202d 2031 3278 2020 2b20 3234 7820 7820   - 12x  + 24x x 
│ │ │ │ -000bee40: 202d 2031 3678 2020 207c 0a7c 2020 3020   - 16x   |.|  0 
│ │ │ │ -000bee50: 3220 3320 2020 2020 2031 2032 2033 2020  2 3      1 2 3  
│ │ │ │ -000bee60: 2020 2032 2033 2020 2020 2020 3020 3320     2 3      0 3 
│ │ │ │ -000bee70: 2020 2020 2031 2033 2020 2020 2032 2033       1 3     2 3
│ │ │ │ -000bee80: 2020 2020 2020 3320 2020 2020 2030 2034        3      0 4
│ │ │ │ -000bee90: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000bedc0: 2020 2020 3220 2020 2020 2020 2020 2032      2          2
│ │ │ │ +000bedd0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +000bede0: 3220 2020 2020 2033 2020 2020 2020 3220  2      3      2 
│ │ │ │ +000bedf0: 2020 2020 2020 2020 207c 0a7c 2037 7820           |.| 7x 
│ │ │ │ +000bee00: 7820 7820 202b 2031 3478 2078 2078 2020  x x  + 14x x x  
│ │ │ │ +000bee10: 2d20 3578 2078 2020 2d20 3130 7820 7820  - 5x x  - 10x x 
│ │ │ │ +000bee20: 202b 2033 3878 2078 2020 2d20 3878 2078   + 38x x  - 8x x
│ │ │ │ +000bee30: 2020 2d20 3132 7820 202b 2032 3478 2078    - 12x  + 24x x
│ │ │ │ +000bee40: 2020 2d20 3136 7820 207c 0a7c 2020 2030    - 16x  |.|   0
│ │ │ │ +000bee50: 2032 2033 2020 2020 2020 3120 3220 3320   2 3      1 2 3 
│ │ │ │ +000bee60: 2020 2020 3220 3320 2020 2020 2030 2033      2 3      0 3
│ │ │ │ +000bee70: 2020 2020 2020 3120 3320 2020 2020 3220        1 3     2 
│ │ │ │ +000bee80: 3320 2020 2020 2033 2020 2020 2020 3020  3      3      0 
│ │ │ │ +000bee90: 3420 2020 2020 2020 207c 0a7c 2020 2020  4        |.|    
│ │ │ │  000beea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000beeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000beec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000beed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000beee0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000beef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000bef00: 2020 2032 2020 2020 2020 2020 2020 3220     2          2 
│ │ │ │ -000bef10: 2020 2020 2020 3220 2020 2020 3320 2020        2     3   
│ │ │ │ -000bef20: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -000bef30: 2020 2020 2020 2020 207c 0a7c 3278 2078           |.|2x x
│ │ │ │ -000bef40: 2078 2020 2d20 3136 7820 7820 7820 202b   x  - 16x x x  +
│ │ │ │ -000bef50: 2034 7820 7820 202d 2032 3878 2078 2020   4x x  - 28x x  
│ │ │ │ -000bef60: 2b20 3878 2078 2020 2b20 3878 2020 2d20  + 8x x  + 8x  - 
│ │ │ │ -000bef70: 3131 7820 7820 202b 2031 3778 2078 2078  11x x  + 17x x x
│ │ │ │ -000bef80: 2020 2b20 3478 2020 207c 0a7c 2020 3020    + 4x   |.|  0 
│ │ │ │ -000bef90: 3220 3320 2020 2020 2031 2032 2033 2020  2 3      1 2 3  
│ │ │ │ -000befa0: 2020 2032 2033 2020 2020 2020 3120 3320     2 3      1 3 
│ │ │ │ -000befb0: 2020 2020 3220 3320 2020 2020 3320 2020      2 3     3   
│ │ │ │ -000befc0: 2020 2030 2034 2020 2020 2020 3020 3120     0 4      0 1 
│ │ │ │ -000befd0: 3420 2020 2020 2020 207c 0a7c 2020 2020  4        |.|    
│ │ │ │ +000bef00: 2020 2020 3220 2020 2020 2020 2020 2032      2          2
│ │ │ │ +000bef10: 2020 2020 2020 2032 2020 2020 2033 2020         2     3  
│ │ │ │ +000bef20: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +000bef30: 2020 2020 2020 2020 207c 0a7c 2032 7820           |.| 2x 
│ │ │ │ +000bef40: 7820 7820 202d 2031 3678 2078 2078 2020  x x  - 16x x x  
│ │ │ │ +000bef50: 2b20 3478 2078 2020 2d20 3238 7820 7820  + 4x x  - 28x x 
│ │ │ │ +000bef60: 202b 2038 7820 7820 202b 2038 7820 202d   + 8x x  + 8x  -
│ │ │ │ +000bef70: 2031 3178 2078 2020 2b20 3137 7820 7820   11x x  + 17x x 
│ │ │ │ +000bef80: 7820 202b 2034 7820 207c 0a7c 2020 2030  x  + 4x  |.|   0
│ │ │ │ +000bef90: 2032 2033 2020 2020 2020 3120 3220 3320   2 3      1 2 3 
│ │ │ │ +000befa0: 2020 2020 3220 3320 2020 2020 2031 2033      2 3      1 3
│ │ │ │ +000befb0: 2020 2020 2032 2033 2020 2020 2033 2020       2 3     3  
│ │ │ │ +000befc0: 2020 2020 3020 3420 2020 2020 2030 2031      0 4      0 1
│ │ │ │ +000befd0: 2034 2020 2020 2020 207c 0a7c 2020 2020   4       |.|    
│ │ │ │  000befe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000beff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bf000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bf010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bf020: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000bf030: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -000bf040: 2020 2020 2020 3220 2020 2020 2020 3220        2       2 
│ │ │ │ -000bf050: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -000bf060: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -000bf070: 2020 2020 2020 2020 207c 0a7c 2d20 3136           |.|- 16
│ │ │ │ -000bf080: 7820 7820 7820 202b 2035 7820 7820 202b  x x x  + 5x x  +
│ │ │ │ -000bf090: 2031 3678 2078 2020 2b20 3878 2078 2020   16x x  + 8x x  
│ │ │ │ -000bf0a0: 2d20 3332 7820 7820 202b 2034 7820 7820  - 32x x  + 4x x 
│ │ │ │ -000bf0b0: 7820 202b 2038 7820 7820 202d 2032 7820  x  + 8x x  - 2x 
│ │ │ │ -000bf0c0: 7820 7820 202d 2020 207c 0a7c 2020 2020  x x  -   |.|    
│ │ │ │ -000bf0d0: 2031 2032 2033 2020 2020 2032 2033 2020   1 2 3     2 3  
│ │ │ │ -000bf0e0: 2020 2020 3020 3320 2020 2020 3220 3320      0 3     2 3 
│ │ │ │ -000bf0f0: 2020 2020 2030 2034 2020 2020 2030 2031       0 4     0 1
│ │ │ │ -000bf100: 2034 2020 2020 2031 2034 2020 2020 2030   4     1 4     0
│ │ │ │ -000bf110: 2032 2034 2020 2020 207c 0a7c 2020 2020   2 4     |.|    
│ │ │ │ +000bf030: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +000bf040: 2020 2020 2020 2032 2020 2020 2020 2032         2       2
│ │ │ │ +000bf050: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +000bf060: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +000bf070: 2020 2020 2020 2020 207c 0a7c 202d 2031           |.| - 1
│ │ │ │ +000bf080: 3678 2078 2078 2020 2b20 3578 2078 2020  6x x x  + 5x x  
│ │ │ │ +000bf090: 2b20 3136 7820 7820 202b 2038 7820 7820  + 16x x  + 8x x 
│ │ │ │ +000bf0a0: 202d 2033 3278 2078 2020 2b20 3478 2078   - 32x x  + 4x x
│ │ │ │ +000bf0b0: 2078 2020 2b20 3878 2078 2020 2d20 3278   x  + 8x x  - 2x
│ │ │ │ +000bf0c0: 2078 2078 2020 2d20 207c 0a7c 2020 2020   x x  -  |.|    
│ │ │ │ +000bf0d0: 2020 3120 3220 3320 2020 2020 3220 3320    1 2 3     2 3 
│ │ │ │ +000bf0e0: 2020 2020 2030 2033 2020 2020 2032 2033       0 3     2 3
│ │ │ │ +000bf0f0: 2020 2020 2020 3020 3420 2020 2020 3020        0 4     0 
│ │ │ │ +000bf100: 3120 3420 2020 2020 3120 3420 2020 2020  1 4     1 4     
│ │ │ │ +000bf110: 3020 3220 3420 2020 207c 0a7c 2020 2020  0 2 4    |.|    
│ │ │ │  000bf120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bf130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bf140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bf150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bf160: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000bf170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000bf180: 2020 2032 2020 2020 2020 2020 2032 2020     2         2  
│ │ │ │ -000bf190: 2020 2020 2020 3220 2020 2020 2020 2032        2        2
│ │ │ │ -000bf1a0: 2020 2020 2020 3320 2020 2020 3220 2020        3     2   
│ │ │ │ -000bf1b0: 2020 2020 2020 2020 207c 0a7c 7820 7820           |.|x x 
│ │ │ │ -000bf1c0: 7820 202d 2034 3778 2078 2078 2020 2b20  x  - 47x x x  + 
│ │ │ │ -000bf1d0: 3131 7820 7820 202d 2032 7820 7820 202d  11x x  - 2x x  -
│ │ │ │ -000bf1e0: 2036 3678 2078 2020 2b20 3232 7820 7820   66x x  + 22x x 
│ │ │ │ -000bf1f0: 202b 2032 3078 2020 2d20 3278 2078 2020   + 20x  - 2x x  
│ │ │ │ -000bf200: 2b20 3138 7820 2020 207c 0a7c 2030 2032  + 18x    |.| 0 2
│ │ │ │ -000bf210: 2033 2020 2020 2020 3120 3220 3320 2020   3      1 2 3   
│ │ │ │ -000bf220: 2020 2032 2033 2020 2020 2030 2033 2020     2 3     0 3  
│ │ │ │ -000bf230: 2020 2020 3120 3320 2020 2020 2032 2033      1 3      2 3
│ │ │ │ -000bf240: 2020 2020 2020 3320 2020 2020 3020 3420        3     0 4 
│ │ │ │ -000bf250: 2020 2020 2030 2020 207c 0a7c 2020 2020       0   |.|    
│ │ │ │ +000bf180: 2020 2020 3220 2020 2020 2020 2020 3220      2         2 
│ │ │ │ +000bf190: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +000bf1a0: 3220 2020 2020 2033 2020 2020 2032 2020  2      3     2  
│ │ │ │ +000bf1b0: 2020 2020 2020 2020 207c 0a7c 2078 2078           |.| x x
│ │ │ │ +000bf1c0: 2078 2020 2d20 3437 7820 7820 7820 202b   x  - 47x x x  +
│ │ │ │ +000bf1d0: 2031 3178 2078 2020 2d20 3278 2078 2020   11x x  - 2x x  
│ │ │ │ +000bf1e0: 2d20 3636 7820 7820 202b 2032 3278 2078  - 66x x  + 22x x
│ │ │ │ +000bf1f0: 2020 2b20 3230 7820 202d 2032 7820 7820    + 20x  - 2x x 
│ │ │ │ +000bf200: 202b 2031 3878 2020 207c 0a7c 2020 3020   + 18x   |.|  0 
│ │ │ │ +000bf210: 3220 3320 2020 2020 2031 2032 2033 2020  2 3      1 2 3  
│ │ │ │ +000bf220: 2020 2020 3220 3320 2020 2020 3020 3320      2 3     0 3 
│ │ │ │ +000bf230: 2020 2020 2031 2033 2020 2020 2020 3220       1 3      2 
│ │ │ │ +000bf240: 3320 2020 2020 2033 2020 2020 2030 2034  3      3     0 4
│ │ │ │ +000bf250: 2020 2020 2020 3020 207c 0a7c 2020 2020        0  |.|    
│ │ │ │  000bf260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bf270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bf280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bf290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bf2a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000bf2b0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -000bf2c0: 2020 2020 3220 2020 2020 2020 2032 2020      2        2  
│ │ │ │ -000bf2d0: 2020 2020 2020 3220 2020 2020 3320 2020        2     3   
│ │ │ │ -000bf2e0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -000bf2f0: 2020 2020 2020 2020 207c 0a7c 2032 3678           |.| 26x
│ │ │ │ -000bf300: 2078 2078 2020 2d20 3678 2078 2020 2b20   x x  - 6x x  + 
│ │ │ │ -000bf310: 3478 2078 2020 2b20 3238 7820 7820 202d  4x x  + 28x x  -
│ │ │ │ -000bf320: 2031 3278 2078 2020 2d20 3878 2020 2d20   12x x  - 8x  - 
│ │ │ │ -000bf330: 3130 7820 7820 7820 202d 2032 7820 7820  10x x x  - 2x x 
│ │ │ │ -000bf340: 202b 2031 3078 2020 207c 0a7c 2020 2020   + 10x   |.|    
│ │ │ │ -000bf350: 3120 3220 3320 2020 2020 3220 3320 2020  1 2 3     2 3   
│ │ │ │ -000bf360: 2020 3020 3320 2020 2020 2031 2033 2020    0 3      1 3  
│ │ │ │ -000bf370: 2020 2020 3220 3320 2020 2020 3320 2020      2 3     3   
│ │ │ │ -000bf380: 2020 2030 2031 2034 2020 2020 2031 2034     0 1 4     1 4
│ │ │ │ -000bf390: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000bf2b0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +000bf2c0: 2020 2020 2032 2020 2020 2020 2020 3220       2        2 
│ │ │ │ +000bf2d0: 2020 2020 2020 2032 2020 2020 2033 2020         2     3  
│ │ │ │ +000bf2e0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +000bf2f0: 2020 2020 2020 2020 207c 0a7c 2b20 3236           |.|+ 26
│ │ │ │ +000bf300: 7820 7820 7820 202d 2036 7820 7820 202b  x x x  - 6x x  +
│ │ │ │ +000bf310: 2034 7820 7820 202b 2032 3878 2078 2020   4x x  + 28x x  
│ │ │ │ +000bf320: 2d20 3132 7820 7820 202d 2038 7820 202d  - 12x x  - 8x  -
│ │ │ │ +000bf330: 2031 3078 2078 2078 2020 2d20 3278 2078   10x x x  - 2x x
│ │ │ │ +000bf340: 2020 2b20 3130 7820 207c 0a7c 2020 2020    + 10x  |.|    
│ │ │ │ +000bf350: 2031 2032 2033 2020 2020 2032 2033 2020   1 2 3     2 3  
│ │ │ │ +000bf360: 2020 2030 2033 2020 2020 2020 3120 3320     0 3      1 3 
│ │ │ │ +000bf370: 2020 2020 2032 2033 2020 2020 2033 2020       2 3     3  
│ │ │ │ +000bf380: 2020 2020 3020 3120 3420 2020 2020 3120      0 1 4     1 
│ │ │ │ +000bf390: 3420 2020 2020 2020 207c 0a7c 2020 2020  4        |.|    
│ │ │ │  000bf3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bf3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bf3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000bf3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000bf3e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000bf3e0: 2020 2020 2020 2020 207c 0a7c 3220 2020           |.|2   
│ │ │ │  000bf3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000bf400: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -000bf410: 2020 2020 2032 2020 2020 2020 2020 3220       2        2 
│ │ │ │ -000bf420: 2020 2020 2020 2032 2020 2020 2020 3320         2      3 
│ │ │ │ -000bf430: 2020 2020 3220 2020 207c 0a7c 7820 202d      2    |.|x  -
│ │ │ │ -000bf440: 2032 3178 2078 2078 2020 2b20 3437 7820   21x x x  + 47x 
│ │ │ │ -000bf450: 7820 7820 202d 2031 3378 2078 2020 2d20  x x  - 13x x  - 
│ │ │ │ -000bf460: 3134 7820 7820 202b 2036 3678 2078 2020  14x x  + 66x x  
│ │ │ │ -000bf470: 2d20 3232 7820 7820 202d 2032 3078 2020  - 22x x  - 20x  
│ │ │ │ -000bf480: 2b20 3278 2078 2020 207c 0a7c 2033 2020  + 2x x   |.| 3  
│ │ │ │ -000bf490: 2020 2020 3020 3220 3320 2020 2020 2031      0 2 3      1
│ │ │ │ -000bf4a0: 2032 2033 2020 2020 2020 3220 3320 2020   2 3      2 3   
│ │ │ │ -000bf4b0: 2020 2030 2033 2020 2020 2020 3120 3320     0 3      1 3 
│ │ │ │ -000bf4c0: 2020 2020 2032 2033 2020 2020 2020 3320       2 3      3 
│ │ │ │ -000bf4d0: 2020 2020 3020 2020 207c 0a7c 2d2d 2d2d      0    |.|----
│ │ │ │ +000bf400: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +000bf410: 2020 2020 2020 3220 2020 2020 2020 2032        2        2
│ │ │ │ +000bf420: 2020 2020 2020 2020 3220 2020 2020 2033          2      3
│ │ │ │ +000bf430: 2020 2020 2032 2020 207c 0a7c 2078 2020       2   |.| x  
│ │ │ │ +000bf440: 2d20 3231 7820 7820 7820 202b 2034 3778  - 21x x x  + 47x
│ │ │ │ +000bf450: 2078 2078 2020 2d20 3133 7820 7820 202d   x x  - 13x x  -
│ │ │ │ +000bf460: 2031 3478 2078 2020 2b20 3636 7820 7820   14x x  + 66x x 
│ │ │ │ +000bf470: 202d 2032 3278 2078 2020 2d20 3230 7820   - 22x x  - 20x 
│ │ │ │ +000bf480: 202b 2032 7820 7820 207c 0a7c 3120 3320   + 2x x  |.|1 3 
│ │ │ │ +000bf490: 2020 2020 2030 2032 2033 2020 2020 2020       0 2 3      
│ │ │ │ +000bf4a0: 3120 3220 3320 2020 2020 2032 2033 2020  1 2 3      2 3  
│ │ │ │ +000bf4b0: 2020 2020 3020 3320 2020 2020 2031 2033      0 3      1 3
│ │ │ │ +000bf4c0: 2020 2020 2020 3220 3320 2020 2020 2033        2 3      3
│ │ │ │ +000bf4d0: 2020 2020 2030 2020 207c 0a7c 2d2d 2d2d       0   |.|----
│ │ │ │  000bf4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000bf4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000bf500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000bf510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000bf520: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2074 2020  ---------|.| t  
│ │ │ │  000bf530: 2d20 3133 7420 7420 202b 2031 3374 2074  - 13t t  + 13t t
│ │ │ │  000bf540: 2020 2b20 3138 7420 7420 202d 2034 7420    + 18t t  - 4t 
│ │ │ │ @@ -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 3138 3239 3834 7320  used 0.0182984s 
│ │ │ │ -000c55d0: 2863 7075 293b 2030 2e30 3138 3238 3537  (cpu); 0.0182857
│ │ │ │ +000c55c0: 7573 6564 2030 2e30 3138 3536 3733 7320  used 0.0185673s 
│ │ │ │ +000c55d0: 2863 7075 293b 2030 2e30 3138 3536 3932  (cpu); 0.0185692
│ │ │ │  000c55e0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │  000c55f0: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │  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 3733 3339 3237  - used 0.0733927
│ │ │ │ -000c6330: 7320 2863 7075 293b 2030 2e30 3733 3339  s (cpu); 0.07339
│ │ │ │ -000c6340: 3231 7320 2874 6872 6561 6429 3b20 3073  21s (thread); 0s
│ │ │ │ +000c6320: 2d20 7573 6564 2030 2e30 3735 3533 3232  - used 0.0755322
│ │ │ │ +000c6330: 7320 2863 7075 293b 2030 2e30 3735 3533  s (cpu); 0.07553
│ │ │ │ +000c6340: 3139 7320 2874 6872 6561 6429 3b20 3073  19s (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 3130 3233 3273  - used 0.110232s
│ │ │ │ -000c85e0: 2028 6370 7529 3b20 302e 3033 3037 3435   (cpu); 0.030745
│ │ │ │ -000c85f0: 3673 2028 7468 7265 6164 293b 2030 7320  6s (thread); 0s 
│ │ │ │ -000c8600: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │ +000c85d0: 2d20 7573 6564 2030 2e31 3132 3631 7320  - used 0.11261s 
│ │ │ │ +000c85e0: 2863 7075 293b 2030 2e30 3331 3131 3773  (cpu); 0.031117s
│ │ │ │ +000c85f0: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ +000c8600: 6329 2020 2020 2020 2020 2020 2020 2020  c)              
│ │ │ │  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 
│ │ │ │ @@ -52398,16 +52398,16 @@
│ │ │ │  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 3334 3139 3773 2028 6370 7529  0.0234197s (cpu)
│ │ │ │ -000ccb50: 3b20 302e 3032 3334 3138 3973 2028 7468  ; 0.0234189s (th
│ │ │ │ +000ccb40: 302e 3032 3532 3533 3173 2028 6370 7529  0.0252531s (cpu)
│ │ │ │ +000ccb50: 3b20 302e 3032 3532 3533 3773 2028 7468  ; 0.0252537s (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
│ │ │ │ @@ -52422,17 +52422,17 @@
│ │ │ │  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 3030 3339 7320 2863 7075 293b 2030  540039s (cpu); 0
│ │ │ │ -000cccd0: 2e30 3035 3430 3037 3873 2028 7468 7265  .00540078s (thre
│ │ │ │ -000ccce0: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │ +000cccc0: 3539 3430 3273 2028 6370 7529 3b20 302e  59402s (cpu); 0.
│ │ │ │ +000cccd0: 3030 3539 3436 3032 7320 2874 6872 6561  00594602s (threa
│ │ │ │ +000ccce0: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 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 = 
│ │ │ │  000ccd50: 7261 7469 6f6e 616c 206d 6170 2064 6566  rational map def
│ │ │ │ @@ -52488,18 +52488,18 @@
│ │ │ │  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 3434 3130 3473 2028  sed 0.0044104s (
│ │ │ │ -000cd0f0: 6370 7529 3b20 302e 3030 3434 3131 3137  cpu); 0.00441117
│ │ │ │ -000cd100: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ -000cd110: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │ +000cd0e0: 7365 6420 302e 3030 3530 3637 3036 7320  sed 0.00506706s 
│ │ │ │ +000cd0f0: 2863 7075 293b 2030 2e30 3035 3037 3232  (cpu); 0.0050722
│ │ │ │ +000cd100: 3773 2028 7468 7265 6164 293b 2030 7320  7s (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
│ │ │ │  000cd180: 206d 6170 2064 6566 696e 6564 2062 7920   map defined by
│ │ ├── ./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 3136 3937   -- used 0.01697
│ │ │ │ -00004d60: 3373 2028 6370 7529 3b20 302e 3031 3630  3s (cpu); 0.0160
│ │ │ │ -00004d70: 3832 3873 2020 2020 2020 7c0a 7c20 2020  828s      |.|   
│ │ │ │ +00004d50: 202d 2d20 7573 6564 2030 2e30 3432 3437   -- used 0.04247
│ │ │ │ +00004d60: 3136 7320 2863 7075 293b 2030 2e30 3234  16s (cpu); 0.024
│ │ │ │ +00004d70: 3232 3132 7320 2020 2020 7c0a 7c20 2020  2212s     |.|   
│ │ │ │  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 3838 3930 3173 2028 6370 7529  0.0188901s (cpu)
│ │ │ │ -00005de0: 3b20 302e 3031 3733 3434 3373 2020 2020  ; 0.0173443s    
│ │ │ │ +00005dd0: 302e 3235 3436 3239 7320 2863 7075 293b  0.254629s (cpu);
│ │ │ │ +00005de0: 2030 2e30 3533 3839 3238 7320 2020 2020   0.0538928s     
│ │ │ │  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 3236  :  -- used 0.126
│ │ │ │ -0000aaa0: 3936 3273 2028 6370 7529 3b20 302e 3035  962s (cpu); 0.05
│ │ │ │ -0000aab0: 3234 3534 3473 2020 2020 2020 7c0a 7c20  24544s      |.| 
│ │ │ │ +0000aa90: 3a20 202d 2d20 7573 6564 2030 2e32 3636  :  -- used 0.266
│ │ │ │ +0000aaa0: 3437 3173 2028 6370 7529 3b20 302e 3036  471s (cpu); 0.06
│ │ │ │ +0000aab0: 3834 3132 3373 2020 2020 2020 7c0a 7c20  84123s      |.| 
│ │ │ │  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 3437  :  -- used 0.547
│ │ │ │ -0000b1d0: 3137 7320 2863 7075 293b 2030 2e34 3732  17s (cpu); 0.472
│ │ │ │ -0000b1e0: 3632 3373 2020 2020 2020 2020 7c0a 7c20  623s        |.| 
│ │ │ │ +0000b1c0: 3a20 202d 2d20 7573 6564 2030 2e36 3435  :  -- used 0.645
│ │ │ │ +0000b1d0: 3535 3673 2028 6370 7529 3b20 302e 3630  556s (cpu); 0.60
│ │ │ │ +0000b1e0: 3936 3234 7320 2020 2020 2020 7c0a 7c20  9624s       |.| 
│ │ │ │  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 3133 3831 3239 7320 2863 7075   0.0138129s (cpu
│ │ │ │ -00011e00: 293b 2030 2e30 3133 3039 3538 7320 2020  ); 0.0130958s   
│ │ │ │ +00011df0: 2030 2e30 3333 3231 3234 7320 2863 7075   0.0332124s (cpu
│ │ │ │ +00011e00: 293b 2030 2e30 3230 3532 3636 7320 2020  ); 0.0205266s   
│ │ │ │  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: 3031 3938 3837 7320 2863 7075 293b 2030  019887s (cpu); 0
│ │ │ │ -000189f0: 2e30 3137 3134 3636 7320 2020 2020 207c  .0171466s      |
│ │ │ │ +000189e0: 3133 3531 3234 7320 2863 7075 293b 2030  135124s (cpu); 0
│ │ │ │ +000189f0: 2e30 3333 3636 3235 7320 2020 2020 207c  .0336625s      |
│ │ │ │  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 3031   :  -- used 0.01
│ │ │ │ -0003bd10: 3435 3639 3773 2028 6370 7529 3b20 302e  45697s (cpu); 0.
│ │ │ │ -0003bd20: 3031 3338 3230 3773 2020 2020 207c 0a7c  0138207s     |.|
│ │ │ │ +0003bd00: 203a 2020 2d2d 2075 7365 6420 302e 3039   :  -- used 0.09
│ │ │ │ +0003bd10: 3934 3439 7320 2863 7075 293b 2030 2e30  9449s (cpu); 0.0
│ │ │ │ +0003bd20: 3237 3738 3673 2020 2020 2020 207c 0a7c  27786s       |.|
│ │ │ │  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 3136 3934   -- used 0.01694
│ │ │ │ -0003d8c0: 3733 7320 2863 7075 293b 2030 2e30 3136  73s (cpu); 0.016
│ │ │ │ -0003d8d0: 3231 3032 7320 2020 2020 7c0a 7c20 2020  2102s     |.|   
│ │ │ │ +0003d8b0: 202d 2d20 7573 6564 2030 2e30 3536 3333   -- used 0.05633
│ │ │ │ +0003d8c0: 3139 7320 2863 7075 293b 2030 2e30 3330  19s (cpu); 0.030
│ │ │ │ +0003d8d0: 3731 3432 7320 2020 2020 7c0a 7c20 2020  7142s     |.|   
│ │ │ │  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 3835 3332 3936 7320 2863 7075 293b  .0853296s (cpu);
│ │ │ │ -0003e2f0: 2030 2e30 3832 3638 3339 7320 2020 2020   0.0826839s     
│ │ │ │ +0003e2e0: 2e31 3039 3530 3173 2028 6370 7529 3b20  .109501s (cpu); 
│ │ │ │ +0003e2f0: 302e 3039 3538 3635 3273 2020 2020 2020  0.0958652s      
│ │ │ │  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 3531 3134 3438 7320 2863 7075 293b  .0511448s (cpu);
│ │ │ │ -0003f9c0: 2030 2e30 3439 3933 3034 7320 2020 2020   0.0499304s     
│ │ │ │ +0003f9b0: 2e30 3733 3832 3934 7320 2863 7075 293b  .0738294s (cpu);
│ │ │ │ +0003f9c0: 2030 2e30 3631 3139 3537 7320 2020 2020   0.0611957s     
│ │ │ │  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 3136  :  -- used 0.016
│ │ │ │ -000406f0: 3935 3838 7320 2863 7075 293b 2030 2e30  9588s (cpu); 0.0
│ │ │ │ -00040700: 3136 3232 3173 2020 2020 2020 7c0a 7c20  16221s      |.| 
│ │ │ │ +000406e0: 3a20 202d 2d20 7573 6564 2030 2e30 3531  :  -- used 0.051
│ │ │ │ +000406f0: 3936 3639 7320 2863 7075 293b 2030 2e30  9669s (cpu); 0.0
│ │ │ │ +00040700: 3237 3731 3935 7320 2020 2020 7c0a 7c20  277195s     |.| 
│ │ │ │  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 3337 3432 3273   used 0.0137422s
│ │ │ │ -00043640: 2028 6370 7529 3b20 302e 3031 3330 3232   (cpu); 0.013022
│ │ │ │ -00043650: 3373 2020 2020 207c 0a7c 2020 2020 2020  3s     |.|      
│ │ │ │ +00043630: 2075 7365 6420 302e 3036 3031 3631 3673   used 0.0601616s
│ │ │ │ +00043640: 2028 6370 7529 3b20 302e 3032 3336 3831   (cpu); 0.023681
│ │ │ │ +00043650: 7320 2020 2020 207c 0a7c 2020 2020 2020  s      |.|      
│ │ │ │  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                  
│ │ │ │ @@ -17611,18 +17611,18 @@
│ │ │ │  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 2e32 3738 3138 7320  - used 0.27818s 
│ │ │ │ -00044d20: 2863 7075 293b 2030 2e32 3032 3336 3273  (cpu); 0.202362s
│ │ │ │ -00044d30: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ -00044d40: 6329 2020 2020 2020 2020 2020 2020 2020  c)              
│ │ │ │ +00044d10: 2d20 7573 6564 2030 2e35 3634 3730 3673  - used 0.564706s
│ │ │ │ +00044d20: 2028 6370 7529 3b20 302e 3137 3439 3331   (cpu); 0.174931
│ │ │ │ +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                  
│ │ │ │  00044da0: 207c 0a7c 6f36 203d 2048 4b52 2020 2020   |.|o6 = HKR    
│ │ │ │  00044db0: 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 3137 3034 3273 2028 6370 7529 3b20  .517042s (cpu); 
│ │ │ │ -00057170: 302e 3434 3530 3832 7320 2020 2020 2020  0.445082s       
│ │ │ │ +00057160: 2e36 3833 3332 3873 2028 6370 7529 3b20  .683328s (cpu); 
│ │ │ │ +00057170: 302e 3538 3133 3834 7320 2020 2020 2020  0.581384s       
│ │ │ │  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               
│ │ │ │ @@ -22792,16 +22792,16 @@
│ │ │ │  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: 3338 3838 3473 2028 6370 7529 3b20 302e  38884s (cpu); 0.
│ │ │ │ -000590f0: 3133 3632 3332 7320 2020 2020 2020 7c0a  136232s       |.
│ │ │ │ +000590e0: 3831 3537 3573 2028 6370 7529 3b20 302e  81575s (cpu); 0.
│ │ │ │ +000590f0: 3136 3738 3733 7320 2020 2020 2020 7c0a  167873s       |.
│ │ │ │  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 3630  |.| -- used 0.60
│ │ │ │ -00066760: 3137 3636 7320 2863 7075 293b 2030 2e35  1766s (cpu); 0.5
│ │ │ │ -00066770: 3036 3830 3473 2028 7468 7265 6164 293b  06804s (thread);
│ │ │ │ +00066750: 7c0a 7c20 2d2d 2075 7365 6420 302e 3631  |.| -- used 0.61
│ │ │ │ +00066760: 3132 3333 7320 2863 7075 293b 2030 2e35  1233s (cpu); 0.5
│ │ │ │ +00066770: 3032 3330 3773 2028 7468 7265 6164 293b  02307s (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
│ │ │ │ @@ -21417,17 +21417,17 @@
│ │ │ │  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 3120 2020 3320 2020 3420 2020      1   3   4   
│ │ │ │ -00053b00: 2020 3120 2020 3520 2020 2020 3220 2020    1   5     2   
│ │ │ │ -00053b10: 3320 2020 3420 2020 3520 2020 2020 2020  3   4   5       
│ │ │ │ +00053af0: 2020 2020 3120 2020 3220 2020 3320 2020      1   2   3   
│ │ │ │ +00053b00: 2020 3320 2020 3520 2020 2020 3120 2020    3   5     1   
│ │ │ │ +00053b10: 3220 2020 3420 2020 3520 2020 2020 2020  2   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     |.|          
│ │ │ │  00053b80: 2020 2020 2020 2276 6572 7469 6365 7322        "vertices"
│ │ │ │ @@ -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 3220 2020 3320 2020 3420 2020      2   3   4   
│ │ │ │ -00053e20: 2020 3120 2020 3220 2020 2020 3120 2020    1   2     1   
│ │ │ │ -00053e30: 3320 2020 3420 2020 3520 2020 2020 2020  3   4   5       
│ │ │ │ +00053e10: 2020 2020 3120 2020 3220 2020 3420 2020      1   2   4   
│ │ │ │ +00053e20: 2020 3420 2020 3520 2020 2020 3120 2020    4   5     1   
│ │ │ │ +00053e30: 3220 2020 3320 2020 3520 2020 2020 2020  2   3   5       
│ │ │ │  00053e40: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00053e50: 2020 2020 2020 2272 696e 6722 203d 3e20        "ring" => 
│ │ │ │  00053e60: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │  00053e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053e90: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00053ea0: 2020 2020 2020 2276 6572 7469 6365 7322        "vertices"
│ │ ├── ./usr/share/info/EigenSolver.info.gz
│ │ │ ├── EigenSolver.info
│ │ │ │ @@ -171,16 +171,16 @@
│ │ │ │  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 3334 3134 3473 2065  |.| -- .34144s e
│ │ │ │ -00000b20: 6c61 7073 6564 2020 2020 2020 2020 2020  lapsed          
│ │ │ │ +00000b10: 7c0a 7c20 2d2d 202e 3231 3930 3833 7320  |.| -- .219083s 
│ │ │ │ +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  ----------------
│ │ │ │  00000b90: 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 3736 3434 3873  used 0.00276448s
│ │ │ │ -00001570: 2028 6370 7529 3b20 302e 3030 3237 3631   (cpu); 0.002761
│ │ │ │ -00001580: 3732 7320 2874 6872 6561 6429 3b20 3073  72s (thread); 0s
│ │ │ │ +00001560: 7573 6564 2030 2e30 3033 3135 3038 3273  used 0.00315082s
│ │ │ │ +00001570: 2028 6370 7529 3b20 302e 3030 3331 3436   (cpu); 0.003146
│ │ │ │ +00001580: 3731 7320 2874 6872 6561 6429 3b20 3073  71s (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 3631 3936 3173  used 0.00161961s
│ │ │ │ -00001750: 2028 6370 7529 3b20 302e 3030 3136 3230   (cpu); 0.001620
│ │ │ │ -00001760: 3133 7320 2874 6872 6561 6429 3b20 3073  13s (thread); 0s
│ │ │ │ +00001740: 7573 6564 2030 2e30 3031 3831 3433 3173  used 0.00181431s
│ │ │ │ +00001750: 2028 6370 7529 3b20 302e 3030 3138 3135   (cpu); 0.001815
│ │ │ │ +00001760: 3633 7320 2874 6872 6561 6429 3b20 3073  63s (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
│ │ │ │ @@ -620,18 +620,18 @@
│ │ │ │  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 3335 3739  -- used 0.003579
│ │ │ │ -00002730: 3036 7320 2863 7075 293b 2030 2e30 3033  06s (cpu); 0.003
│ │ │ │ -00002740: 3537 3730 3973 2028 7468 7265 6164 293b  57709s (thread);
│ │ │ │ -00002750: 2030 7320 2867 6329 7c0a 7c20 2020 2020   0s (gc)|.|     
│ │ │ │ +00002720: 2d2d 2075 7365 6420 302e 3030 3331 3432  -- used 0.003142
│ │ │ │ +00002730: 3137 7320 2863 7075 293b 2030 2e30 3033  17s (cpu); 0.003
│ │ │ │ +00002740: 3133 3873 2028 7468 7265 6164 293b 2030  138s (thread); 0
│ │ │ │ +00002750: 7320 2867 6329 2020 7c0a 7c20 2020 2020  s (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             
│ │ │ │  000027c0: 3220 2020 2020 2020 2020 2020 3220 2020  2           2   
│ │ │ │ @@ -650,18 +650,18 @@
│ │ │ │  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 3231 3137  -- used 0.002117
│ │ │ │ -00002910: 3836 7320 2863 7075 293b 2030 2e30 3032  86s (cpu); 0.002
│ │ │ │ -00002920: 3132 3136 7320 2874 6872 6561 6429 3b20  1216s (thread); 
│ │ │ │ -00002930: 3073 2028 6763 2920 7c0a 7c20 2020 2020  0s (gc) |.|     
│ │ │ │ +00002900: 2d2d 2075 7365 6420 302e 3030 3136 3731  -- used 0.001671
│ │ │ │ +00002910: 3035 7320 2863 7075 293b 2030 2e30 3031  05s (cpu); 0.001
│ │ │ │ +00002920: 3637 3234 3373 2028 7468 7265 6164 293b  67243s (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           
│ │ │ │  000029a0: 2020 3220 2020 2020 2020 2020 2020 3220    2           2 
│ │ │ │ @@ -995,16 +995,16 @@
│ │ │ │  00003e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003e30: 2d2d 2b0a 7c69 3420 3a20 7469 6d65 2065  --+.|i4 : time e
│ │ │ │  00003e40: 6c69 6d69 6e61 7465 2869 6465 616c 2866  liminate(ideal(f
│ │ │ │  00003e50: 2c67 292c 7829 2020 2020 2020 2020 2020  ,g),x)          
│ │ │ │  00003e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003e80: 2020 7c0a 7c20 2d2d 2075 7365 6420 312e    |.| -- used 1.
│ │ │ │ -00003e90: 3839 3036 3873 2028 6370 7529 3b20 312e  89068s (cpu); 1.
│ │ │ │ -00003ea0: 3637 3038 3573 2028 7468 7265 6164 293b  67085s (thread);
│ │ │ │ +00003e90: 3338 3734 3373 2028 6370 7529 3b20 312e  38743s (cpu); 1.
│ │ │ │ +00003ea0: 3235 3837 3673 2028 7468 7265 6164 293b  25876s (thread);
│ │ │ │  00003eb0: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │  00003ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003ed0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00003ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -1275,17 +1275,17 @@
│ │ │ │  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: 3032 3430 3235 3373 2028 6370 7529 3b20  0240253s (cpu); 
│ │ │ │ -00005020: 302e 3032 3430 3238 3373 2028 7468 7265  0.0240283s (thre
│ │ │ │ -00005030: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │ +00005010: 3031 3538 3735 7320 2863 7075 293b 2030  015875s (cpu); 0
│ │ │ │ +00005020: 2e30 3135 3837 3638 7320 2874 6872 6561  .0158768s (threa
│ │ │ │ +00005030: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 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                  
│ │ │ │  000050a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ @@ -1917,16 +1917,16 @@
│ │ │ │  000077c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000077d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520  ----------+.|i5 
│ │ │ │  000077e0: 3a20 7469 6d65 2065 6c69 6d69 6e61 7465  : time eliminate
│ │ │ │  000077f0: 2869 6465 616c 2866 2c67 292c 7829 2020  (ideal(f,g),x)  
│ │ │ │  00007800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007820: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ -00007830: 2075 7365 6420 312e 3637 3633 3473 2028   used 1.67634s (
│ │ │ │ -00007840: 6370 7529 3b20 312e 3435 3039 3973 2028  cpu); 1.45099s (
│ │ │ │ +00007830: 2075 7365 6420 312e 3436 3538 3973 2028   used 1.46589s (
│ │ │ │ +00007840: 6370 7529 3b20 312e 3335 3032 3873 2028  cpu); 1.35028s (
│ │ │ │  00007850: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │  00007860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007870: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00007880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000078a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000078b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -2197,18 +2197,18 @@
│ │ │ │  00008940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00008950: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620  ----------+.|i6 
│ │ │ │  00008960: 3a20 7469 6d65 2069 6465 616c 2072 6573  : time ideal res
│ │ │ │  00008970: 756c 7461 6e74 2866 2c67 2c78 2920 2020  ultant(f,g,x)   
│ │ │ │  00008980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000089a0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ -000089b0: 2075 7365 6420 302e 3031 3631 3173 2028   used 0.01611s (
│ │ │ │ -000089c0: 6370 7529 3b20 302e 3031 3631 3132 7320  cpu); 0.016112s 
│ │ │ │ -000089d0: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ -000089e0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +000089b0: 2075 7365 6420 302e 3031 3533 3938 3173   used 0.0153981s
│ │ │ │ +000089c0: 2028 6370 7529 3b20 302e 3031 3534 3032   (cpu); 0.015402
│ │ │ │ +000089d0: 3173 2028 7468 7265 6164 293b 2030 7320  1s (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 3830 3833 3573 2028 6370  d 0.0280835s (cp
│ │ │ │ -00001070: 7529 3b20 302e 3032 3830 3836 3873 2028  u); 0.0280868s (
│ │ │ │ +00001060: 6420 302e 3032 3839 3830 3373 2028 6370  d 0.0289803s (cp
│ │ │ │ +00001070: 7529 3b20 302e 3032 3839 3830 3673 2028  u); 0.0289806s (
│ │ │ │  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 3332 3837 3937 7320 2863  sed 0.328797s (c
│ │ │ │ -00002900: 7075 293b 2030 2e32 3736 3933 3673 2028  pu); 0.276936s (
│ │ │ │ -00002910: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │ +000028f0: 7365 6420 302e 3334 3538 3734 7320 2863  sed 0.345874s (c
│ │ │ │ +00002900: 7075 293b 2030 2e32 3931 3573 2028 7468  pu); 0.2915s (th
│ │ │ │ +00002910: 7265 6164 293b 2030 7320 2867 6329 2020  read); 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
│ │ │ │ @@ -685,17 +685,17 @@
│ │ │ │  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 3238 3134 3273 2028 6370  ed 0.228142s (cp
│ │ │ │ -00002b40: 7529 3b20 302e 3136 3838 3437 7320 2874  u); 0.168847s (t
│ │ │ │ -00002b50: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ +00002b30: 6564 2030 2e31 3330 3037 3373 2028 6370  ed 0.130073s (cp
│ │ │ │ +00002b40: 7529 3b20 302e 3133 3030 3873 2028 7468  u); 0.13008s (th
│ │ │ │ +00002b50: 7265 6164 293b 2030 7320 2867 6329 2020  read); 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                  
│ │ │ │  00002bc0: 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 3332 3938 3973 2028   used 5.32989s (
│ │ │ │ -00002d50: 6370 7529 3b20 342e 3631 3639 3773 2028  cpu); 4.61697s (
│ │ │ │ +00002d40: 2075 7365 6420 342e 3931 3937 3973 2028   used 4.91979s (
│ │ │ │ +00002d50: 6370 7529 3b20 342e 3339 3538 3173 2028  cpu); 4.39581s (
│ │ │ │  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  
│ │ │ │ @@ -757,275 +757,274 @@
│ │ │ │  00002f40: 6e20 6120 6765 6e65 7261 6c20 7175 696e  n a general quin
│ │ │ │  00002f50: 7469 6320 7468 7265 6566 6f6c 6420 6361  tic threefold ca
│ │ │ │  00002f60: 6e20 6265 0a63 6f6d 7075 7465 6420 6173  n be.computed as
│ │ │ │  00002f70: 2066 6f6c 6c6f 7773 3a0a 0a0a 0a2b 2d2d   follows:....+--
│ │ │ │  00002f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00002fb0: 2d2d 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 2e32 3137 3231 3473  - used 0.217214s
│ │ │ │ -00003000: 2028 6370 7529 3b20 302e 3136 3735 3434   (cpu); 0.167544
│ │ │ │ -00003010: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ -00003020: 6763 297c 0a7c 2020 2020 2020 2020 2020  gc)|.|          
│ │ │ │ +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 2e31 3330 3173 2028 6370 7529  ed 0.1301s (cpu)
│ │ │ │ +00003000: 3b20 302e 3133 3031 3037 7320 2874 6872  ; 0.130107s (thr
│ │ │ │ +00003010: 6561 6429 3b20 3073 2028 6763 297c 0a7c  ead); 0s (gc)|.|
│ │ │ │ +00003020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003050: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -00003060: 3020 3d20 3331 3732 3036 3337 3520 2020  0 = 317206375   
│ │ │ │ +00003050: 2020 207c 0a7c 6f31 3020 3d20 3331 3732     |.|o10 = 3172
│ │ │ │ +00003060: 3036 3337 3520 2020 2020 2020 2020 2020  06375           
│ │ │ │  00003070: 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     |.|          
│ │ │ │ +00003080: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00003090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000030a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000030b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000030c0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -000030d0: 3020 3a20 5151 2020 2020 2020 2020 2020  0 : QQ          
│ │ │ │ +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                  
│ │ │ │  000030e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000030f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003100: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +000030f0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00003100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003130: 2d2d 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:....+
│ │ │ │ +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  ----------------
│ │ │ │  000031c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000031d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000031e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000031f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003200: 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 7469  -----+.|i11 : ti
│ │ │ │ -00003210: 6d65 2066 6f72 2044 2069 6e20 5420 6c69  me for D in T li
│ │ │ │ -00003220: 7374 2072 6174 696f 6e61 6c43 7572 7665  st rationalCurve
│ │ │ │ -00003230: 2833 2c44 2920 2d20 7261 7469 6f6e 616c  (3,D) - rational
│ │ │ │ -00003240: 4375 7276 6528 312c 4429 2f32 377c 0a7c  Curve(1,D)/27|.|
│ │ │ │ -00003250: 202d 2d20 7573 6564 2035 2e33 3133 3432   -- used 5.31342
│ │ │ │ -00003260: 7320 2863 7075 293b 2034 2e36 3432 3038  s (cpu); 4.64208
│ │ │ │ -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       |.|        
│ │ │ │ +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 2034 2e39 3638 3038   -- used 4.96808
│ │ │ │ +00003250: 7320 2863 7075 293b 2034 2e34 3032 3233  s (cpu); 4.40223
│ │ │ │ +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                  
│ │ │ │  000032a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000032b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000032c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000032d0: 2020 2020 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       |.|        
│ │ │ │ +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                  
│ │ │ │  00003330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003360: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00003370: 6f31 3120 3a20 4c69 7374 2020 2020 2020  o11 : List      
│ │ │ │ +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                  
│ │ │ │  00003380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000033a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000033b0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000033a0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000033b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000033c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000033d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000033e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000033f0: 2d2d 2d2d 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  ....+-----------
│ │ │ │ +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  ----------------
│ │ │ │  00003430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 3634 3937  | -- used 1.6497
│ │ │ │ -000034a0: 3673 2028 6370 7529 3b20 312e 3434 3231  6s (cpu); 1.4421
│ │ │ │ -000034b0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ -000034c0: 6763 2920 7c0a 7c20 2020 2020 2020 2020  gc) |.|         
│ │ │ │ +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 3432 3338  | -- used 1.4238
│ │ │ │ +00003490: 3773 2028 6370 7529 3b20 312e 3330 3630  7s (cpu); 1.3060
│ │ │ │ +000034a0: 3473 2028 7468 7265 6164 293b 2030 7320  4s (thread); 0s 
│ │ │ │ +000034b0: 2867 6329 7c0a 7c20 2020 2020 2020 2020  (gc)|.|         
│ │ │ │ +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 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              |.| 
│ │ │ │ +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                  
│ │ │ │  000035a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000035b0: 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   
│ │ │ │ +000035c0: 2020 7c0a 7c6f 3132 203a 2051 5120 2020    |.|o12 : QQ   
│ │ │ │ +000035d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000035e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000035f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003600: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +000035f0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00003600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 3436 3738 3773 2028 6370 7529 3b20  7.46787s (cpu); 
│ │ │ │ -00003690: 352e 3739 3430 3473 2028 7468 7265 6164  5.79404s (thread
│ │ │ │ -000036a0: 293b 2030 7320 2867 6329 7c0a 7c20 2020  ); 0s (gc)|.|   
│ │ │ │ +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: 362e 3736 3339 3173 2028 6370 7529 3b20  6.76391s (cpu); 
│ │ │ │ +00003680: 352e 3637 3634 3173 2028 7468 7265 6164  5.67641s (thread
│ │ │ │ +00003690: 293b 2030 7320 2867 6329 7c0a 7c20 2020  ); 0s (gc)|.|   
│ │ │ │ +000036a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000036b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000036c0: 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        |.|       
│ │ │ │ +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                  
│ │ │ │  00003720: 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         
│ │ │ │ +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                  
│ │ │ │  00003760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003780: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00003770: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00003780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003790: 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  ...+------------
│ │ │ │ +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  ----------------
│ │ │ │  00003830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003850: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20  -------+.|i14 : 
│ │ │ │ -00003860: 7469 6d65 2072 6174 696f 6e61 6c43 7572  time rationalCur
│ │ │ │ -00003870: 7665 2834 2920 2d20 7261 7469 6f6e 616c  ve(4) - rational
│ │ │ │ -00003880: 4375 7276 6528 3229 2f38 2020 207c 0a7c  Curve(2)/8   |.|
│ │ │ │ -00003890: 202d 2d20 7573 6564 2031 2e36 3539 3331   -- used 1.65931
│ │ │ │ -000038a0: 7320 2863 7075 293b 2031 2e34 3239 3732  s (cpu); 1.42972
│ │ │ │ -000038b0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ -000038c0: 6763 297c 0a7c 2020 2020 2020 2020 2020  gc)|.|          
│ │ │ │ +00003840: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20  -------+.|i14 : 
│ │ │ │ +00003850: 7469 6d65 2072 6174 696f 6e61 6c43 7572  time rationalCur
│ │ │ │ +00003860: 7665 2834 2920 2d20 7261 7469 6f6e 616c  ve(4) - rational
│ │ │ │ +00003870: 4375 7276 6528 3229 2f38 2020 207c 0a7c  Curve(2)/8   |.|
│ │ │ │ +00003880: 202d 2d20 7573 6564 2031 2e35 3432 3532   -- used 1.54252
│ │ │ │ +00003890: 7320 2863 7075 293b 2031 2e33 3631 3235  s (cpu); 1.36125
│ │ │ │ +000038a0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ +000038b0: 6763 297c 0a7c 2020 2020 2020 2020 2020  gc)|.|          
│ │ │ │ +000038c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000038d0: 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  .|              
│ │ │ │ +000038e0: 2020 2020 2020 2020 207c 0a7c 6f31 3420           |.|o14 
│ │ │ │ +000038f0: 3d20 3234 3234 3637 3533 3030 3030 2020  = 242467530000  
│ │ │ │ +00003900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003910: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00003920: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00003930: 2020 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
│ │ │ │ +00003950: 2020 2020 207c 0a7c 6f31 3420 3a20 5151       |.|o14 : QQ
│ │ │ │ +00003960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003990: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00003980: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00003990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000039a0: 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:....+-------
│ │ │ │ +000039c0: 2d2b 0a0a 5468 6520 6e75 6d62 6572 7320  -+..The numbers 
│ │ │ │ +000039d0: 6f66 2072 6174 696f 6e61 6c20 6375 7276  of rational curv
│ │ │ │ +000039e0: 6573 206f 6620 6465 6772 6565 2034 206f  es of degree 4 o
│ │ │ │ +000039f0: 6e20 6765 6e65 7261 6c20 636f 6d70 6c65  n general comple
│ │ │ │ +00003a00: 7465 2069 6e74 6572 7365 6374 696f 6e73  te intersections
│ │ │ │ +00003a10: 206f 660a 7479 7065 7320 2834 2c32 2920   of.types (4,2) 
│ │ │ │ +00003a20: 616e 6420 2833 2c33 2920 696e 205c 6d61  and (3,3) in \ma
│ │ │ │ +00003a30: 7468 6262 2050 5e35 2063 616e 2062 6520  thbb P^5 can be 
│ │ │ │ +00003a40: 636f 6d70 7574 6564 2061 7320 666f 6c6c  computed as foll
│ │ │ │ +00003a50: 6f77 733a 0a0a 0a0a 2b2d 2d2d 2d2d 2d2d  ows:....+-------
│ │ │ │ +00003a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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 3638 3037 3573 2028 6370 7529 3b20  7.68075s (cpu); 
│ │ │ │ -00003b00: 362e 3133 3632 3373 2028 7468 7265 6164  6.13623s (thread
│ │ │ │ -00003b10: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ -00003b20: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00003a90: 2d2d 2d2d 2d2b 0a7c 6931 3520 3a20 7469  -----+.|i15 : ti
│ │ │ │ +00003aa0: 6d65 2072 6174 696f 6e61 6c43 7572 7665  me rationalCurve
│ │ │ │ +00003ab0: 2834 2c7b 342c 327d 2920 2d20 7261 7469  (4,{4,2}) - rati
│ │ │ │ +00003ac0: 6f6e 616c 4375 7276 6528 322c 7b34 2c32  onalCurve(2,{4,2
│ │ │ │ +00003ad0: 7d29 2f38 7c0a 7c20 2d2d 2075 7365 6420  })/8|.| -- used 
│ │ │ │ +00003ae0: 362e 3538 3935 3673 2028 6370 7529 3b20  6.58956s (cpu); 
│ │ │ │ +00003af0: 352e 3437 3932 3373 2028 7468 7265 6164  5.47923s (thread
│ │ │ │ +00003b00: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ +00003b10: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00003b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003b60: 2020 7c0a 7c6f 3135 203d 2033 3838 3339    |.|o15 = 38839
│ │ │ │ -00003b70: 3032 3532 3820 2020 2020 2020 2020 2020  02528           
│ │ │ │ +00003b50: 2020 7c0a 7c6f 3135 203d 2033 3838 3339    |.|o15 = 38839
│ │ │ │ +00003b60: 3032 3532 3820 2020 2020 2020 2020 2020  02528           
│ │ │ │ +00003b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003ba0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00003b90: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00003ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003be0: 7c0a 7c6f 3135 203a 2051 5120 2020 2020  |.|o15 : QQ     
│ │ │ │ +00003bd0: 7c0a 7c6f 3135 203a 2051 5120 2020 2020  |.|o15 : QQ     
│ │ │ │ +00003be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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  .+--------------
│ │ │ │ +00003c00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00003c10: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00003c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003c50: 2d2d 2d2d 2d2d 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 2e39 3236 3273   -- used 7.9262s
│ │ │ │ -00003cb0: 2028 6370 7529 3b20 362e 3031 3639 3273   (cpu); 6.01692s
│ │ │ │ -00003cc0: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ -00003cd0: 6329 2020 2020 2020 2020 2020 7c0a 7c20  c)          |.| 
│ │ │ │ +00003c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00003c50: 7c69 3136 203a 2074 696d 6520 7261 7469  |i16 : time rati
│ │ │ │ +00003c60: 6f6e 616c 4375 7276 6528 342c 7b33 2c33  onalCurve(4,{3,3
│ │ │ │ +00003c70: 7d29 202d 2072 6174 696f 6e61 6c43 7572  }) - rationalCur
│ │ │ │ +00003c80: 7665 2832 2c7b 332c 337d 292f 387c 0a7c  ve(2,{3,3})/8|.|
│ │ │ │ +00003c90: 202d 2d20 7573 6564 2036 2e38 3033 3234   -- used 6.80324
│ │ │ │ +00003ca0: 7320 2863 7075 293b 2035 2e36 3738 3736  s (cpu); 5.67876
│ │ │ │ +00003cb0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ +00003cc0: 6763 2920 2020 2020 2020 2020 7c0a 7c20  gc)         |.| 
│ │ │ │ +00003cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003d10: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -00003d20: 3620 3d20 3131 3339 3434 3833 3834 2020  6 = 1139448384  
│ │ │ │ +00003d00: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00003d10: 3620 3d20 3131 3339 3434 3833 3834 2020  6 = 1139448384  
│ │ │ │ +00003d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003d30: 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            |.|   
│ │ │ │ +00003d40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00003d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003d90: 2020 2020 2020 2020 207c 0a7c 6f31 3620           |.|o16 
│ │ │ │ -00003da0: 3a20 5151 2020 2020 2020 2020 2020 2020  : QQ            
│ │ │ │ +00003d80: 2020 2020 2020 2020 207c 0a7c 6f31 3620           |.|o16 
│ │ │ │ +00003d90: 3a20 5151 2020 2020 2020 2020 2020 2020  : QQ            
│ │ │ │ +00003da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003dd0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00003dc0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00003dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003e10: 2d2d 2d2d 2d2d 2d2b 0a0a 5761 7973 2074  -------+..Ways t
│ │ │ │ -00003e20: 6f20 7573 6520 7261 7469 6f6e 616c 4375  o use rationalCu
│ │ │ │ -00003e30: 7276 653a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  rve:.===========
│ │ │ │ -00003e40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -00003e50: 0a20 202a 2022 7261 7469 6f6e 616c 4375  .  * "rationalCu
│ │ │ │ -00003e60: 7276 6528 5a5a 2922 0a20 202a 2022 7261  rve(ZZ)".  * "ra
│ │ │ │ -00003e70: 7469 6f6e 616c 4375 7276 6528 5a5a 2c4c  tionalCurve(ZZ,L
│ │ │ │ -00003e80: 6973 7429 220a 0a46 6f72 2074 6865 2070  ist)"..For the p
│ │ │ │ -00003e90: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ -00003ea0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ -00003eb0: 6520 6f62 6a65 6374 202a 6e6f 7465 2072  e object *note r
│ │ │ │ -00003ec0: 6174 696f 6e61 6c43 7572 7665 3a20 7261  ationalCurve: ra
│ │ │ │ -00003ed0: 7469 6f6e 616c 4375 7276 652c 2069 7320  tionalCurve, is 
│ │ │ │ -00003ee0: 6120 2a6e 6f74 6520 6d65 7468 6f64 2066  a *note method f
│ │ │ │ -00003ef0: 756e 6374 696f 6e3a 0a28 4d61 6361 756c  unction:.(Macaul
│ │ │ │ -00003f00: 6179 3244 6f63 294d 6574 686f 6446 756e  ay2Doc)MethodFun
│ │ │ │ -00003f10: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d  ction,...-------
│ │ │ │ +00003e00: 2d2d 2d2d 2d2d 2d2b 0a0a 5761 7973 2074  -------+..Ways t
│ │ │ │ +00003e10: 6f20 7573 6520 7261 7469 6f6e 616c 4375  o use rationalCu
│ │ │ │ +00003e20: 7276 653a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  rve:.===========
│ │ │ │ +00003e30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +00003e40: 0a20 202a 2022 7261 7469 6f6e 616c 4375  .  * "rationalCu
│ │ │ │ +00003e50: 7276 6528 5a5a 2922 0a20 202a 2022 7261  rve(ZZ)".  * "ra
│ │ │ │ +00003e60: 7469 6f6e 616c 4375 7276 6528 5a5a 2c4c  tionalCurve(ZZ,L
│ │ │ │ +00003e70: 6973 7429 220a 0a46 6f72 2074 6865 2070  ist)"..For the p
│ │ │ │ +00003e80: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ +00003e90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ +00003ea0: 6520 6f62 6a65 6374 202a 6e6f 7465 2072  e object *note r
│ │ │ │ +00003eb0: 6174 696f 6e61 6c43 7572 7665 3a20 7261  ationalCurve: ra
│ │ │ │ +00003ec0: 7469 6f6e 616c 4375 7276 652c 2069 7320  tionalCurve, is 
│ │ │ │ +00003ed0: 6120 2a6e 6f74 6520 6d65 7468 6f64 2066  a *note method f
│ │ │ │ +00003ee0: 756e 6374 696f 6e3a 0a28 4d61 6361 756c  unction:.(Macaul
│ │ │ │ +00003ef0: 6179 3244 6f63 294d 6574 686f 6446 756e  ay2Doc)MethodFun
│ │ │ │ +00003f00: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d  ction,...-------
│ │ │ │ +00003f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003f60: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f  --------..The so
│ │ │ │ -00003f70: 7572 6365 206f 6620 7468 6973 2064 6f63  urce of this doc
│ │ │ │ -00003f80: 756d 656e 7420 6973 2069 6e0a 2f62 7569  ument is in./bui
│ │ │ │ -00003f90: 6c64 2f72 6570 726f 6475 6369 626c 652d  ld/reproducible-
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│ │ │ ├── EquivariantGB.info
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│ │ │ │ @@ -1967,61 +1967,61 @@
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│ │ │ │ +00007bb0: 3620 7365 636f 6e64 7320 2020 2020 2020  6 seconds       
│ │ │ │  00007bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007be0: 2020 2020 2020 2020 2020 7c0a 7c28 3235            |.|(25
│ │ │ │  00007bf0: 2c20 3630 2920 2020 2020 2020 2020 2020  , 60)           
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│ │ │ │  00007cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00007cf0: 3620 7365 636f 6e64 7320 2020 2020 2020  6 seconds       
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│ │ │ │  00007d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00007df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007e10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
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│ │ │ │  00007e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007ea0: 2020 2020 2020 2020 2020 2020 2020 2020
│ │ ├── ./usr/share/info/FastMinors.info.gz
│ │ │ ├── FastMinors.info
│ │ │ │ @@ -4406,17 +4406,17 @@
│ │ │ │  00011350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011360: 2d2d 2d2d 2d2d 2b0a 7c69 3238 203a 2074  ------+.|i28 : t
│ │ │ │  00011370: 696d 6520 6469 6d20 284a 202b 2063 686f  ime dim (J + cho
│ │ │ │  00011380: 6f73 6547 6f6f 644d 696e 6f72 7328 382c  oseGoodMinors(8,
│ │ │ │  00011390: 2036 2c20 4d2c 204a 2c20 5374 7261 7465   6, M, J, Strate
│ │ │ │  000113a0: 6779 3d3e 5261 6e64 6f6d 2929 2020 2020  gy=>Random))    
│ │ │ │  000113b0: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ -000113c0: 6420 302e 3137 3739 3639 7320 2863 7075  d 0.177969s (cpu
│ │ │ │ -000113d0: 293b 2030 2e31 3232 3030 3473 2028 7468  ); 0.122004s (th
│ │ │ │ -000113e0: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │ +000113c0: 6420 302e 3231 3939 3538 7320 2863 7075  d 0.219958s (cpu
│ │ │ │ +000113d0: 293b 2030 2e31 3533 3632 7320 2874 6872  ); 0.15362s (thr
│ │ │ │ +000113e0: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │  000113f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011400: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00011410: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00011440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011450: 2020 2020 2020 7c0a 7c6f 3238 203d 2032        |.|o28 = 2
│ │ │ │ @@ -4431,16 +4431,16 @@
│ │ │ │  000114e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000114f0: 2d2d 2d2d 2d2d 2b0a 7c69 3239 203a 2074  ------+.|i29 : t
│ │ │ │  00011500: 696d 6520 6469 6d20 284a 202b 2063 686f  ime dim (J + cho
│ │ │ │  00011510: 6f73 6547 6f6f 644d 696e 6f72 7328 382c  oseGoodMinors(8,
│ │ │ │  00011520: 2036 2c20 4d2c 204a 2c20 5374 7261 7465   6, M, J, Strate
│ │ │ │  00011530: 6779 3d3e 4c65 7853 6d61 6c6c 6573 7429  gy=>LexSmallest)
│ │ │ │  00011540: 2920 2020 2020 7c0a 7c20 2d2d 2075 7365  )     |.| -- use
│ │ │ │ -00011550: 6420 302e 3332 3437 3632 7320 2863 7075  d 0.324762s (cpu
│ │ │ │ -00011560: 293b 2030 2e32 3132 3635 3273 2028 7468  ); 0.212652s (th
│ │ │ │ +00011550: 6420 302e 3335 3538 3737 7320 2863 7075  d 0.355877s (cpu
│ │ │ │ +00011560: 293b 2030 2e32 3234 3031 3773 2028 7468  ); 0.224017s (th
│ │ │ │  00011570: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │  00011580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011590: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000115a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000115b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000115c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000115d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4456,17 +4456,17 @@
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│ │ │ │  00011690: 696d 6520 6469 6d20 284a 202b 2063 686f  ime dim (J + cho
│ │ │ │  000116a0: 6f73 6547 6f6f 644d 696e 6f72 7328 382c  oseGoodMinors(8,
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│ │ │ │ -000116e0: 6420 302e 3530 3530 3735 7320 2863 7075  d 0.505075s (cpu
│ │ │ │ -000116f0: 293b 2030 2e33 3238 3334 7320 2874 6872  ); 0.32834s (thr
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│ │ │ │ +000116e0: 6420 302e 3536 3335 3839 7320 2863 7075  d 0.563589s (cpu
│ │ │ │ +000116f0: 293b 2030 2e33 3630 3931 3373 2028 7468  ); 0.360913s (th
│ │ │ │ +00011700: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │  00011710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011720: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00011730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011770: 2020 2020 2020 7c0a 7c6f 3330 203d 2031        |.|o30 = 1
│ │ │ │ @@ -4481,16 +4481,16 @@
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│ │ │ │  00011810: 2d2d 2d2d 2d2d 2b0a 7c69 3331 203a 2074  ------+.|i31 : t
│ │ │ │  00011820: 696d 6520 6469 6d20 284a 202b 2063 686f  ime dim (J + cho
│ │ │ │  00011830: 6f73 6547 6f6f 644d 696e 6f72 7328 382c  oseGoodMinors(8,
│ │ │ │  00011840: 2036 2c20 4d2c 204a 2c20 5374 7261 7465   6, M, J, Strate
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│ │ │ │  00011860: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ -00011870: 6420 302e 3232 3638 3235 7320 2863 7075  d 0.226825s (cpu
│ │ │ │ -00011880: 293b 2030 2e31 3835 3538 3773 2028 7468  ); 0.185587s (th
│ │ │ │ +00011870: 6420 302e 3238 3831 3534 7320 2863 7075  d 0.288154s (cpu
│ │ │ │ +00011880: 293b 2030 2e32 3235 3136 3273 2028 7468  ); 0.225162s (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 3338 3633 3738 7320 2863 7075  d 0.386378s (cpu
│ │ │ │ -00011a10: 293b 2030 2e32 3132 3731 7320 2874 6872  ); 0.21271s (thr
│ │ │ │ +00011a00: 6420 302e 3434 3033 3273 2028 6370 7529  d 0.44032s (cpu)
│ │ │ │ +00011a10: 3b20 302e 3233 3333 3939 7320 2874 6872  ; 0.233399s (thr
│ │ │ │  00011a20: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 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                  
│ │ │ │ @@ -4528,20 +4528,20 @@
│ │ │ │  00011af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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=>            
│ │ │ │ +00011b60: 2036 2c20 4d2c 204a 2c20 2020 2020 2020   6, M, J,       
│ │ │ │ +00011b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011b80: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ -00011b90: 6420 302e 3335 3630 3333 7320 2863 7075  d 0.356033s (cpu
│ │ │ │ -00011ba0: 293b 2030 2e32 3439 3031 7320 2874 6872  ); 0.24901s (thr
│ │ │ │ -00011bb0: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │ +00011b90: 6420 302e 3337 3132 3137 7320 2863 7075  d 0.371217s (cpu
│ │ │ │ +00011ba0: 293b 2030 2e32 3339 3536 3873 2028 7468  ); 0.239568s (th
│ │ │ │ +00011bb0: 7265 6164 293b 2030 7320 2020 2020 2020  read); 0s       
│ │ │ │  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
│ │ │ │ @@ -4550,6019 +4550,6019 @@
│ │ │ │  00011c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011c70: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │  00011c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011cc0: 2d2d 2d2d 2d2d 7c0a 7c47 5265 764c 6578  ------|.|GRevLex
│ │ │ │ -00011cd0: 536d 616c 6c65 7374 5465 726d 2929 2020  SmallestTerm))  
│ │ │ │ -00011ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011cc0: 2d2d 2d2d 2d2d 7c0a 7c53 7472 6174 6567  ------|.|Strateg
│ │ │ │ +00011cd0: 793d 3e47 5265 764c 6578 536d 616c 6c65  y=>GRevLexSmalle
│ │ │ │ +00011ce0: 7374 5465 726d 2929 2020 2020 2020 2020  stTerm))        
│ │ │ │  00011cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011d10: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -00011d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 3239 3938 3835 7320 2863 7075  d 0.299885s (cpu
│ │ │ │ -00011dd0: 293b 2030 2e31 3838 3633 3673 2028 7468  ); 0.188636s (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                  
│ │ │ │ +00011d10: 2020 2020 2020 7c0a 7c28 6763 2920 2020        |.|(gc)   
│ │ │ │ +00011d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011d60: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00011d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011db0: 2d2d 2d2d 2d2d 2b0a 7c69 3334 203a 2074  ------+.|i34 : t
│ │ │ │ +00011dc0: 696d 6520 6469 6d20 284a 202b 2063 686f  ime dim (J + cho
│ │ │ │ +00011dd0: 6f73 6547 6f6f 644d 696e 6f72 7328 382c  oseGoodMinors(8,
│ │ │ │ +00011de0: 2036 2c20 4d2c 204a 2c20 5374 7261 7465   6, M, J, Strate
│ │ │ │ +00011df0: 6779 3d3e 4752 6576 4c65 784c 6172 6765  gy=>GRevLexLarge
│ │ │ │ +00011e00: 7374 2929 2020 7c0a 7c20 2d2d 2075 7365  st))  |.| -- use
│ │ │ │ +00011e10: 6420 302e 3334 3539 3832 7320 2863 7075  d 0.345982s (cpu
│ │ │ │ +00011e20: 293b 2030 2e31 3938 3873 2028 7468 7265  ); 0.1988s (thre
│ │ │ │ +00011e30: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │  00011e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011e50: 2020 2020 2020 7c0a 7c6f 3334 203d 2033        |.|o34 = 3
│ │ │ │ +00011e50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00011e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011ea0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -00011eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00011ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 2e34 3736 3673 2028 6370 7529  d 15.4766s (cpu)
│ │ │ │ -00011f60: 3b20 3130 2e35 3337 3573 2028 7468 7265  ; 10.5375s (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                  
│ │ │ │ +00011ea0: 2020 2020 2020 7c0a 7c6f 3334 203d 2033        |.|o34 = 3
│ │ │ │ +00011eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011ef0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00011f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00011f40: 2d2d 2d2d 2d2d 2b0a 7c69 3335 203a 2074  ------+.|i35 : t
│ │ │ │ +00011f50: 696d 6520 6469 6d20 284a 202b 2063 686f  ime dim (J + cho
│ │ │ │ +00011f60: 6f73 6547 6f6f 644d 696e 6f72 7328 382c  oseGoodMinors(8,
│ │ │ │ +00011f70: 2036 2c20 4d2c 204a 2c20 5374 7261 7465   6, M, J, Strate
│ │ │ │ +00011f80: 6779 3d3e 506f 696e 7473 2929 2020 2020  gy=>Points))    
│ │ │ │ +00011f90: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ +00011fa0: 6420 3137 2e35 3232 3373 2028 6370 7529  d 17.5223s (cpu)
│ │ │ │ +00011fb0: 3b20 3131 2e34 3932 3173 2028 7468 7265  ; 11.4921s (thre
│ │ │ │ +00011fc0: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │  00011fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011fe0: 2020 2020 2020 7c0a 7c6f 3335 203d 2031        |.|o35 = 1
│ │ │ │ +00011fe0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00011ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012030: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -00012040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012080: 2d2d 2d2d 2d2d 2b0a 0a49 6e64 6565 642c  ------+..Indeed,
│ │ │ │ -00012090: 2069 6e20 7468 6973 2065 7861 6d70 6c65   in this example
│ │ │ │ -000120a0: 2c20 6576 656e 2063 6f6d 7075 7469 6e67  , even computing
│ │ │ │ -000120b0: 2064 6574 6572 6d69 6e61 6e74 7320 6f66   determinants of
│ │ │ │ -000120c0: 2031 2c30 3030 2072 616e 646f 6d0a 7375   1,000 random.su
│ │ │ │ -000120d0: 626d 6174 7269 6365 7320 6973 206e 6f74  bmatrices is not
│ │ │ │ -000120e0: 2074 7970 6963 616c 6c79 2065 6e6f 7567   typically enoug
│ │ │ │ -000120f0: 6820 746f 2076 6572 6966 7920 7468 6174  h to verify that
│ │ │ │ -00012100: 2024 5628 4a29 2420 6973 2072 6567 756c   $V(J)$ is regul
│ │ │ │ -00012110: 6172 2069 6e0a 636f 6469 6d65 6e73 696f  ar in.codimensio
│ │ │ │ -00012120: 6e20 312e 2020 4f6e 2074 6865 206f 7468  n 1.  On the oth
│ │ │ │ -00012130: 6572 2068 616e 642c 2050 6f69 6e74 7320  er hand, Points 
│ │ │ │ -00012140: 6973 2061 6c6d 6f73 7420 616c 7761 7973  is almost always
│ │ │ │ -00012150: 2071 7569 7465 2065 6666 6563 7469 7665   quite effective
│ │ │ │ -00012160: 2061 740a 6669 6e64 696e 6720 7661 6c75   at.finding valu
│ │ │ │ -00012170: 6162 6c65 2073 7562 6d61 7472 6963 6573  able submatrices
│ │ │ │ -00012180: 2c20 6275 7420 6361 6e20 6265 2071 7569  , but can be qui
│ │ │ │ -00012190: 7465 2073 6c6f 772e 2020 496e 2074 6869  te slow.  In thi
│ │ │ │ -000121a0: 7320 7061 7274 6963 756c 6172 0a65 7861  s particular.exa
│ │ │ │ -000121b0: 6d70 6c65 2c20 7765 2063 616e 2073 6565  mple, we can see
│ │ │ │ -000121c0: 2074 6861 7420 4c65 7853 6d61 6c6c 6573   that LexSmalles
│ │ │ │ -000121d0: 7454 6572 6d20 616c 736f 2070 6572 666f  tTerm also perfo
│ │ │ │ -000121e0: 726d 7320 7665 7279 2077 656c 6c20 2861  rms very well (a
│ │ │ │ -000121f0: 6e64 2064 6f65 7320 6974 0a71 7569 636b  nd does it.quick
│ │ │ │ -00012200: 6c79 292e 2053 696e 6365 2064 6966 6665  ly). Since diffe
│ │ │ │ -00012210: 7265 6e74 2073 7472 6174 6567 6965 7320  rent strategies 
│ │ │ │ -00012220: 776f 726b 2062 6574 7465 7220 6f72 2077  work better or w
│ │ │ │ -00012230: 6f72 7365 206f 6e20 6469 6666 6572 656e  orse on differen
│ │ │ │ -00012240: 740a 6578 616d 706c 6573 2c20 7468 6520  t.examples, the 
│ │ │ │ -00012250: 6465 6661 756c 7420 7374 7261 7465 6779  default strategy
│ │ │ │ -00012260: 2061 6374 7561 6c6c 7920 6d69 7865 7320   actually mixes 
│ │ │ │ -00012270: 616e 6420 6d61 7463 6865 7320 7661 7269  and matches vari
│ │ │ │ -00012280: 6f75 7320 7374 7261 7465 6769 6573 2e0a  ous strategies..
│ │ │ │ -00012290: 5468 6520 6465 6661 756c 7420 7374 7261  The default stra
│ │ │ │ -000122a0: 7465 6779 2c20 7768 6963 6820 7765 206e  tegy, which we n
│ │ │ │ -000122b0: 6f77 2065 6c75 6369 6461 7465 2c0a 0a2b  ow elucidate,..+
│ │ │ │ -000122c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000122d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000122e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -000122f0: 3336 203a 2070 6565 6b20 5374 7261 7465  36 : peek Strate
│ │ │ │ -00012300: 6779 4465 6661 756c 7420 2020 2020 2020  gyDefault       
│ │ │ │ -00012310: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00012320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012340: 2020 2020 2020 2020 2020 7c0a 7c6f 3336            |.|o36
│ │ │ │ -00012350: 203d 204f 7074 696f 6e54 6162 6c65 7b47   = OptionTable{G
│ │ │ │ -00012360: 5265 764c 6578 4c61 7267 6573 7420 3d3e  RevLexLargest =>
│ │ │ │ -00012370: 2030 2020 2020 2020 7d7c 0a7c 2020 2020   0      }|.|    
│ │ │ │ -00012380: 2020 2020 2020 2020 2020 2020 2020 4752                GR
│ │ │ │ -00012390: 6576 4c65 7853 6d61 6c6c 6573 7420 3d3e  evLexSmallest =>
│ │ │ │ -000123a0: 2031 3620 2020 2020 7c0a 7c20 2020 2020   16     |.|     
│ │ │ │ -000123b0: 2020 2020 2020 2020 2020 2020 2047 5265               GRe
│ │ │ │ -000123c0: 764c 6578 536d 616c 6c65 7374 5465 726d  vLexSmallestTerm
│ │ │ │ -000123d0: 203d 3e20 3136 207c 0a7c 2020 2020 2020   => 16 |.|      
│ │ │ │ -000123e0: 2020 2020 2020 2020 2020 2020 4c65 784c              LexL
│ │ │ │ -000123f0: 6172 6765 7374 203d 3e20 3020 2020 2020  argest => 0     
│ │ │ │ -00012400: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00012410: 2020 2020 2020 2020 2020 204c 6578 536d             LexSm
│ │ │ │ -00012420: 616c 6c65 7374 203d 3e20 3136 2020 2020  allest => 16    
│ │ │ │ -00012430: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00012440: 2020 2020 2020 2020 2020 4c65 7853 6d61            LexSma
│ │ │ │ -00012450: 6c6c 6573 7454 6572 6d20 3d3e 2031 3620  llestTerm => 16 
│ │ │ │ -00012460: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00012470: 2020 2020 2020 2020 2050 6f69 6e74 7320           Points 
│ │ │ │ -00012480: 3d3e 2030 2020 2020 2020 2020 2020 2020  => 0            
│ │ │ │ -00012490: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -000124a0: 2020 2020 2020 2020 5261 6e64 6f6d 203d          Random =
│ │ │ │ -000124b0: 3e20 3136 2020 2020 2020 2020 2020 2020  > 16            
│ │ │ │ -000124c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -000124d0: 2020 2020 2020 2052 616e 646f 6d4e 6f6e         RandomNon
│ │ │ │ -000124e0: 7a65 726f 203d 3e20 3136 2020 2020 2020  zero => 16      
│ │ │ │ -000124f0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -00012500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012520: 2b0a 0a73 6179 7320 7468 6174 2077 6520  +..says that we 
│ │ │ │ -00012530: 7368 6f75 6c64 2075 7365 2047 5265 764c  should use GRevL
│ │ │ │ -00012540: 6578 536d 616c 6c65 7374 2c20 4752 6576  exSmallest, GRev
│ │ │ │ -00012550: 4c65 7853 6d61 6c6c 6573 7454 6572 6d2c  LexSmallestTerm,
│ │ │ │ -00012560: 204c 6578 536d 616c 6c65 7374 2c0a 4c65   LexSmallest,.Le
│ │ │ │ -00012570: 7853 6d61 6c6c 6573 7454 6572 6d2c 2052  xSmallestTerm, R
│ │ │ │ -00012580: 616e 646f 6d2c 2052 616e 646f 6d4e 6f6e  andom, RandomNon
│ │ │ │ -00012590: 7a65 726f 2061 6c6c 2077 6974 6820 6571  zero all with eq
│ │ │ │ -000125a0: 7561 6c20 7072 6f62 6162 696c 6974 7920  ual probability 
│ │ │ │ -000125b0: 286e 6f74 650a 5261 6e64 6f6d 4e6f 6e7a  (note.RandomNonz
│ │ │ │ -000125c0: 6572 6f2c 2077 6869 6368 2077 6520 6861  ero, which we ha
│ │ │ │ -000125d0: 7665 206e 6f74 2079 6574 2064 6973 6375  ve not yet discu
│ │ │ │ -000125e0: 7373 6564 2063 686f 6f73 6573 2072 616e  ssed chooses ran
│ │ │ │ -000125f0: 646f 6d20 7375 626d 6174 7269 6365 7320  dom submatrices 
│ │ │ │ -00012600: 7768 6572 650a 6e6f 2072 6f77 206f 7220  where.no row or 
│ │ │ │ -00012610: 636f 6c75 6d6e 2069 7320 7a65 726f 2c20  column is zero, 
│ │ │ │ -00012620: 7768 6963 6820 6973 2067 6f6f 6420 666f  which is good fo
│ │ │ │ -00012630: 7220 776f 726b 696e 6720 696e 2073 7061  r working in spa
│ │ │ │ -00012640: 7273 6520 6d61 7472 6963 6573 292e 2020  rse matrices).  
│ │ │ │ -00012650: 466f 720a 696e 7374 616e 6365 2c20 6966  For.instance, if
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│ │ │ │ -00012670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 3338 3331 3336 7320 2863  sed 0.383136s (c
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│ │ │ │ -00012730: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 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                  
│ │ │ │ +00012030: 2020 2020 2020 7c0a 7c6f 3335 203d 2031        |.|o35 = 1
│ │ │ │ +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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012080: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00012090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000120a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000120b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000120c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00012310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00012340: 3336 203a 2070 6565 6b20 5374 7261 7465  36 : peek Strate
│ │ │ │ +00012350: 6779 4465 6661 756c 7420 2020 2020 2020  gyDefault       
│ │ │ │ +00012360: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00012370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012390: 2020 2020 2020 2020 2020 7c0a 7c6f 3336            |.|o36
│ │ │ │ +000123a0: 203d 204f 7074 696f 6e54 6162 6c65 7b47   = OptionTable{G
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│ │ │ │ +000123d0: 2020 2020 2020 2020 2020 2020 2020 4752                GR
│ │ │ │ +000123e0: 6576 4c65 7853 6d61 6c6c 6573 7420 3d3e  evLexSmallest =>
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│ │ │ │ +00012400: 2020 2020 2020 2020 2020 2020 2047 5265               GRe
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│ │ │ │ +00012420: 203d 3e20 3136 207c 0a7c 2020 2020 2020   => 16 |.|      
│ │ │ │ +00012430: 2020 2020 2020 2020 2020 2020 4c65 784c              LexL
│ │ │ │ +00012440: 6172 6765 7374 203d 3e20 3020 2020 2020  argest => 0     
│ │ │ │ +00012450: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00012460: 2020 2020 2020 2020 2020 204c 6578 536d             LexSm
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│ │ │ │ +00012480: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00012490: 2020 2020 2020 2020 2020 4c65 7853 6d61            LexSma
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│ │ │ │ +000124b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000124c0: 2020 2020 2020 2020 2050 6f69 6e74 7320           Points 
│ │ │ │ +000124d0: 3d3e 2030 2020 2020 2020 2020 2020 2020  => 0            
│ │ │ │ +000124e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000124f0: 2020 2020 2020 2020 5261 6e64 6f6d 203d          Random =
│ │ │ │ +00012500: 3e20 3136 2020 2020 2020 2020 2020 2020  > 16            
│ │ │ │ +00012510: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
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│ │ │ │ +00012530: 7a65 726f 203d 3e20 3136 2020 2020 2020  zero => 16      
│ │ │ │ +00012540: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00012550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012570: 2b0a 0a73 6179 7320 7468 6174 2077 6520  +..says that we 
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│ │ │ │ +000126a0: 466f 720a 696e 7374 616e 6365 2c20 6966  For.instance, if
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│ │ │ │ +000126c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000126d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000126e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000126f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012700: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3337 203a  --------+.|i37 :
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│ │ │ │ +00012720: 4d69 6e6f 7273 2832 302c 2036 2c20 4d2c  Minors(20, 6, M,
│ │ │ │ +00012730: 204a 2c20 5374 7261 7465 6779 3d3e 5374   J, Strategy=>St
│ │ │ │ +00012740: 7261 7465 6779 4465 6661 756c 742c 2020  rategyDefault,  
│ │ │ │ +00012750: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ +00012760: 7365 6420 302e 3435 3033 3831 7320 2863  sed 0.450381s (c
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│ │ │ │ +00012780: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │  00012790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000127a0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  000127b0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -000127c0: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ -000127d0: 6c65 7374 2020 2020 2020 2020 2020 2020  lest            
│ │ │ │ +000127c0: 4368 6f6f 7369 6e67 2052 616e 646f 6d20  Choosing Random 
│ │ │ │ +000127d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000127e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000127f0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  00012800: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -00012810: 4368 6f6f 7369 6e67 2052 616e 646f 6d20  Choosing Random 
│ │ │ │ -00012820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012810: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +00012820: 6c65 7374 2020 2020 2020 2020 2020 2020  lest            
│ │ │ │  00012830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012840: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  00012850: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -00012860: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ -00012870: 536d 616c 6c65 7374 5465 726d 2020 2020  SmallestTerm    
│ │ │ │ +00012860: 4368 6f6f 7369 6e67 2052 616e 646f 6d20  Choosing Random 
│ │ │ │ +00012870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012890: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  000128a0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
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│ │ │ │ -000128c0: 6f6e 5a65 726f 2020 2020 2020 2020 2020  onZero          
│ │ │ │ +000128b0: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +000128c0: 536d 616c 6c65 7374 5465 726d 2020 2020  SmallestTerm    
│ │ │ │  000128d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000128e0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  000128f0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │  00012900: 4368 6f6f 7369 6e67 2052 616e 646f 6d4e  Choosing RandomN
│ │ │ │  00012910: 6f6e 5a65 726f 2020 2020 2020 2020 2020  onZero          
│ │ │ │  00012920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012930: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  00012940: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -00012950: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ -00012960: 6c65 7374 2020 2020 2020 2020 2020 2020  lest            
│ │ │ │ +00012950: 4368 6f6f 7369 6e67 2052 616e 646f 6d4e  Choosing RandomN
│ │ │ │ +00012960: 6f6e 5a65 726f 2020 2020 2020 2020 2020  onZero          
│ │ │ │  00012970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012980: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  00012990: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -000129a0: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ -000129b0: 536d 616c 6c65 7374 5465 726d 2020 2020  SmallestTerm    
│ │ │ │ +000129a0: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +000129b0: 6c65 7374 2020 2020 2020 2020 2020 2020  lest            
│ │ │ │  000129c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  000129e0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -000129f0: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ -00012a00: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │ +000129f0: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +00012a00: 536d 616c 6c65 7374 5465 726d 2020 2020  SmallestTerm    
│ │ │ │  00012a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012a20: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  00012a30: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -00012a40: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ -00012a50: 536d 616c 6c65 7374 5465 726d 2020 2020  SmallestTerm    
│ │ │ │ +00012a40: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +00012a50: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │  00012a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012a70: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  00012a80: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -00012a90: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ -00012aa0: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │ +00012a90: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +00012aa0: 536d 616c 6c65 7374 5465 726d 2020 2020  SmallestTerm    
│ │ │ │  00012ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012ac0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  00012ad0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -00012ae0: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ -00012af0: 536d 616c 6c65 7374 2020 2020 2020 2020  Smallest        
│ │ │ │ +00012ae0: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +00012af0: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │  00012b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012b10: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  00012b20: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │  00012b30: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │  00012b40: 536d 616c 6c65 7374 2020 2020 2020 2020  Smallest        
│ │ │ │  00012b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012b60: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  00012b70: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -00012b80: 4368 6f6f 7369 6e67 2052 616e 646f 6d20  Choosing Random 
│ │ │ │ -00012b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012b80: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +00012b90: 536d 616c 6c65 7374 2020 2020 2020 2020  Smallest        
│ │ │ │  00012ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012bb0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  00012bc0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -00012bd0: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ -00012be0: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │ +00012bd0: 4368 6f6f 7369 6e67 2052 616e 646f 6d20  Choosing Random 
│ │ │ │ +00012be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012c00: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  00012c10: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │  00012c20: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │  00012c30: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │  00012c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012c50: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  00012c60: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -00012c70: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ -00012c80: 536d 616c 6c65 7374 5465 726d 2020 2020  SmallestTerm    
│ │ │ │ +00012c70: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +00012c80: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │  00012c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012ca0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  00012cb0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -00012cc0: 4368 6f6f 7369 6e67 2052 616e 646f 6d4e  Choosing RandomN
│ │ │ │ -00012cd0: 6f6e 5a65 726f 2020 2020 2020 2020 2020  onZero          
│ │ │ │ +00012cc0: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +00012cd0: 536d 616c 6c65 7374 5465 726d 2020 2020  SmallestTerm    
│ │ │ │  00012ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012cf0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  00012d00: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -00012d10: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ -00012d20: 536d 616c 6c65 7374 2020 2020 2020 2020  Smallest        
│ │ │ │ +00012d10: 4368 6f6f 7369 6e67 2052 616e 646f 6d4e  Choosing RandomN
│ │ │ │ +00012d20: 6f6e 5a65 726f 2020 2020 2020 2020 2020  onZero          
│ │ │ │  00012d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012d40: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │  00012d50: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ -00012d60: 4368 6f6f 7369 6e67 2052 616e 646f 6d20  Choosing Random 
│ │ │ │ -00012d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012d60: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +00012d70: 536d 616c 6c65 7374 2020 2020 2020 2020  Smallest        
│ │ │ │  00012d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012d90: 2020 2020 2020 2020 7c0a 7c63 686f 6f73          |.|choos
│ │ │ │ -00012da0: 6547 6f6f 644d 696e 6f72 733a 2066 6f75  eGoodMinors: fou
│ │ │ │ -00012db0: 6e64 203d 3230 2c20 6174 7465 6d70 7465  nd =20, attempte
│ │ │ │ -00012dc0: 6420 3d20 3230 2020 2020 2020 2020 2020  d = 20          
│ │ │ │ +00012d90: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00012da0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00012db0: 4368 6f6f 7369 6e67 2052 616e 646f 6d20  Choosing Random 
│ │ │ │ +00012dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012de0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00012df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012de0: 2020 2020 2020 2020 7c0a 7c63 686f 6f73          |.|choos
│ │ │ │ +00012df0: 6547 6f6f 644d 696e 6f72 733a 2066 6f75  eGoodMinors: fou
│ │ │ │ +00012e00: 6e64 203d 3230 2c20 6174 7465 6d70 7465  nd =20, attempte
│ │ │ │ +00012e10: 6420 3d20 3230 2020 2020 2020 2020 2020  d = 20          
│ │ │ │  00012e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012e30: 2020 2020 2020 2020 7c0a 7c6f 3337 203a          |.|o37 :
│ │ │ │ -00012e40: 2049 6465 616c 206f 6620 5320 2020 2020   Ideal of S     
│ │ │ │ +00012e30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00012e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012e80: 2020 2020 2020 2020 7c0a 7c2d 2d2d 2d2d          |.|-----
│ │ │ │ -00012e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012ed0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c56 6572 626f  --------|.|Verbo
│ │ │ │ -00012ee0: 7365 3d3e 7472 7565 293b 2020 2020 2020  se=>true);      
│ │ │ │ -00012ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012f20: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -00012f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00012f70: 2d2d 2d2d 2d2d 2d2d 2b0a 0a77 6520 6361  --------+..we ca
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│ │ │ │ -00012fa0: 7365 6e20 7669 6120 6469 6666 6572 656e  sen via differen
│ │ │ │ -00012fb0: 7420 7374 7261 7465 6769 6573 2e0a 0a4e  t strategies...N
│ │ │ │ -00012fc0: 6f74 652c 2069 6620 6f6e 6520 6173 6b73  ote, if one asks
│ │ │ │ -00012fd0: 2063 686f 6f73 6547 6f6f 644d 696e 6f72   chooseGoodMinor
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│ │ │ │ -00013000: 616e 7920 7469 6d65 2061 0a50 6f69 6e74  any time a.Point
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│ │ │ │ -00013020: 6c65 6374 6564 2c20 7468 6520 706f 696e  lected, the poin
│ │ │ │ -00013030: 7420 6973 2066 6f75 6e64 206f 6e20 244a  t is found on $J
│ │ │ │ -00013040: 2420 706c 7573 2074 6865 2069 6465 616c  $ plus the ideal
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│ │ │ │ -00013070: 2e0a 0a4c 6574 2075 7320 7461 6b65 2061  ...Let us take a
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│ │ │ │ -00013090: 6865 7220 6275 696c 742d 696e 2073 7472  her built-in str
│ │ │ │ -000130a0: 6174 6567 6965 732e 0a0a 2b2d 2d2d 2d2d  ategies...+-----
│ │ │ │ -000130b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000130c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000130d0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 3820 3a20  -------+.|i38 : 
│ │ │ │ -000130e0: 7065 656b 2053 7472 6174 6567 7944 6566  peek StrategyDef
│ │ │ │ -000130f0: 6175 6c74 4e6f 6e52 616e 646f 6d20 2020  aultNonRandom   
│ │ │ │ -00013100: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00013110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013130: 2020 2020 207c 0a7c 6f33 3820 3d20 4f70       |.|o38 = Op
│ │ │ │ -00013140: 7469 6f6e 5461 626c 657b 4752 6576 4c65  tionTable{GRevLe
│ │ │ │ -00013150: 784c 6172 6765 7374 203d 3e20 3020 2020  xLargest => 0   
│ │ │ │ -00013160: 2020 207d 7c0a 7c20 2020 2020 2020 2020     }|.|         
│ │ │ │ -00013170: 2020 2020 2020 2020 2047 5265 764c 6578           GRevLex
│ │ │ │ -00013180: 536d 616c 6c65 7374 203d 3e20 3235 2020  Smallest => 25  
│ │ │ │ -00013190: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -000131a0: 2020 2020 2020 2020 4752 6576 4c65 7853          GRevLexS
│ │ │ │ -000131b0: 6d61 6c6c 6573 7454 6572 6d20 3d3e 2032  mallestTerm => 2
│ │ │ │ -000131c0: 3520 7c0a 7c20 2020 2020 2020 2020 2020  5 |.|           
│ │ │ │ -000131d0: 2020 2020 2020 204c 6578 4c61 7267 6573         LexLarges
│ │ │ │ -000131e0: 7420 3d3e 2030 2020 2020 2020 2020 2020  t => 0          
│ │ │ │ -000131f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00013200: 2020 2020 2020 4c65 7853 6d61 6c6c 6573        LexSmalles
│ │ │ │ -00013210: 7420 3d3e 2032 3520 2020 2020 2020 2020  t => 25         
│ │ │ │ -00013220: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00013230: 2020 2020 204c 6578 536d 616c 6c65 7374       LexSmallest
│ │ │ │ -00013240: 5465 726d 203d 3e20 3235 2020 2020 207c  Term => 25     |
│ │ │ │ -00013250: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00013260: 2020 2020 506f 696e 7473 203d 3e20 3020      Points => 0 
│ │ │ │ -00013270: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00013280: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00013290: 2020 2052 616e 646f 6d20 3d3e 2030 2020     Random => 0  
│ │ │ │ -000132a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000132b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000132c0: 2020 5261 6e64 6f6d 4e6f 6e7a 6572 6f20    RandomNonzero 
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│ │ │ │ -000132e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000132f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │ -00013310: 3920 3a20 7065 656b 2053 7472 6174 6567  9 : peek Strateg
│ │ │ │ -00013320: 7944 6566 6175 6c74 5769 7468 506f 696e  yDefaultWithPoin
│ │ │ │ -00013330: 7473 2020 2020 2020 2020 7c0a 7c20 2020  ts        |.|   
│ │ │ │ -00013340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013360: 2020 2020 2020 2020 207c 0a7c 6f33 3920           |.|o39 
│ │ │ │ -00013370: 3d20 4f70 7469 6f6e 5461 626c 657b 4752  = OptionTable{GR
│ │ │ │ -00013380: 6576 4c65 784c 6172 6765 7374 203d 3e20  evLexLargest => 
│ │ │ │ -00013390: 3020 2020 2020 207d 7c0a 7c20 2020 2020  0      }|.|     
│ │ │ │ -000133a0: 2020 2020 2020 2020 2020 2020 2047 5265               GRe
│ │ │ │ -000133b0: 764c 6578 536d 616c 6c65 7374 203d 3e20  vLexSmallest => 
│ │ │ │ -000133c0: 3136 2020 2020 207c 0a7c 2020 2020 2020  16     |.|      
│ │ │ │ -000133d0: 2020 2020 2020 2020 2020 2020 4752 6576              GRev
│ │ │ │ -000133e0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ -000133f0: 3d3e 2031 3620 7c0a 7c20 2020 2020 2020  => 16 |.|       
│ │ │ │ -00013400: 2020 2020 2020 2020 2020 204c 6578 4c61             LexLa
│ │ │ │ -00013410: 7267 6573 7420 3d3e 2030 2020 2020 2020  rgest => 0      
│ │ │ │ -00013420: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00013430: 2020 2020 2020 2020 2020 4c65 7853 6d61            LexSma
│ │ │ │ -00013440: 6c6c 6573 7420 3d3e 2031 3620 2020 2020  llest => 16     
│ │ │ │ -00013450: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00013460: 2020 2020 2020 2020 204c 6578 536d 616c           LexSmal
│ │ │ │ -00013470: 6c65 7374 5465 726d 203d 3e20 3136 2020  lestTerm => 16  
│ │ │ │ -00013480: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00013490: 2020 2020 2020 2020 506f 696e 7473 203d          Points =
│ │ │ │ -000134a0: 3e20 3332 2020 2020 2020 2020 2020 2020  > 32            
│ │ │ │ -000134b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -000134c0: 2020 2020 2020 2052 616e 646f 6d20 3d3e         Random =>
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│ │ │ │ -000134e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000134f0: 2020 2020 2020 5261 6e64 6f6d 4e6f 6e7a        RandomNonz
│ │ │ │ -00013500: 6572 6f20 3d3e 2030 2020 2020 2020 2020  ero => 0        
│ │ │ │ -00013510: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -00013520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00013540: 0a7c 6934 3020 3a20 7065 656b 2053 7472  .|i40 : peek Str
│ │ │ │ -00013550: 6174 6567 7950 6f69 6e74 7320 2020 2020  ategyPoints     
│ │ │ │ -00013560: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00013570: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00013580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013590: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000135a0: 6f34 3020 3d20 4f70 7469 6f6e 5461 626c  o40 = OptionTabl
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│ │ │ │ +00012e80: 2020 2020 2020 2020 7c0a 7c6f 3337 203a          |.|o37 :
│ │ │ │ +00012e90: 2049 6465 616c 206f 6620 5320 2020 2020   Ideal of S     
│ │ │ │ +00012ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00012eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00012ed0: 2020 2020 2020 2020 7c0a 7c2d 2d2d 2d2d          |.|-----
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│ │ │ │ +00012ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012f20: 2d2d 2d2d 2d2d 2d2d 7c0a 7c56 6572 626f  --------|.|Verbo
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│ │ │ │ +00012f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00012fc0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a77 6520 6361  --------+..we ca
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│ │ │ │ +000130c0: 2e0a 0a4c 6574 2075 7320 7461 6b65 2061  ...Let us take a
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│ │ │ │ +000130e0: 6865 7220 6275 696c 742d 696e 2073 7472  her built-in str
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│ │ │ │ +00013100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013120: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 3820 3a20  -------+.|i38 : 
│ │ │ │ +00013130: 7065 656b 2053 7472 6174 6567 7944 6566  peek StrategyDef
│ │ │ │ +00013140: 6175 6c74 4e6f 6e52 616e 646f 6d20 2020  aultNonRandom   
│ │ │ │ +00013150: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00013160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013180: 2020 2020 207c 0a7c 6f33 3820 3d20 4f70       |.|o38 = Op
│ │ │ │ +00013190: 7469 6f6e 5461 626c 657b 4752 6576 4c65  tionTable{GRevLe
│ │ │ │ +000131a0: 784c 6172 6765 7374 203d 3e20 3020 2020  xLargest => 0   
│ │ │ │ +000131b0: 2020 207d 7c0a 7c20 2020 2020 2020 2020     }|.|         
│ │ │ │ +000131c0: 2020 2020 2020 2020 2047 5265 764c 6578           GRevLex
│ │ │ │ +000131d0: 536d 616c 6c65 7374 203d 3e20 3235 2020  Smallest => 25  
│ │ │ │ +000131e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000131f0: 2020 2020 2020 2020 4752 6576 4c65 7853          GRevLexS
│ │ │ │ +00013200: 6d61 6c6c 6573 7454 6572 6d20 3d3e 2032  mallestTerm => 2
│ │ │ │ +00013210: 3520 7c0a 7c20 2020 2020 2020 2020 2020  5 |.|           
│ │ │ │ +00013220: 2020 2020 2020 204c 6578 4c61 7267 6573         LexLarges
│ │ │ │ +00013230: 7420 3d3e 2030 2020 2020 2020 2020 2020  t => 0          
│ │ │ │ +00013240: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00013250: 2020 2020 2020 4c65 7853 6d61 6c6c 6573        LexSmalles
│ │ │ │ +00013260: 7420 3d3e 2032 3520 2020 2020 2020 2020  t => 25         
│ │ │ │ +00013270: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00013280: 2020 2020 204c 6578 536d 616c 6c65 7374       LexSmallest
│ │ │ │ +00013290: 5465 726d 203d 3e20 3235 2020 2020 207c  Term => 25     |
│ │ │ │ +000132a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000132b0: 2020 2020 506f 696e 7473 203d 3e20 3020      Points => 0 
│ │ │ │ +000132c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000132d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000132e0: 2020 2052 616e 646f 6d20 3d3e 2030 2020     Random => 0  
│ │ │ │ +000132f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00013300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013310: 2020 5261 6e64 6f6d 4e6f 6e7a 6572 6f20    RandomNonzero 
│ │ │ │ +00013320: 3d3e 2030 2020 2020 2020 2020 7c0a 2b2d  => 0        |.+-
│ │ │ │ +00013330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │ +00013360: 3920 3a20 7065 656b 2053 7472 6174 6567  9 : peek Strateg
│ │ │ │ +00013370: 7944 6566 6175 6c74 5769 7468 506f 696e  yDefaultWithPoin
│ │ │ │ +00013380: 7473 2020 2020 2020 2020 7c0a 7c20 2020  ts        |.|   
│ │ │ │ +00013390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000133a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000133b0: 2020 2020 2020 2020 207c 0a7c 6f33 3920           |.|o39 
│ │ │ │ +000133c0: 3d20 4f70 7469 6f6e 5461 626c 657b 4752  = OptionTable{GR
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│ │ │ │ +000133e0: 3020 2020 2020 207d 7c0a 7c20 2020 2020  0      }|.|     
│ │ │ │ +000133f0: 2020 2020 2020 2020 2020 2020 2047 5265               GRe
│ │ │ │ +00013400: 764c 6578 536d 616c 6c65 7374 203d 3e20  vLexSmallest => 
│ │ │ │ +00013410: 3136 2020 2020 207c 0a7c 2020 2020 2020  16     |.|      
│ │ │ │ +00013420: 2020 2020 2020 2020 2020 2020 4752 6576              GRev
│ │ │ │ +00013430: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ +00013440: 3d3e 2031 3620 7c0a 7c20 2020 2020 2020  => 16 |.|       
│ │ │ │ +00013450: 2020 2020 2020 2020 2020 204c 6578 4c61             LexLa
│ │ │ │ +00013460: 7267 6573 7420 3d3e 2030 2020 2020 2020  rgest => 0      
│ │ │ │ +00013470: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00013480: 2020 2020 2020 2020 2020 4c65 7853 6d61            LexSma
│ │ │ │ +00013490: 6c6c 6573 7420 3d3e 2031 3620 2020 2020  llest => 16     
│ │ │ │ +000134a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000134b0: 2020 2020 2020 2020 204c 6578 536d 616c           LexSmal
│ │ │ │ +000134c0: 6c65 7374 5465 726d 203d 3e20 3136 2020  lestTerm => 16  
│ │ │ │ +000134d0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000134e0: 2020 2020 2020 2020 506f 696e 7473 203d          Points =
│ │ │ │ +000134f0: 3e20 3332 2020 2020 2020 2020 2020 2020  > 32            
│ │ │ │ +00013500: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00013510: 2020 2020 2020 2052 616e 646f 6d20 3d3e         Random =>
│ │ │ │ +00013520: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ +00013530: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00013540: 2020 2020 2020 5261 6e64 6f6d 4e6f 6e7a        RandomNonz
│ │ │ │ +00013550: 6572 6f20 3d3e 2030 2020 2020 2020 2020  ero => 0        
│ │ │ │ +00013560: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00013570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00013590: 0a7c 6934 3020 3a20 7065 656b 2053 7472  .|i40 : peek Str
│ │ │ │ +000135a0: 6174 6567 7950 6f69 6e74 7320 2020 2020  ategyPoints     
│ │ │ │ +000135b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000135c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000135d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000135e0: 2047 5265 764c 6578 536d 616c 6c65 7374   GRevLexSmallest
│ │ │ │ -000135f0: 203d 3e20 3020 2020 2020 207c 0a7c 2020   => 0      |.|  
│ │ │ │ -00013600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013610: 4752 6576 4c65 7853 6d61 6c6c 6573 7454  GRevLexSmallestT
│ │ │ │ -00013620: 6572 6d20 3d3e 2030 2020 7c0a 7c20 2020  erm => 0  |.|   
│ │ │ │ -00013630: 2020 2020 2020 2020 2020 2020 2020 204c                 L
│ │ │ │ -00013640: 6578 4c61 7267 6573 7420 3d3e 2030 2020  exLargest => 0  
│ │ │ │ -00013650: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00013660: 2020 2020 2020 2020 2020 2020 2020 4c65                Le
│ │ │ │ -00013670: 7853 6d61 6c6c 6573 7420 3d3e 2030 2020  xSmallest => 0  
│ │ │ │ -00013680: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00013690: 2020 2020 2020 2020 2020 2020 204c 6578               Lex
│ │ │ │ -000136a0: 536d 616c 6c65 7374 5465 726d 203d 3e20  SmallestTerm => 
│ │ │ │ -000136b0: 3020 2020 2020 207c 0a7c 2020 2020 2020  0      |.|      
│ │ │ │ -000136c0: 2020 2020 2020 2020 2020 2020 506f 696e              Poin
│ │ │ │ -000136d0: 7473 203d 3e20 3130 3020 2020 2020 2020  ts => 100       
│ │ │ │ -000136e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000136f0: 2020 2020 2020 2020 2020 2052 616e 646f             Rando
│ │ │ │ -00013700: 6d20 3d3e 2030 2020 2020 2020 2020 2020  m => 0          
│ │ │ │ -00013710: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00013720: 2020 2020 2020 2020 2020 5261 6e64 6f6d            Random
│ │ │ │ -00013730: 4e6f 6e7a 6572 6f20 3d3e 2030 2020 2020  Nonzero => 0    
│ │ │ │ -00013740: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00013750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013770: 2d2d 2d2b 0a0a 5374 7261 7465 6779 4465  ---+..StrategyDe
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│ │ │ │ -000137a0: 6566 6175 6c74 2062 7574 2072 656d 6f76  efault but remov
│ │ │ │ -000137b0: 6573 2072 616e 646f 6d20 7375 626d 6174  es random submat
│ │ │ │ -000137c0: 7269 6365 730a 2877 6869 6368 2063 616e  rices.(which can
│ │ │ │ -000137d0: 2062 6520 7375 7270 7269 7369 6e67 6c79   be surprisingly
│ │ │ │ -000137e0: 2062 656e 6566 6963 6961 6c20 696e 2073   beneficial in s
│ │ │ │ -000137f0: 6f6d 6520 6361 7365 7329 2e0a 5374 7261  ome cases)..Stra
│ │ │ │ -00013800: 7465 6779 4465 6661 756c 7457 6974 6850  tegyDefaultWithP
│ │ │ │ -00013810: 6f69 6e74 7320 7265 6d6f 7665 7320 7261  oints removes ra
│ │ │ │ -00013820: 6e64 6f6d 6e65 7373 2062 7574 2061 6464  ndomness but add
│ │ │ │ -00013830: 7320 696e 2070 6f69 6e74 7320 696e 7374  s in points inst
│ │ │ │ -00013840: 6561 642e 0a0a 4120 7761 726e 696e 6720  ead...A warning 
│ │ │ │ -00013850: 6f6e 2063 686f 6f73 6547 6f6f 644d 696e  on chooseGoodMin
│ │ │ │ -00013860: 6f72 733a 2020 5468 6520 7374 7261 7465  ors:  The strate
│ │ │ │ -00013870: 6769 6573 204c 6578 536d 616c 6c65 7374  gies LexSmallest
│ │ │ │ -00013880: 2061 6e64 204c 6578 536d 616c 6c65 7374   and LexSmallest
│ │ │ │ -00013890: 5465 726d 0a77 696c 6c20 7665 7279 2066  Term.will very f
│ │ │ │ -000138a0: 7265 7175 656e 746c 7920 7265 7065 6174  requently repeat
│ │ │ │ -000138b0: 6564 6c79 2063 686f 6f73 6520 7468 6520  edly choose the 
│ │ │ │ -000138c0: 7361 6d65 2073 7562 6d61 7472 6978 206f  same submatrix o
│ │ │ │ -000138d0: 6620 7468 6520 6769 7665 6e20 6d61 7472  f the given matr
│ │ │ │ -000138e0: 6978 2e0a 4865 6e63 6520 6966 206f 6e65  ix..Hence if one
│ │ │ │ -000138f0: 2074 7269 6573 2074 6f20 7275 6e20 6368   tries to run ch
│ │ │ │ -00013900: 6f6f 7365 476f 6f64 4d69 6e6f 7273 2061  ooseGoodMinors a
│ │ │ │ -00013910: 6e64 2063 686f 6f73 6520 746f 6f20 6d61  nd choose too ma
│ │ │ │ -00013920: 6e79 206d 696e 6f72 7320 7769 7468 2073  ny minors with s
│ │ │ │ -00013930: 7563 680a 6120 7374 7261 7465 6779 2c20  uch.a strategy, 
│ │ │ │ -00013940: 6f6e 6520 6361 6e20 6765 7420 696e 746f  one can get into
│ │ │ │ -00013950: 2061 206c 6f6e 6720 6c6f 6f70 2028 7468   a long loop (th
│ │ │ │ -00013960: 6520 6675 6e63 7469 6f6e 2067 6976 6520  e function give 
│ │ │ │ -00013970: 7570 2065 7665 6e74 7561 6c6c 792c 2062  up eventually, b
│ │ │ │ -00013980: 7574 0a6f 6e6c 7920 6166 7465 7220 646f  ut.only after do
│ │ │ │ -00013990: 696e 6720 7761 7920 746f 6f20 6d75 6368  ing way too much
│ │ │ │ -000139a0: 2077 6f72 6b29 2e20 2054 6865 2047 5265   work).  The GRe
│ │ │ │ -000139b0: 764c 6578 2073 7472 6174 6567 6965 7320  vLex strategies 
│ │ │ │ -000139c0: 7065 7269 6f64 6963 616c 6c79 0a74 656d  periodically.tem
│ │ │ │ -000139d0: 706f 7261 7269 6c79 2063 6861 6e67 6520  porarily change 
│ │ │ │ -000139e0: 7468 6520 756e 6465 726c 7969 6e67 206d  the underlying m
│ │ │ │ -000139f0: 6174 7269 7820 746f 2061 766f 6964 2074  atrix to avoid t
│ │ │ │ -00013a00: 6869 7320 736f 7274 206f 6620 6c6f 6f70  his sort of loop
│ │ │ │ -00013a10: 2e0a 0a50 6f69 6e74 733a 204e 6f74 6963  ...Points: Notic
│ │ │ │ -00013a20: 6520 7468 6174 2053 7472 6174 6567 7920  e that Strategy 
│ │ │ │ -00013a30: 3d3e 2053 7472 6174 6567 7950 6f69 6e74  => StrategyPoint
│ │ │ │ -00013a40: 7320 616e 6420 5374 7261 7465 6779 203d  s and Strategy =
│ │ │ │ -00013a50: 3e20 506f 696e 7473 2064 6f20 7468 650a  > Points do the.
│ │ │ │ -00013a60: 7361 6d65 2074 6869 6e67 2e20 5765 2062  same thing. We b
│ │ │ │ -00013a70: 7269 6566 6c79 2064 6573 6372 6962 6520  riefly describe 
│ │ │ │ -00013a80: 686f 7720 6368 6f6f 7365 476f 6f64 4d69  how chooseGoodMi
│ │ │ │ -00013a90: 6e6f 7273 2069 6e74 6572 6163 7473 2077  nors interacts w
│ │ │ │ -00013aa0: 6974 6820 506f 696e 7473 2e0a 496e 6465  ith Points..Inde
│ │ │ │ -00013ab0: 6564 2050 6f69 6e74 7320 666f 726d 7320  ed Points forms 
│ │ │ │ -00013ac0: 7468 6520 6964 6561 6c20 6f66 206d 696e  the ideal of min
│ │ │ │ -00013ad0: 6f72 7320 636f 6d70 7574 6564 2073 6f20  ors computed so 
│ │ │ │ -00013ae0: 6661 7220 2870 6c75 7320 244a 2429 2c20  far (plus $J$), 
│ │ │ │ -00013af0: 6669 6e64 7320 610a 706f 696e 7420 7768  finds a.point wh
│ │ │ │ -00013b00: 6572 6520 7468 6174 2069 6465 616c 2076  ere that ideal v
│ │ │ │ -00013b10: 616e 6973 6865 7320 2877 6869 6368 2063  anishes (which c
│ │ │ │ -00013b20: 616e 2062 6520 736c 6f77 292c 2065 7661  an be slow), eva
│ │ │ │ -00013b30: 6c75 6174 6573 2074 6865 206d 6174 7269  luates the matri
│ │ │ │ -00013b40: 7820 244d 240a 6174 2074 6861 7420 706f  x $M$.at that po
│ │ │ │ -00013b50: 696e 742c 2061 6e64 2074 6865 6e20 6669  int, and then fi
│ │ │ │ -00013b60: 6e61 6c6c 7920 636f 6d70 7574 6573 2074  nally computes t
│ │ │ │ -00013b70: 6865 2063 6f72 7265 7370 6f6e 6469 6e67  he corresponding
│ │ │ │ -00013b80: 2064 6574 6572 6d69 6e61 6e74 206f 6620   determinant of 
│ │ │ │ -00013b90: 7468 650a 7375 626d 6174 7269 782e 2020  the.submatrix.  
│ │ │ │ -00013ba0: 5468 6973 2073 7562 6d61 7472 6978 2077  This submatrix w
│ │ │ │ -00013bb0: 696c 6c20 616c 7761 7973 2070 726f 6475  ill always produ
│ │ │ │ -00013bc0: 6365 2061 206d 696e 6f72 2077 6869 6368  ce a minor which
│ │ │ │ -00013bd0: 2073 6872 696e 6b73 206f 7572 0a76 616e   shrinks our.van
│ │ │ │ -00013be0: 6973 6869 6e67 206c 6f63 7573 2e0a 0a42  ishing locus...B
│ │ │ │ -00013bf0: 7920 6465 6661 756c 742c 2074 6865 2050  y default, the P
│ │ │ │ -00013c00: 6f69 6e74 7320 7374 7261 7465 6779 2061  oints strategy a
│ │ │ │ -00013c10: 6374 7561 6c6c 7920 6669 6e64 7320 6765  ctually finds ge
│ │ │ │ -00013c20: 6f6d 6574 7269 6320 706f 696e 7473 2e20  ometric points. 
│ │ │ │ -00013c30: 2057 6869 6368 2063 616e 2062 650a 736f   Which can be.so
│ │ │ │ -00013c40: 6d65 7469 6d65 7320 736c 6f77 6572 2028  metimes slower (
│ │ │ │ -00013c50: 6275 7420 7768 6963 6820 6172 6520 616c  but which are al
│ │ │ │ -00013c60: 6d6f 7374 2063 6572 7461 696e 2074 6f20  most certain to 
│ │ │ │ -00013c70: 6578 6973 742c 2061 6e64 2061 7265 206c  exist, and are l
│ │ │ │ -00013c80: 6573 7320 6c69 6b65 6c79 2074 6f0a 6861  ess likely to.ha
│ │ │ │ -00013c90: 6e67 2069 6620 7468 6520 6675 6e63 7469  ng if the functi
│ │ │ │ -00013ca0: 6f6e 2068 6173 2074 726f 7562 6c65 2066  on has trouble f
│ │ │ │ -00013cb0: 696e 6469 6e67 2061 2070 6f69 6e74 292e  inding a point).
│ │ │ │ -00013cc0: 2020 466f 7220 696e 7374 616e 6365 2c20    For instance, 
│ │ │ │ -00013cd0: 7765 2063 616e 0a63 6f6e 7472 6f6c 2074  we can.control t
│ │ │ │ -00013ce0: 6861 7420 6173 2066 6f6c 6c6f 7773 2e0a  hat as follows..
│ │ │ │ -00013cf0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00013d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00013d40: 0a7c 6934 3120 3a20 7074 7353 7472 6174  .|i41 : ptsStrat
│ │ │ │ -00013d50: 4765 6f6d 6574 7269 6320 3d20 6e65 7720  Geometric = new 
│ │ │ │ -00013d60: 4f70 7469 6f6e 5461 626c 6520 6672 6f6d  OptionTable from
│ │ │ │ -00013d70: 2028 6f70 7469 6f6e 7320 2020 2020 2020   (options       
│ │ │ │ -00013d80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00013d90: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ -00013da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00013de0: 0a7c 6368 6f6f 7365 476f 6f64 4d69 6e6f  .|chooseGoodMino
│ │ │ │ -00013df0: 7273 2923 506f 696e 744f 7074 696f 6e73  rs)#PointOptions
│ │ │ │ -00013e00: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ -00013e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013e20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00013e30: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00013e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00013e80: 0a7c 6934 3220 3a20 7074 7353 7472 6174  .|i42 : ptsStrat
│ │ │ │ -00013e90: 4765 6f6d 6574 7269 6323 4578 7465 6e64  Geometric#Extend
│ │ │ │ -00013ea0: 4669 656c 6420 2d2d 6c6f 6f6b 2061 7420  Field --look at 
│ │ │ │ -00013eb0: 7468 6520 6465 6661 756c 7420 7661 6c75  the default valu
│ │ │ │ -00013ec0: 6520 2020 2020 2020 2020 2020 2020 207c  e              |
│ │ │ │ -00013ed0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00013ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013f10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00013f20: 0a7c 6f34 3220 3d20 7472 7565 2020 2020  .|o42 = true    
│ │ │ │ +000135e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000135f0: 6f34 3020 3d20 4f70 7469 6f6e 5461 626c  o40 = OptionTabl
│ │ │ │ +00013600: 657b 4752 6576 4c65 784c 6172 6765 7374  e{GRevLexLargest
│ │ │ │ +00013610: 203d 3e20 3020 2020 2020 7d20 7c0a 7c20   => 0     } |.| 
│ │ │ │ +00013620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013630: 2047 5265 764c 6578 536d 616c 6c65 7374   GRevLexSmallest
│ │ │ │ +00013640: 203d 3e20 3020 2020 2020 207c 0a7c 2020   => 0      |.|  
│ │ │ │ +00013650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013660: 4752 6576 4c65 7853 6d61 6c6c 6573 7454  GRevLexSmallestT
│ │ │ │ +00013670: 6572 6d20 3d3e 2030 2020 7c0a 7c20 2020  erm => 0  |.|   
│ │ │ │ +00013680: 2020 2020 2020 2020 2020 2020 2020 204c                 L
│ │ │ │ +00013690: 6578 4c61 7267 6573 7420 3d3e 2030 2020  exLargest => 0  
│ │ │ │ +000136a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000136b0: 2020 2020 2020 2020 2020 2020 2020 4c65                Le
│ │ │ │ +000136c0: 7853 6d61 6c6c 6573 7420 3d3e 2030 2020  xSmallest => 0  
│ │ │ │ +000136d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000136e0: 2020 2020 2020 2020 2020 2020 204c 6578               Lex
│ │ │ │ +000136f0: 536d 616c 6c65 7374 5465 726d 203d 3e20  SmallestTerm => 
│ │ │ │ +00013700: 3020 2020 2020 207c 0a7c 2020 2020 2020  0      |.|      
│ │ │ │ +00013710: 2020 2020 2020 2020 2020 2020 506f 696e              Poin
│ │ │ │ +00013720: 7473 203d 3e20 3130 3020 2020 2020 2020  ts => 100       
│ │ │ │ +00013730: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00013740: 2020 2020 2020 2020 2020 2052 616e 646f             Rando
│ │ │ │ +00013750: 6d20 3d3e 2030 2020 2020 2020 2020 2020  m => 0          
│ │ │ │ +00013760: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00013770: 2020 2020 2020 2020 2020 5261 6e64 6f6d            Random
│ │ │ │ +00013780: 4e6f 6e7a 6572 6f20 3d3e 2030 2020 2020  Nonzero => 0    
│ │ │ │ +00013790: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000137a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000137b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000137c0: 2d2d 2d2b 0a0a 5374 7261 7465 6779 4465  ---+..StrategyDe
│ │ │ │ +000137d0: 6661 756c 744e 6f6e 5261 6e64 6f6d 2069  faultNonRandom i
│ │ │ │ +000137e0: 7320 6c69 6b65 2053 7472 6174 6567 7944  s like StrategyD
│ │ │ │ +000137f0: 6566 6175 6c74 2062 7574 2072 656d 6f76  efault but remov
│ │ │ │ +00013800: 6573 2072 616e 646f 6d20 7375 626d 6174  es random submat
│ │ │ │ +00013810: 7269 6365 730a 2877 6869 6368 2063 616e  rices.(which can
│ │ │ │ +00013820: 2062 6520 7375 7270 7269 7369 6e67 6c79   be surprisingly
│ │ │ │ +00013830: 2062 656e 6566 6963 6961 6c20 696e 2073   beneficial in s
│ │ │ │ +00013840: 6f6d 6520 6361 7365 7329 2e0a 5374 7261  ome cases)..Stra
│ │ │ │ +00013850: 7465 6779 4465 6661 756c 7457 6974 6850  tegyDefaultWithP
│ │ │ │ +00013860: 6f69 6e74 7320 7265 6d6f 7665 7320 7261  oints removes ra
│ │ │ │ +00013870: 6e64 6f6d 6e65 7373 2062 7574 2061 6464  ndomness but add
│ │ │ │ +00013880: 7320 696e 2070 6f69 6e74 7320 696e 7374  s in points inst
│ │ │ │ +00013890: 6561 642e 0a0a 4120 7761 726e 696e 6720  ead...A warning 
│ │ │ │ +000138a0: 6f6e 2063 686f 6f73 6547 6f6f 644d 696e  on chooseGoodMin
│ │ │ │ +000138b0: 6f72 733a 2020 5468 6520 7374 7261 7465  ors:  The strate
│ │ │ │ +000138c0: 6769 6573 204c 6578 536d 616c 6c65 7374  gies LexSmallest
│ │ │ │ +000138d0: 2061 6e64 204c 6578 536d 616c 6c65 7374   and LexSmallest
│ │ │ │ +000138e0: 5465 726d 0a77 696c 6c20 7665 7279 2066  Term.will very f
│ │ │ │ +000138f0: 7265 7175 656e 746c 7920 7265 7065 6174  requently repeat
│ │ │ │ +00013900: 6564 6c79 2063 686f 6f73 6520 7468 6520  edly choose the 
│ │ │ │ +00013910: 7361 6d65 2073 7562 6d61 7472 6978 206f  same submatrix o
│ │ │ │ +00013920: 6620 7468 6520 6769 7665 6e20 6d61 7472  f the given matr
│ │ │ │ +00013930: 6978 2e0a 4865 6e63 6520 6966 206f 6e65  ix..Hence if one
│ │ │ │ +00013940: 2074 7269 6573 2074 6f20 7275 6e20 6368   tries to run ch
│ │ │ │ +00013950: 6f6f 7365 476f 6f64 4d69 6e6f 7273 2061  ooseGoodMinors a
│ │ │ │ +00013960: 6e64 2063 686f 6f73 6520 746f 6f20 6d61  nd choose too ma
│ │ │ │ +00013970: 6e79 206d 696e 6f72 7320 7769 7468 2073  ny minors with s
│ │ │ │ +00013980: 7563 680a 6120 7374 7261 7465 6779 2c20  uch.a strategy, 
│ │ │ │ +00013990: 6f6e 6520 6361 6e20 6765 7420 696e 746f  one can get into
│ │ │ │ +000139a0: 2061 206c 6f6e 6720 6c6f 6f70 2028 7468   a long loop (th
│ │ │ │ +000139b0: 6520 6675 6e63 7469 6f6e 2067 6976 6520  e function give 
│ │ │ │ +000139c0: 7570 2065 7665 6e74 7561 6c6c 792c 2062  up eventually, b
│ │ │ │ +000139d0: 7574 0a6f 6e6c 7920 6166 7465 7220 646f  ut.only after do
│ │ │ │ +000139e0: 696e 6720 7761 7920 746f 6f20 6d75 6368  ing way too much
│ │ │ │ +000139f0: 2077 6f72 6b29 2e20 2054 6865 2047 5265   work).  The GRe
│ │ │ │ +00013a00: 764c 6578 2073 7472 6174 6567 6965 7320  vLex strategies 
│ │ │ │ +00013a10: 7065 7269 6f64 6963 616c 6c79 0a74 656d  periodically.tem
│ │ │ │ +00013a20: 706f 7261 7269 6c79 2063 6861 6e67 6520  porarily change 
│ │ │ │ +00013a30: 7468 6520 756e 6465 726c 7969 6e67 206d  the underlying m
│ │ │ │ +00013a40: 6174 7269 7820 746f 2061 766f 6964 2074  atrix to avoid t
│ │ │ │ +00013a50: 6869 7320 736f 7274 206f 6620 6c6f 6f70  his sort of loop
│ │ │ │ +00013a60: 2e0a 0a50 6f69 6e74 733a 204e 6f74 6963  ...Points: Notic
│ │ │ │ +00013a70: 6520 7468 6174 2053 7472 6174 6567 7920  e that Strategy 
│ │ │ │ +00013a80: 3d3e 2053 7472 6174 6567 7950 6f69 6e74  => StrategyPoint
│ │ │ │ +00013a90: 7320 616e 6420 5374 7261 7465 6779 203d  s and Strategy =
│ │ │ │ +00013aa0: 3e20 506f 696e 7473 2064 6f20 7468 650a  > Points do the.
│ │ │ │ +00013ab0: 7361 6d65 2074 6869 6e67 2e20 5765 2062  same thing. We b
│ │ │ │ +00013ac0: 7269 6566 6c79 2064 6573 6372 6962 6520  riefly describe 
│ │ │ │ +00013ad0: 686f 7720 6368 6f6f 7365 476f 6f64 4d69  how chooseGoodMi
│ │ │ │ +00013ae0: 6e6f 7273 2069 6e74 6572 6163 7473 2077  nors interacts w
│ │ │ │ +00013af0: 6974 6820 506f 696e 7473 2e0a 496e 6465  ith Points..Inde
│ │ │ │ +00013b00: 6564 2050 6f69 6e74 7320 666f 726d 7320  ed Points forms 
│ │ │ │ +00013b10: 7468 6520 6964 6561 6c20 6f66 206d 696e  the ideal of min
│ │ │ │ +00013b20: 6f72 7320 636f 6d70 7574 6564 2073 6f20  ors computed so 
│ │ │ │ +00013b30: 6661 7220 2870 6c75 7320 244a 2429 2c20  far (plus $J$), 
│ │ │ │ +00013b40: 6669 6e64 7320 610a 706f 696e 7420 7768  finds a.point wh
│ │ │ │ +00013b50: 6572 6520 7468 6174 2069 6465 616c 2076  ere that ideal v
│ │ │ │ +00013b60: 616e 6973 6865 7320 2877 6869 6368 2063  anishes (which c
│ │ │ │ +00013b70: 616e 2062 6520 736c 6f77 292c 2065 7661  an be slow), eva
│ │ │ │ +00013b80: 6c75 6174 6573 2074 6865 206d 6174 7269  luates the matri
│ │ │ │ +00013b90: 7820 244d 240a 6174 2074 6861 7420 706f  x $M$.at that po
│ │ │ │ +00013ba0: 696e 742c 2061 6e64 2074 6865 6e20 6669  int, and then fi
│ │ │ │ +00013bb0: 6e61 6c6c 7920 636f 6d70 7574 6573 2074  nally computes t
│ │ │ │ +00013bc0: 6865 2063 6f72 7265 7370 6f6e 6469 6e67  he corresponding
│ │ │ │ +00013bd0: 2064 6574 6572 6d69 6e61 6e74 206f 6620   determinant of 
│ │ │ │ +00013be0: 7468 650a 7375 626d 6174 7269 782e 2020  the.submatrix.  
│ │ │ │ +00013bf0: 5468 6973 2073 7562 6d61 7472 6978 2077  This submatrix w
│ │ │ │ +00013c00: 696c 6c20 616c 7761 7973 2070 726f 6475  ill always produ
│ │ │ │ +00013c10: 6365 2061 206d 696e 6f72 2077 6869 6368  ce a minor which
│ │ │ │ +00013c20: 2073 6872 696e 6b73 206f 7572 0a76 616e   shrinks our.van
│ │ │ │ +00013c30: 6973 6869 6e67 206c 6f63 7573 2e0a 0a42  ishing locus...B
│ │ │ │ +00013c40: 7920 6465 6661 756c 742c 2074 6865 2050  y default, the P
│ │ │ │ +00013c50: 6f69 6e74 7320 7374 7261 7465 6779 2061  oints strategy a
│ │ │ │ +00013c60: 6374 7561 6c6c 7920 6669 6e64 7320 6765  ctually finds ge
│ │ │ │ +00013c70: 6f6d 6574 7269 6320 706f 696e 7473 2e20  ometric points. 
│ │ │ │ +00013c80: 2057 6869 6368 2063 616e 2062 650a 736f   Which can be.so
│ │ │ │ +00013c90: 6d65 7469 6d65 7320 736c 6f77 6572 2028  metimes slower (
│ │ │ │ +00013ca0: 6275 7420 7768 6963 6820 6172 6520 616c  but which are al
│ │ │ │ +00013cb0: 6d6f 7374 2063 6572 7461 696e 2074 6f20  most certain to 
│ │ │ │ +00013cc0: 6578 6973 742c 2061 6e64 2061 7265 206c  exist, and are l
│ │ │ │ +00013cd0: 6573 7320 6c69 6b65 6c79 2074 6f0a 6861  ess likely to.ha
│ │ │ │ +00013ce0: 6e67 2069 6620 7468 6520 6675 6e63 7469  ng if the functi
│ │ │ │ +00013cf0: 6f6e 2068 6173 2074 726f 7562 6c65 2066  on has trouble f
│ │ │ │ +00013d00: 696e 6469 6e67 2061 2070 6f69 6e74 292e  inding a point).
│ │ │ │ +00013d10: 2020 466f 7220 696e 7374 616e 6365 2c20    For instance, 
│ │ │ │ +00013d20: 7765 2063 616e 0a63 6f6e 7472 6f6c 2074  we can.control t
│ │ │ │ +00013d30: 6861 7420 6173 2066 6f6c 6c6f 7773 2e0a  hat as follows..
│ │ │ │ +00013d40: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00013d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00013d90: 0a7c 6934 3120 3a20 7074 7353 7472 6174  .|i41 : ptsStrat
│ │ │ │ +00013da0: 4765 6f6d 6574 7269 6320 3d20 6e65 7720  Geometric = new 
│ │ │ │ +00013db0: 4f70 7469 6f6e 5461 626c 6520 6672 6f6d  OptionTable from
│ │ │ │ +00013dc0: 2028 6f70 7469 6f6e 7320 2020 2020 2020   (options       
│ │ │ │ +00013dd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00013de0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00013df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00013e30: 0a7c 6368 6f6f 7365 476f 6f64 4d69 6e6f  .|chooseGoodMino
│ │ │ │ +00013e40: 7273 2923 506f 696e 744f 7074 696f 6e73  rs)#PointOptions
│ │ │ │ +00013e50: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ +00013e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013e70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00013e80: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00013e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00013ed0: 0a7c 6934 3220 3a20 7074 7353 7472 6174  .|i42 : ptsStrat
│ │ │ │ +00013ee0: 4765 6f6d 6574 7269 6323 4578 7465 6e64  Geometric#Extend
│ │ │ │ +00013ef0: 4669 656c 6420 2d2d 6c6f 6f6b 2061 7420  Field --look at 
│ │ │ │ +00013f00: 7468 6520 6465 6661 756c 7420 7661 6c75  the default valu
│ │ │ │ +00013f10: 6520 2020 2020 2020 2020 2020 2020 207c  e              |
│ │ │ │ +00013f20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00013f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013f60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00013f70: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00013f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 3837  .| -- used 0.487
│ │ │ │ -00014020: 3231 3473 2028 6370 7529 3b20 302e 3432  214s (cpu); 0.42
│ │ │ │ -00014030: 3633 3333 7320 2874 6872 6561 6429 3b20  6333s (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                  
│ │ │ │ +00013f70: 0a7c 6f34 3220 3d20 7472 7565 2020 2020  .|o42 = true    
│ │ │ │ +00013f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013fb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00013fc0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00013fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00014010: 0a7c 6934 3320 3a20 7469 6d65 2064 696d  .|i43 : time dim
│ │ │ │ +00014020: 2028 4a20 2b20 6368 6f6f 7365 476f 6f64   (J + chooseGood
│ │ │ │ +00014030: 4d69 6e6f 7273 2831 2c20 362c 204d 2c20  Minors(1, 6, M, 
│ │ │ │ +00014040: 4a2c 2053 7472 6174 6567 793d 3e50 6f69  J, Strategy=>Poi
│ │ │ │ +00014050: 6e74 732c 2020 2020 2020 2020 2020 207c  nts,           |
│ │ │ │ +00014060: 0a7c 202d 2d20 7573 6564 2030 2e37 3435  .| -- used 0.745
│ │ │ │ +00014070: 3733 3273 2028 6370 7529 3b20 302e 3630  732s (cpu); 0.60
│ │ │ │ +00014080: 3038 3933 7320 2874 6872 6561 6429 3b20  0893s (thread); 
│ │ │ │ +00014090: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │  000140a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000140b0: 0a7c 6f34 3320 3d20 3220 2020 2020 2020  .|o43 = 2       
│ │ │ │ +000140b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000140c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000140d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000140e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000140f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014100: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ -00014110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00014150: 0a7c 506f 696e 744f 7074 696f 6e73 3d3e  .|PointOptions=>
│ │ │ │ -00014160: 7074 7353 7472 6174 4765 6f6d 6574 7269  ptsStratGeometri
│ │ │ │ -00014170: 6329 2920 2020 2020 2020 2020 2020 2020  c))             
│ │ │ │ -00014180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014190: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000141a0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -000141b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000141c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000141d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000141e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000141f0: 0a7c 6934 3420 3a20 7074 7353 7472 6174  .|i44 : ptsStrat
│ │ │ │ -00014200: 5261 7469 6f6e 616c 203d 2070 7473 5374  Rational = ptsSt
│ │ │ │ -00014210: 7261 7447 656f 6d65 7472 6963 2b2b 7b45  ratGeometric++{E
│ │ │ │ -00014220: 7874 656e 6446 6965 6c64 3d3e 6661 6c73  xtendField=>fals
│ │ │ │ -00014230: 657d 202d 2d63 6861 6e67 6520 2020 207c  e} --change    |
│ │ │ │ -00014240: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00014250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014280: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014290: 0a7c 6f34 3420 3d20 4f70 7469 6f6e 5461  .|o44 = OptionTa
│ │ │ │ -000142a0: 626c 657b 4465 636f 6d70 6f73 6974 696f  ble{Decompositio
│ │ │ │ -000142b0: 6e53 7472 6174 6567 7920 3d3e 2044 6563  nStrategy => Dec
│ │ │ │ -000142c0: 6f6d 706f 7365 7d20 2020 2020 2020 2020  ompose}         
│ │ │ │ +00014100: 0a7c 6f34 3320 3d20 3220 2020 2020 2020  .|o43 = 2       
│ │ │ │ +00014110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014140: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00014150: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00014160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +000141a0: 0a7c 506f 696e 744f 7074 696f 6e73 3d3e  .|PointOptions=>
│ │ │ │ +000141b0: 7074 7353 7472 6174 4765 6f6d 6574 7269  ptsStratGeometri
│ │ │ │ +000141c0: 6329 2920 2020 2020 2020 2020 2020 2020  c))             
│ │ │ │ +000141d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000141e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000141f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00014200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00014240: 0a7c 6934 3420 3a20 7074 7353 7472 6174  .|i44 : ptsStrat
│ │ │ │ +00014250: 5261 7469 6f6e 616c 203d 2070 7473 5374  Rational = ptsSt
│ │ │ │ +00014260: 7261 7447 656f 6d65 7472 6963 2b2b 7b45  ratGeometric++{E
│ │ │ │ +00014270: 7874 656e 6446 6965 6c64 3d3e 6661 6c73  xtendField=>fals
│ │ │ │ +00014280: 657d 202d 2d63 6861 6e67 6520 2020 207c  e} --change    |
│ │ │ │ +00014290: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000142a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000142b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000142c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000142d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000142e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000142f0: 2020 2020 4469 6d65 6e73 696f 6e46 756e      DimensionFun
│ │ │ │ -00014300: 6374 696f 6e20 3d3e 2064 696d 2020 2020  ction => dim    
│ │ │ │ -00014310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000142e0: 0a7c 6f34 3420 3d20 4f70 7469 6f6e 5461  .|o44 = OptionTa
│ │ │ │ +000142f0: 626c 657b 4465 636f 6d70 6f73 6974 696f  ble{Decompositio
│ │ │ │ +00014300: 6e53 7472 6174 6567 7920 3d3e 2044 6563  nStrategy => Dec
│ │ │ │ +00014310: 6f6d 706f 7365 7d20 2020 2020 2020 2020  ompose}         
│ │ │ │  00014320: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00014330: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00014340: 2020 2020 4578 7465 6e64 4669 656c 6420      ExtendField 
│ │ │ │ -00014350: 3d3e 2066 616c 7365 2020 2020 2020 2020  => false        
│ │ │ │ +00014340: 2020 2020 4469 6d65 6e73 696f 6e46 756e      DimensionFun
│ │ │ │ +00014350: 6374 696f 6e20 3d3e 2064 696d 2020 2020  ction => dim    
│ │ │ │  00014360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014370: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00014380: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00014390: 2020 2020 486f 6d6f 6765 6e65 6f75 7320      Homogeneous 
│ │ │ │ +00014390: 2020 2020 4578 7465 6e64 4669 656c 6420      ExtendField 
│ │ │ │  000143a0: 3d3e 2066 616c 7365 2020 2020 2020 2020  => false        
│ │ │ │  000143b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000143c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  000143d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000143e0: 2020 2020 4e75 6d54 6872 6561 6473 546f      NumThreadsTo
│ │ │ │ -000143f0: 5573 6520 3d3e 2031 2020 2020 2020 2020  Use => 1        
│ │ │ │ +000143e0: 2020 2020 486f 6d6f 6765 6e65 6f75 7320      Homogeneous 
│ │ │ │ +000143f0: 3d3e 2066 616c 7365 2020 2020 2020 2020  => false        
│ │ │ │  00014400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014410: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00014420: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00014430: 2020 2020 506f 696e 7443 6865 636b 4174      PointCheckAt
│ │ │ │ -00014440: 7465 6d70 7473 203d 3e20 3020 2020 2020  tempts => 0     
│ │ │ │ +00014430: 2020 2020 4e75 6d54 6872 6561 6473 546f      NumThreadsTo
│ │ │ │ +00014440: 5573 6520 3d3e 2031 2020 2020 2020 2020  Use => 1        
│ │ │ │  00014450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014460: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00014470: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00014480: 2020 2020 5265 706c 6163 656d 656e 7420      Replacement 
│ │ │ │ -00014490: 3d3e 2042 696e 6f6d 6961 6c20 2020 2020  => Binomial     
│ │ │ │ +00014480: 2020 2020 506f 696e 7443 6865 636b 4174      PointCheckAt
│ │ │ │ +00014490: 7465 6d70 7473 203d 3e20 3020 2020 2020  tempts => 0     
│ │ │ │  000144a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000144b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  000144c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000144d0: 2020 2020 5374 7261 7465 6779 203d 3e20      Strategy => 
│ │ │ │ -000144e0: 4465 6661 756c 7420 2020 2020 2020 2020  Default         
│ │ │ │ +000144d0: 2020 2020 5265 706c 6163 656d 656e 7420      Replacement 
│ │ │ │ +000144e0: 3d3e 2042 696e 6f6d 6961 6c20 2020 2020  => Binomial     
│ │ │ │  000144f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014500: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00014510: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00014520: 2020 2020 5665 7262 6f73 6520 3d3e 2066      Verbose => f
│ │ │ │ -00014530: 616c 7365 2020 2020 2020 2020 2020 2020  alse            
│ │ │ │ +00014520: 2020 2020 5374 7261 7465 6779 203d 3e20      Strategy => 
│ │ │ │ +00014530: 4465 6661 756c 7420 2020 2020 2020 2020  Default         
│ │ │ │  00014540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014550: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00014560: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00014570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014570: 2020 2020 5665 7262 6f73 6520 3d3e 2066      Verbose => f
│ │ │ │ +00014580: 616c 7365 2020 2020 2020 2020 2020 2020  alse            
│ │ │ │  00014590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000145a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000145b0: 0a7c 6f34 3420 3a20 4f70 7469 6f6e 5461  .|o44 : OptionTa
│ │ │ │ -000145c0: 626c 6520 2020 2020 2020 2020 2020 2020  ble             
│ │ │ │ +000145b0: 0a7c 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 2020                  
│ │ │ │  000145f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014600: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ -00014610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00014650: 0a7c 7468 6174 2076 616c 7565 2020 2020  .|that value    
│ │ │ │ -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 2020                  
│ │ │ │ -00014690: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000146a0: 0a2b 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 2d2d  ----------------
│ │ │ │ -000146e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000146f0: 0a7c 6934 3520 3a20 7074 7353 7472 6174  .|i45 : ptsStrat
│ │ │ │ -00014700: 5261 7469 6f6e 616c 2e45 7874 656e 6446  Rational.ExtendF
│ │ │ │ -00014710: 6965 6c64 202d 2d6c 6f6f 6b20 6174 206f  ield --look at o
│ │ │ │ -00014720: 7572 2063 6861 6e67 6564 2076 616c 7565  ur changed value
│ │ │ │ -00014730: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014740: 0a7c 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 2020                  
│ │ │ │ +00014600: 0a7c 6f34 3420 3a20 4f70 7469 6f6e 5461  .|o44 : OptionTa
│ │ │ │ +00014610: 626c 6520 2020 2020 2020 2020 2020 2020  ble             
│ │ │ │ +00014620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014640: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00014650: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00014660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +000146a0: 0a7c 7468 6174 2076 616c 7565 2020 2020  .|that value    
│ │ │ │ +000146b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000146c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000146d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000146e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000146f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00014700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00014740: 0a7c 6934 3520 3a20 7074 7353 7472 6174  .|i45 : ptsStrat
│ │ │ │ +00014750: 5261 7469 6f6e 616c 2e45 7874 656e 6446  Rational.ExtendF
│ │ │ │ +00014760: 6965 6c64 202d 2d6c 6f6f 6b20 6174 206f  ield --look at o
│ │ │ │ +00014770: 7572 2063 6861 6e67 6564 2076 616c 7565  ur changed value
│ │ │ │  00014780: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014790: 0a7c 6f34 3520 3d20 6661 6c73 6520 2020  .|o45 = false   
│ │ │ │ +00014790: 0a7c 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 2020                  
│ │ │ │  000147d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000147e0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -000147f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 2e35 3036  .| -- used 0.506
│ │ │ │ -00014890: 3638 3373 2028 6370 7529 3b20 302e 3337  683s (cpu); 0.37
│ │ │ │ -000148a0: 3936 3938 7320 2874 6872 6561 6429 3b20  9698s (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                  
│ │ │ │ +000147e0: 0a7c 6f34 3520 3d20 6661 6c73 6520 2020  .|o45 = false   
│ │ │ │ +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 2020                  
│ │ │ │ +00014820: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00014830: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00014840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00014880: 0a7c 6934 3620 3a20 7469 6d65 2064 696d  .|i46 : time dim
│ │ │ │ +00014890: 2028 4a20 2b20 6368 6f6f 7365 476f 6f64   (J + chooseGood
│ │ │ │ +000148a0: 4d69 6e6f 7273 2831 2c20 362c 204d 2c20  Minors(1, 6, M, 
│ │ │ │ +000148b0: 4a2c 2053 7472 6174 6567 793d 3e50 6f69  J, Strategy=>Poi
│ │ │ │ +000148c0: 6e74 732c 2020 2020 2020 2020 2020 207c  nts,           |
│ │ │ │ +000148d0: 0a7c 202d 2d20 7573 6564 2030 2e34 3838  .| -- used 0.488
│ │ │ │ +000148e0: 3473 2028 6370 7529 3b20 302e 3431 3938  4s (cpu); 0.4198
│ │ │ │ +000148f0: 3731 7320 2874 6872 6561 6429 3b20 3073  71s (thread); 0s
│ │ │ │ +00014900: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │  00014910: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014920: 0a7c 6f34 3620 3d20 3220 2020 2020 2020  .|o46 = 2       
│ │ │ │ +00014920: 0a7c 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 2020                  
│ │ │ │  00014960: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014970: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ -00014980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000149a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000149b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000149c0: 0a7c 506f 696e 744f 7074 696f 6e73 3d3e  .|PointOptions=>
│ │ │ │ -000149d0: 7074 7353 7472 6174 5261 7469 6f6e 616c  ptsStratRational
│ │ │ │ -000149e0: 2929 2020 2020 2020 2020 2020 2020 2020  ))              
│ │ │ │ -000149f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014a00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014a10: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00014a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00014a60: 0a0a 4f74 6865 7220 6f70 7469 6f6e 7320  ..Other options 
│ │ │ │ -00014a70: 6d61 7920 616c 736f 2062 6520 7061 7373  may also be pass
│ │ │ │ -00014a80: 6564 2074 6f20 7468 6520 2a6e 6f74 6520  ed to the *note 
│ │ │ │ -00014a90: 5261 6e64 6f6d 506f 696e 7473 3a20 2852  RandomPoints: (R
│ │ │ │ -00014aa0: 616e 646f 6d50 6f69 6e74 7329 546f 702c  andomPoints)Top,
│ │ │ │ -00014ab0: 0a70 6163 6b61 6765 2076 6961 2074 6865  .package via the
│ │ │ │ -00014ac0: 202a 6e6f 7465 2050 6f69 6e74 4f70 7469   *note PointOpti
│ │ │ │ -00014ad0: 6f6e 733a 2050 6f69 6e74 4f70 7469 6f6e  ons: PointOption
│ │ │ │ -00014ae0: 732c 206f 7074 696f 6e2e 0a0a 7265 6775  s, option...regu
│ │ │ │ -00014af0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ -00014b00: 3a20 2049 7420 6973 2072 6561 736f 6e61  :  It is reasona
│ │ │ │ -00014b10: 626c 6520 746f 2074 6869 6e6b 2074 6861  ble to think tha
│ │ │ │ -00014b20: 7420 796f 7520 7368 6f75 6c64 2066 696e  t you should fin
│ │ │ │ -00014b30: 6420 6120 6665 770a 6d69 6e6f 7273 2028  d a few.minors (
│ │ │ │ -00014b40: 7769 7468 206f 6e65 2073 7472 6174 6567  with one strateg
│ │ │ │ -00014b50: 7920 6f72 2061 6e6f 7468 6572 292c 2061  y or another), a
│ │ │ │ -00014b60: 6e64 2073 6565 2069 6620 7065 7268 6170  nd see if perhap
│ │ │ │ -00014b70: 7320 7468 6520 6d69 6e6f 7273 2079 6f75  s the minors you
│ │ │ │ -00014b80: 2068 6176 650a 636f 6d70 7574 6564 2073   have.computed s
│ │ │ │ -00014b90: 6f20 6661 7220 6172 6520 656e 6f75 6768  o far are enough
│ │ │ │ -00014ba0: 2074 6f20 7665 7269 6679 206f 7572 2072   to verify our r
│ │ │ │ -00014bb0: 696e 6720 6973 2072 6567 756c 6172 2069  ing is regular i
│ │ │ │ -00014bc0: 6e20 636f 6469 6d65 6e73 696f 6e20 312e  n codimension 1.
│ │ │ │ -00014bd0: 0a54 6869 7320 6973 2065 7861 6374 6c79  .This is exactly
│ │ │ │ -00014be0: 2077 6861 7420 7265 6775 6c61 7249 6e43   what regularInC
│ │ │ │ -00014bf0: 6f64 696d 656e 7369 6f6e 2064 6f65 732e  odimension does.
│ │ │ │ -00014c00: 2020 4f6e 6520 6361 6e20 636f 6e74 726f    One can contro
│ │ │ │ -00014c10: 6c20 6174 2061 2066 696e 650a 6c65 7665  l at a fine.leve
│ │ │ │ -00014c20: 6c20 686f 7720 6672 6571 7565 6e74 6c79  l how frequently
│ │ │ │ -00014c30: 206e 6577 206d 696e 6f72 7320 6172 6520   new minors are 
│ │ │ │ -00014c40: 636f 6d70 7574 6564 2c20 616e 6420 686f  computed, and ho
│ │ │ │ -00014c50: 7720 6672 6571 7565 6e74 6c79 2074 6865  w frequently the
│ │ │ │ -00014c60: 2064 696d 656e 7369 6f6e 0a6f 6620 7768   dimension.of wh
│ │ │ │ -00014c70: 6174 2077 6520 6861 7665 2063 6f6d 7075  at we have compu
│ │ │ │ -00014c80: 7465 6420 736f 2066 6172 2069 7320 6368  ted so far is ch
│ │ │ │ -00014c90: 6563 6b65 642c 2062 7920 7468 6520 6f70  ecked, by the op
│ │ │ │ -00014ca0: 7469 6f6e 2063 6f64 696d 4368 6563 6b46  tion codimCheckF
│ │ │ │ -00014cb0: 756e 6374 696f 6e2e 0a46 6f72 206d 6f72  unction..For mor
│ │ │ │ -00014cc0: 6520 6f6e 2074 6861 742c 2073 6565 202a  e on that, see *
│ │ │ │ -00014cd0: 6e6f 7465 2052 6567 756c 6172 496e 436f  note RegularInCo
│ │ │ │ -00014ce0: 6469 6d65 6e73 696f 6e54 7574 6f72 6961  dimensionTutoria
│ │ │ │ -00014cf0: 6c3a 0a52 6567 756c 6172 496e 436f 6469  l:.RegularInCodi
│ │ │ │ -00014d00: 6d65 6e73 696f 6e54 7574 6f72 6961 6c2c  mensionTutorial,
│ │ │ │ -00014d10: 2061 6e64 202a 6e6f 7465 2072 6567 756c   and *note regul
│ │ │ │ -00014d20: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ -00014d30: 0a72 6567 756c 6172 496e 436f 6469 6d65  .regularInCodime
│ │ │ │ -00014d40: 6e73 696f 6e2c 2e20 204c 6574 2075 7320  nsion,.  Let us 
│ │ │ │ -00014d50: 6669 6e69 7368 2072 756e 6e69 6e67 2072  finish running r
│ │ │ │ -00014d60: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ -00014d70: 696f 6e20 6f6e 206f 7572 0a65 7861 6d70  ion on our.examp
│ │ │ │ -00014d80: 6c65 2077 6974 6820 7365 7665 7261 6c20  le with several 
│ │ │ │ -00014d90: 6469 6666 6572 656e 7420 7374 7261 7465  different strate
│ │ │ │ -00014da0: 6769 6573 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  gies...+--------
│ │ │ │ -00014db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 2e32 3839 3434 7320 2863 7075 293b   3.28944s (cpu);
│ │ │ │ -00014e60: 2033 2e30 3132 3873 2028 7468 7265 6164   3.0128s (thread
│ │ │ │ -00014e70: 293b 2030 7320 2867 6329 2020 2020 2020  ); 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  ----------------
│ │ │ │ -00014ee0: 2d2d 2d2d 2d7c 0a7c 5374 7261 7465 6779  -----|.|Strategy
│ │ │ │ -00014ef0: 3d3e 5374 7261 7465 6779 4465 6661 756c  =>StrategyDefaul
│ │ │ │ -00014f00: 7429 2020 2020 2020 2020 2020 2020 2020  t)              
│ │ │ │ -00014f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014f30: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00014f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 2e39 3035 3730 3373 2028 6370 7529   0.905703s (cpu)
│ │ │ │ -00014ff0: 3b20 302e 3738 3934 3733 7320 2874 6872  ; 0.789473s (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                  
│ │ │ │ +00014970: 0a7c 6f34 3620 3d20 3220 2020 2020 2020  .|o46 = 2       
│ │ │ │ +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 2020                  
│ │ │ │ +000149b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000149c0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +000149d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000149e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000149f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00014a10: 0a7c 506f 696e 744f 7074 696f 6e73 3d3e  .|PointOptions=>
│ │ │ │ +00014a20: 7074 7353 7472 6174 5261 7469 6f6e 616c  ptsStratRational
│ │ │ │ +00014a30: 2929 2020 2020 2020 2020 2020 2020 2020  ))              
│ │ │ │ +00014a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014a50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00014a60: 0a2b 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 2d2d  ----------------
│ │ │ │ +00014aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00014ab0: 0a0a 4f74 6865 7220 6f70 7469 6f6e 7320  ..Other options 
│ │ │ │ +00014ac0: 6d61 7920 616c 736f 2062 6520 7061 7373  may also be pass
│ │ │ │ +00014ad0: 6564 2074 6f20 7468 6520 2a6e 6f74 6520  ed to the *note 
│ │ │ │ +00014ae0: 5261 6e64 6f6d 506f 696e 7473 3a20 2852  RandomPoints: (R
│ │ │ │ +00014af0: 616e 646f 6d50 6f69 6e74 7329 546f 702c  andomPoints)Top,
│ │ │ │ +00014b00: 0a70 6163 6b61 6765 2076 6961 2074 6865  .package via the
│ │ │ │ +00014b10: 202a 6e6f 7465 2050 6f69 6e74 4f70 7469   *note PointOpti
│ │ │ │ +00014b20: 6f6e 733a 2050 6f69 6e74 4f70 7469 6f6e  ons: PointOption
│ │ │ │ +00014b30: 732c 206f 7074 696f 6e2e 0a0a 7265 6775  s, option...regu
│ │ │ │ +00014b40: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ +00014b50: 3a20 2049 7420 6973 2072 6561 736f 6e61  :  It is reasona
│ │ │ │ +00014b60: 626c 6520 746f 2074 6869 6e6b 2074 6861  ble to think tha
│ │ │ │ +00014b70: 7420 796f 7520 7368 6f75 6c64 2066 696e  t you should fin
│ │ │ │ +00014b80: 6420 6120 6665 770a 6d69 6e6f 7273 2028  d a few.minors (
│ │ │ │ +00014b90: 7769 7468 206f 6e65 2073 7472 6174 6567  with one strateg
│ │ │ │ +00014ba0: 7920 6f72 2061 6e6f 7468 6572 292c 2061  y or another), a
│ │ │ │ +00014bb0: 6e64 2073 6565 2069 6620 7065 7268 6170  nd see if perhap
│ │ │ │ +00014bc0: 7320 7468 6520 6d69 6e6f 7273 2079 6f75  s the minors you
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│ │ │ │ +00014bf0: 2074 6f20 7665 7269 6679 206f 7572 2072   to verify our r
│ │ │ │ +00014c00: 696e 6720 6973 2072 6567 756c 6172 2069  ing is regular i
│ │ │ │ +00014c10: 6e20 636f 6469 6d65 6e73 696f 6e20 312e  n codimension 1.
│ │ │ │ +00014c20: 0a54 6869 7320 6973 2065 7861 6374 6c79  .This is exactly
│ │ │ │ +00014c30: 2077 6861 7420 7265 6775 6c61 7249 6e43   what regularInC
│ │ │ │ +00014c40: 6f64 696d 656e 7369 6f6e 2064 6f65 732e  odimension does.
│ │ │ │ +00014c50: 2020 4f6e 6520 6361 6e20 636f 6e74 726f    One can contro
│ │ │ │ +00014c60: 6c20 6174 2061 2066 696e 650a 6c65 7665  l at a fine.leve
│ │ │ │ +00014c70: 6c20 686f 7720 6672 6571 7565 6e74 6c79  l how frequently
│ │ │ │ +00014c80: 206e 6577 206d 696e 6f72 7320 6172 6520   new minors are 
│ │ │ │ +00014c90: 636f 6d70 7574 6564 2c20 616e 6420 686f  computed, and ho
│ │ │ │ +00014ca0: 7720 6672 6571 7565 6e74 6c79 2074 6865  w frequently the
│ │ │ │ +00014cb0: 2064 696d 656e 7369 6f6e 0a6f 6620 7768   dimension.of wh
│ │ │ │ +00014cc0: 6174 2077 6520 6861 7665 2063 6f6d 7075  at we have compu
│ │ │ │ +00014cd0: 7465 6420 736f 2066 6172 2069 7320 6368  ted so far is ch
│ │ │ │ +00014ce0: 6563 6b65 642c 2062 7920 7468 6520 6f70  ecked, by the op
│ │ │ │ +00014cf0: 7469 6f6e 2063 6f64 696d 4368 6563 6b46  tion codimCheckF
│ │ │ │ +00014d00: 756e 6374 696f 6e2e 0a46 6f72 206d 6f72  unction..For mor
│ │ │ │ +00014d10: 6520 6f6e 2074 6861 742c 2073 6565 202a  e on that, see *
│ │ │ │ +00014d20: 6e6f 7465 2052 6567 756c 6172 496e 436f  note RegularInCo
│ │ │ │ +00014d30: 6469 6d65 6e73 696f 6e54 7574 6f72 6961  dimensionTutoria
│ │ │ │ +00014d40: 6c3a 0a52 6567 756c 6172 496e 436f 6469  l:.RegularInCodi
│ │ │ │ +00014d50: 6d65 6e73 696f 6e54 7574 6f72 6961 6c2c  mensionTutorial,
│ │ │ │ +00014d60: 2061 6e64 202a 6e6f 7465 2072 6567 756c   and *note regul
│ │ │ │ +00014d70: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00014d80: 0a72 6567 756c 6172 496e 436f 6469 6d65  .regularInCodime
│ │ │ │ +00014d90: 6e73 696f 6e2c 2e20 204c 6574 2075 7320  nsion,.  Let us 
│ │ │ │ +00014da0: 6669 6e69 7368 2072 756e 6e69 6e67 2072  finish running r
│ │ │ │ +00014db0: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ +00014dc0: 696f 6e20 6f6e 206f 7572 0a65 7861 6d70  ion on our.examp
│ │ │ │ +00014dd0: 6c65 2077 6974 6820 7365 7665 7261 6c20  le with several 
│ │ │ │ +00014de0: 6469 6666 6572 656e 7420 7374 7261 7465  different strate
│ │ │ │ +00014df0: 6769 6573 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  gies...+--------
│ │ │ │ +00014e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014e40: 2d2d 2d2d 2d2b 0a7c 6934 3720 3a20 7469  -----+.|i47 : ti
│ │ │ │ +00014e50: 6d65 2072 6567 756c 6172 496e 436f 6469  me regularInCodi
│ │ │ │ +00014e60: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20  mension(1, S/J, 
│ │ │ │ +00014e70: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ +00014e80: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +00014e90: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ +00014ea0: 2033 2e39 3834 3037 7320 2863 7075 293b   3.98407s (cpu);
│ │ │ │ +00014eb0: 2033 2e35 3339 3539 7320 2874 6872 6561   3.53959s (threa
│ │ │ │ +00014ec0: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +00014ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014ee0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +00014ef0: 2d2d 2d2d 2d2d 2d2d 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 2d7c 0a7c 5374 7261 7465 6779  -----|.|Strategy
│ │ │ │ +00014f40: 3d3e 5374 7261 7465 6779 4465 6661 756c  =>StrategyDefaul
│ │ │ │ +00014f50: 7429 2020 2020 2020 2020 2020 2020 2020  t)              
│ │ │ │ +00014f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014f80: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00014f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014fd0: 2d2d 2d2d 2d2b 0a7c 6934 3820 3a20 7469  -----+.|i48 : ti
│ │ │ │ +00014fe0: 6d65 2072 6567 756c 6172 496e 436f 6469  me regularInCodi
│ │ │ │ +00014ff0: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20  mension(1, S/J, 
│ │ │ │ +00015000: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ +00015010: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +00015020: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ +00015030: 2030 2e38 3634 3431 3673 2028 6370 7529   0.864416s (cpu)
│ │ │ │ +00015040: 3b20 302e 3732 3937 3739 7320 2874 6872  ; 0.729779s (thr
│ │ │ │ +00015050: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │  00015060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015070: 2020 2020 207c 0a7c 6f34 3820 3d20 7472       |.|o48 = tr
│ │ │ │ -00015080: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ +00015070: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00015080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000150a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000150b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000150c0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ -000150d0: 2d2d 2d2d 2d2d 2d2d 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 2d7c 0a7c 5374 7261 7465 6779  -----|.|Strategy
│ │ │ │ -00015120: 3d3e 5374 7261 7465 6779 4465 6661 756c  =>StrategyDefaul
│ │ │ │ -00015130: 744e 6f6e 5261 6e64 6f6d 2920 2020 2020  tNonRandom)     
│ │ │ │ -00015140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015160: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00015170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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: 2033 2e35 3832 3634 7320 2863 7075 293b   3.58264s (cpu);
│ │ │ │ -00015220: 2033 2e33 3634 3534 7320 2874 6872 6561   3.36454s (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 2e33 3832 3739 7320 2863 7075 293b   2.38279s (cpu);
│ │ │ │ -00015310: 2031 2e39 3838 3373 2028 7468 7265 6164   1.9883s (thread
│ │ │ │ -00015320: 293b 2030 7320 2867 6329 2020 2020 2020  ); 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
│ │ │ │ -000153a0: 3d3e 4c65 7853 6d61 6c6c 6573 7429 2020  =>LexSmallest)  
│ │ │ │ -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 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -000153f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 3331 3137 3473 2028 6370 7529   0.831174s (cpu)
│ │ │ │ -000154a0: 3b20 302e 3731 3537 3873 2028 7468 7265  ; 0.71578s (thre
│ │ │ │ -000154b0: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 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                  
│ │ │ │ +000150c0: 2020 2020 207c 0a7c 6f34 3820 3d20 7472       |.|o48 = tr
│ │ │ │ +000150d0: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ +000150e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000150f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015110: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +00015120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015160: 2d2d 2d2d 2d7c 0a7c 5374 7261 7465 6779  -----|.|Strategy
│ │ │ │ +00015170: 3d3e 5374 7261 7465 6779 4465 6661 756c  =>StrategyDefaul
│ │ │ │ +00015180: 744e 6f6e 5261 6e64 6f6d 2920 2020 2020  tNonRandom)     
│ │ │ │ +00015190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000151a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000151b0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000151c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000151d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000151e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000151f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015200: 2d2d 2d2d 2d2b 0a7c 6934 3920 3a20 7469  -----+.|i49 : ti
│ │ │ │ +00015210: 6d65 2072 6567 756c 6172 496e 436f 6469  me regularInCodi
│ │ │ │ +00015220: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20  mension(1, S/J, 
│ │ │ │ +00015230: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ +00015240: 2c20 5374 7261 7465 6779 3d3e 5261 6e64  , Strategy=>Rand
│ │ │ │ +00015250: 6f6d 2920 207c 0a7c 202d 2d20 7573 6564  om)  |.| -- used
│ │ │ │ +00015260: 2033 2e32 3436 3539 7320 2863 7075 293b   3.24659s (cpu);
│ │ │ │ +00015270: 2033 2e30 3239 3739 7320 2874 6872 6561   3.02979s (threa
│ │ │ │ +00015280: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +00015290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000152a0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000152b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000152c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000152d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000152e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000152f0: 2d2d 2d2d 2d2b 0a7c 6935 3020 3a20 7469  -----+.|i50 : ti
│ │ │ │ +00015300: 6d65 2072 6567 756c 6172 496e 436f 6469  me regularInCodi
│ │ │ │ +00015310: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20  mension(1, S/J, 
│ │ │ │ +00015320: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ +00015330: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +00015340: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ +00015350: 2032 2e37 3532 3033 7320 2863 7075 293b   2.75203s (cpu);
│ │ │ │ +00015360: 2032 2e32 3434 3937 7320 2874 6872 6561   2.24497s (threa
│ │ │ │ +00015370: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +00015380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015390: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +000153a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000153b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000153c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000153d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000153e0: 2d2d 2d2d 2d7c 0a7c 5374 7261 7465 6779  -----|.|Strategy
│ │ │ │ +000153f0: 3d3e 4c65 7853 6d61 6c6c 6573 7429 2020  =>LexSmallest)  
│ │ │ │ +00015400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015430: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00015440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015480: 2d2d 2d2d 2d2b 0a7c 6935 3120 3a20 7469  -----+.|i51 : ti
│ │ │ │ +00015490: 6d65 2072 6567 756c 6172 496e 436f 6469  me regularInCodi
│ │ │ │ +000154a0: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20  mension(1, S/J, 
│ │ │ │ +000154b0: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ +000154c0: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +000154d0: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ +000154e0: 2030 2e38 3737 3136 7320 2863 7075 293b   0.87716s (cpu);
│ │ │ │ +000154f0: 2030 2e38 3135 3039 3173 2028 7468 7265   0.815091s (thre
│ │ │ │ +00015500: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │  00015510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015520: 2020 2020 207c 0a7c 6f35 3120 3d20 7472       |.|o51 = tr
│ │ │ │ -00015530: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ +00015520: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00015530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ -00015580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000155a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000155b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000155c0: 2d2d 2d2d 2d7c 0a7c 5374 7261 7465 6779  -----|.|Strategy
│ │ │ │ -000155d0: 3d3e 4c65 7853 6d61 6c6c 6573 7454 6572  =>LexSmallestTer
│ │ │ │ -000155e0: 6d29 2020 2020 2020 2020 2020 2020 2020  m)              
│ │ │ │ -000155f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015610: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00015620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 2e36 3031 3873 2028 6370 7529 3b20   2.6018s (cpu); 
│ │ │ │ -000156d0: 322e 3135 3936 3173 2028 7468 7265 6164  2.15961s (thread
│ │ │ │ -000156e0: 293b 2030 7320 2867 6329 2020 2020 2020  ); 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  ----------------
│ │ │ │ -00015750: 2d2d 2d2d 2d7c 0a7c 5374 7261 7465 6779  -----|.|Strategy
│ │ │ │ -00015760: 3d3e 4752 6576 4c65 7853 6d61 6c6c 6573  =>GRevLexSmalles
│ │ │ │ -00015770: 7429 2020 2020 2020 2020 2020 2020 2020  t)              
│ │ │ │ -00015780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000157a0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -000157b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000157c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000157d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 2e30 3330 3033 7320 2863 7075 293b   3.03003s (cpu);
│ │ │ │ -00015860: 2032 2e36 3233 3639 7320 2874 6872 6561   2.62369s (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
│ │ │ │ -000158f0: 3d3e 4752 6576 4c65 7853 6d61 6c6c 6573  =>GRevLexSmalles
│ │ │ │ -00015900: 7454 6572 6d29 2020 2020 2020 2020 2020  tTerm)          
│ │ │ │ -00015910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015930: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00015940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 2e31 3535 3331 7320 2863 7075 293b   9.15531s (cpu);
│ │ │ │ -000159f0: 2037 2e35 3834 3435 7320 2874 6872 6561   7.58445s (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                  
│ │ │ │ +00015570: 2020 2020 207c 0a7c 6f35 3120 3d20 7472       |.|o51 = tr
│ │ │ │ +00015580: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ +00015590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000155a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000155b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000155c0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +000155d0: 2d2d 2d2d 2d2d 2d2d 2d2d 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 2d7c 0a7c 5374 7261 7465 6779  -----|.|Strategy
│ │ │ │ +00015620: 3d3e 4c65 7853 6d61 6c6c 6573 7454 6572  =>LexSmallestTer
│ │ │ │ +00015630: 6d29 2020 2020 2020 2020 2020 2020 2020  m)              
│ │ │ │ +00015640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015660: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00015670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000156a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000156b0: 2d2d 2d2d 2d2b 0a7c 6935 3220 3a20 7469  -----+.|i52 : ti
│ │ │ │ +000156c0: 6d65 2072 6567 756c 6172 496e 436f 6469  me regularInCodi
│ │ │ │ +000156d0: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20  mension(1, S/J, 
│ │ │ │ +000156e0: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ +000156f0: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +00015700: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ +00015710: 2033 2e30 3636 3035 7320 2863 7075 293b   3.06605s (cpu);
│ │ │ │ +00015720: 2032 2e35 3139 3236 7320 2874 6872 6561   2.51926s (threa
│ │ │ │ +00015730: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +00015740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015750: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +00015760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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 2d7c 0a7c 5374 7261 7465 6779  -----|.|Strategy
│ │ │ │ +000157b0: 3d3e 4752 6576 4c65 7853 6d61 6c6c 6573  =>GRevLexSmalles
│ │ │ │ +000157c0: 7429 2020 2020 2020 2020 2020 2020 2020  t)              
│ │ │ │ +000157d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000157e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000157f0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00015800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015840: 2d2d 2d2d 2d2b 0a7c 6935 3320 3a20 7469  -----+.|i53 : ti
│ │ │ │ +00015850: 6d65 2072 6567 756c 6172 496e 436f 6469  me regularInCodi
│ │ │ │ +00015860: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20  mension(1, S/J, 
│ │ │ │ +00015870: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ +00015880: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +00015890: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ +000158a0: 2033 2e35 3639 3437 7320 2863 7075 293b   3.56947s (cpu);
│ │ │ │ +000158b0: 2033 2e30 3731 3635 7320 2874 6872 6561   3.07165s (threa
│ │ │ │ +000158c0: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +000158d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000158e0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +000158f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015930: 2d2d 2d2d 2d7c 0a7c 5374 7261 7465 6779  -----|.|Strategy
│ │ │ │ +00015940: 3d3e 4752 6576 4c65 7853 6d61 6c6c 6573  =>GRevLexSmalles
│ │ │ │ +00015950: 7454 6572 6d29 2020 2020 2020 2020 2020  tTerm)          
│ │ │ │ +00015960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015980: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00015990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000159a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000159b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000159c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000159d0: 2d2d 2d2d 2d2b 0a7c 6935 3420 3a20 7469  -----+.|i54 : ti
│ │ │ │ +000159e0: 6d65 2072 6567 756c 6172 496e 436f 6469  me regularInCodi
│ │ │ │ +000159f0: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20  mension(1, S/J, 
│ │ │ │ +00015a00: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ +00015a10: 2c20 5374 7261 7465 6779 3d3e 506f 696e  , Strategy=>Poin
│ │ │ │ +00015a20: 7473 2920 207c 0a7c 202d 2d20 7573 6564  ts)  |.| -- used
│ │ │ │ +00015a30: 2031 302e 3733 3636 7320 2863 7075 293b   10.7366s (cpu);
│ │ │ │ +00015a40: 2038 2e38 3735 3539 7320 2874 6872 6561   8.87559s (threa
│ │ │ │ +00015a50: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │  00015a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a70: 2020 2020 207c 0a7c 6f35 3420 3d20 7472       |.|o54 = tr
│ │ │ │ -00015a80: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ +00015a70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00015a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015ac0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00015ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 2e31 3734 3536 7320 2863 7075 293b   7.17456s (cpu);
│ │ │ │ -00015b80: 2035 2e38 3535 3332 7320 2874 6872 6561   5.85532s (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                  
│ │ │ │ +00015ac0: 2020 2020 207c 0a7c 6f35 3420 3d20 7472       |.|o54 = tr
│ │ │ │ +00015ad0: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ +00015ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015b10: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00015b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015b60: 2d2d 2d2d 2d2b 0a7c 6935 3520 3a20 7469  -----+.|i55 : ti
│ │ │ │ +00015b70: 6d65 2072 6567 756c 6172 496e 436f 6469  me regularInCodi
│ │ │ │ +00015b80: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20  mension(1, S/J, 
│ │ │ │ +00015b90: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ +00015ba0: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +00015bb0: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ +00015bc0: 2037 2e39 3830 3932 7320 2863 7075 293b   7.98092s (cpu);
│ │ │ │ +00015bd0: 2036 2e35 3932 3138 7320 2874 6872 6561   6.59218s (threa
│ │ │ │ +00015be0: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │  00015bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015c00: 2020 2020 207c 0a7c 6f35 3520 3d20 7472       |.|o55 = tr
│ │ │ │ -00015c10: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ +00015c00: 2020 2020 207c 0a7c 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 2020 2020 2020                  
│ │ │ │  00015c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015c50: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ -00015c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015ca0: 2d2d 2d2d 2d7c 0a7c 5374 7261 7465 6779  -----|.|Strategy
│ │ │ │ -00015cb0: 3d3e 5374 7261 7465 6779 4465 6661 756c  =>StrategyDefaul
│ │ │ │ -00015cc0: 7457 6974 6850 6f69 6e74 7329 2020 2020  tWithPoints)    
│ │ │ │ -00015cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015cf0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00015d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015d40: 2d2d 2d2d 2d2b 0a0a 4966 2072 6567 756c  -----+..If regul
│ │ │ │ -00015d50: 6172 496e 436f 6469 6d65 6e73 696f 6e20  arInCodimension 
│ │ │ │ -00015d60: 6f75 7470 7574 7320 6e6f 7468 696e 672c  outputs nothing,
│ │ │ │ -00015d70: 2074 6865 6e20 6974 2063 6f75 6c64 6e27   then it couldn'
│ │ │ │ -00015d80: 7420 7665 7269 6679 2074 6861 7420 7468  t verify that th
│ │ │ │ -00015d90: 6520 7269 6e67 0a77 6173 2072 6567 756c  e ring.was regul
│ │ │ │ -00015da0: 6172 2069 6e20 7468 6174 2063 6f64 696d  ar in that codim
│ │ │ │ -00015db0: 656e 7369 6f6e 2e20 2057 6520 7365 7420  ension.  We set 
│ │ │ │ -00015dc0: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ -00015dd0: 2074 6f20 6b65 6570 2069 7420 6672 6f6d   to keep it from
│ │ │ │ -00015de0: 0a72 756e 6e69 6e67 2074 6f6f 206c 6f6e  .running too lon
│ │ │ │ -00015df0: 6720 7769 7468 2061 6e20 696e 6566 6665  g with an ineffe
│ │ │ │ -00015e00: 6374 6976 6520 7374 7261 7465 6779 2e20  ctive strategy. 
│ │ │ │ -00015e10: 2041 6761 696e 2c20 6576 656e 2074 686f   Again, even tho
│ │ │ │ -00015e20: 7567 680a 4752 6576 4c65 7853 6d61 6c6c  ugh.GRevLexSmall
│ │ │ │ -00015e30: 6573 7420 616e 6420 4752 6576 4c65 7853  est and GRevLexS
│ │ │ │ -00015e40: 6d61 6c6c 6573 7454 6572 6d20 6172 6520  mallestTerm are 
│ │ │ │ -00015e50: 6e6f 7420 6566 6665 6374 6976 6520 696e  not effective in
│ │ │ │ -00015e60: 2074 6869 7320 7061 7274 6963 756c 6172   this particular
│ │ │ │ -00015e70: 0a65 7861 6d70 6c65 2c20 696e 206f 7468  .example, in oth
│ │ │ │ -00015e80: 6572 7320 7468 6579 2070 6572 666f 726d  ers they perform
│ │ │ │ -00015e90: 2062 6574 7465 7220 7468 616e 206f 7468   better than oth
│ │ │ │ -00015ea0: 6572 2073 7472 6174 6567 6965 732e 2020  er strategies.  
│ │ │ │ -00015eb0: 4e6f 7465 2073 696d 696c 6172 0a63 6f6e  Note similar.con
│ │ │ │ -00015ec0: 7369 6465 7261 7469 6f6e 7320 616c 736f  siderations also
│ │ │ │ -00015ed0: 2061 7070 6c79 2074 6f20 2a6e 6f74 6520   apply to *note 
│ │ │ │ -00015ee0: 7072 6f6a 4469 6d3a 2070 726f 6a44 696d  projDim: projDim
│ │ │ │ -00015ef0: 2c2e 0a0a 5365 6520 616c 736f 0a3d 3d3d  ,...See also.===
│ │ │ │ -00015f00: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ -00015f10: 2063 686f 6f73 6547 6f6f 644d 696e 6f72   chooseGoodMinor
│ │ │ │ -00015f20: 7328 2e2e 2e2c 5374 7261 7465 6779 3d3e  s(...,Strategy=>
│ │ │ │ -00015f30: 2e2e 2e29 3a20 5374 7261 7465 6779 4465  ...): StrategyDe
│ │ │ │ -00015f40: 6661 756c 742c 202d 2d20 7374 7261 7465  fault, -- strate
│ │ │ │ -00015f50: 6769 6573 0a20 2020 2066 6f72 2063 686f  gies.    for cho
│ │ │ │ -00015f60: 6f73 696e 6720 7375 626d 6174 7269 6365  osing submatrice
│ │ │ │ -00015f70: 730a 2020 2a20 2a6e 6f74 6520 5265 6775  s.  * *note Regu
│ │ │ │ -00015f80: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ -00015f90: 5475 746f 7269 616c 3a20 5265 6775 6c61  Tutorial: Regula
│ │ │ │ -00015fa0: 7249 6e43 6f64 696d 656e 7369 6f6e 5475  rInCodimensionTu
│ │ │ │ -00015fb0: 746f 7269 616c 2c20 2d2d 2041 0a20 2020  torial, -- A.   
│ │ │ │ -00015fc0: 2074 7574 6f72 6961 6c20 666f 7220 686f   tutorial for ho
│ │ │ │ -00015fd0: 7720 746f 2075 7365 2074 6865 2061 6476  w to use the adv
│ │ │ │ -00015fe0: 616e 6365 6420 6f70 7469 6f6e 7320 6f66  anced options of
│ │ │ │ -00015ff0: 2074 6865 2072 6567 756c 6172 496e 436f   the regularInCo
│ │ │ │ -00016000: 6469 6d65 6e73 696f 6e0a 2020 2020 6675  dimension.    fu
│ │ │ │ -00016010: 6e63 7469 6f6e 0a0a 466f 7220 7468 6520  nction..For the 
│ │ │ │ -00016020: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
│ │ │ │ -00016030: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
│ │ │ │ -00016040: 6865 206f 626a 6563 7420 2a6e 6f74 6520  he object *note 
│ │ │ │ -00016050: 4661 7374 4d69 6e6f 7273 5374 7261 7465  FastMinorsStrate
│ │ │ │ -00016060: 6779 5475 746f 7269 616c 3a20 4661 7374  gyTutorial: Fast
│ │ │ │ -00016070: 4d69 6e6f 7273 5374 7261 7465 6779 5475  MinorsStrategyTu
│ │ │ │ -00016080: 746f 7269 616c 2c20 6973 2061 0a2a 6e6f  torial, is a.*no
│ │ │ │ -00016090: 7465 2073 796d 626f 6c3a 2028 4d61 6361  te symbol: (Maca
│ │ │ │ -000160a0: 756c 6179 3244 6f63 2953 796d 626f 6c2c  ulay2Doc)Symbol,
│ │ │ │ -000160b0: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...-------------
│ │ │ │ -000160c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000160d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000160e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000160f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016100: 2d2d 0a0a 5468 6520 736f 7572 6365 206f  --..The source o
│ │ │ │ -00016110: 6620 7468 6973 2064 6f63 756d 656e 7420  f this document 
│ │ │ │ -00016120: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570  is in./build/rep
│ │ │ │ -00016130: 726f 6475 6369 626c 652d 7061 7468 2f6d  roducible-path/m
│ │ │ │ -00016140: 6163 6175 6c61 7932 2d31 2e32 352e 3131  acaulay2-1.25.11
│ │ │ │ -00016150: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932  +ds/M2/Macaulay2
│ │ │ │ -00016160: 2f70 6163 6b61 6765 732f 4661 7374 4d69  /packages/FastMi
│ │ │ │ -00016170: 6e6f 7273 2e0a 6d32 3a31 3532 353a 302e  nors..m2:1525:0.
│ │ │ │ -00016180: 0a1f 0a46 696c 653a 2046 6173 744d 696e  ...File: FastMin
│ │ │ │ -00016190: 6f72 732e 696e 666f 2c20 4e6f 6465 3a20  ors.info, Node: 
│ │ │ │ -000161a0: 6765 7453 7562 6d61 7472 6978 4f66 5261  getSubmatrixOfRa
│ │ │ │ -000161b0: 6e6b 2c20 4e65 7874 3a20 6973 436f 6469  nk, Next: isCodi
│ │ │ │ -000161c0: 6d41 744c 6561 7374 2c20 5072 6576 3a20  mAtLeast, Prev: 
│ │ │ │ -000161d0: 4661 7374 4d69 6e6f 7273 5374 7261 7465  FastMinorsStrate
│ │ │ │ -000161e0: 6779 5475 746f 7269 616c 2c20 5570 3a20  gyTutorial, Up: 
│ │ │ │ -000161f0: 546f 700a 0a67 6574 5375 626d 6174 7269  Top..getSubmatri
│ │ │ │ -00016200: 784f 6652 616e 6b20 2d2d 2074 7269 6573  xOfRank -- tries
│ │ │ │ -00016210: 2074 6f20 6669 6e64 2061 2073 7562 6d61   to find a subma
│ │ │ │ -00016220: 7472 6978 206f 6620 7468 6520 6769 7665  trix of the give
│ │ │ │ -00016230: 6e20 7261 6e6b 0a2a 2a2a 2a2a 2a2a 2a2a  n rank.*********
│ │ │ │ -00016240: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00016250: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00016260: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00016270: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573  ********..  * Us
│ │ │ │ -00016280: 6167 653a 200a 2020 2020 2020 2020 6765  age: .        ge
│ │ │ │ -00016290: 7453 7562 6d61 7472 6978 4f66 5261 6e6b  tSubmatrixOfRank
│ │ │ │ -000162a0: 286e 312c 204d 3129 0a20 202a 2049 6e70  (n1, M1).  * Inp
│ │ │ │ -000162b0: 7574 733a 0a20 2020 2020 202a 206e 312c  uts:.      * n1,
│ │ │ │ -000162c0: 2061 6e20 2a6e 6f74 6520 696e 7465 6765   an *note intege
│ │ │ │ -000162d0: 723a 2028 4d61 6361 756c 6179 3244 6f63  r: (Macaulay2Doc
│ │ │ │ -000162e0: 295a 5a2c 2c20 0a20 2020 2020 202a 204d  )ZZ,, .      * M
│ │ │ │ -000162f0: 312c 2061 202a 6e6f 7465 206d 6174 7269  1, a *note matri
│ │ │ │ -00016300: 783a 2028 4d61 6361 756c 6179 3244 6f63  x: (Macaulay2Doc
│ │ │ │ -00016310: 294d 6174 7269 782c 2c20 0a20 202a 202a  )Matrix,, .  * *
│ │ │ │ -00016320: 6e6f 7465 204f 7074 696f 6e61 6c20 696e  note Optional in
│ │ │ │ -00016330: 7075 7473 3a20 284d 6163 6175 6c61 7932  puts: (Macaulay2
│ │ │ │ -00016340: 446f 6329 7573 696e 6720 6675 6e63 7469  Doc)using functi
│ │ │ │ -00016350: 6f6e 7320 7769 7468 206f 7074 696f 6e61  ons with optiona
│ │ │ │ -00016360: 6c20 696e 7075 7473 2c3a 0a20 2020 2020  l inputs,:.     
│ │ │ │ -00016370: 202a 202a 6e6f 7465 2044 6574 5374 7261   * *note DetStra
│ │ │ │ -00016380: 7465 6779 3a20 4465 7453 7472 6174 6567  tegy: DetStrateg
│ │ │ │ -00016390: 792c 203d 3e20 2e2e 2e2c 2064 6566 6175  y, => ..., defau
│ │ │ │ -000163a0: 6c74 2076 616c 7565 2052 616e 6b2c 2044  lt value Rank, D
│ │ │ │ -000163b0: 6574 5374 7261 7465 6779 0a20 2020 2020  etStrategy.     
│ │ │ │ -000163c0: 2020 2069 7320 6120 7374 7261 7465 6779     is a strategy
│ │ │ │ -000163d0: 2066 6f72 2061 6c6c 6f77 696e 6720 7468   for allowing th
│ │ │ │ -000163e0: 6520 7573 6572 2074 6f20 6368 6f6f 7365  e user to choose
│ │ │ │ -000163f0: 2068 6f77 2064 6574 6572 6d69 6e61 6e74   how determinant
│ │ │ │ -00016400: 7320 286f 720a 2020 2020 2020 2020 7261  s (or.        ra
│ │ │ │ -00016410: 6e6b 292c 2069 7320 636f 6d70 7574 6564  nk), is computed
│ │ │ │ -00016420: 0a20 2020 2020 202a 202a 6e6f 7465 204d  .      * *note M
│ │ │ │ -00016430: 6178 4d69 6e6f 7273 3a20 4d61 784d 696e  axMinors: MaxMin
│ │ │ │ -00016440: 6f72 732c 203d 3e20 2e2e 2e2c 2064 6566  ors, => ..., def
│ │ │ │ -00016450: 6175 6c74 2076 616c 7565 206e 756c 6c2c  ault value null,
│ │ │ │ -00016460: 2061 6e20 6f70 7469 6f6e 2074 6f0a 2020   an option to.  
│ │ │ │ -00016470: 2020 2020 2020 636f 6e74 726f 6c20 6465        control de
│ │ │ │ -00016480: 7074 6820 6f66 2073 6561 7263 680a 2020  pth of search.  
│ │ │ │ -00016490: 2020 2020 2a20 2a6e 6f74 6520 506f 696e      * *note Poin
│ │ │ │ -000164a0: 744f 7074 696f 6e73 3a20 506f 696e 744f  tOptions: PointO
│ │ │ │ -000164b0: 7074 696f 6e73 2c20 3d3e 202e 2e2e 2c20  ptions, => ..., 
│ │ │ │ -000164c0: 6465 6661 756c 7420 7661 6c75 6520 7b53  default value {S
│ │ │ │ -000164d0: 7472 6174 6567 7920 3d3e 0a20 2020 2020  trategy =>.     
│ │ │ │ -000164e0: 2020 2044 6566 6175 6c74 2c20 486f 6d6f     Default, Homo
│ │ │ │ -000164f0: 6765 6e65 6f75 7320 3d3e 2066 616c 7365  geneous => false
│ │ │ │ -00016500: 2c20 5265 706c 6163 656d 656e 7420 3d3e  , Replacement =>
│ │ │ │ -00016510: 2042 696e 6f6d 6961 6c2c 2045 7874 656e   Binomial, Exten
│ │ │ │ -00016520: 6446 6965 6c64 203d 3e0a 2020 2020 2020  dField =>.      
│ │ │ │ -00016530: 2020 7472 7565 2c20 506f 696e 7443 6865    true, PointChe
│ │ │ │ -00016540: 636b 4174 7465 6d70 7473 203d 3e20 302c  ckAttempts => 0,
│ │ │ │ -00016550: 2044 6563 6f6d 706f 7369 7469 6f6e 5374   DecompositionSt
│ │ │ │ -00016560: 7261 7465 6779 203d 3e20 4465 636f 6d70  rategy => Decomp
│ │ │ │ -00016570: 6f73 652c 0a20 2020 2020 2020 204e 756d  ose,.        Num
│ │ │ │ -00016580: 5468 7265 6164 7354 6f55 7365 203d 3e20  ThreadsToUse => 
│ │ │ │ -00016590: 312c 2044 696d 656e 7369 6f6e 4675 6e63  1, DimensionFunc
│ │ │ │ -000165a0: 7469 6f6e 203d 3e20 6469 6d2c 2056 6572  tion => dim, Ver
│ │ │ │ -000165b0: 626f 7365 203d 3e20 6661 6c73 657d 2c0a  bose => false},.
│ │ │ │ -000165c0: 2020 2020 2020 2020 6f70 7469 6f6e 7320          options 
│ │ │ │ -000165d0: 746f 2070 6173 7320 746f 2066 756e 6374  to pass to funct
│ │ │ │ -000165e0: 696f 6e73 2069 6e20 7468 6520 7061 636b  ions in the pack
│ │ │ │ -000165f0: 6167 6520 5261 6e64 6f6d 506f 696e 7473  age RandomPoints
│ │ │ │ -00016600: 0a20 2020 2020 202a 202a 6e6f 7465 2053  .      * *note S
│ │ │ │ -00016610: 7472 6174 6567 793a 2053 7472 6174 6567  trategy: Strateg
│ │ │ │ -00016620: 7944 6566 6175 6c74 2c20 3d3e 202e 2e2e  yDefault, => ...
│ │ │ │ -00016630: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ -00016640: 6e65 7720 4f70 7469 6f6e 5461 626c 650a  new OptionTable.
│ │ │ │ -00016650: 2020 2020 2020 2020 6672 6f6d 207b 506f          from {Po
│ │ │ │ -00016660: 696e 7473 203d 3e20 302c 2052 616e 646f  ints => 0, Rando
│ │ │ │ -00016670: 6d20 3d3e 2030 2c20 4752 6576 4c65 784c  m => 0, GRevLexL
│ │ │ │ -00016680: 6172 6765 7374 203d 3e20 302c 204c 6578  argest => 0, Lex
│ │ │ │ -00016690: 536d 616c 6c65 7374 5465 726d 203d 3e0a  SmallestTerm =>.
│ │ │ │ -000166a0: 2020 2020 2020 2020 3235 2c20 4c65 784c          25, LexL
│ │ │ │ -000166b0: 6172 6765 7374 203d 3e20 302c 204c 6578  argest => 0, Lex
│ │ │ │ -000166c0: 536d 616c 6c65 7374 203d 3e20 3235 2c20  Smallest => 25, 
│ │ │ │ -000166d0: 4752 6576 4c65 7853 6d61 6c6c 6573 7454  GRevLexSmallestT
│ │ │ │ -000166e0: 6572 6d20 3d3e 2032 352c 0a20 2020 2020  erm => 25,.     
│ │ │ │ -000166f0: 2020 2052 616e 646f 6d4e 6f6e 7a65 726f     RandomNonzero
│ │ │ │ -00016700: 203d 3e20 302c 2047 5265 764c 6578 536d   => 0, GRevLexSm
│ │ │ │ -00016710: 616c 6c65 7374 203d 3e20 3235 7d2c 2073  allest => 25}, s
│ │ │ │ -00016720: 7472 6174 6567 6965 7320 666f 7220 6368  trategies for ch
│ │ │ │ -00016730: 6f6f 7369 6e67 0a20 2020 2020 2020 2073  oosing.        s
│ │ │ │ -00016740: 7562 6d61 7472 6963 6573 0a20 2020 2020  ubmatrices.     
│ │ │ │ -00016750: 202a 202a 6e6f 7465 2054 6872 6561 6473   * *note Threads
│ │ │ │ -00016760: 3a20 6973 5261 6e6b 4174 4c65 6173 745f  : isRankAtLeast_
│ │ │ │ -00016770: 6c70 5f70 645f 7064 5f70 645f 636d 5468  lp_pd_pd_pd_cmTh
│ │ │ │ -00016780: 7265 6164 733d 3e5f 7064 5f70 645f 7064  reads=>_pd_pd_pd
│ │ │ │ -00016790: 5f72 702c 203d 3e0a 2020 2020 2020 2020  _rp, =>.        
│ │ │ │ -000167a0: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
│ │ │ │ -000167b0: 7565 2031 2c20 616e 206f 7074 696f 6e20  ue 1, an option 
│ │ │ │ -000167c0: 666f 7220 7661 7269 6f75 7320 6675 6e63  for various func
│ │ │ │ -000167d0: 7469 6f6e 730a 2020 2020 2020 2a20 5665  tions.      * Ve
│ │ │ │ -000167e0: 7262 6f73 6520 3d3e 202e 2e2e 2c20 6465  rbose => ..., de
│ │ │ │ -000167f0: 6661 756c 7420 7661 6c75 6520 6661 6c73  fault value fals
│ │ │ │ -00016800: 650a 2020 2a20 4f75 7470 7574 733a 0a20  e.  * Outputs:. 
│ │ │ │ -00016810: 2020 2020 202a 2061 202a 6e6f 7465 206c       * a *note l
│ │ │ │ -00016820: 6973 743a 2028 4d61 6361 756c 6179 3244  ist: (Macaulay2D
│ │ │ │ -00016830: 6f63 294c 6973 742c 2c20 7468 6520 6669  oc)List,, the fi
│ │ │ │ -00016840: 7273 7420 656e 7472 7920 6973 2061 206c  rst entry is a l
│ │ │ │ -00016850: 6973 7420 6f66 2072 6f77 0a20 2020 2020  ist of row.     
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│ │ │ │ -00016870: 7365 636f 6e64 2069 7320 6120 6c69 7374  second is a list
│ │ │ │ -00016880: 206f 6620 636f 6c75 6d6e 2069 6e64 6963   of column indic
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│ │ │ │ -000168a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869  ===========..Thi
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│ │ │ │ -000168e0: 7269 782c 2061 6e64 2074 7269 6573 2074  rix, and tries t
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│ │ │ │ -00016920: 732c 2069 7420 7265 7475 726e 7320 6120  s, it returns a 
│ │ │ │ -00016930: 6c69 7374 206f 6620 7477 6f20 6c69 7374  list of two list
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│ │ │ │ -00016970: 6f6e 6420 6973 2074 6865 206c 6973 7420  ond is the list 
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│ │ │ │ -000169a0: 7375 626d 6174 7269 782e 2049 6620 6974  submatrix. If it
│ │ │ │ -000169b0: 2066 6169 6c73 2074 6f20 6669 6e64 2073   fails to find s
│ │ │ │ -000169c0: 7563 6820 6120 6d61 7472 6978 2c20 7468  uch a matrix, th
│ │ │ │ -000169d0: 6520 6675 6e63 7469 6f6e 2072 6574 7572  e function retur
│ │ │ │ -000169e0: 6e73 0a6e 756c 6c2e 2020 5468 6520 6f70  ns.null.  The op
│ │ │ │ -000169f0: 7469 6f6e 204d 6178 4d69 6e6f 7273 2069  tion MaxMinors i
│ │ │ │ -00016a00: 7320 7573 6564 2074 6f20 636f 6e74 726f  s used to contro
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│ │ │ │ -00016a20: 7320 746f 2063 6f6e 7369 6465 722e 2020  s to consider.  
│ │ │ │ -00016a30: 4966 0a6c 6566 7420 6e75 6c6c 2c20 7468  If.left null, th
│ │ │ │ -00016a40: 6520 6e75 6d62 6572 2063 6f6e 7369 6465  e number conside
│ │ │ │ -00016a50: 7265 6420 6973 2062 6173 6564 206f 6e20  red is based on 
│ │ │ │ -00016a60: 7468 6520 7369 7a65 206f 6620 7468 6520  the size of the 
│ │ │ │ -00016a70: 6d61 7472 6978 2e0a 0a2b 2d2d 2d2d 2d2d  matrix...+------
│ │ │ │ -00016a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016ab0: 2d2d 2d2d 2d2b 0a7c 6931 203a 2052 203d  -----+.|i1 : R =
│ │ │ │ -00016ac0: 2051 515b 782c 795d 3b20 2020 2020 2020   QQ[x,y];       
│ │ │ │ -00016ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016af0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -00016b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016b30: 2d2b 0a7c 6932 203a 204d 203d 206d 6174  -+.|i2 : M = mat
│ │ │ │ -00016b40: 7269 787b 7b78 2c79 2c32 2c30 2c32 2a78  rix{{x,y,2,0,2*x
│ │ │ │ -00016b50: 2b79 7d2c 207b 302c 302c 312c 302c 787d  +y}, {0,0,1,0,x}
│ │ │ │ -00016b60: 2c20 7b78 2c79 2c30 2c30 2c79 7d7d 3b7c  , {x,y,0,0,y}};|
│ │ │ │ -00016b70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00016b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ba0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00016bb0: 2020 2020 2020 2020 2020 2020 2033 2020               3  
│ │ │ │ -00016bc0: 2020 2020 3520 2020 2020 2020 2020 2020      5           
│ │ │ │ +00015c50: 2020 2020 207c 0a7c 6f35 3520 3d20 7472       |.|o55 = tr
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│ │ │ │ +00015c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015ca0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +00015cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015cf0: 2d2d 2d2d 2d7c 0a7c 5374 7261 7465 6779  -----|.|Strategy
│ │ │ │ +00015d00: 3d3e 5374 7261 7465 6779 4465 6661 756c  =>StrategyDefaul
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│ │ │ │ +00015d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015d40: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00015d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015d90: 2d2d 2d2d 2d2b 0a0a 4966 2072 6567 756c  -----+..If regul
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│ │ │ │ +00015db0: 6f75 7470 7574 7320 6e6f 7468 696e 672c  outputs nothing,
│ │ │ │ +00015dc0: 2074 6865 6e20 6974 2063 6f75 6c64 6e27   then it couldn'
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│ │ │ │ +00015de0: 6520 7269 6e67 0a77 6173 2072 6567 756c  e ring.was regul
│ │ │ │ +00015df0: 6172 2069 6e20 7468 6174 2063 6f64 696d  ar in that codim
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│ │ │ │ +00015e10: 4d61 784d 696e 6f72 7320 3d3e 2031 3030  MaxMinors => 100
│ │ │ │ +00015e20: 2074 6f20 6b65 6570 2069 7420 6672 6f6d   to keep it from
│ │ │ │ +00015e30: 0a72 756e 6e69 6e67 2074 6f6f 206c 6f6e  .running too lon
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│ │ │ │ +00015e60: 2041 6761 696e 2c20 6576 656e 2074 686f   Again, even tho
│ │ │ │ +00015e70: 7567 680a 4752 6576 4c65 7853 6d61 6c6c  ugh.GRevLexSmall
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│ │ │ │ +00015e90: 6d61 6c6c 6573 7454 6572 6d20 6172 6520  mallestTerm are 
│ │ │ │ +00015ea0: 6e6f 7420 6566 6665 6374 6976 6520 696e  not effective in
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│ │ │ │ +00015ec0: 0a65 7861 6d70 6c65 2c20 696e 206f 7468  .example, in oth
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│ │ │ │ +00015f00: 4e6f 7465 2073 696d 696c 6172 0a63 6f6e  Note similar.con
│ │ │ │ +00015f10: 7369 6465 7261 7469 6f6e 7320 616c 736f  siderations also
│ │ │ │ +00015f20: 2061 7070 6c79 2074 6f20 2a6e 6f74 6520   apply to *note 
│ │ │ │ +00015f30: 7072 6f6a 4469 6d3a 2070 726f 6a44 696d  projDim: projDim
│ │ │ │ +00015f40: 2c2e 0a0a 5365 6520 616c 736f 0a3d 3d3d  ,...See also.===
│ │ │ │ +00015f50: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ +00015f60: 2063 686f 6f73 6547 6f6f 644d 696e 6f72   chooseGoodMinor
│ │ │ │ +00015f70: 7328 2e2e 2e2c 5374 7261 7465 6779 3d3e  s(...,Strategy=>
│ │ │ │ +00015f80: 2e2e 2e29 3a20 5374 7261 7465 6779 4465  ...): StrategyDe
│ │ │ │ +00015f90: 6661 756c 742c 202d 2d20 7374 7261 7465  fault, -- strate
│ │ │ │ +00015fa0: 6769 6573 0a20 2020 2066 6f72 2063 686f  gies.    for cho
│ │ │ │ +00015fb0: 6f73 696e 6720 7375 626d 6174 7269 6365  osing submatrice
│ │ │ │ +00015fc0: 730a 2020 2a20 2a6e 6f74 6520 5265 6775  s.  * *note Regu
│ │ │ │ +00015fd0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ +00015fe0: 5475 746f 7269 616c 3a20 5265 6775 6c61  Tutorial: Regula
│ │ │ │ +00015ff0: 7249 6e43 6f64 696d 656e 7369 6f6e 5475  rInCodimensionTu
│ │ │ │ +00016000: 746f 7269 616c 2c20 2d2d 2041 0a20 2020  torial, -- A.   
│ │ │ │ +00016010: 2074 7574 6f72 6961 6c20 666f 7220 686f   tutorial for ho
│ │ │ │ +00016020: 7720 746f 2075 7365 2074 6865 2061 6476  w to use the adv
│ │ │ │ +00016030: 616e 6365 6420 6f70 7469 6f6e 7320 6f66  anced options of
│ │ │ │ +00016040: 2074 6865 2072 6567 756c 6172 496e 436f   the regularInCo
│ │ │ │ +00016050: 6469 6d65 6e73 696f 6e0a 2020 2020 6675  dimension.    fu
│ │ │ │ +00016060: 6e63 7469 6f6e 0a0a 466f 7220 7468 6520  nction..For the 
│ │ │ │ +00016070: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
│ │ │ │ +00016080: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
│ │ │ │ +00016090: 6865 206f 626a 6563 7420 2a6e 6f74 6520  he object *note 
│ │ │ │ +000160a0: 4661 7374 4d69 6e6f 7273 5374 7261 7465  FastMinorsStrate
│ │ │ │ +000160b0: 6779 5475 746f 7269 616c 3a20 4661 7374  gyTutorial: Fast
│ │ │ │ +000160c0: 4d69 6e6f 7273 5374 7261 7465 6779 5475  MinorsStrategyTu
│ │ │ │ +000160d0: 746f 7269 616c 2c20 6973 2061 0a2a 6e6f  torial, is a.*no
│ │ │ │ +000160e0: 7465 2073 796d 626f 6c3a 2028 4d61 6361  te symbol: (Maca
│ │ │ │ +000160f0: 756c 6179 3244 6f63 2953 796d 626f 6c2c  ulay2Doc)Symbol,
│ │ │ │ +00016100: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...-------------
│ │ │ │ +00016110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016150: 2d2d 0a0a 5468 6520 736f 7572 6365 206f  --..The source o
│ │ │ │ +00016160: 6620 7468 6973 2064 6f63 756d 656e 7420  f this document 
│ │ │ │ +00016170: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570  is in./build/rep
│ │ │ │ +00016180: 726f 6475 6369 626c 652d 7061 7468 2f6d  roducible-path/m
│ │ │ │ +00016190: 6163 6175 6c61 7932 2d31 2e32 352e 3131  acaulay2-1.25.11
│ │ │ │ +000161a0: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932  +ds/M2/Macaulay2
│ │ │ │ +000161b0: 2f70 6163 6b61 6765 732f 4661 7374 4d69  /packages/FastMi
│ │ │ │ +000161c0: 6e6f 7273 2e0a 6d32 3a31 3532 353a 302e  nors..m2:1525:0.
│ │ │ │ +000161d0: 0a1f 0a46 696c 653a 2046 6173 744d 696e  ...File: FastMin
│ │ │ │ +000161e0: 6f72 732e 696e 666f 2c20 4e6f 6465 3a20  ors.info, Node: 
│ │ │ │ +000161f0: 6765 7453 7562 6d61 7472 6978 4f66 5261  getSubmatrixOfRa
│ │ │ │ +00016200: 6e6b 2c20 4e65 7874 3a20 6973 436f 6469  nk, Next: isCodi
│ │ │ │ +00016210: 6d41 744c 6561 7374 2c20 5072 6576 3a20  mAtLeast, Prev: 
│ │ │ │ +00016220: 4661 7374 4d69 6e6f 7273 5374 7261 7465  FastMinorsStrate
│ │ │ │ +00016230: 6779 5475 746f 7269 616c 2c20 5570 3a20  gyTutorial, Up: 
│ │ │ │ +00016240: 546f 700a 0a67 6574 5375 626d 6174 7269  Top..getSubmatri
│ │ │ │ +00016250: 784f 6652 616e 6b20 2d2d 2074 7269 6573  xOfRank -- tries
│ │ │ │ +00016260: 2074 6f20 6669 6e64 2061 2073 7562 6d61   to find a subma
│ │ │ │ +00016270: 7472 6978 206f 6620 7468 6520 6769 7665  trix of the give
│ │ │ │ +00016280: 6e20 7261 6e6b 0a2a 2a2a 2a2a 2a2a 2a2a  n rank.*********
│ │ │ │ +00016290: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000162a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000162b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000162c0: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573  ********..  * Us
│ │ │ │ +000162d0: 6167 653a 200a 2020 2020 2020 2020 6765  age: .        ge
│ │ │ │ +000162e0: 7453 7562 6d61 7472 6978 4f66 5261 6e6b  tSubmatrixOfRank
│ │ │ │ +000162f0: 286e 312c 204d 3129 0a20 202a 2049 6e70  (n1, M1).  * Inp
│ │ │ │ +00016300: 7574 733a 0a20 2020 2020 202a 206e 312c  uts:.      * n1,
│ │ │ │ +00016310: 2061 6e20 2a6e 6f74 6520 696e 7465 6765   an *note intege
│ │ │ │ +00016320: 723a 2028 4d61 6361 756c 6179 3244 6f63  r: (Macaulay2Doc
│ │ │ │ +00016330: 295a 5a2c 2c20 0a20 2020 2020 202a 204d  )ZZ,, .      * M
│ │ │ │ +00016340: 312c 2061 202a 6e6f 7465 206d 6174 7269  1, a *note matri
│ │ │ │ +00016350: 783a 2028 4d61 6361 756c 6179 3244 6f63  x: (Macaulay2Doc
│ │ │ │ +00016360: 294d 6174 7269 782c 2c20 0a20 202a 202a  )Matrix,, .  * *
│ │ │ │ +00016370: 6e6f 7465 204f 7074 696f 6e61 6c20 696e  note Optional in
│ │ │ │ +00016380: 7075 7473 3a20 284d 6163 6175 6c61 7932  puts: (Macaulay2
│ │ │ │ +00016390: 446f 6329 7573 696e 6720 6675 6e63 7469  Doc)using functi
│ │ │ │ +000163a0: 6f6e 7320 7769 7468 206f 7074 696f 6e61  ons with optiona
│ │ │ │ +000163b0: 6c20 696e 7075 7473 2c3a 0a20 2020 2020  l inputs,:.     
│ │ │ │ +000163c0: 202a 202a 6e6f 7465 2044 6574 5374 7261   * *note DetStra
│ │ │ │ +000163d0: 7465 6779 3a20 4465 7453 7472 6174 6567  tegy: DetStrateg
│ │ │ │ +000163e0: 792c 203d 3e20 2e2e 2e2c 2064 6566 6175  y, => ..., defau
│ │ │ │ +000163f0: 6c74 2076 616c 7565 2052 616e 6b2c 2044  lt value Rank, D
│ │ │ │ +00016400: 6574 5374 7261 7465 6779 0a20 2020 2020  etStrategy.     
│ │ │ │ +00016410: 2020 2069 7320 6120 7374 7261 7465 6779     is a strategy
│ │ │ │ +00016420: 2066 6f72 2061 6c6c 6f77 696e 6720 7468   for allowing th
│ │ │ │ +00016430: 6520 7573 6572 2074 6f20 6368 6f6f 7365  e user to choose
│ │ │ │ +00016440: 2068 6f77 2064 6574 6572 6d69 6e61 6e74   how determinant
│ │ │ │ +00016450: 7320 286f 720a 2020 2020 2020 2020 7261  s (or.        ra
│ │ │ │ +00016460: 6e6b 292c 2069 7320 636f 6d70 7574 6564  nk), is computed
│ │ │ │ +00016470: 0a20 2020 2020 202a 202a 6e6f 7465 204d  .      * *note M
│ │ │ │ +00016480: 6178 4d69 6e6f 7273 3a20 4d61 784d 696e  axMinors: MaxMin
│ │ │ │ +00016490: 6f72 732c 203d 3e20 2e2e 2e2c 2064 6566  ors, => ..., def
│ │ │ │ +000164a0: 6175 6c74 2076 616c 7565 206e 756c 6c2c  ault value null,
│ │ │ │ +000164b0: 2061 6e20 6f70 7469 6f6e 2074 6f0a 2020   an option to.  
│ │ │ │ +000164c0: 2020 2020 2020 636f 6e74 726f 6c20 6465        control de
│ │ │ │ +000164d0: 7074 6820 6f66 2073 6561 7263 680a 2020  pth of search.  
│ │ │ │ +000164e0: 2020 2020 2a20 2a6e 6f74 6520 506f 696e      * *note Poin
│ │ │ │ +000164f0: 744f 7074 696f 6e73 3a20 506f 696e 744f  tOptions: PointO
│ │ │ │ +00016500: 7074 696f 6e73 2c20 3d3e 202e 2e2e 2c20  ptions, => ..., 
│ │ │ │ +00016510: 6465 6661 756c 7420 7661 6c75 6520 7b53  default value {S
│ │ │ │ +00016520: 7472 6174 6567 7920 3d3e 0a20 2020 2020  trategy =>.     
│ │ │ │ +00016530: 2020 2044 6566 6175 6c74 2c20 486f 6d6f     Default, Homo
│ │ │ │ +00016540: 6765 6e65 6f75 7320 3d3e 2066 616c 7365  geneous => false
│ │ │ │ +00016550: 2c20 5265 706c 6163 656d 656e 7420 3d3e  , Replacement =>
│ │ │ │ +00016560: 2042 696e 6f6d 6961 6c2c 2045 7874 656e   Binomial, Exten
│ │ │ │ +00016570: 6446 6965 6c64 203d 3e0a 2020 2020 2020  dField =>.      
│ │ │ │ +00016580: 2020 7472 7565 2c20 506f 696e 7443 6865    true, PointChe
│ │ │ │ +00016590: 636b 4174 7465 6d70 7473 203d 3e20 302c  ckAttempts => 0,
│ │ │ │ +000165a0: 2044 6563 6f6d 706f 7369 7469 6f6e 5374   DecompositionSt
│ │ │ │ +000165b0: 7261 7465 6779 203d 3e20 4465 636f 6d70  rategy => Decomp
│ │ │ │ +000165c0: 6f73 652c 0a20 2020 2020 2020 204e 756d  ose,.        Num
│ │ │ │ +000165d0: 5468 7265 6164 7354 6f55 7365 203d 3e20  ThreadsToUse => 
│ │ │ │ +000165e0: 312c 2044 696d 656e 7369 6f6e 4675 6e63  1, DimensionFunc
│ │ │ │ +000165f0: 7469 6f6e 203d 3e20 6469 6d2c 2056 6572  tion => dim, Ver
│ │ │ │ +00016600: 626f 7365 203d 3e20 6661 6c73 657d 2c0a  bose => false},.
│ │ │ │ +00016610: 2020 2020 2020 2020 6f70 7469 6f6e 7320          options 
│ │ │ │ +00016620: 746f 2070 6173 7320 746f 2066 756e 6374  to pass to funct
│ │ │ │ +00016630: 696f 6e73 2069 6e20 7468 6520 7061 636b  ions in the pack
│ │ │ │ +00016640: 6167 6520 5261 6e64 6f6d 506f 696e 7473  age RandomPoints
│ │ │ │ +00016650: 0a20 2020 2020 202a 202a 6e6f 7465 2053  .      * *note S
│ │ │ │ +00016660: 7472 6174 6567 793a 2053 7472 6174 6567  trategy: Strateg
│ │ │ │ +00016670: 7944 6566 6175 6c74 2c20 3d3e 202e 2e2e  yDefault, => ...
│ │ │ │ +00016680: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ +00016690: 6e65 7720 4f70 7469 6f6e 5461 626c 650a  new OptionTable.
│ │ │ │ +000166a0: 2020 2020 2020 2020 6672 6f6d 207b 506f          from {Po
│ │ │ │ +000166b0: 696e 7473 203d 3e20 302c 2052 616e 646f  ints => 0, Rando
│ │ │ │ +000166c0: 6d20 3d3e 2030 2c20 4752 6576 4c65 784c  m => 0, GRevLexL
│ │ │ │ +000166d0: 6172 6765 7374 203d 3e20 302c 204c 6578  argest => 0, Lex
│ │ │ │ +000166e0: 536d 616c 6c65 7374 5465 726d 203d 3e0a  SmallestTerm =>.
│ │ │ │ +000166f0: 2020 2020 2020 2020 3235 2c20 4c65 784c          25, LexL
│ │ │ │ +00016700: 6172 6765 7374 203d 3e20 302c 204c 6578  argest => 0, Lex
│ │ │ │ +00016710: 536d 616c 6c65 7374 203d 3e20 3235 2c20  Smallest => 25, 
│ │ │ │ +00016720: 4752 6576 4c65 7853 6d61 6c6c 6573 7454  GRevLexSmallestT
│ │ │ │ +00016730: 6572 6d20 3d3e 2032 352c 0a20 2020 2020  erm => 25,.     
│ │ │ │ +00016740: 2020 2052 616e 646f 6d4e 6f6e 7a65 726f     RandomNonzero
│ │ │ │ +00016750: 203d 3e20 302c 2047 5265 764c 6578 536d   => 0, GRevLexSm
│ │ │ │ +00016760: 616c 6c65 7374 203d 3e20 3235 7d2c 2073  allest => 25}, s
│ │ │ │ +00016770: 7472 6174 6567 6965 7320 666f 7220 6368  trategies for ch
│ │ │ │ +00016780: 6f6f 7369 6e67 0a20 2020 2020 2020 2073  oosing.        s
│ │ │ │ +00016790: 7562 6d61 7472 6963 6573 0a20 2020 2020  ubmatrices.     
│ │ │ │ +000167a0: 202a 202a 6e6f 7465 2054 6872 6561 6473   * *note Threads
│ │ │ │ +000167b0: 3a20 6973 5261 6e6b 4174 4c65 6173 745f  : isRankAtLeast_
│ │ │ │ +000167c0: 6c70 5f70 645f 7064 5f70 645f 636d 5468  lp_pd_pd_pd_cmTh
│ │ │ │ +000167d0: 7265 6164 733d 3e5f 7064 5f70 645f 7064  reads=>_pd_pd_pd
│ │ │ │ +000167e0: 5f72 702c 203d 3e0a 2020 2020 2020 2020  _rp, =>.        
│ │ │ │ +000167f0: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
│ │ │ │ +00016800: 7565 2031 2c20 616e 206f 7074 696f 6e20  ue 1, an option 
│ │ │ │ +00016810: 666f 7220 7661 7269 6f75 7320 6675 6e63  for various func
│ │ │ │ +00016820: 7469 6f6e 730a 2020 2020 2020 2a20 5665  tions.      * Ve
│ │ │ │ +00016830: 7262 6f73 6520 3d3e 202e 2e2e 2c20 6465  rbose => ..., de
│ │ │ │ +00016840: 6661 756c 7420 7661 6c75 6520 6661 6c73  fault value fals
│ │ │ │ +00016850: 650a 2020 2a20 4f75 7470 7574 733a 0a20  e.  * Outputs:. 
│ │ │ │ +00016860: 2020 2020 202a 2061 202a 6e6f 7465 206c       * a *note l
│ │ │ │ +00016870: 6973 743a 2028 4d61 6361 756c 6179 3244  ist: (Macaulay2D
│ │ │ │ +00016880: 6f63 294c 6973 742c 2c20 7468 6520 6669  oc)List,, the fi
│ │ │ │ +00016890: 7273 7420 656e 7472 7920 6973 2061 206c  rst entry is a l
│ │ │ │ +000168a0: 6973 7420 6f66 2072 6f77 0a20 2020 2020  ist of row.     
│ │ │ │ +000168b0: 2020 2069 6e64 6963 6573 2c20 7468 6520     indices, the 
│ │ │ │ +000168c0: 7365 636f 6e64 2069 7320 6120 6c69 7374  second is a list
│ │ │ │ +000168d0: 206f 6620 636f 6c75 6d6e 2069 6e64 6963   of column indic
│ │ │ │ +000168e0: 6573 0a0a 4465 7363 7269 7074 696f 6e0a  es..Description.
│ │ │ │ +000168f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869  ===========..Thi
│ │ │ │ +00016900: 7320 6675 6e63 7469 6f6e 206c 6f6f 6b73  s function looks
│ │ │ │ +00016910: 2061 7420 7375 626d 6174 7269 6365 7320   at submatrices 
│ │ │ │ +00016920: 6f66 2074 6865 2067 6976 656e 206d 6174  of the given mat
│ │ │ │ +00016930: 7269 782c 2061 6e64 2074 7269 6573 2074  rix, and tries t
│ │ │ │ +00016940: 6f20 6669 6e64 206f 6e65 0a6f 6620 7468  o find one.of th
│ │ │ │ +00016950: 6520 7370 6563 6966 6965 6420 7261 6e6b  e specified rank
│ │ │ │ +00016960: 2e20 2049 6620 6974 2073 7563 6365 6564  .  If it succeed
│ │ │ │ +00016970: 732c 2069 7420 7265 7475 726e 7320 6120  s, it returns a 
│ │ │ │ +00016980: 6c69 7374 206f 6620 7477 6f20 6c69 7374  list of two list
│ │ │ │ +00016990: 732e 2054 6865 0a66 6972 7374 2069 7320  s. The.first is 
│ │ │ │ +000169a0: 7468 6520 6c69 7374 206f 6620 726f 7720  the list of row 
│ │ │ │ +000169b0: 696e 6469 6365 732c 2074 6865 2073 6563  indices, the sec
│ │ │ │ +000169c0: 6f6e 6420 6973 2074 6865 206c 6973 7420  ond is the list 
│ │ │ │ +000169d0: 6f66 2063 6f6c 756d 6e73 2c20 6f66 2074  of columns, of t
│ │ │ │ +000169e0: 6865 0a64 6573 6972 6564 2072 616e 6b20  he.desired rank 
│ │ │ │ +000169f0: 7375 626d 6174 7269 782e 2049 6620 6974  submatrix. If it
│ │ │ │ +00016a00: 2066 6169 6c73 2074 6f20 6669 6e64 2073   fails to find s
│ │ │ │ +00016a10: 7563 6820 6120 6d61 7472 6978 2c20 7468  uch a matrix, th
│ │ │ │ +00016a20: 6520 6675 6e63 7469 6f6e 2072 6574 7572  e function retur
│ │ │ │ +00016a30: 6e73 0a6e 756c 6c2e 2020 5468 6520 6f70  ns.null.  The op
│ │ │ │ +00016a40: 7469 6f6e 204d 6178 4d69 6e6f 7273 2069  tion MaxMinors i
│ │ │ │ +00016a50: 7320 7573 6564 2074 6f20 636f 6e74 726f  s used to contro
│ │ │ │ +00016a60: 6c20 686f 7720 6d61 6e79 206d 696e 6f72  l how many minor
│ │ │ │ +00016a70: 7320 746f 2063 6f6e 7369 6465 722e 2020  s to consider.  
│ │ │ │ +00016a80: 4966 0a6c 6566 7420 6e75 6c6c 2c20 7468  If.left null, th
│ │ │ │ +00016a90: 6520 6e75 6d62 6572 2063 6f6e 7369 6465  e number conside
│ │ │ │ +00016aa0: 7265 6420 6973 2062 6173 6564 206f 6e20  red is based on 
│ │ │ │ +00016ab0: 7468 6520 7369 7a65 206f 6620 7468 6520  the size of the 
│ │ │ │ +00016ac0: 6d61 7472 6978 2e0a 0a2b 2d2d 2d2d 2d2d  matrix...+------
│ │ │ │ +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 2d2b 0a7c 6931 203a 2052 203d  -----+.|i1 : R =
│ │ │ │ +00016b10: 2051 515b 782c 795d 3b20 2020 2020 2020   QQ[x,y];       
│ │ │ │ +00016b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016b40: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00016b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016b80: 2d2b 0a7c 6932 203a 204d 203d 206d 6174  -+.|i2 : M = mat
│ │ │ │ +00016b90: 7269 787b 7b78 2c79 2c32 2c30 2c32 2a78  rix{{x,y,2,0,2*x
│ │ │ │ +00016ba0: 2b79 7d2c 207b 302c 302c 312c 302c 787d  +y}, {0,0,1,0,x}
│ │ │ │ +00016bb0: 2c20 7b78 2c79 2c30 2c30 2c79 7d7d 3b7c  , {x,y,0,0,y}};|
│ │ │ │ +00016bc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00016bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016be0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -00016bf0: 203a 204d 6174 7269 7820 5220 203c 2d2d   : Matrix R  <--
│ │ │ │ -00016c00: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ -00016c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016c20: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -00016c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016c60: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 206c  -------+.|i3 : l
│ │ │ │ -00016c70: 203d 2067 6574 5375 626d 6174 7269 784f   = getSubmatrixO
│ │ │ │ -00016c80: 6652 616e 6b28 322c 204d 2920 2020 2020  fRank(2, M)     
│ │ │ │ -00016c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ca0: 2020 2020 207c 0a7c 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 207c 0a7c 6f33 203d 207b 7b31 2c20     |.|o3 = {{1, 
│ │ │ │ -00016cf0: 327d 2c20 7b32 2c20 317d 7d20 2020 2020  2}, {2, 1}}     
│ │ │ │ +00016be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016bf0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00016c00: 2020 2020 2020 2020 2020 2020 2033 2020               3  
│ │ │ │ +00016c10: 2020 2020 3520 2020 2020 2020 2020 2020      5           
│ │ │ │ +00016c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016c30: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00016c40: 203a 204d 6174 7269 7820 5220 203c 2d2d   : Matrix R  <--
│ │ │ │ +00016c50: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ +00016c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016c70: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00016c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016cb0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 206c  -------+.|i3 : l
│ │ │ │ +00016cc0: 203d 2067 6574 5375 626d 6174 7269 784f   = getSubmatrixO
│ │ │ │ +00016cd0: 6652 616e 6b28 322c 204d 2920 2020 2020  fRank(2, M)     
│ │ │ │ +00016ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016cf0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00016d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016d20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00016d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016d50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016d60: 0a7c 6f33 203a 204c 6973 7420 2020 2020  .|o3 : List     
│ │ │ │ -00016d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016d30: 2020 207c 0a7c 6f33 203d 207b 7b31 2c20     |.|o3 = {{1, 
│ │ │ │ +00016d40: 327d 2c20 7b32 2c20 317d 7d20 2020 2020  2}, {2, 1}}     
│ │ │ │ +00016d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016d70: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00016d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016d90: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00016da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ -00016de0: 203a 2028 4d5e 286c 2330 2929 5f28 6c23   : (M^(l#0))_(l#
│ │ │ │ -00016df0: 3129 2020 2020 2020 2020 2020 2020 2020  1)              
│ │ │ │ -00016e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e10: 2020 2020 2020 2020 207c 0a7c 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 207c 0a7c 6f34 203d 207c         |.|o4 = |
│ │ │ │ -00016e60: 2031 2030 207c 2020 2020 2020 2020 2020   1 0 |          
│ │ │ │ +00016d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016da0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00016db0: 0a7c 6f33 203a 204c 6973 7420 2020 2020  .|o3 : List     
│ │ │ │ +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 207c 0a2b               |.+
│ │ │ │ +00016df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ +00016e30: 203a 2028 4d5e 286c 2330 2929 5f28 6c23   : (M^(l#0))_(l#
│ │ │ │ +00016e40: 3129 2020 2020 2020 2020 2020 2020 2020  1)              
│ │ │ │ +00016e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016e60: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00016e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e90: 2020 2020 207c 0a7c 2020 2020 207c 2030       |.|     | 0
│ │ │ │ -00016ea0: 2079 207c 2020 2020 2020 2020 2020 2020   y |            
│ │ │ │ -00016eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016ea0: 2020 2020 2020 207c 0a7c 6f34 203d 207c         |.|o4 = |
│ │ │ │ +00016eb0: 2031 2030 207c 2020 2020 2020 2020 2020   1 0 |          
│ │ │ │  00016ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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                  
│ │ │ │ +00016ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016ee0: 2020 2020 207c 0a7c 2020 2020 207c 2030       |.|     | 0
│ │ │ │ +00016ef0: 2079 207c 2020 2020 2020 2020 2020 2020   y |            
│ │ │ │  00016f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016f10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00016f20: 2032 2020 2020 2020 3220 2020 2020 2020   2      2       
│ │ │ │ +00016f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016f20: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00016f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016f40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016f50: 0a7c 6f34 203a 204d 6174 7269 7820 5220  .|o4 : Matrix R 
│ │ │ │ -00016f60: 203c 2d2d 2052 2020 2020 2020 2020 2020   <-- R          
│ │ │ │ -00016f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016f80: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00016f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935  -----------+.|i5
│ │ │ │ -00016fd0: 203a 206c 203d 2067 6574 5375 626d 6174   : l = getSubmat
│ │ │ │ -00016fe0: 7269 784f 6652 616e 6b28 322c 204d 2920  rixOfRank(2, M) 
│ │ │ │ -00016ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017000: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00017010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017040: 2020 2020 2020 207c 0a7c 6f35 203d 207b         |.|o5 = {
│ │ │ │ -00017050: 7b31 2c20 307d 2c20 7b32 2c20 317d 7d20  {1, 0}, {2, 1}} 
│ │ │ │ +00016f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016f60: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00016f70: 2032 2020 2020 2020 3220 2020 2020 2020   2      2       
│ │ │ │ +00016f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016f90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00016fa0: 0a7c 6f34 203a 204d 6174 7269 7820 5220  .|o4 : Matrix R 
│ │ │ │ +00016fb0: 203c 2d2d 2052 2020 2020 2020 2020 2020   <-- R          
│ │ │ │ +00016fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016fd0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00016fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935  -----------+.|i5
│ │ │ │ +00017020: 203a 206c 203d 2067 6574 5375 626d 6174   : l = getSubmat
│ │ │ │ +00017030: 7269 784f 6652 616e 6b28 322c 204d 2920  rixOfRank(2, M) 
│ │ │ │ +00017040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017050: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00017060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017080: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00017090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000170a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017090: 2020 2020 2020 207c 0a7c 6f35 203d 207b         |.|o5 = {
│ │ │ │ +000170a0: 7b31 2c20 307d 2c20 7b32 2c20 317d 7d20  {1, 0}, {2, 1}} 
│ │ │ │  000170b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000170c0: 2020 207c 0a7c 6f35 203a 204c 6973 7420     |.|o5 : List 
│ │ │ │ -000170d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000170c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000170d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000170e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000170f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017100: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -00017110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00017140: 0a7c 6936 203a 2028 4d5e 286c 2330 2929  .|i6 : (M^(l#0))
│ │ │ │ -00017150: 5f28 6c23 3129 2020 2020 2020 2020 2020  _(l#1)          
│ │ │ │ -00017160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017170: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00017180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000171a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000171b0: 2020 2020 2020 2020 2020 207c 0a7c 6f36             |.|o6
│ │ │ │ -000171c0: 203d 207c 2031 2030 207c 2020 2020 2020   = | 1 0 |      
│ │ │ │ +00017100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017110: 2020 207c 0a7c 6f35 203a 204c 6973 7420     |.|o5 : List 
│ │ │ │ +00017120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017150: 207c 0a2b 2d2d 2d2d 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 2d2b  ---------------+
│ │ │ │ +00017190: 0a7c 6936 203a 2028 4d5e 286c 2330 2929  .|i6 : (M^(l#0))
│ │ │ │ +000171a0: 5f28 6c23 3129 2020 2020 2020 2020 2020  _(l#1)          
│ │ │ │ +000171b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000171c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000171d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000171e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000171f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00017200: 207c 2032 2079 207c 2020 2020 2020 2020   | 2 y |        
│ │ │ │ -00017210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000171f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017200: 2020 2020 2020 2020 2020 207c 0a7c 6f36             |.|o6
│ │ │ │ +00017210: 203d 207c 2031 2030 207c 2020 2020 2020   = | 1 0 |      
│ │ │ │  00017220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017230: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00017240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017240: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00017250: 207c 2032 2079 207c 2020 2020 2020 2020   | 2 y |        
│ │ │ │  00017260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017270: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00017280: 2020 2020 2032 2020 2020 2020 3220 2020       2      2   
│ │ │ │ +00017270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017280: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00017290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000172a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000172b0: 2020 207c 0a7c 6f36 203a 204d 6174 7269     |.|o6 : Matri
│ │ │ │ -000172c0: 7820 5220 203c 2d2d 2052 2020 2020 2020  x R  <-- R      
│ │ │ │ -000172d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000172b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000172c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000172d0: 2020 2020 2032 2020 2020 2020 3220 2020       2      2   
│ │ │ │  000172e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000172f0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -00017300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00017330: 0a7c 6937 203a 2067 6574 5375 626d 6174  .|i7 : getSubmat
│ │ │ │ -00017340: 7269 784f 6652 616e 6b28 332c 204d 2920  rixOfRank(3, M) 
│ │ │ │ -00017350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -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 2d2b 0a0a 5468  -----------+..Th
│ │ │ │ -000173b0: 6520 6f70 7469 6f6e 2053 7472 6174 6567  e option Strateg
│ │ │ │ -000173c0: 7920 6973 2075 7365 6420 746f 2075 7365  y is used to use
│ │ │ │ -000173d0: 6420 746f 2063 6f6e 7472 6f6c 2068 6f77  d to control how
│ │ │ │ -000173e0: 2074 6865 2066 756e 6374 696f 6e20 636f   the function co
│ │ │ │ -000173f0: 6d70 7574 6573 2074 6865 0a72 616e 6b20  mputes the.rank 
│ │ │ │ -00017400: 6f66 2074 6865 2073 7562 6d61 7472 6963  of the submatric
│ │ │ │ -00017410: 6573 2063 6f6e 7369 6465 7265 642e 2020  es considered.  
│ │ │ │ -00017420: 5365 6520 2a6e 6f74 650a 6765 7453 7562  See *note.getSub
│ │ │ │ -00017430: 6d61 7472 6978 4f66 5261 6e6b 282e 2e2e  matrixOfRank(...
│ │ │ │ -00017440: 2c53 7472 6174 6567 793d 3e2e 2e2e 293a  ,Strategy=>...):
│ │ │ │ -00017450: 2053 7472 6174 6567 7944 6566 6175 6c74   StrategyDefault
│ │ │ │ -00017460: 2c2e 2049 6e20 7468 6520 6675 7475 7265  ,. In the future
│ │ │ │ -00017470: 2c20 7765 2068 6f70 650a 746f 2073 7065  , we hope.to spe
│ │ │ │ -00017480: 6564 2075 7020 7468 6520 6675 6e63 7469  ed up the functi
│ │ │ │ -00017490: 6f6e 2074 6f20 7573 6520 6d75 6c74 6970  on to use multip
│ │ │ │ -000174a0: 6c65 2074 6872 6561 6473 206f 6620 6578  le threads of ex
│ │ │ │ -000174b0: 6563 7574 696f 6e2c 2069 6e20 7768 6963  ecution, in whic
│ │ │ │ -000174c0: 6820 6361 7365 0a74 6865 2074 6872 6561  h case.the threa
│ │ │ │ -000174d0: 6469 6e67 2077 6f75 6c64 2062 6520 636f  ding would be co
│ │ │ │ -000174e0: 6e74 726f 6c6c 6564 2062 7920 7468 6520  ntrolled by the 
│ │ │ │ -000174f0: 6f70 7469 6f6e 2054 6872 6561 6473 2e0a  option Threads..
│ │ │ │ -00017500: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ -00017510: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6973  ==..  * *note is
│ │ │ │ -00017520: 5261 6e6b 4174 4c65 6173 743a 2069 7352  RankAtLeast: isR
│ │ │ │ -00017530: 616e 6b41 744c 6561 7374 2c20 2d2d 2064  ankAtLeast, -- d
│ │ │ │ -00017540: 6574 6572 6d69 6e65 7320 6966 2074 6865  etermines if the
│ │ │ │ -00017550: 206d 6174 7269 7820 6861 7320 7261 6e6b   matrix has rank
│ │ │ │ -00017560: 2061 740a 2020 2020 6c65 6173 7420 6120   at.    least a 
│ │ │ │ -00017570: 6e75 6d62 6572 0a20 202a 202a 6e6f 7465  number.  * *note
│ │ │ │ -00017580: 2067 6574 5375 626d 6174 7269 784f 6652   getSubmatrixOfR
│ │ │ │ -00017590: 616e 6b28 2e2e 2e2c 5374 7261 7465 6779  ank(...,Strategy
│ │ │ │ -000175a0: 3d3e 2e2e 2e29 3a20 5374 7261 7465 6779  =>...): Strategy
│ │ │ │ -000175b0: 4465 6661 756c 742c 202d 2d20 7374 7261  Default, -- stra
│ │ │ │ -000175c0: 7465 6769 6573 0a20 2020 2066 6f72 2063  tegies.    for c
│ │ │ │ -000175d0: 686f 6f73 696e 6720 7375 626d 6174 7269  hoosing submatri
│ │ │ │ -000175e0: 6365 730a 0a57 6179 7320 746f 2075 7365  ces..Ways to use
│ │ │ │ -000175f0: 2067 6574 5375 626d 6174 7269 784f 6652   getSubmatrixOfR
│ │ │ │ -00017600: 616e 6b3a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  ank:.===========
│ │ │ │ -00017610: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00017620: 3d3d 3d3d 0a0a 2020 2a20 2267 6574 5375  ====..  * "getSu
│ │ │ │ -00017630: 626d 6174 7269 784f 6652 616e 6b28 5a5a  bmatrixOfRank(ZZ
│ │ │ │ -00017640: 2c4d 6174 7269 7829 220a 0a46 6f72 2074  ,Matrix)"..For t
│ │ │ │ -00017650: 6865 2070 726f 6772 616d 6d65 720a 3d3d  he programmer.==
│ │ │ │ +000172f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017300: 2020 207c 0a7c 6f36 203a 204d 6174 7269     |.|o6 : Matri
│ │ │ │ +00017310: 7820 5220 203c 2d2d 2052 2020 2020 2020  x R  <-- R      
│ │ │ │ +00017320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017340: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00017350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00017380: 0a7c 6937 203a 2067 6574 5375 626d 6174  .|i7 : getSubmat
│ │ │ │ +00017390: 7269 784f 6652 616e 6b28 332c 204d 2920  rixOfRank(3, M) 
│ │ │ │ +000173a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000173b0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000173c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000173d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000173e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000173f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5468  -----------+..Th
│ │ │ │ +00017400: 6520 6f70 7469 6f6e 2053 7472 6174 6567  e option Strateg
│ │ │ │ +00017410: 7920 6973 2075 7365 6420 746f 2075 7365  y is used to use
│ │ │ │ +00017420: 6420 746f 2063 6f6e 7472 6f6c 2068 6f77  d to control how
│ │ │ │ +00017430: 2074 6865 2066 756e 6374 696f 6e20 636f   the function co
│ │ │ │ +00017440: 6d70 7574 6573 2074 6865 0a72 616e 6b20  mputes the.rank 
│ │ │ │ +00017450: 6f66 2074 6865 2073 7562 6d61 7472 6963  of the submatric
│ │ │ │ +00017460: 6573 2063 6f6e 7369 6465 7265 642e 2020  es considered.  
│ │ │ │ +00017470: 5365 6520 2a6e 6f74 650a 6765 7453 7562  See *note.getSub
│ │ │ │ +00017480: 6d61 7472 6978 4f66 5261 6e6b 282e 2e2e  matrixOfRank(...
│ │ │ │ +00017490: 2c53 7472 6174 6567 793d 3e2e 2e2e 293a  ,Strategy=>...):
│ │ │ │ +000174a0: 2053 7472 6174 6567 7944 6566 6175 6c74   StrategyDefault
│ │ │ │ +000174b0: 2c2e 2049 6e20 7468 6520 6675 7475 7265  ,. In the future
│ │ │ │ +000174c0: 2c20 7765 2068 6f70 650a 746f 2073 7065  , we hope.to spe
│ │ │ │ +000174d0: 6564 2075 7020 7468 6520 6675 6e63 7469  ed up the functi
│ │ │ │ +000174e0: 6f6e 2074 6f20 7573 6520 6d75 6c74 6970  on to use multip
│ │ │ │ +000174f0: 6c65 2074 6872 6561 6473 206f 6620 6578  le threads of ex
│ │ │ │ +00017500: 6563 7574 696f 6e2c 2069 6e20 7768 6963  ecution, in whic
│ │ │ │ +00017510: 6820 6361 7365 0a74 6865 2074 6872 6561  h case.the threa
│ │ │ │ +00017520: 6469 6e67 2077 6f75 6c64 2062 6520 636f  ding would be co
│ │ │ │ +00017530: 6e74 726f 6c6c 6564 2062 7920 7468 6520  ntrolled by the 
│ │ │ │ +00017540: 6f70 7469 6f6e 2054 6872 6561 6473 2e0a  option Threads..
│ │ │ │ +00017550: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ +00017560: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6973  ==..  * *note is
│ │ │ │ +00017570: 5261 6e6b 4174 4c65 6173 743a 2069 7352  RankAtLeast: isR
│ │ │ │ +00017580: 616e 6b41 744c 6561 7374 2c20 2d2d 2064  ankAtLeast, -- d
│ │ │ │ +00017590: 6574 6572 6d69 6e65 7320 6966 2074 6865  etermines if the
│ │ │ │ +000175a0: 206d 6174 7269 7820 6861 7320 7261 6e6b   matrix has rank
│ │ │ │ +000175b0: 2061 740a 2020 2020 6c65 6173 7420 6120   at.    least a 
│ │ │ │ +000175c0: 6e75 6d62 6572 0a20 202a 202a 6e6f 7465  number.  * *note
│ │ │ │ +000175d0: 2067 6574 5375 626d 6174 7269 784f 6652   getSubmatrixOfR
│ │ │ │ +000175e0: 616e 6b28 2e2e 2e2c 5374 7261 7465 6779  ank(...,Strategy
│ │ │ │ +000175f0: 3d3e 2e2e 2e29 3a20 5374 7261 7465 6779  =>...): Strategy
│ │ │ │ +00017600: 4465 6661 756c 742c 202d 2d20 7374 7261  Default, -- stra
│ │ │ │ +00017610: 7465 6769 6573 0a20 2020 2066 6f72 2063  tegies.    for c
│ │ │ │ +00017620: 686f 6f73 696e 6720 7375 626d 6174 7269  hoosing submatri
│ │ │ │ +00017630: 6365 730a 0a57 6179 7320 746f 2075 7365  ces..Ways to use
│ │ │ │ +00017640: 2067 6574 5375 626d 6174 7269 784f 6652   getSubmatrixOfR
│ │ │ │ +00017650: 616e 6b3a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  ank:.===========
│ │ │ │  00017660: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00017670: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f  ..The object *no
│ │ │ │ -00017680: 7465 2067 6574 5375 626d 6174 7269 784f  te getSubmatrixO
│ │ │ │ -00017690: 6652 616e 6b3a 2067 6574 5375 626d 6174  fRank: getSubmat
│ │ │ │ -000176a0: 7269 784f 6652 616e 6b2c 2069 7320 6120  rixOfRank, is a 
│ │ │ │ -000176b0: 2a6e 6f74 6520 6d65 7468 6f64 0a66 756e  *note method.fun
│ │ │ │ -000176c0: 6374 696f 6e20 7769 7468 206f 7074 696f  ction with optio
│ │ │ │ -000176d0: 6e73 3a20 284d 6163 6175 6c61 7932 446f  ns: (Macaulay2Do
│ │ │ │ -000176e0: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e  c)MethodFunction
│ │ │ │ -000176f0: 5769 7468 4f70 7469 6f6e 732c 2e0a 0a2d  WithOptions,...-
│ │ │ │ -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 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ -00017750: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
│ │ │ │ -00017760: 6973 2064 6f63 756d 656e 7420 6973 2069  is document is i
│ │ │ │ -00017770: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475  n./build/reprodu
│ │ │ │ -00017780: 6369 626c 652d 7061 7468 2f6d 6163 6175  cible-path/macau
│ │ │ │ -00017790: 6c61 7932 2d31 2e32 352e 3131 2b64 732f  lay2-1.25.11+ds/
│ │ │ │ -000177a0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163  M2/Macaulay2/pac
│ │ │ │ -000177b0: 6b61 6765 732f 4661 7374 4d69 6e6f 7273  kages/FastMinors
│ │ │ │ -000177c0: 2e0a 6d32 3a31 3737 323a 302e 0a1f 0a46  ..m2:1772:0....F
│ │ │ │ -000177d0: 696c 653a 2046 6173 744d 696e 6f72 732e  ile: FastMinors.
│ │ │ │ -000177e0: 696e 666f 2c20 4e6f 6465 3a20 6973 436f  info, Node: isCo
│ │ │ │ -000177f0: 6469 6d41 744c 6561 7374 2c20 4e65 7874  dimAtLeast, Next
│ │ │ │ -00017800: 3a20 6973 4469 6d41 744d 6f73 742c 2050  : isDimAtMost, P
│ │ │ │ -00017810: 7265 763a 2067 6574 5375 626d 6174 7269  rev: getSubmatri
│ │ │ │ -00017820: 784f 6652 616e 6b2c 2055 703a 2054 6f70  xOfRank, Up: Top
│ │ │ │ -00017830: 0a0a 6973 436f 6469 6d41 744c 6561 7374  ..isCodimAtLeast
│ │ │ │ -00017840: 202d 2d20 7265 7475 726e 7320 7472 7565   -- returns true
│ │ │ │ -00017850: 2069 6620 7765 2063 616e 2071 7569 636b   if we can quick
│ │ │ │ -00017860: 6c79 2073 6565 2077 6865 7468 6572 2074  ly see whether t
│ │ │ │ -00017870: 6865 2063 6f64 696d 2069 7320 6174 206c  he codim is at l
│ │ │ │ -00017880: 6561 7374 2061 2067 6976 656e 206e 756d  east a given num
│ │ │ │ -00017890: 6265 720a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ber.************
│ │ │ │ -000178a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000178b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000178c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000178d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000178e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000178f0: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765  *****..  * Usage
│ │ │ │ -00017900: 3a20 0a20 2020 2020 2020 2069 7343 6f64  : .        isCod
│ │ │ │ -00017910: 696d 4174 4c65 6173 7428 6e2c 2049 290a  imAtLeast(n, I).
│ │ │ │ -00017920: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ -00017930: 2020 2a20 6e2c 2061 6e20 2a6e 6f74 6520    * n, an *note 
│ │ │ │ -00017940: 696e 7465 6765 723a 2028 4d61 6361 756c  integer: (Macaul
│ │ │ │ -00017950: 6179 3244 6f63 295a 5a2c 2c20 616e 2069  ay2Doc)ZZ,, an i
│ │ │ │ -00017960: 6e74 6567 6572 0a20 2020 2020 202a 2049  nteger.      * I
│ │ │ │ -00017970: 2c20 616e 202a 6e6f 7465 2069 6465 616c  , an *note ideal
│ │ │ │ -00017980: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -00017990: 4964 6561 6c2c 2c20 616e 2069 6465 616c  Ideal,, an ideal
│ │ │ │ -000179a0: 2069 6e20 6120 706f 6c79 6e6f 6d69 616c   in a polynomial
│ │ │ │ -000179b0: 2072 696e 670a 2020 2020 2020 2020 6f76   ring.        ov
│ │ │ │ -000179c0: 6572 2061 2066 6965 6c64 2c20 6f72 2061  er a field, or a
│ │ │ │ -000179d0: 2071 756f 7469 656e 7420 7269 6e67 0a20   quotient ring. 
│ │ │ │ -000179e0: 202a 202a 6e6f 7465 204f 7074 696f 6e61   * *note Optiona
│ │ │ │ -000179f0: 6c20 696e 7075 7473 3a20 284d 6163 6175  l inputs: (Macau
│ │ │ │ -00017a00: 6c61 7932 446f 6329 7573 696e 6720 6675  lay2Doc)using fu
│ │ │ │ -00017a10: 6e63 7469 6f6e 7320 7769 7468 206f 7074  nctions with opt
│ │ │ │ -00017a20: 696f 6e61 6c20 696e 7075 7473 2c3a 0a20  ional inputs,:. 
│ │ │ │ -00017a30: 2020 2020 202a 2053 5061 6972 7346 756e       * SPairsFun
│ │ │ │ -00017a40: 6374 696f 6e20 3d3e 2061 202a 6e6f 7465  ction => a *note
│ │ │ │ -00017a50: 2066 756e 6374 696f 6e3a 2028 4d61 6361   function: (Maca
│ │ │ │ -00017a60: 756c 6179 3244 6f63 2946 756e 6374 696f  ulay2Doc)Functio
│ │ │ │ -00017a70: 6e2c 2c20 6465 6661 756c 740a 2020 2020  n,, default.    
│ │ │ │ -00017a80: 2020 2020 7661 6c75 6520 4675 6e63 7469      value Functi
│ │ │ │ -00017a90: 6f6e 436c 6f73 7572 655b 2e2e 2f46 6173  onClosure[../Fas
│ │ │ │ -00017aa0: 744d 696e 6f72 732e 6d32 3a32 3131 3a32  tMinors.m2:211:2
│ │ │ │ -00017ab0: 332d 3231 313a 3432 5d2c 2061 2066 756e  3-211:42], a fun
│ │ │ │ -00017ac0: 6374 696f 6e20 746f 0a20 2020 2020 2020  ction to.       
│ │ │ │ -00017ad0: 2063 6f6e 7472 6f6c 2068 6f77 2077 6865   control how whe
│ │ │ │ -00017ae0: 6e20 7468 6520 636f 6469 6d65 6e73 696f  n the codimensio
│ │ │ │ -00017af0: 6e20 6f66 206d 696e 6f72 7320 6973 2063  n of minors is c
│ │ │ │ -00017b00: 6f6d 7075 7465 642c 2064 6566 6175 6c74  omputed, default
│ │ │ │ -00017b10: 2069 730a 2020 2020 2020 2020 692d 3e63   is.        i->c
│ │ │ │ -00017b20: 6569 6c69 6e67 2831 2e35 5e69 290a 2020  eiling(1.5^i).  
│ │ │ │ -00017b30: 2020 2020 2a20 5061 6972 4c69 6d69 7420      * PairLimit 
│ │ │ │ -00017b40: 3d3e 2061 202a 6e6f 7465 206e 756d 6265  => a *note numbe
│ │ │ │ -00017b50: 723a 2028 4d61 6361 756c 6179 3244 6f63  r: (Macaulay2Doc
│ │ │ │ -00017b60: 294e 756d 6265 722c 2c20 6465 6661 756c  )Number,, defaul
│ │ │ │ -00017b70: 7420 7661 6c75 6520 3130 302c 0a20 2020  t value 100,.   
│ │ │ │ -00017b80: 2020 2020 2074 6865 206d 6178 2076 616c       the max val
│ │ │ │ -00017b90: 7565 2074 6f20 6265 2070 6c75 6767 6564  ue to be plugged
│ │ │ │ -00017ba0: 2069 6e74 6f20 5350 6169 7273 4675 6e63   into SPairsFunc
│ │ │ │ -00017bb0: 7469 6f6e 0a20 2020 2020 202a 2056 6572  tion.      * Ver
│ │ │ │ -00017bc0: 626f 7365 203d 3e20 2e2e 2e2c 2064 6566  bose => ..., def
│ │ │ │ -00017bd0: 6175 6c74 2076 616c 7565 2066 616c 7365  ault value false
│ │ │ │ -00017be0: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ -00017bf0: 2020 2020 2a20 7472 7565 2069 6620 7468      * true if th
│ │ │ │ -00017c00: 6520 636f 6469 6d65 6e73 696f 6e20 6f66  e codimension of
│ │ │ │ -00017c10: 2049 2069 7320 6174 206c 6561 7374 206e   I is at least n
│ │ │ │ -00017c20: 206f 7220 6e75 6c6c 2069 6620 7468 6520   or null if the 
│ │ │ │ -00017c30: 6675 6e63 7469 6f6e 0a20 2020 2020 2020  function.       
│ │ │ │ -00017c40: 2063 616e 6e6f 7420 7465 6c6c 2077 6865   cannot tell whe
│ │ │ │ -00017c50: 7468 6572 2074 6865 2063 6f64 696d 656e  ther the codimen
│ │ │ │ -00017c60: 7369 6f6e 2069 7320 6174 206c 6561 7374  sion is at least
│ │ │ │ -00017c70: 206e 0a0a 4465 7363 7269 7074 696f 6e0a   n..Description.
│ │ │ │ -00017c80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869  ===========..Thi
│ │ │ │ -00017c90: 7320 636f 6d70 7574 6573 2061 2070 6172  s computes a par
│ │ │ │ -00017ca0: 7469 616c 2047 726f 6562 6e65 7220 6261  tial Groebner ba
│ │ │ │ -00017cb0: 7369 732c 2074 616b 6573 2074 6865 2069  sis, takes the i
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│ │ │ │ -00017cd0: 6420 6368 6563 6b73 0a77 6865 7468 6572  d checks.whether
│ │ │ │ -00017ce0: 2074 6861 7420 2870 6172 7469 616c 2920   that (partial) 
│ │ │ │ -00017cf0: 696e 6974 6961 6c20 6964 6561 6c20 6861  initial ideal ha
│ │ │ │ -00017d00: 7320 636f 6469 6d65 6e73 696f 6e20 6174  s codimension at
│ │ │ │ -00017d10: 206c 6561 7374 206e 2e20 436f 6e73 6964   least n. Consid
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│ │ │ │ -00017d30: 2065 7861 6d70 6c65 2e20 2057 6520 6372   example.  We cr
│ │ │ │ -00017d40: 6561 7465 2061 6e20 6964 6561 6c20 6f66  eate an ideal of
│ │ │ │ -00017d50: 2031 3520 6d69 6e6f 7273 206f 6620 7468   15 minors of th
│ │ │ │ -00017d60: 6520 6d61 7472 6978 206d 7944 6966 6620  e matrix myDiff 
│ │ │ │ -00017d70: 2861 0a6d 6174 7269 7820 636f 6e73 7472  (a.matrix constr
│ │ │ │ -00017d80: 7563 7465 6420 696e 2061 2077 6179 2074  ucted in a way t
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│ │ │ │ -00017dc0: 790a 7468 6174 2074 6865 2063 6f64 696d  y.that the codim
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│ │ │ │ -00017de0: 6465 616c 2069 7320 6174 206c 6561 7374  deal is at least
│ │ │ │ -00017df0: 2033 2e20 2054 6865 2062 7569 6c74 2d69   3.  The built-i
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│ │ │ │ -00017e20: 6e6f 7420 7465 726d 696e 6174 652e 2048  not terminate. H
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│ │ │ │ -00017e40: 744c 6561 7374 2069 7320 6e6f 726d 616c  tLeast is normal
│ │ │ │ -00017e50: 6c79 2076 6572 7920 6661 7374 2e0a 0a2b  ly very fast...+
│ │ │ │ -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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00017eb0: 6931 203a 2052 203d 205a 5a2f 3132 375b  i1 : R = ZZ/127[
│ │ │ │ -00017ec0: 785f 3120 2e2e 2078 5f28 3132 295d 3b20  x_1 .. x_(12)]; 
│ │ │ │ -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 207c 0a2b               |.+
│ │ │ │ -00017f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00017f50: 6932 203a 2050 203d 206d 696e 6f72 7328  i2 : P = minors(
│ │ │ │ -00017f60: 332c 6765 6e65 7269 634d 6174 7269 7828  3,genericMatrix(
│ │ │ │ -00017f70: 522c 785f 312c 332c 3429 293b 2020 2020  R,x_1,3,4));    
│ │ │ │ -00017f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017f90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00017fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017670: 3d3d 3d3d 0a0a 2020 2a20 2267 6574 5375  ====..  * "getSu
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│ │ │ │ +000176a0: 6865 2070 726f 6772 616d 6d65 720a 3d3d  he programmer.==
│ │ │ │ +000176b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000176c0: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f  ..The object *no
│ │ │ │ +000176d0: 7465 2067 6574 5375 626d 6174 7269 784f  te getSubmatrixO
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│ │ │ │ +00017700: 2a6e 6f74 6520 6d65 7468 6f64 0a66 756e  *note method.fun
│ │ │ │ +00017710: 6374 696f 6e20 7769 7468 206f 7074 696f  ction with optio
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│ │ │ │ +00017730: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e  c)MethodFunction
│ │ │ │ +00017740: 5769 7468 4f70 7469 6f6e 732c 2e0a 0a2d  WithOptions,...-
│ │ │ │ +00017750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ +000177a0: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
│ │ │ │ +000177b0: 6973 2064 6f63 756d 656e 7420 6973 2069  is document is i
│ │ │ │ +000177c0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475  n./build/reprodu
│ │ │ │ +000177d0: 6369 626c 652d 7061 7468 2f6d 6163 6175  cible-path/macau
│ │ │ │ +000177e0: 6c61 7932 2d31 2e32 352e 3131 2b64 732f  lay2-1.25.11+ds/
│ │ │ │ +000177f0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163  M2/Macaulay2/pac
│ │ │ │ +00017800: 6b61 6765 732f 4661 7374 4d69 6e6f 7273  kages/FastMinors
│ │ │ │ +00017810: 2e0a 6d32 3a31 3737 323a 302e 0a1f 0a46  ..m2:1772:0....F
│ │ │ │ +00017820: 696c 653a 2046 6173 744d 696e 6f72 732e  ile: FastMinors.
│ │ │ │ +00017830: 696e 666f 2c20 4e6f 6465 3a20 6973 436f  info, Node: isCo
│ │ │ │ +00017840: 6469 6d41 744c 6561 7374 2c20 4e65 7874  dimAtLeast, Next
│ │ │ │ +00017850: 3a20 6973 4469 6d41 744d 6f73 742c 2050  : isDimAtMost, P
│ │ │ │ +00017860: 7265 763a 2067 6574 5375 626d 6174 7269  rev: getSubmatri
│ │ │ │ +00017870: 784f 6652 616e 6b2c 2055 703a 2054 6f70  xOfRank, Up: Top
│ │ │ │ +00017880: 0a0a 6973 436f 6469 6d41 744c 6561 7374  ..isCodimAtLeast
│ │ │ │ +00017890: 202d 2d20 7265 7475 726e 7320 7472 7565   -- returns true
│ │ │ │ +000178a0: 2069 6620 7765 2063 616e 2071 7569 636b   if we can quick
│ │ │ │ +000178b0: 6c79 2073 6565 2077 6865 7468 6572 2074  ly see whether t
│ │ │ │ +000178c0: 6865 2063 6f64 696d 2069 7320 6174 206c  he codim is at l
│ │ │ │ +000178d0: 6561 7374 2061 2067 6976 656e 206e 756d  east a given num
│ │ │ │ +000178e0: 6265 720a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ber.************
│ │ │ │ +000178f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00017900: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00017910: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00017920: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00017930: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00017940: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765  *****..  * Usage
│ │ │ │ +00017950: 3a20 0a20 2020 2020 2020 2069 7343 6f64  : .        isCod
│ │ │ │ +00017960: 696d 4174 4c65 6173 7428 6e2c 2049 290a  imAtLeast(n, I).
│ │ │ │ +00017970: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ +00017980: 2020 2a20 6e2c 2061 6e20 2a6e 6f74 6520    * n, an *note 
│ │ │ │ +00017990: 696e 7465 6765 723a 2028 4d61 6361 756c  integer: (Macaul
│ │ │ │ +000179a0: 6179 3244 6f63 295a 5a2c 2c20 616e 2069  ay2Doc)ZZ,, an i
│ │ │ │ +000179b0: 6e74 6567 6572 0a20 2020 2020 202a 2049  nteger.      * I
│ │ │ │ +000179c0: 2c20 616e 202a 6e6f 7465 2069 6465 616c  , an *note ideal
│ │ │ │ +000179d0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +000179e0: 4964 6561 6c2c 2c20 616e 2069 6465 616c  Ideal,, an ideal
│ │ │ │ +000179f0: 2069 6e20 6120 706f 6c79 6e6f 6d69 616c   in a polynomial
│ │ │ │ +00017a00: 2072 696e 670a 2020 2020 2020 2020 6f76   ring.        ov
│ │ │ │ +00017a10: 6572 2061 2066 6965 6c64 2c20 6f72 2061  er a field, or a
│ │ │ │ +00017a20: 2071 756f 7469 656e 7420 7269 6e67 0a20   quotient ring. 
│ │ │ │ +00017a30: 202a 202a 6e6f 7465 204f 7074 696f 6e61   * *note Optiona
│ │ │ │ +00017a40: 6c20 696e 7075 7473 3a20 284d 6163 6175  l inputs: (Macau
│ │ │ │ +00017a50: 6c61 7932 446f 6329 7573 696e 6720 6675  lay2Doc)using fu
│ │ │ │ +00017a60: 6e63 7469 6f6e 7320 7769 7468 206f 7074  nctions with opt
│ │ │ │ +00017a70: 696f 6e61 6c20 696e 7075 7473 2c3a 0a20  ional inputs,:. 
│ │ │ │ +00017a80: 2020 2020 202a 2053 5061 6972 7346 756e       * SPairsFun
│ │ │ │ +00017a90: 6374 696f 6e20 3d3e 2061 202a 6e6f 7465  ction => a *note
│ │ │ │ +00017aa0: 2066 756e 6374 696f 6e3a 2028 4d61 6361   function: (Maca
│ │ │ │ +00017ab0: 756c 6179 3244 6f63 2946 756e 6374 696f  ulay2Doc)Functio
│ │ │ │ +00017ac0: 6e2c 2c20 6465 6661 756c 740a 2020 2020  n,, default.    
│ │ │ │ +00017ad0: 2020 2020 7661 6c75 6520 4675 6e63 7469      value Functi
│ │ │ │ +00017ae0: 6f6e 436c 6f73 7572 655b 2e2e 2f46 6173  onClosure[../Fas
│ │ │ │ +00017af0: 744d 696e 6f72 732e 6d32 3a32 3131 3a32  tMinors.m2:211:2
│ │ │ │ +00017b00: 332d 3231 313a 3432 5d2c 2061 2066 756e  3-211:42], a fun
│ │ │ │ +00017b10: 6374 696f 6e20 746f 0a20 2020 2020 2020  ction to.       
│ │ │ │ +00017b20: 2063 6f6e 7472 6f6c 2068 6f77 2077 6865   control how whe
│ │ │ │ +00017b30: 6e20 7468 6520 636f 6469 6d65 6e73 696f  n the codimensio
│ │ │ │ +00017b40: 6e20 6f66 206d 696e 6f72 7320 6973 2063  n of minors is c
│ │ │ │ +00017b50: 6f6d 7075 7465 642c 2064 6566 6175 6c74  omputed, default
│ │ │ │ +00017b60: 2069 730a 2020 2020 2020 2020 692d 3e63   is.        i->c
│ │ │ │ +00017b70: 6569 6c69 6e67 2831 2e35 5e69 290a 2020  eiling(1.5^i).  
│ │ │ │ +00017b80: 2020 2020 2a20 5061 6972 4c69 6d69 7420      * PairLimit 
│ │ │ │ +00017b90: 3d3e 2061 202a 6e6f 7465 206e 756d 6265  => a *note numbe
│ │ │ │ +00017ba0: 723a 2028 4d61 6361 756c 6179 3244 6f63  r: (Macaulay2Doc
│ │ │ │ +00017bb0: 294e 756d 6265 722c 2c20 6465 6661 756c  )Number,, defaul
│ │ │ │ +00017bc0: 7420 7661 6c75 6520 3130 302c 0a20 2020  t value 100,.   
│ │ │ │ +00017bd0: 2020 2020 2074 6865 206d 6178 2076 616c       the max val
│ │ │ │ +00017be0: 7565 2074 6f20 6265 2070 6c75 6767 6564  ue to be plugged
│ │ │ │ +00017bf0: 2069 6e74 6f20 5350 6169 7273 4675 6e63   into SPairsFunc
│ │ │ │ +00017c00: 7469 6f6e 0a20 2020 2020 202a 2056 6572  tion.      * Ver
│ │ │ │ +00017c10: 626f 7365 203d 3e20 2e2e 2e2c 2064 6566  bose => ..., def
│ │ │ │ +00017c20: 6175 6c74 2076 616c 7565 2066 616c 7365  ault value false
│ │ │ │ +00017c30: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ +00017c40: 2020 2020 2a20 7472 7565 2069 6620 7468      * true if th
│ │ │ │ +00017c50: 6520 636f 6469 6d65 6e73 696f 6e20 6f66  e codimension of
│ │ │ │ +00017c60: 2049 2069 7320 6174 206c 6561 7374 206e   I is at least n
│ │ │ │ +00017c70: 206f 7220 6e75 6c6c 2069 6620 7468 6520   or null if the 
│ │ │ │ +00017c80: 6675 6e63 7469 6f6e 0a20 2020 2020 2020  function.       
│ │ │ │ +00017c90: 2063 616e 6e6f 7420 7465 6c6c 2077 6865   cannot tell whe
│ │ │ │ +00017ca0: 7468 6572 2074 6865 2063 6f64 696d 656e  ther the codimen
│ │ │ │ +00017cb0: 7369 6f6e 2069 7320 6174 206c 6561 7374  sion is at least
│ │ │ │ +00017cc0: 206e 0a0a 4465 7363 7269 7074 696f 6e0a   n..Description.
│ │ │ │ +00017cd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869  ===========..Thi
│ │ │ │ +00017ce0: 7320 636f 6d70 7574 6573 2061 2070 6172  s computes a par
│ │ │ │ +00017cf0: 7469 616c 2047 726f 6562 6e65 7220 6261  tial Groebner ba
│ │ │ │ +00017d00: 7369 732c 2074 616b 6573 2074 6865 2069  sis, takes the i
│ │ │ │ +00017d10: 6e69 7469 616c 2074 6572 6d73 2c20 616e  nitial terms, an
│ │ │ │ +00017d20: 6420 6368 6563 6b73 0a77 6865 7468 6572  d checks.whether
│ │ │ │ +00017d30: 2074 6861 7420 2870 6172 7469 616c 2920   that (partial) 
│ │ │ │ +00017d40: 696e 6974 6961 6c20 6964 6561 6c20 6861  initial ideal ha
│ │ │ │ +00017d50: 7320 636f 6469 6d65 6e73 696f 6e20 6174  s codimension at
│ │ │ │ +00017d60: 206c 6561 7374 206e 2e20 436f 6e73 6964   least n. Consid
│ │ │ │ +00017d70: 6572 2074 6865 0a66 6f6c 6c6f 7769 6e67  er the.following
│ │ │ │ +00017d80: 2065 7861 6d70 6c65 2e20 2057 6520 6372   example.  We cr
│ │ │ │ +00017d90: 6561 7465 2061 6e20 6964 6561 6c20 6f66  eate an ideal of
│ │ │ │ +00017da0: 2031 3520 6d69 6e6f 7273 206f 6620 7468   15 minors of th
│ │ │ │ +00017db0: 6520 6d61 7472 6978 206d 7944 6966 6620  e matrix myDiff 
│ │ │ │ +00017dc0: 2861 0a6d 6174 7269 7820 636f 6e73 7472  (a.matrix constr
│ │ │ │ +00017dd0: 7563 7465 6420 696e 2061 2077 6179 2074  ucted in a way t
│ │ │ │ +00017de0: 7970 6963 616c 206f 6620 6170 706c 6963  ypical of applic
│ │ │ │ +00017df0: 6174 696f 6e73 292e 2020 5765 2077 6f75  ations).  We wou
│ │ │ │ +00017e00: 6c64 206c 696b 6520 746f 2076 6572 6966  ld like to verif
│ │ │ │ +00017e10: 790a 7468 6174 2074 6865 2063 6f64 696d  y.that the codim
│ │ │ │ +00017e20: 656e 7369 6f6e 206f 6620 7468 6973 2069  ension of this i
│ │ │ │ +00017e30: 6465 616c 2069 7320 6174 206c 6561 7374  deal is at least
│ │ │ │ +00017e40: 2033 2e20 2054 6865 2062 7569 6c74 2d69   3.  The built-i
│ │ │ │ +00017e50: 6e20 636f 6469 6d20 6675 6e63 7469 6f6e  n codim function
│ │ │ │ +00017e60: 0a74 7970 6963 616c 6c79 2064 6f65 7320  .typically does 
│ │ │ │ +00017e70: 6e6f 7420 7465 726d 696e 6174 652e 2048  not terminate. H
│ │ │ │ +00017e80: 6f77 6576 6572 2c20 6973 436f 6469 6d41  owever, isCodimA
│ │ │ │ +00017e90: 744c 6561 7374 2069 7320 6e6f 726d 616c  tLeast is normal
│ │ │ │ +00017ea0: 6c79 2076 6572 7920 6661 7374 2e0a 0a2b  ly very fast...+
│ │ │ │ +00017eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00017f00: 6931 203a 2052 203d 205a 5a2f 3132 375b  i1 : R = ZZ/127[
│ │ │ │ +00017f10: 785f 3120 2e2e 2078 5f28 3132 295d 3b20  x_1 .. x_(12)]; 
│ │ │ │ +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 207c 0a2b               |.+
│ │ │ │ +00017f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00017fa0: 6932 203a 2050 203d 206d 696e 6f72 7328  i2 : P = minors(
│ │ │ │ +00017fb0: 332c 6765 6e65 7269 634d 6174 7269 7828  3,genericMatrix(
│ │ │ │ +00017fc0: 522c 785f 312c 332c 3429 293b 2020 2020  R,x_1,3,4));    
│ │ │ │  00017fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017fe0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00017ff0: 6f32 203a 2049 6465 616c 206f 6620 5220  o2 : Ideal of R 
│ │ │ │ +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 2020 2020                  
│ │ │ │  00018020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018030: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00018040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00018090: 6933 203a 2043 203d 2072 6573 2028 525e  i3 : C = res (R^
│ │ │ │ -000180a0: 312f 2850 5e33 2929 3b20 2020 2020 2020  1/(P^3));       
│ │ │ │ -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 207c 0a2b               |.+
│ │ │ │ -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 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00018130: 6934 203a 206d 7944 6966 6620 3d20 432e  i4 : myDiff = C.
│ │ │ │ -00018140: 6464 5f33 3b20 2020 2020 2020 2020 2020  dd_3;           
│ │ │ │ -00018150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018170: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00018180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018030: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00018040: 6f32 203a 2049 6465 616c 206f 6620 5220  o2 : Ideal of R 
│ │ │ │ +00018050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018080: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00018090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000180a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000180b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000180c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000180d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +000180e0: 6933 203a 2043 203d 2072 6573 2028 525e  i3 : C = res (R^
│ │ │ │ +000180f0: 312f 2850 5e33 2929 3b20 2020 2020 2020  1/(P^3));       
│ │ │ │ +00018100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018120: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00018180: 6934 203a 206d 7944 6966 6620 3d20 432e  i4 : myDiff = C.
│ │ │ │ +00018190: 6464 5f33 3b20 2020 2020 2020 2020 2020  dd_3;           
│ │ │ │  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 207c 0a7c               |.|
│ │ │ │ -000181d0: 2020 2020 2020 2020 2020 2020 2033 3020               30 
│ │ │ │ -000181e0: 2020 2020 2031 3220 2020 2020 2020 2020       12         
│ │ │ │ +000181d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000181e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000181f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018210: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00018220: 6f34 203a 204d 6174 7269 7820 5220 2020  o4 : Matrix R   
│ │ │ │ -00018230: 3c2d 2d20 5220 2020 2020 2020 2020 2020  <-- R           
│ │ │ │ +00018220: 2020 2020 2020 2020 2020 2020 2033 3020               30 
│ │ │ │ +00018230: 2020 2020 2031 3220 2020 2020 2020 2020       12         
│ │ │ │  00018240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018260: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00018270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000182a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000182b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -000182c0: 6935 203a 2072 203d 2072 616e 6b20 6d79  i5 : r = rank my
│ │ │ │ -000182d0: 4469 6666 3b20 2020 2020 2020 2020 2020  Diff;           
│ │ │ │ -000182e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000182f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018300: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00018310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00018360: 6936 203a 204a 203d 2063 686f 6f73 6547  i6 : J = chooseG
│ │ │ │ -00018370: 6f6f 644d 696e 6f72 7328 3135 2c20 722c  oodMinors(15, r,
│ │ │ │ -00018380: 206d 7944 6966 662c 2053 7472 6174 6567   myDiff, Strateg
│ │ │ │ -00018390: 793d 3e53 7472 6174 6567 7944 6566 6175  y=>StrategyDefau
│ │ │ │ -000183a0: 6c74 4e6f 6e52 616e 646f 6d29 3b7c 0a7c  ltNonRandom);|.|
│ │ │ │ -000183b0: 2020 2020 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 2020 2020 207c 0a7c               |.|
│ │ │ │ -00018400: 6f36 203a 2049 6465 616c 206f 6620 5220  o6 : Ideal of R 
│ │ │ │ +00018260: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00018270: 6f34 203a 204d 6174 7269 7820 5220 2020  o4 : Matrix R   
│ │ │ │ +00018280: 3c2d 2d20 5220 2020 2020 2020 2020 2020  <-- R           
│ │ │ │ +00018290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000182a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000182b0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000182c0: 2d2d 2d2d 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 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00018310: 6935 203a 2072 203d 2072 616e 6b20 6d79  i5 : r = rank my
│ │ │ │ +00018320: 4469 6666 3b20 2020 2020 2020 2020 2020  Diff;           
│ │ │ │ +00018330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018350: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00018360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000183a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +000183b0: 6936 203a 204a 203d 2063 686f 6f73 6547  i6 : J = chooseG
│ │ │ │ +000183c0: 6f6f 644d 696e 6f72 7328 3135 2c20 722c  oodMinors(15, r,
│ │ │ │ +000183d0: 206d 7944 6966 662c 2053 7472 6174 6567   myDiff, Strateg
│ │ │ │ +000183e0: 793d 3e53 7472 6174 6567 7944 6566 6175  y=>StrategyDefau
│ │ │ │ +000183f0: 6c74 4e6f 6e52 616e 646f 6d29 3b7c 0a7c  ltNonRandom);|.|
│ │ │ │ +00018400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018440: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00018450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 3033 3934   -- used 0.00394
│ │ │ │ -00018500: 3539 3373 2028 6370 7529 3b20 302e 3030  593s (cpu); 0.00
│ │ │ │ -00018510: 3238 3031 3031 7320 2874 6872 6561 6429  280101s (thread)
│ │ │ │ -00018520: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ +00018440: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00018450: 6f36 203a 2049 6465 616c 206f 6620 5220  o6 : Ideal of R 
│ │ │ │ +00018460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018490: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000184a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000184b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000184c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000184d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000184e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +000184f0: 6937 203a 2074 696d 6520 6973 436f 6469  i7 : time isCodi
│ │ │ │ +00018500: 6d41 744c 6561 7374 2833 2c20 4a29 2020  mAtLeast(3, J)  
│ │ │ │ +00018510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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                  
│ │ │ │ +00018540: 202d 2d20 7573 6564 2030 2e30 3034 3035   -- used 0.00405
│ │ │ │ +00018550: 3033 3373 2028 6370 7529 3b20 302e 3030  033s (cpu); 0.00
│ │ │ │ +00018560: 3533 3233 3439 7320 2874 6872 6561 6429  532349s (thread)
│ │ │ │ +00018570: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │  00018580: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00018590: 6f37 203d 2074 7275 6520 2020 2020 2020  o7 = true       
│ │ │ │ +00018590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000185a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000185b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000185c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000185d0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ -00018630: 5468 6520 6675 6e63 7469 6f6e 2077 6f72  The function wor
│ │ │ │ -00018640: 6b73 2062 7920 636f 6d70 7574 696e 6720  ks by computing 
│ │ │ │ -00018650: 6762 2849 2c20 5061 6972 4c69 6d69 743d  gb(I, PairLimit=
│ │ │ │ -00018660: 3e66 2869 2929 2066 6f72 2073 7563 6365  >f(i)) for succe
│ │ │ │ -00018670: 7373 6976 6520 7661 6c75 6573 206f 660a  ssive values of.
│ │ │ │ -00018680: 692e 2020 4865 7265 2066 2869 2920 6973  i.  Here f(i) is
│ │ │ │ -00018690: 2061 2066 756e 6374 696f 6e20 7468 6174   a function that
│ │ │ │ -000186a0: 2074 616b 6573 2074 2c20 736f 6d65 2061   takes t, some a
│ │ │ │ -000186b0: 7070 726f 7869 6d61 7469 6f6e 206f 6620  pproximation of 
│ │ │ │ -000186c0: 7468 6520 6261 7365 2064 6567 7265 650a  the base degree.
│ │ │ │ -000186d0: 7661 6c75 6520 6f66 2074 6865 2070 6f6c  value of the pol
│ │ │ │ -000186e0: 796e 6f6d 6961 6c20 7269 6e67 2028 666f  ynomial ring (fo
│ │ │ │ -000186f0: 7220 6578 616d 706c 652c 2069 6e20 6120  r example, in a 
│ │ │ │ -00018700: 7374 616e 6461 7264 2067 7261 6465 6420  standard graded 
│ │ │ │ -00018710: 706f 6c79 6e6f 6d69 616c 0a72 696e 672c  polynomial.ring,
│ │ │ │ -00018720: 2074 6869 7320 6973 2070 726f 6261 626c   this is probabl
│ │ │ │ -00018730: 7920 6578 7065 6374 6564 2074 6f20 6265  y expected to be
│ │ │ │ -00018740: 205c 7b31 5c7d 292e 2020 416e 6420 6920   \{1\}).  And i 
│ │ │ │ -00018750: 6973 2061 2063 6f75 6e74 696e 6720 7661  is a counting va
│ │ │ │ -00018760: 7269 6162 6c65 2e0a 596f 7520 6361 6e20  riable..You can 
│ │ │ │ -00018770: 7072 6f76 6964 6520 796f 7572 206f 776e  provide your own
│ │ │ │ -00018780: 2066 756e 6374 696f 6e20 6279 2063 616c   function by cal
│ │ │ │ -00018790: 6c69 6e67 2069 7343 6f64 696d 4174 4c65  ling isCodimAtLe
│ │ │ │ -000187a0: 6173 7428 6e2c 2049 2c0a 5350 6169 7273  ast(n, I,.SPairs
│ │ │ │ -000187b0: 4675 6e63 7469 6f6e 3d3e 2820 2869 2920  Function=>( (i) 
│ │ │ │ -000187c0: 2d3e 2066 2869 2920 292c 2074 6865 2064  -> f(i) ), the d
│ │ │ │ -000187d0: 6566 6175 6c74 2066 756e 6374 696f 6e20  efault function 
│ │ │ │ -000187e0: 6973 0a53 5061 6972 7346 756e 6374 696f  is.SPairsFunctio
│ │ │ │ -000187f0: 6e3d 3e69 2d3e 6365 696c 696e 6728 312e  n=>i->ceiling(1.
│ │ │ │ -00018800: 355e 6929 2020 2050 6572 6861 7073 206d  5^i)   Perhaps m
│ │ │ │ -00018810: 6f72 6520 636f 6d6d 6f6e 6c79 2068 6f77  ore commonly how
│ │ │ │ -00018820: 6576 6572 2c20 7468 6520 7573 6572 206d  ever, the user m
│ │ │ │ -00018830: 6179 0a77 616e 7420 746f 2069 6e73 7465  ay.want to inste
│ │ │ │ -00018840: 6164 2074 656c 6c20 7468 6520 6675 6e63  ad tell the func
│ │ │ │ -00018850: 7469 6f6e 2074 6f20 636f 6d70 7574 6520  tion to compute 
│ │ │ │ -00018860: 666f 7220 6c61 7267 6572 2076 616c 7565  for larger value
│ │ │ │ -00018870: 7320 6f66 2069 2e20 2054 6869 7320 6973  s of i.  This is
│ │ │ │ -00018880: 0a64 6f6e 6520 7669 6120 7468 6520 6f70  .done via the op
│ │ │ │ -00018890: 7469 6f6e 2050 6169 724c 696d 6974 2e20  tion PairLimit. 
│ │ │ │ -000188a0: 2054 6869 7320 6973 2074 6865 206d 6178   This is the max
│ │ │ │ -000188b0: 2076 616c 7565 206f 6620 6920 746f 2062   value of i to b
│ │ │ │ -000188c0: 6520 706c 7567 6765 6420 696e 746f 0a53  e plugged into.S
│ │ │ │ -000188d0: 5061 6972 7346 756e 6374 696f 6e20 6265  PairsFunction be
│ │ │ │ -000188e0: 666f 7265 2074 6865 2066 756e 6374 696f  fore the functio
│ │ │ │ -000188f0: 6e20 6769 7665 7320 7570 2e20 2049 6e20  n gives up.  In 
│ │ │ │ -00018900: 6f74 6865 7220 776f 7264 732c 2050 6169  other words, Pai
│ │ │ │ -00018910: 724c 696d 6974 3d3e 3520 7769 6c6c 0a74  rLimit=>5 will.t
│ │ │ │ -00018920: 656c 6c20 7468 6520 6675 6e63 7469 6f6e  ell the function
│ │ │ │ -00018930: 2074 6f20 6368 6563 6b20 636f 6469 6d65   to check codime
│ │ │ │ -00018940: 6e73 696f 6e20 3520 7469 6d65 732e 0a0a  nsion 5 times...
│ │ │ │ -00018950: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -00018960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -000189a0: 7c69 3820 3a20 4920 3d20 6964 6561 6c28  |i8 : I = ideal(
│ │ │ │ -000189b0: 785f 325e 382a 785f 3130 5e33 2d33 2a78  x_2^8*x_10^3-3*x
│ │ │ │ -000189c0: 5f31 2a78 5f32 5e37 2a78 5f31 305e 322a  _1*x_2^7*x_10^2*
│ │ │ │ -000189d0: 785f 3131 2b33 2a78 5f31 5e32 2a78 5f32  x_11+3*x_1^2*x_2
│ │ │ │ -000189e0: 5e36 2a78 5f31 302a 785f 3131 5e32 7c0a  ^6*x_10*x_11^2|.
│ │ │ │ -000189f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00018a00: 2020 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 7c0a                |.
│ │ │ │ +000185d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000185e0: 6f37 203d 2074 7275 6520 2020 2020 2020  o7 = true       
│ │ │ │ +000185f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018620: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00018630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ +00018680: 5468 6520 6675 6e63 7469 6f6e 2077 6f72  The function wor
│ │ │ │ +00018690: 6b73 2062 7920 636f 6d70 7574 696e 6720  ks by computing 
│ │ │ │ +000186a0: 6762 2849 2c20 5061 6972 4c69 6d69 743d  gb(I, PairLimit=
│ │ │ │ +000186b0: 3e66 2869 2929 2066 6f72 2073 7563 6365  >f(i)) for succe
│ │ │ │ +000186c0: 7373 6976 6520 7661 6c75 6573 206f 660a  ssive values of.
│ │ │ │ +000186d0: 692e 2020 4865 7265 2066 2869 2920 6973  i.  Here f(i) is
│ │ │ │ +000186e0: 2061 2066 756e 6374 696f 6e20 7468 6174   a function that
│ │ │ │ +000186f0: 2074 616b 6573 2074 2c20 736f 6d65 2061   takes t, some a
│ │ │ │ +00018700: 7070 726f 7869 6d61 7469 6f6e 206f 6620  pproximation of 
│ │ │ │ +00018710: 7468 6520 6261 7365 2064 6567 7265 650a  the base degree.
│ │ │ │ +00018720: 7661 6c75 6520 6f66 2074 6865 2070 6f6c  value of the pol
│ │ │ │ +00018730: 796e 6f6d 6961 6c20 7269 6e67 2028 666f  ynomial ring (fo
│ │ │ │ +00018740: 7220 6578 616d 706c 652c 2069 6e20 6120  r example, in a 
│ │ │ │ +00018750: 7374 616e 6461 7264 2067 7261 6465 6420  standard graded 
│ │ │ │ +00018760: 706f 6c79 6e6f 6d69 616c 0a72 696e 672c  polynomial.ring,
│ │ │ │ +00018770: 2074 6869 7320 6973 2070 726f 6261 626c   this is probabl
│ │ │ │ +00018780: 7920 6578 7065 6374 6564 2074 6f20 6265  y expected to be
│ │ │ │ +00018790: 205c 7b31 5c7d 292e 2020 416e 6420 6920   \{1\}).  And i 
│ │ │ │ +000187a0: 6973 2061 2063 6f75 6e74 696e 6720 7661  is a counting va
│ │ │ │ +000187b0: 7269 6162 6c65 2e0a 596f 7520 6361 6e20  riable..You can 
│ │ │ │ +000187c0: 7072 6f76 6964 6520 796f 7572 206f 776e  provide your own
│ │ │ │ +000187d0: 2066 756e 6374 696f 6e20 6279 2063 616c   function by cal
│ │ │ │ +000187e0: 6c69 6e67 2069 7343 6f64 696d 4174 4c65  ling isCodimAtLe
│ │ │ │ +000187f0: 6173 7428 6e2c 2049 2c0a 5350 6169 7273  ast(n, I,.SPairs
│ │ │ │ +00018800: 4675 6e63 7469 6f6e 3d3e 2820 2869 2920  Function=>( (i) 
│ │ │ │ +00018810: 2d3e 2066 2869 2920 292c 2074 6865 2064  -> f(i) ), the d
│ │ │ │ +00018820: 6566 6175 6c74 2066 756e 6374 696f 6e20  efault function 
│ │ │ │ +00018830: 6973 0a53 5061 6972 7346 756e 6374 696f  is.SPairsFunctio
│ │ │ │ +00018840: 6e3d 3e69 2d3e 6365 696c 696e 6728 312e  n=>i->ceiling(1.
│ │ │ │ +00018850: 355e 6929 2020 2050 6572 6861 7073 206d  5^i)   Perhaps m
│ │ │ │ +00018860: 6f72 6520 636f 6d6d 6f6e 6c79 2068 6f77  ore commonly how
│ │ │ │ +00018870: 6576 6572 2c20 7468 6520 7573 6572 206d  ever, the user m
│ │ │ │ +00018880: 6179 0a77 616e 7420 746f 2069 6e73 7465  ay.want to inste
│ │ │ │ +00018890: 6164 2074 656c 6c20 7468 6520 6675 6e63  ad tell the func
│ │ │ │ +000188a0: 7469 6f6e 2074 6f20 636f 6d70 7574 6520  tion to compute 
│ │ │ │ +000188b0: 666f 7220 6c61 7267 6572 2076 616c 7565  for larger value
│ │ │ │ +000188c0: 7320 6f66 2069 2e20 2054 6869 7320 6973  s of i.  This is
│ │ │ │ +000188d0: 0a64 6f6e 6520 7669 6120 7468 6520 6f70  .done via the op
│ │ │ │ +000188e0: 7469 6f6e 2050 6169 724c 696d 6974 2e20  tion PairLimit. 
│ │ │ │ +000188f0: 2054 6869 7320 6973 2074 6865 206d 6178   This is the max
│ │ │ │ +00018900: 2076 616c 7565 206f 6620 6920 746f 2062   value of i to b
│ │ │ │ +00018910: 6520 706c 7567 6765 6420 696e 746f 0a53  e plugged into.S
│ │ │ │ +00018920: 5061 6972 7346 756e 6374 696f 6e20 6265  PairsFunction be
│ │ │ │ +00018930: 666f 7265 2074 6865 2066 756e 6374 696f  fore the functio
│ │ │ │ +00018940: 6e20 6769 7665 7320 7570 2e20 2049 6e20  n gives up.  In 
│ │ │ │ +00018950: 6f74 6865 7220 776f 7264 732c 2050 6169  other words, Pai
│ │ │ │ +00018960: 724c 696d 6974 3d3e 3520 7769 6c6c 0a74  rLimit=>5 will.t
│ │ │ │ +00018970: 656c 6c20 7468 6520 6675 6e63 7469 6f6e  ell the function
│ │ │ │ +00018980: 2074 6f20 6368 6563 6b20 636f 6469 6d65   to check codime
│ │ │ │ +00018990: 6e73 696f 6e20 3520 7469 6d65 732e 0a0a  nsion 5 times...
│ │ │ │ +000189a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +000189b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000189c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000189d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000189e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000189f0: 7c69 3820 3a20 4920 3d20 6964 6561 6c28  |i8 : I = ideal(
│ │ │ │ +00018a00: 785f 325e 382a 785f 3130 5e33 2d33 2a78  x_2^8*x_10^3-3*x
│ │ │ │ +00018a10: 5f31 2a78 5f32 5e37 2a78 5f31 305e 322a  _1*x_2^7*x_10^2*
│ │ │ │ +00018a20: 785f 3131 2b33 2a78 5f31 5e32 2a78 5f32  x_11+3*x_1^2*x_2
│ │ │ │ +00018a30: 5e36 2a78 5f31 302a 785f 3131 5e32 7c0a  ^6*x_10*x_11^2|.
│ │ │ │  00018a40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00018a50: 5a5a 2020 2020 2020 2020 2020 2020 2020  ZZ              
│ │ │ │ +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 2020 2020 2020 7c0a                |.
│ │ │ │ -00018a90: 7c6f 3820 3a20 4964 6561 6c20 6f66 202d  |o8 : Ideal of -
│ │ │ │ -00018aa0: 2d2d 5b78 2020 2c20 7820 2c20 7820 2c20  --[x  , x , x , 
│ │ │ │ -00018ab0: 7820 2c20 7820 202c 2078 202c 2078 202c  x , x  , x , x ,
│ │ │ │ -00018ac0: 2078 2020 2c20 7820 2c20 7820 2c20 7820   x  , x , x , x 
│ │ │ │ -00018ad0: 2c20 7820 5d20 2020 2020 2020 2020 7c0a  , x ]         |.
│ │ │ │ -00018ae0: 7c20 2020 2020 2020 2020 2020 2020 2031  |              1
│ │ │ │ -00018af0: 3237 2020 3131 2020 2038 2020 2031 2020  27  11   8   1  
│ │ │ │ -00018b00: 2039 2020 2031 3220 2020 3620 2020 3520   9   12   6   5 
│ │ │ │ -00018b10: 2020 3130 2020 2032 2020 2034 2020 2033    10   2   4   3
│ │ │ │ -00018b20: 2020 2037 2020 2020 2020 2020 2020 7c0a     7          |.
│ │ │ │ -00018b30: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ -00018b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00018b80: 7c2d 785f 315e 332a 785f 325e 352a 785f  |-x_1^3*x_2^5*x_
│ │ │ │ -00018b90: 3131 5e33 2c78 5f35 5e35 2a78 5f36 5e33  11^3,x_5^5*x_6^3
│ │ │ │ -00018ba0: 2a78 5f31 315e 332d 332a 785f 355e 362a  *x_11^3-3*x_5^6*
│ │ │ │ -00018bb0: 785f 365e 322a 785f 3131 5e32 2a78 5f31  x_6^2*x_11^2*x_1
│ │ │ │ -00018bc0: 322b 332a 785f 355e 372a 785f 362a 7c0a  2+3*x_5^7*x_6*|.
│ │ │ │ -00018bd0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ -00018be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00018c20: 7c78 5f31 312a 785f 3132 5e32 2d78 5f35  |x_11*x_12^2-x_5
│ │ │ │ -00018c30: 5e38 2a78 5f31 325e 332c 785f 315e 352a  ^8*x_12^3,x_1^5*
│ │ │ │ -00018c40: 785f 325e 332a 785f 345e 332d 332a 785f  x_2^3*x_4^3-3*x_
│ │ │ │ -00018c50: 315e 362a 785f 325e 322a 785f 345e 322a  1^6*x_2^2*x_4^2*
│ │ │ │ -00018c60: 785f 352b 332a 785f 315e 372a 785f 7c0a  x_5+3*x_1^7*x_|.
│ │ │ │ -00018c70: 7c2d 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 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00018cc0: 7c32 2a78 5f34 2a78 5f35 5e32 2d78 5f31  |2*x_4*x_5^2-x_1
│ │ │ │ -00018cd0: 5e38 2a78 5f35 5e33 2c78 5f36 5e38 2a78  ^8*x_5^3,x_6^8*x
│ │ │ │ -00018ce0: 5f31 315e 332d 332a 785f 352a 785f 365e  _11^3-3*x_5*x_6^
│ │ │ │ -00018cf0: 372a 785f 3131 5e32 2a78 5f31 322b 332a  7*x_11^2*x_12+3*
│ │ │ │ -00018d00: 785f 355e 322a 785f 365e 362a 785f 7c0a  x_5^2*x_6^6*x_|.
│ │ │ │ -00018d10: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ -00018d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00018d60: 7c31 312a 785f 3132 5e32 2d78 5f35 5e33  |11*x_12^2-x_5^3
│ │ │ │ -00018d70: 2a78 5f36 5e35 2a78 5f31 325e 332c 785f  *x_6^5*x_12^3,x_
│ │ │ │ -00018d80: 385e 332a 785f 3130 5e38 2d33 2a78 5f37  8^3*x_10^8-3*x_7
│ │ │ │ -00018d90: 2a78 5f38 5e32 2a78 5f31 305e 372a 785f  *x_8^2*x_10^7*x_
│ │ │ │ -00018da0: 3131 2b33 2a78 5f37 5e32 2a78 5f38 7c0a  11+3*x_7^2*x_8|.
│ │ │ │ -00018db0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ -00018dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00018e00: 7c2a 785f 3130 5e36 2a78 5f31 315e 322d  |*x_10^6*x_11^2-
│ │ │ │ -00018e10: 785f 375e 332a 785f 3130 5e35 2a78 5f31  x_7^3*x_10^5*x_1
│ │ │ │ -00018e20: 315e 332c 785f 325e 382a 785f 345e 332d  1^3,x_2^8*x_4^3-
│ │ │ │ -00018e30: 332a 785f 312a 785f 325e 372a 785f 345e  3*x_1*x_2^7*x_4^
│ │ │ │ -00018e40: 322a 785f 352b 332a 785f 315e 322a 7c0a  2*x_5+3*x_1^2*|.
│ │ │ │ -00018e50: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ -00018e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00018ea0: 7c78 5f32 5e36 2a78 5f34 2a78 5f35 5e32  |x_2^6*x_4*x_5^2
│ │ │ │ -00018eb0: 2d78 5f31 5e33 2a78 5f32 5e35 2a78 5f35  -x_1^3*x_2^5*x_5
│ │ │ │ -00018ec0: 5e33 2c2d 785f 365e 332a 785f 3131 5e38  ^3,-x_6^3*x_11^8
│ │ │ │ -00018ed0: 2b33 2a78 5f35 2a78 5f36 5e32 2a78 5f31  +3*x_5*x_6^2*x_1
│ │ │ │ -00018ee0: 315e 372a 785f 3132 2d33 2a78 5f35 7c0a  1^7*x_12-3*x_5|.
│ │ │ │ -00018ef0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ -00018f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00018f40: 7c5e 322a 785f 362a 785f 3131 5e36 2a78  |^2*x_6*x_11^6*x
│ │ │ │ -00018f50: 5f31 325e 322b 785f 355e 332a 785f 3131  _12^2+x_5^3*x_11
│ │ │ │ -00018f60: 5e35 2a78 5f31 325e 332c 2d78 5f36 5e33  ^5*x_12^3,-x_6^3
│ │ │ │ -00018f70: 2a78 5f37 5e33 2a78 5f39 5e35 2b33 2a78  *x_7^3*x_9^5+3*x
│ │ │ │ -00018f80: 5f34 2a78 5f36 5e32 2a78 5f37 5e32 7c0a  _4*x_6^2*x_7^2|.
│ │ │ │ -00018f90: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ -00018fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00018fe0: 7c2a 785f 395e 362d 332a 785f 345e 322a  |*x_9^6-3*x_4^2*
│ │ │ │ -00018ff0: 785f 362a 785f 372a 785f 395e 372b 785f  x_6*x_7*x_9^7+x_
│ │ │ │ -00019000: 345e 332a 785f 395e 382c 785f 385e 382a  4^3*x_9^8,x_8^8*
│ │ │ │ -00019010: 785f 3130 5e33 2d33 2a78 5f37 2a78 5f38  x_10^3-3*x_7*x_8
│ │ │ │ -00019020: 5e37 2a78 5f31 305e 322a 785f 3131 7c0a  ^7*x_10^2*x_11|.
│ │ │ │ -00019030: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ -00019040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00019080: 7c2b 332a 785f 375e 322a 785f 385e 362a  |+3*x_7^2*x_8^6*
│ │ │ │ -00019090: 785f 3130 2a78 5f31 315e 322d 785f 375e  x_10*x_11^2-x_7^
│ │ │ │ -000190a0: 332a 785f 385e 352a 785f 3131 5e33 2c78  3*x_8^5*x_11^3,x
│ │ │ │ -000190b0: 5f32 5e35 2a78 5f33 5e33 2a78 5f31 315e  _2^5*x_3^3*x_11^
│ │ │ │ -000190c0: 332d 332a 785f 325e 362a 785f 335e 7c0a  3-3*x_2^6*x_3^|.
│ │ │ │ -000190d0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ -000190e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000190f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -00019120: 7c32 2a78 5f31 315e 322a 785f 3132 2b33  |2*x_11^2*x_12+3
│ │ │ │ -00019130: 2a78 5f32 5e37 2a78 5f33 2a78 5f31 312a  *x_2^7*x_3*x_11*
│ │ │ │ -00019140: 785f 3132 5e32 2d78 5f32 5e38 2a78 5f31  x_12^2-x_2^8*x_1
│ │ │ │ -00019150: 325e 3329 3b20 2020 2020 2020 2020 2020  2^3);           
│ │ │ │ -00019160: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00019170: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -00019180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 3233  | -- used 0.0023
│ │ │ │ -00019220: 3133 3638 7320 2863 7075 293b 2030 2e30  1368s (cpu); 0.0
│ │ │ │ -00019230: 3032 3533 3933 3873 2028 7468 7265 6164  0253938s (thread
│ │ │ │ -00019240: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ +00018a90: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00018aa0: 5a5a 2020 2020 2020 2020 2020 2020 2020  ZZ              
│ │ │ │ +00018ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018ad0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00018ae0: 7c6f 3820 3a20 4964 6561 6c20 6f66 202d  |o8 : Ideal of -
│ │ │ │ +00018af0: 2d2d 5b78 2020 2c20 7820 2c20 7820 2c20  --[x  , x , x , 
│ │ │ │ +00018b00: 7820 2c20 7820 202c 2078 202c 2078 202c  x , x  , x , x ,
│ │ │ │ +00018b10: 2078 2020 2c20 7820 2c20 7820 2c20 7820   x  , x , x , x 
│ │ │ │ +00018b20: 2c20 7820 5d20 2020 2020 2020 2020 7c0a  , x ]         |.
│ │ │ │ +00018b30: 7c20 2020 2020 2020 2020 2020 2020 2031  |              1
│ │ │ │ +00018b40: 3237 2020 3131 2020 2038 2020 2031 2020  27  11   8   1  
│ │ │ │ +00018b50: 2039 2020 2031 3220 2020 3620 2020 3520   9   12   6   5 
│ │ │ │ +00018b60: 2020 3130 2020 2032 2020 2034 2020 2033    10   2   4   3
│ │ │ │ +00018b70: 2020 2037 2020 2020 2020 2020 2020 7c0a     7          |.
│ │ │ │ +00018b80: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00018b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00018bd0: 7c2d 785f 315e 332a 785f 325e 352a 785f  |-x_1^3*x_2^5*x_
│ │ │ │ +00018be0: 3131 5e33 2c78 5f35 5e35 2a78 5f36 5e33  11^3,x_5^5*x_6^3
│ │ │ │ +00018bf0: 2a78 5f31 315e 332d 332a 785f 355e 362a  *x_11^3-3*x_5^6*
│ │ │ │ +00018c00: 785f 365e 322a 785f 3131 5e32 2a78 5f31  x_6^2*x_11^2*x_1
│ │ │ │ +00018c10: 322b 332a 785f 355e 372a 785f 362a 7c0a  2+3*x_5^7*x_6*|.
│ │ │ │ +00018c20: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00018c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00018c70: 7c78 5f31 312a 785f 3132 5e32 2d78 5f35  |x_11*x_12^2-x_5
│ │ │ │ +00018c80: 5e38 2a78 5f31 325e 332c 785f 315e 352a  ^8*x_12^3,x_1^5*
│ │ │ │ +00018c90: 785f 325e 332a 785f 345e 332d 332a 785f  x_2^3*x_4^3-3*x_
│ │ │ │ +00018ca0: 315e 362a 785f 325e 322a 785f 345e 322a  1^6*x_2^2*x_4^2*
│ │ │ │ +00018cb0: 785f 352b 332a 785f 315e 372a 785f 7c0a  x_5+3*x_1^7*x_|.
│ │ │ │ +00018cc0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00018cd0: 2d2d 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 7c0a  --------------|.
│ │ │ │ +00018d10: 7c32 2a78 5f34 2a78 5f35 5e32 2d78 5f31  |2*x_4*x_5^2-x_1
│ │ │ │ +00018d20: 5e38 2a78 5f35 5e33 2c78 5f36 5e38 2a78  ^8*x_5^3,x_6^8*x
│ │ │ │ +00018d30: 5f31 315e 332d 332a 785f 352a 785f 365e  _11^3-3*x_5*x_6^
│ │ │ │ +00018d40: 372a 785f 3131 5e32 2a78 5f31 322b 332a  7*x_11^2*x_12+3*
│ │ │ │ +00018d50: 785f 355e 322a 785f 365e 362a 785f 7c0a  x_5^2*x_6^6*x_|.
│ │ │ │ +00018d60: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00018d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00018db0: 7c31 312a 785f 3132 5e32 2d78 5f35 5e33  |11*x_12^2-x_5^3
│ │ │ │ +00018dc0: 2a78 5f36 5e35 2a78 5f31 325e 332c 785f  *x_6^5*x_12^3,x_
│ │ │ │ +00018dd0: 385e 332a 785f 3130 5e38 2d33 2a78 5f37  8^3*x_10^8-3*x_7
│ │ │ │ +00018de0: 2a78 5f38 5e32 2a78 5f31 305e 372a 785f  *x_8^2*x_10^7*x_
│ │ │ │ +00018df0: 3131 2b33 2a78 5f37 5e32 2a78 5f38 7c0a  11+3*x_7^2*x_8|.
│ │ │ │ +00018e00: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00018e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00018e50: 7c2a 785f 3130 5e36 2a78 5f31 315e 322d  |*x_10^6*x_11^2-
│ │ │ │ +00018e60: 785f 375e 332a 785f 3130 5e35 2a78 5f31  x_7^3*x_10^5*x_1
│ │ │ │ +00018e70: 315e 332c 785f 325e 382a 785f 345e 332d  1^3,x_2^8*x_4^3-
│ │ │ │ +00018e80: 332a 785f 312a 785f 325e 372a 785f 345e  3*x_1*x_2^7*x_4^
│ │ │ │ +00018e90: 322a 785f 352b 332a 785f 315e 322a 7c0a  2*x_5+3*x_1^2*|.
│ │ │ │ +00018ea0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00018eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00018ef0: 7c78 5f32 5e36 2a78 5f34 2a78 5f35 5e32  |x_2^6*x_4*x_5^2
│ │ │ │ +00018f00: 2d78 5f31 5e33 2a78 5f32 5e35 2a78 5f35  -x_1^3*x_2^5*x_5
│ │ │ │ +00018f10: 5e33 2c2d 785f 365e 332a 785f 3131 5e38  ^3,-x_6^3*x_11^8
│ │ │ │ +00018f20: 2b33 2a78 5f35 2a78 5f36 5e32 2a78 5f31  +3*x_5*x_6^2*x_1
│ │ │ │ +00018f30: 315e 372a 785f 3132 2d33 2a78 5f35 7c0a  1^7*x_12-3*x_5|.
│ │ │ │ +00018f40: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00018f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00018f90: 7c5e 322a 785f 362a 785f 3131 5e36 2a78  |^2*x_6*x_11^6*x
│ │ │ │ +00018fa0: 5f31 325e 322b 785f 355e 332a 785f 3131  _12^2+x_5^3*x_11
│ │ │ │ +00018fb0: 5e35 2a78 5f31 325e 332c 2d78 5f36 5e33  ^5*x_12^3,-x_6^3
│ │ │ │ +00018fc0: 2a78 5f37 5e33 2a78 5f39 5e35 2b33 2a78  *x_7^3*x_9^5+3*x
│ │ │ │ +00018fd0: 5f34 2a78 5f36 5e32 2a78 5f37 5e32 7c0a  _4*x_6^2*x_7^2|.
│ │ │ │ +00018fe0: 7c2d 2d2d 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 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00019030: 7c2a 785f 395e 362d 332a 785f 345e 322a  |*x_9^6-3*x_4^2*
│ │ │ │ +00019040: 785f 362a 785f 372a 785f 395e 372b 785f  x_6*x_7*x_9^7+x_
│ │ │ │ +00019050: 345e 332a 785f 395e 382c 785f 385e 382a  4^3*x_9^8,x_8^8*
│ │ │ │ +00019060: 785f 3130 5e33 2d33 2a78 5f37 2a78 5f38  x_10^3-3*x_7*x_8
│ │ │ │ +00019070: 5e37 2a78 5f31 305e 322a 785f 3131 7c0a  ^7*x_10^2*x_11|.
│ │ │ │ +00019080: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00019090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000190a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000190b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000190c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +000190d0: 7c2b 332a 785f 375e 322a 785f 385e 362a  |+3*x_7^2*x_8^6*
│ │ │ │ +000190e0: 785f 3130 2a78 5f31 315e 322d 785f 375e  x_10*x_11^2-x_7^
│ │ │ │ +000190f0: 332a 785f 385e 352a 785f 3131 5e33 2c78  3*x_8^5*x_11^3,x
│ │ │ │ +00019100: 5f32 5e35 2a78 5f33 5e33 2a78 5f31 315e  _2^5*x_3^3*x_11^
│ │ │ │ +00019110: 332d 332a 785f 325e 362a 785f 335e 7c0a  3-3*x_2^6*x_3^|.
│ │ │ │ +00019120: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ +00019130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00019170: 7c32 2a78 5f31 315e 322a 785f 3132 2b33  |2*x_11^2*x_12+3
│ │ │ │ +00019180: 2a78 5f32 5e37 2a78 5f33 2a78 5f31 312a  *x_2^7*x_3*x_11*
│ │ │ │ +00019190: 785f 3132 5e32 2d78 5f32 5e38 2a78 5f31  x_12^2-x_2^8*x_1
│ │ │ │ +000191a0: 325e 3329 3b20 2020 2020 2020 2020 2020  2^3);           
│ │ │ │ +000191b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000191c0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +000191d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000191e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000191f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00019210: 7c69 3920 3a20 7469 6d65 2069 7343 6f64  |i9 : time isCod
│ │ │ │ +00019220: 696d 4174 4c65 6173 7428 352c 2049 2c20  imAtLeast(5, I, 
│ │ │ │ +00019230: 5061 6972 4c69 6d69 7420 3d3e 2035 2c20  PairLimit => 5, 
│ │ │ │ +00019240: 5665 7262 6f73 653d 3e74 7275 6529 2020  Verbose=>true)  
│ │ │ │  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  |               
│ │ │ │ -000192c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000192d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000192e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000192f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00019300: 7c6f 3920 3d20 7472 7565 2020 2020 2020  |o9 = true      
│ │ │ │ +00019260: 7c20 2d2d 2075 7365 6420 302e 3030 3339  | -- used 0.0039
│ │ │ │ +00019270: 3632 3235 7320 2863 7075 293b 2030 2e30  6225s (cpu); 0.0
│ │ │ │ +00019280: 3035 3130 3536 3273 2028 7468 7265 6164  0510562s (thread
│ │ │ │ +00019290: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ +000192a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000192b0: 7c69 7343 6f64 696d 4174 4c65 6173 743a  |isCodimAtLeast:
│ │ │ │ +000192c0: 2043 6f6d 7075 7469 6e67 2063 6f64 696d   Computing codim
│ │ │ │ +000192d0: 206f 6620 6d6f 6e6f 6d69 616c 7320 6261   of monomials ba
│ │ │ │ +000192e0: 7365 6420 6f6e 2069 6465 616c 2067 656e  sed on ideal gen
│ │ │ │ +000192f0: 6572 6174 6f72 732e 2020 2020 2020 7c0a  erators.      |.
│ │ │ │ +00019300: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00019310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019340: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00019350: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -00019360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 362e 3036 3034  | -- used 6.0604
│ │ │ │ -00019400: 652d 3035 7320 2863 7075 293b 2030 2e30  e-05s (cpu); 0.0
│ │ │ │ -00019410: 3032 3433 3835 3173 2028 7468 7265 6164  0243851s (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                  
│ │ │ │ +00019350: 7c6f 3920 3d20 7472 7565 2020 2020 2020  |o9 = true      
│ │ │ │ +00019360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019390: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000193a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +000193b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000193c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000193d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000193e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000193f0: 7c69 3130 203a 2074 696d 6520 6973 436f  |i10 : time isCo
│ │ │ │ +00019400: 6469 6d41 744c 6561 7374 2835 2c20 492c  dimAtLeast(5, I,
│ │ │ │ +00019410: 2050 6169 724c 696d 6974 203d 3e20 3230   PairLimit => 20
│ │ │ │ +00019420: 302c 2056 6572 626f 7365 3d3e 6661 6c73  0, Verbose=>fals
│ │ │ │ +00019430: 6529 2020 2020 2020 2020 2020 2020 7c0a  e)            |.
│ │ │ │ +00019440: 7c20 2d2d 2075 7365 6420 302e 3030 3438  | -- used 0.0048
│ │ │ │ +00019450: 3930 3831 7320 2863 7075 293b 2030 2e30  9081s (cpu); 0.0
│ │ │ │ +00019460: 3034 3833 3438 3673 2028 7468 7265 6164  0483486s (thread
│ │ │ │ +00019470: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │  00019480: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00019490: 7c6f 3130 203d 2074 7275 6520 2020 2020  |o10 = true     
│ │ │ │ +00019490: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000194a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000194b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000194c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000194d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000194e0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -000194f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00019530: 0a4e 6f74 6963 6520 696e 2074 6865 2066  .Notice in the f
│ │ │ │ -00019540: 6972 7374 2063 6173 6520 7468 6520 6675  irst case the fu
│ │ │ │ -00019550: 6e63 7469 6f6e 2072 6574 7572 6e65 6420  nction returned 
│ │ │ │ -00019560: 6e75 6c6c 2c20 6265 6361 7573 6520 7468  null, because th
│ │ │ │ -00019570: 6520 6465 7074 6820 6f66 0a73 6561 7263  e depth of.searc
│ │ │ │ -00019580: 6820 7761 7320 6e6f 7420 6869 6768 2065  h was not high e
│ │ │ │ -00019590: 6e6f 7567 682e 2020 4974 206f 6e6c 7920  nough.  It only 
│ │ │ │ -000195a0: 636f 6d70 7574 6564 2063 6f64 696d 2035  computed codim 5
│ │ │ │ -000195b0: 2074 696d 6573 2e20 2054 6865 2073 6563   times.  The sec
│ │ │ │ -000195c0: 6f6e 640a 7265 7475 726e 6564 2074 7275  ond.returned tru
│ │ │ │ -000195d0: 652c 2062 7574 2069 7420 6469 6420 736f  e, but it did so
│ │ │ │ -000195e0: 2061 7320 736f 6f6e 2061 7320 7468 6520   as soon as the 
│ │ │ │ -000195f0: 616e 7377 6572 2077 6173 2066 6f75 6e64  answer was found
│ │ │ │ -00019600: 2028 616e 6420 6265 666f 7265 2077 6520   (and before we 
│ │ │ │ -00019610: 6869 740a 7468 6520 5061 6972 4c69 6d69  hit.the PairLimi
│ │ │ │ -00019620: 7420 6c69 6d69 7429 2e0a 0a57 6179 7320  t limit)...Ways 
│ │ │ │ -00019630: 746f 2075 7365 2069 7343 6f64 696d 4174  to use isCodimAt
│ │ │ │ -00019640: 4c65 6173 743a 0a3d 3d3d 3d3d 3d3d 3d3d  Least:.=========
│ │ │ │ -00019650: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00019660: 3d3d 0a0a 2020 2a20 2269 7343 6f64 696d  ==..  * "isCodim
│ │ │ │ -00019670: 4174 4c65 6173 7428 5a5a 2c49 6465 616c  AtLeast(ZZ,Ideal
│ │ │ │ -00019680: 2922 0a0a 466f 7220 7468 6520 7072 6f67  )"..For the prog
│ │ │ │ -00019690: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d  rammer.=========
│ │ │ │ -000196a0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f  =========..The o
│ │ │ │ -000196b0: 626a 6563 7420 2a6e 6f74 6520 6973 436f  bject *note isCo
│ │ │ │ -000196c0: 6469 6d41 744c 6561 7374 3a20 6973 436f  dimAtLeast: isCo
│ │ │ │ -000196d0: 6469 6d41 744c 6561 7374 2c20 6973 2061  dimAtLeast, is a
│ │ │ │ -000196e0: 202a 6e6f 7465 206d 6574 686f 6420 6675   *note method fu
│ │ │ │ -000196f0: 6e63 7469 6f6e 0a77 6974 6820 6f70 7469  nction.with opti
│ │ │ │ -00019700: 6f6e 733a 2028 4d61 6361 756c 6179 3244  ons: (Macaulay2D
│ │ │ │ -00019710: 6f63 294d 6574 686f 6446 756e 6374 696f  oc)MethodFunctio
│ │ │ │ -00019720: 6e57 6974 684f 7074 696f 6e73 2c2e 0a0a  nWithOptions,...
│ │ │ │ -00019730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a  ---------------.
│ │ │ │ -00019780: 0a54 6865 2073 6f75 7263 6520 6f66 2074  .The source of t
│ │ │ │ -00019790: 6869 7320 646f 6375 6d65 6e74 2069 7320  his document is 
│ │ │ │ -000197a0: 696e 0a2f 6275 696c 642f 7265 7072 6f64  in./build/reprod
│ │ │ │ -000197b0: 7563 6962 6c65 2d70 6174 682f 6d61 6361  ucible-path/maca
│ │ │ │ -000197c0: 756c 6179 322d 312e 3235 2e31 312b 6473  ulay2-1.25.11+ds
│ │ │ │ -000197d0: 2f4d 322f 4d61 6361 756c 6179 322f 7061  /M2/Macaulay2/pa
│ │ │ │ -000197e0: 636b 6167 6573 2f46 6173 744d 696e 6f72  ckages/FastMinor
│ │ │ │ -000197f0: 732e 0a6d 323a 3232 3436 3a30 2e0a 1f0a  s..m2:2246:0....
│ │ │ │ -00019800: 4669 6c65 3a20 4661 7374 4d69 6e6f 7273  File: FastMinors
│ │ │ │ -00019810: 2e69 6e66 6f2c 204e 6f64 653a 2069 7344  .info, Node: isD
│ │ │ │ -00019820: 696d 4174 4d6f 7374 2c20 4e65 7874 3a20  imAtMost, Next: 
│ │ │ │ -00019830: 6973 5261 6e6b 4174 4c65 6173 742c 2050  isRankAtLeast, P
│ │ │ │ -00019840: 7265 763a 2069 7343 6f64 696d 4174 4c65  rev: isCodimAtLe
│ │ │ │ -00019850: 6173 742c 2055 703a 2054 6f70 0a0a 6973  ast, Up: Top..is
│ │ │ │ -00019860: 4469 6d41 744d 6f73 7420 2d2d 2072 6574  DimAtMost -- ret
│ │ │ │ -00019870: 7572 6e73 2074 7275 6520 6966 2077 6520  urns true if we 
│ │ │ │ -00019880: 6361 6e20 7175 6963 6b6c 7920 7365 6520  can quickly see 
│ │ │ │ -00019890: 7768 6574 6865 7220 7468 6520 6469 6d20  whether the dim 
│ │ │ │ -000198a0: 6973 2061 7420 6d6f 7374 2061 2067 6976  is at most a giv
│ │ │ │ -000198b0: 656e 206e 756d 6265 720a 2a2a 2a2a 2a2a  en number.******
│ │ │ │ -000198c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000198d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000198e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000198f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00019900: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00019910: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765  *****..  * Usage
│ │ │ │ -00019920: 3a20 0a20 2020 2020 2020 2069 7344 696d  : .        isDim
│ │ │ │ -00019930: 4174 4d6f 7374 286e 2c20 4929 0a20 202a  AtMost(n, I).  *
│ │ │ │ -00019940: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ -00019950: 206e 2c20 616e 202a 6e6f 7465 2069 6e74   n, an *note int
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│ │ │ │ -00019970: 446f 6329 5a5a 2c2c 2061 6e20 696e 7465  Doc)ZZ,, an inte
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│ │ │ │ -00019990: 6e20 2a6e 6f74 6520 6964 6561 6c3a 2028  n *note ideal: (
│ │ │ │ -000199a0: 4d61 6361 756c 6179 3244 6f63 2949 6465  Macaulay2Doc)Ide
│ │ │ │ -000199b0: 616c 2c2c 2061 6e20 6964 6561 6c20 696e  al,, an ideal in
│ │ │ │ -000199c0: 2061 2070 6f6c 796e 6f6d 6961 6c20 7269   a polynomial ri
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│ │ │ │ -000199e0: 6120 6669 656c 642c 206f 7220 6120 7175  a field, or a qu
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│ │ │ │ -00019a10: 7469 6f6e 616c 2069 6e70 7574 733a 2028  tional inputs: (
│ │ │ │ -00019a20: 4d61 6361 756c 6179 3244 6f63 2975 7369  Macaulay2Doc)usi
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│ │ │ │ -00019a60: 4c69 6d69 7420 3d3e 202e 2e2e 2c20 6465  Limit => ..., de
│ │ │ │ -00019a70: 6661 756c 7420 7661 6c75 6520 3130 300a  fault value 100.
│ │ │ │ -00019a80: 2020 2020 2020 2a20 5350 6169 7273 4675        * SPairsFu
│ │ │ │ -00019a90: 6e63 7469 6f6e 203d 3e20 2e2e 2e2c 2064  nction => ..., d
│ │ │ │ -00019aa0: 6566 6175 6c74 2076 616c 7565 0a20 2020  efault value.   
│ │ │ │ -00019ab0: 2020 2020 2046 756e 6374 696f 6e43 6c6f       FunctionClo
│ │ │ │ -00019ac0: 7375 7265 5b2e 2e2f 4661 7374 4d69 6e6f  sure[../FastMino
│ │ │ │ -00019ad0: 7273 2e6d 323a 3231 313a 3233 2d32 3131  rs.m2:211:23-211
│ │ │ │ -00019ae0: 3a34 325d 0a20 2020 2020 202a 2056 6572  :42].      * Ver
│ │ │ │ -00019af0: 626f 7365 203d 3e20 2e2e 2e2c 2064 6566  bose => ..., def
│ │ │ │ -00019b00: 6175 6c74 2076 616c 7565 2066 616c 7365  ault value false
│ │ │ │ -00019b10: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ -00019b20: 2020 2020 2a20 7472 7565 2069 6620 7468      * true if th
│ │ │ │ -00019b30: 6520 6469 6d65 6e73 696f 6e20 6f66 2049  e dimension of I
│ │ │ │ -00019b40: 2069 7320 6174 206d 6f73 7420 6e20 6f72   is at most n or
│ │ │ │ -00019b50: 206e 756c 6c20 6966 2074 6865 2066 756e   null if the fun
│ │ │ │ -00019b60: 6374 696f 6e20 6361 6e6e 6f74 0a20 2020  ction cannot.   
│ │ │ │ -00019b70: 2020 2020 2074 656c 6c20 7768 6574 6865       tell whethe
│ │ │ │ -00019b80: 7220 7468 6520 6469 6d65 6e73 696f 6e20  r the dimension 
│ │ │ │ -00019b90: 6973 2061 7420 6d6f 7374 206e 0a0a 4465  is at most n..De
│ │ │ │ -00019ba0: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ -00019bb0: 3d3d 3d3d 3d0a 0a54 6869 7320 7369 6d70  =====..This simp
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│ │ │ │ -00019bd0: 4174 4c65 6173 742c 2070 6173 7369 6e67  AtLeast, passing
│ │ │ │ -00019be0: 206f 7074 696f 6e73 2061 7320 6465 7363   options as desc
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│ │ │ │ -00019c10: 0a20 202a 202a 6e6f 7465 2069 7343 6f64  .  * *note isCod
│ │ │ │ -00019c20: 696d 4174 4c65 6173 743a 2069 7343 6f64  imAtLeast: isCod
│ │ │ │ -00019c30: 696d 4174 4c65 6173 742c 202d 2d20 7265  imAtLeast, -- re
│ │ │ │ -00019c40: 7475 726e 7320 7472 7565 2069 6620 7765  turns true if we
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│ │ │ │ -00019c60: 0a20 2020 2077 6865 7468 6572 2074 6865  .    whether the
│ │ │ │ -00019c70: 2063 6f64 696d 2069 7320 6174 206c 6561   codim is at lea
│ │ │ │ -00019c80: 7374 2061 2067 6976 656e 206e 756d 6265  st a given numbe
│ │ │ │ -00019c90: 720a 0a57 6179 7320 746f 2075 7365 2069  r..Ways to use i
│ │ │ │ -00019ca0: 7344 696d 4174 4d6f 7374 3a0a 3d3d 3d3d  sDimAtMost:.====
│ │ │ │ -00019cb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00019cc0: 3d3d 3d3d 0a0a 2020 2a20 2269 7344 696d  ====..  * "isDim
│ │ │ │ -00019cd0: 4174 4d6f 7374 285a 5a2c 4964 6561 6c29  AtMost(ZZ,Ideal)
│ │ │ │ -00019ce0: 220a 0a46 6f72 2074 6865 2070 726f 6772  "..For the progr
│ │ │ │ -00019cf0: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d  ammer.==========
│ │ │ │ -00019d00: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62  ========..The ob
│ │ │ │ -00019d10: 6a65 6374 202a 6e6f 7465 2069 7344 696d  ject *note isDim
│ │ │ │ -00019d20: 4174 4d6f 7374 3a20 6973 4469 6d41 744d  AtMost: isDimAtM
│ │ │ │ -00019d30: 6f73 742c 2069 7320 6120 2a6e 6f74 6520  ost, is a *note 
│ │ │ │ -00019d40: 6d65 7468 6f64 2066 756e 6374 696f 6e20  method function 
│ │ │ │ -00019d50: 7769 7468 0a6f 7074 696f 6e73 3a20 284d  with.options: (M
│ │ │ │ -00019d60: 6163 6175 6c61 7932 446f 6329 4d65 7468  acaulay2Doc)Meth
│ │ │ │ -00019d70: 6f64 4675 6e63 7469 6f6e 5769 7468 4f70  odFunctionWithOp
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│ │ │ │ -00019d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019dd0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f  --------..The so
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│ │ │ │ -00019e10: 7061 7468 2f6d 6163 6175 6c61 7932 2d31  path/macaulay2-1
│ │ │ │ -00019e20: 2e32 352e 3131 2b64 732f 4d32 2f4d 6163  .25.11+ds/M2/Mac
│ │ │ │ -00019e30: 6175 6c61 7932 2f70 6163 6b61 6765 732f  aulay2/packages/
│ │ │ │ -00019e40: 4661 7374 4d69 6e6f 7273 2e0a 6d32 3a32  FastMinors..m2:2
│ │ │ │ -00019e50: 3237 323a 302e 0a1f 0a46 696c 653a 2046  272:0....File: F
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│ │ │ │ -00019ea0: 645f 7064 5f63 6d54 6872 6561 6473 3d3e  d_pd_cmThreads=>
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│ │ │ │ -00019ec0: 6576 3a20 6973 4469 6d41 744d 6f73 742c  ev: isDimAtMost,
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│ │ │ │ -00019ee0: 4174 4c65 6173 7420 2d2d 2064 6574 6572  AtLeast -- deter
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│ │ │ │ -00019f20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00019f30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00019f40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00019f50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00019f60: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20  ***..  * Usage: 
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│ │ │ │ -00019f90: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ -00019fa0: 202a 206e 312c 2061 6e20 2a6e 6f74 6520   * n1, an *note 
│ │ │ │ -00019fb0: 696e 7465 6765 723a 2028 4d61 6361 756c  integer: (Macaul
│ │ │ │ -00019fc0: 6179 3244 6f63 295a 5a2c 2c20 0a20 2020  ay2Doc)ZZ,, .   
│ │ │ │ -00019fd0: 2020 202a 204d 312c 2061 202a 6e6f 7465     * M1, a *note
│ │ │ │ -00019fe0: 206d 6174 7269 783a 2028 4d61 6361 756c   matrix: (Macaul
│ │ │ │ -00019ff0: 6179 3244 6f63 294d 6174 7269 782c 2c20  ay2Doc)Matrix,, 
│ │ │ │ -0001a000: 0a20 202a 202a 6e6f 7465 204f 7074 696f  .  * *note Optio
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│ │ │ │ -0001a030: 6675 6e63 7469 6f6e 7320 7769 7468 206f  functions with o
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│ │ │ │ -0001a060: 6574 5374 7261 7465 6779 3a20 4465 7453  etStrategy: DetS
│ │ │ │ -0001a070: 7472 6174 6567 792c 203d 3e20 2e2e 2e2c  trategy, => ...,
│ │ │ │ -0001a080: 2064 6566 6175 6c74 2076 616c 7565 2052   default value R
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│ │ │ │ -0001a0d0: 6368 6f6f 7365 2068 6f77 2064 6574 6572  choose how deter
│ │ │ │ -0001a0e0: 6d69 6e61 6e74 7320 286f 720a 2020 2020  minants (or.    
│ │ │ │ -0001a0f0: 2020 2020 7261 6e6b 292c 2069 7320 636f      rank), is co
│ │ │ │ -0001a100: 6d70 7574 6564 0a20 2020 2020 202a 202a  mputed.      * *
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│ │ │ │ -0001a120: 4d61 784d 696e 6f72 732c 203d 3e20 2e2e  MaxMinors, => ..
│ │ │ │ -0001a130: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ -0001a140: 206e 756c 6c2c 2061 6e20 6f70 7469 6f6e   null, an option
│ │ │ │ -0001a150: 2074 6f0a 2020 2020 2020 2020 636f 6e74   to.        cont
│ │ │ │ -0001a160: 726f 6c20 6465 7074 6820 6f66 2073 6561  rol depth of sea
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│ │ │ │ -0001a180: 6520 506f 696e 744f 7074 696f 6e73 3a20  e PointOptions: 
│ │ │ │ -0001a190: 506f 696e 744f 7074 696f 6e73 2c20 3d3e  PointOptions, =>
│ │ │ │ -0001a1a0: 202e 2e2e 2c20 6465 6661 756c 7420 7661   ..., default va
│ │ │ │ -0001a1b0: 6c75 6520 7b53 7472 6174 6567 7920 3d3e  lue {Strategy =>
│ │ │ │ -0001a1c0: 0a20 2020 2020 2020 2044 6566 6175 6c74  .        Default
│ │ │ │ -0001a1d0: 2c20 486f 6d6f 6765 6e65 6f75 7320 3d3e  , Homogeneous =>
│ │ │ │ -0001a1e0: 2066 616c 7365 2c20 5265 706c 6163 656d   false, Replacem
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│ │ │ │ -0001a200: 2045 7874 656e 6446 6965 6c64 203d 3e0a   ExtendField =>.
│ │ │ │ -0001a210: 2020 2020 2020 2020 7472 7565 2c20 506f          true, Po
│ │ │ │ -0001a220: 696e 7443 6865 636b 4174 7465 6d70 7473  intCheckAttempts
│ │ │ │ -0001a230: 203d 3e20 302c 2044 6563 6f6d 706f 7369   => 0, Decomposi
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│ │ │ │ -0001a250: 4465 636f 6d70 6f73 652c 0a20 2020 2020  Decompose,.     
│ │ │ │ -0001a260: 2020 204e 756d 5468 7265 6164 7354 6f55     NumThreadsToU
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│ │ │ │ -0001a280: 6f6e 4675 6e63 7469 6f6e 203d 3e20 6469  onFunction => di
│ │ │ │ -0001a290: 6d2c 2056 6572 626f 7365 203d 3e20 6661  m, Verbose => fa
│ │ │ │ -0001a2a0: 6c73 657d 2c0a 2020 2020 2020 2020 6f70  lse},.        op
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│ │ │ │ -0001a2c0: 2066 756e 6374 696f 6e73 2069 6e20 7468   functions in th
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│ │ │ │ -0001a2e0: 506f 696e 7473 0a20 2020 2020 202a 202a  Points.      * *
│ │ │ │ -0001a2f0: 6e6f 7465 2053 7472 6174 6567 793a 2053  note Strategy: S
│ │ │ │ -0001a300: 7472 6174 6567 7944 6566 6175 6c74 2c20  trategyDefault, 
│ │ │ │ -0001a310: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ -0001a320: 7661 6c75 6520 6e65 7720 4f70 7469 6f6e  value new Option
│ │ │ │ -0001a330: 5461 626c 650a 2020 2020 2020 2020 6672  Table.        fr
│ │ │ │ -0001a340: 6f6d 207b 506f 696e 7473 203d 3e20 302c  om {Points => 0,
│ │ │ │ -0001a350: 2052 616e 646f 6d20 3d3e 2030 2c20 4752   Random => 0, GR
│ │ │ │ -0001a360: 6576 4c65 784c 6172 6765 7374 203d 3e20  evLexLargest => 
│ │ │ │ -0001a370: 302c 204c 6578 536d 616c 6c65 7374 5465  0, LexSmallestTe
│ │ │ │ -0001a380: 726d 203d 3e0a 2020 2020 2020 2020 3235  rm =>.        25
│ │ │ │ -0001a390: 2c20 4c65 784c 6172 6765 7374 203d 3e20  , LexLargest => 
│ │ │ │ -0001a3a0: 302c 204c 6578 536d 616c 6c65 7374 203d  0, LexSmallest =
│ │ │ │ -0001a3b0: 3e20 3235 2c20 4752 6576 4c65 7853 6d61  > 25, GRevLexSma
│ │ │ │ -0001a3c0: 6c6c 6573 7454 6572 6d20 3d3e 2032 352c  llestTerm => 25,
│ │ │ │ -0001a3d0: 0a20 2020 2020 2020 2052 616e 646f 6d4e  .        RandomN
│ │ │ │ -0001a3e0: 6f6e 7a65 726f 203d 3e20 302c 2047 5265  onzero => 0, GRe
│ │ │ │ -0001a3f0: 764c 6578 536d 616c 6c65 7374 203d 3e20  vLexSmallest => 
│ │ │ │ -0001a400: 3235 7d2c 2073 7472 6174 6567 6965 7320  25}, strategies 
│ │ │ │ -0001a410: 666f 7220 6368 6f6f 7369 6e67 0a20 2020  for choosing.   
│ │ │ │ -0001a420: 2020 2020 2073 7562 6d61 7472 6963 6573       submatrices
│ │ │ │ -0001a430: 0a20 2020 2020 202a 202a 6e6f 7465 2054  .      * *note T
│ │ │ │ -0001a440: 6872 6561 6473 3a20 6973 5261 6e6b 4174  hreads: isRankAt
│ │ │ │ -0001a450: 4c65 6173 745f 6c70 5f70 645f 7064 5f70  Least_lp_pd_pd_p
│ │ │ │ -0001a460: 645f 636d 5468 7265 6164 733d 3e5f 7064  d_cmThreads=>_pd
│ │ │ │ -0001a470: 5f70 645f 7064 5f72 702c 203d 3e0a 2020  _pd_pd_rp, =>.  
│ │ │ │ -0001a480: 2020 2020 2020 2e2e 2e2c 2064 6566 6175        ..., defau
│ │ │ │ -0001a490: 6c74 2076 616c 7565 2031 2c20 616e 206f  lt value 1, an o
│ │ │ │ -0001a4a0: 7074 696f 6e20 666f 7220 7661 7269 6f75  ption for variou
│ │ │ │ -0001a4b0: 7320 6675 6e63 7469 6f6e 730a 2020 2020  s functions.    
│ │ │ │ -0001a4c0: 2020 2a20 5665 7262 6f73 6520 3d3e 202e    * Verbose => .
│ │ │ │ -0001a4d0: 2e2e 2c20 6465 6661 756c 7420 7661 6c75  .., default valu
│ │ │ │ -0001a4e0: 6520 6661 6c73 650a 2020 2a20 4f75 7470  e false.  * Outp
│ │ │ │ -0001a4f0: 7574 733a 0a20 2020 2020 202a 2061 202a  uts:.      * a *
│ │ │ │ -0001a500: 6e6f 7465 2042 6f6f 6c65 616e 2076 616c  note Boolean val
│ │ │ │ -0001a510: 7565 3a20 284d 6163 6175 6c61 7932 446f  ue: (Macaulay2Do
│ │ │ │ -0001a520: 6329 426f 6f6c 6561 6e2c 2c20 0a0a 4465  c)Boolean,, ..De
│ │ │ │ -0001a530: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ -0001a540: 3d3d 3d3d 3d0a 0a54 6869 7320 6675 6e63  =====..This func
│ │ │ │ -0001a550: 7469 6f6e 2074 7269 6573 2074 6f20 7175  tion tries to qu
│ │ │ │ -0001a560: 6963 6b6c 7920 6465 7465 726d 696e 6520  ickly determine 
│ │ │ │ -0001a570: 7768 6574 6865 7220 7468 6520 6d61 7472  whether the matr
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│ │ │ │ -0001a590: 616e 6b2e 0a69 7352 616e 6b41 744c 6561  ank..isRankAtLea
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│ │ │ │ +00019510: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00019790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00019870: 696d 4174 4d6f 7374 2c20 4e65 7874 3a20  imAtMost, Next: 
│ │ │ │ +00019880: 6973 5261 6e6b 4174 4c65 6173 742c 2050  isRankAtLeast, P
│ │ │ │ +00019890: 7265 763a 2069 7343 6f64 696d 4174 4c65  rev: isCodimAtLe
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│ │ │ │ +000198b0: 4469 6d41 744d 6f73 7420 2d2d 2072 6574  DimAtMost -- ret
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│ │ │ │ +000198d0: 6361 6e20 7175 6963 6b6c 7920 7365 6520  can quickly see 
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│ │ │ │ +00019900: 656e 206e 756d 6265 720a 2a2a 2a2a 2a2a  en number.******
│ │ │ │ +00019910: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00019920: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00019930: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00019940: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00019950: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00019960: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765  *****..  * Usage
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│ │ │ │ +00019980: 4174 4d6f 7374 286e 2c20 4929 0a20 202a  AtMost(n, I).  *
│ │ │ │ +00019990: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
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│ │ │ │ +00019a00: 616c 2c2c 2061 6e20 6964 6561 6c20 696e  al,, an ideal in
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│ │ │ │ +00019d00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ +00019ee0: 6b41 744c 6561 7374 5f6c 705f 7064 5f70  kAtLeast_lp_pd_p
│ │ │ │ +00019ef0: 645f 7064 5f63 6d54 6872 6561 6473 3d3e  d_pd_cmThreads=>
│ │ │ │ +00019f00: 5f70 645f 7064 5f70 645f 7270 2c20 5072  _pd_pd_pd_rp, Pr
│ │ │ │ +00019f10: 6576 3a20 6973 4469 6d41 744d 6f73 742c  ev: isDimAtMost,
│ │ │ │ +00019f20: 2055 703a 2054 6f70 0a0a 6973 5261 6e6b   Up: Top..isRank
│ │ │ │ +00019f30: 4174 4c65 6173 7420 2d2d 2064 6574 6572  AtLeast -- deter
│ │ │ │ +00019f40: 6d69 6e65 7320 6966 2074 6865 206d 6174  mines if the mat
│ │ │ │ +00019f50: 7269 7820 6861 7320 7261 6e6b 2061 7420  rix has rank at 
│ │ │ │ +00019f60: 6c65 6173 7420 6120 6e75 6d62 6572 0a2a  least a number.*
│ │ │ │ +00019f70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00019f80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00019f90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00019fa0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00019fb0: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20  ***..  * Usage: 
│ │ │ │ +00019fc0: 0a20 2020 2020 2020 2069 7352 616e 6b41  .        isRankA
│ │ │ │ +00019fd0: 744c 6561 7374 286e 312c 204d 3129 0a20  tLeast(n1, M1). 
│ │ │ │ +00019fe0: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ +00019ff0: 202a 206e 312c 2061 6e20 2a6e 6f74 6520   * n1, an *note 
│ │ │ │ +0001a000: 696e 7465 6765 723a 2028 4d61 6361 756c  integer: (Macaul
│ │ │ │ +0001a010: 6179 3244 6f63 295a 5a2c 2c20 0a20 2020  ay2Doc)ZZ,, .   
│ │ │ │ +0001a020: 2020 202a 204d 312c 2061 202a 6e6f 7465     * M1, a *note
│ │ │ │ +0001a030: 206d 6174 7269 783a 2028 4d61 6361 756c   matrix: (Macaul
│ │ │ │ +0001a040: 6179 3244 6f63 294d 6174 7269 782c 2c20  ay2Doc)Matrix,, 
│ │ │ │ +0001a050: 0a20 202a 202a 6e6f 7465 204f 7074 696f  .  * *note Optio
│ │ │ │ +0001a060: 6e61 6c20 696e 7075 7473 3a20 284d 6163  nal inputs: (Mac
│ │ │ │ +0001a070: 6175 6c61 7932 446f 6329 7573 696e 6720  aulay2Doc)using 
│ │ │ │ +0001a080: 6675 6e63 7469 6f6e 7320 7769 7468 206f  functions with o
│ │ │ │ +0001a090: 7074 696f 6e61 6c20 696e 7075 7473 2c3a  ptional inputs,:
│ │ │ │ +0001a0a0: 0a20 2020 2020 202a 202a 6e6f 7465 2044  .      * *note D
│ │ │ │ +0001a0b0: 6574 5374 7261 7465 6779 3a20 4465 7453  etStrategy: DetS
│ │ │ │ +0001a0c0: 7472 6174 6567 792c 203d 3e20 2e2e 2e2c  trategy, => ...,
│ │ │ │ +0001a0d0: 2064 6566 6175 6c74 2076 616c 7565 2052   default value R
│ │ │ │ +0001a0e0: 616e 6b2c 2044 6574 5374 7261 7465 6779  ank, DetStrategy
│ │ │ │ +0001a0f0: 0a20 2020 2020 2020 2069 7320 6120 7374  .        is a st
│ │ │ │ +0001a100: 7261 7465 6779 2066 6f72 2061 6c6c 6f77  rategy for allow
│ │ │ │ +0001a110: 696e 6720 7468 6520 7573 6572 2074 6f20  ing the user to 
│ │ │ │ +0001a120: 6368 6f6f 7365 2068 6f77 2064 6574 6572  choose how deter
│ │ │ │ +0001a130: 6d69 6e61 6e74 7320 286f 720a 2020 2020  minants (or.    
│ │ │ │ +0001a140: 2020 2020 7261 6e6b 292c 2069 7320 636f      rank), is co
│ │ │ │ +0001a150: 6d70 7574 6564 0a20 2020 2020 202a 202a  mputed.      * *
│ │ │ │ +0001a160: 6e6f 7465 204d 6178 4d69 6e6f 7273 3a20  note MaxMinors: 
│ │ │ │ +0001a170: 4d61 784d 696e 6f72 732c 203d 3e20 2e2e  MaxMinors, => ..
│ │ │ │ +0001a180: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ +0001a190: 206e 756c 6c2c 2061 6e20 6f70 7469 6f6e   null, an option
│ │ │ │ +0001a1a0: 2074 6f0a 2020 2020 2020 2020 636f 6e74   to.        cont
│ │ │ │ +0001a1b0: 726f 6c20 6465 7074 6820 6f66 2073 6561  rol depth of sea
│ │ │ │ +0001a1c0: 7263 680a 2020 2020 2020 2a20 2a6e 6f74  rch.      * *not
│ │ │ │ +0001a1d0: 6520 506f 696e 744f 7074 696f 6e73 3a20  e PointOptions: 
│ │ │ │ +0001a1e0: 506f 696e 744f 7074 696f 6e73 2c20 3d3e  PointOptions, =>
│ │ │ │ +0001a1f0: 202e 2e2e 2c20 6465 6661 756c 7420 7661   ..., default va
│ │ │ │ +0001a200: 6c75 6520 7b53 7472 6174 6567 7920 3d3e  lue {Strategy =>
│ │ │ │ +0001a210: 0a20 2020 2020 2020 2044 6566 6175 6c74  .        Default
│ │ │ │ +0001a220: 2c20 486f 6d6f 6765 6e65 6f75 7320 3d3e  , Homogeneous =>
│ │ │ │ +0001a230: 2066 616c 7365 2c20 5265 706c 6163 656d   false, Replacem
│ │ │ │ +0001a240: 656e 7420 3d3e 2042 696e 6f6d 6961 6c2c  ent => Binomial,
│ │ │ │ +0001a250: 2045 7874 656e 6446 6965 6c64 203d 3e0a   ExtendField =>.
│ │ │ │ +0001a260: 2020 2020 2020 2020 7472 7565 2c20 506f          true, Po
│ │ │ │ +0001a270: 696e 7443 6865 636b 4174 7465 6d70 7473  intCheckAttempts
│ │ │ │ +0001a280: 203d 3e20 302c 2044 6563 6f6d 706f 7369   => 0, Decomposi
│ │ │ │ +0001a290: 7469 6f6e 5374 7261 7465 6779 203d 3e20  tionStrategy => 
│ │ │ │ +0001a2a0: 4465 636f 6d70 6f73 652c 0a20 2020 2020  Decompose,.     
│ │ │ │ +0001a2b0: 2020 204e 756d 5468 7265 6164 7354 6f55     NumThreadsToU
│ │ │ │ +0001a2c0: 7365 203d 3e20 312c 2044 696d 656e 7369  se => 1, Dimensi
│ │ │ │ +0001a2d0: 6f6e 4675 6e63 7469 6f6e 203d 3e20 6469  onFunction => di
│ │ │ │ +0001a2e0: 6d2c 2056 6572 626f 7365 203d 3e20 6661  m, Verbose => fa
│ │ │ │ +0001a2f0: 6c73 657d 2c0a 2020 2020 2020 2020 6f70  lse},.        op
│ │ │ │ +0001a300: 7469 6f6e 7320 746f 2070 6173 7320 746f  tions to pass to
│ │ │ │ +0001a310: 2066 756e 6374 696f 6e73 2069 6e20 7468   functions in th
│ │ │ │ +0001a320: 6520 7061 636b 6167 6520 5261 6e64 6f6d  e package Random
│ │ │ │ +0001a330: 506f 696e 7473 0a20 2020 2020 202a 202a  Points.      * *
│ │ │ │ +0001a340: 6e6f 7465 2053 7472 6174 6567 793a 2053  note Strategy: S
│ │ │ │ +0001a350: 7472 6174 6567 7944 6566 6175 6c74 2c20  trategyDefault, 
│ │ │ │ +0001a360: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ +0001a370: 7661 6c75 6520 6e65 7720 4f70 7469 6f6e  value new Option
│ │ │ │ +0001a380: 5461 626c 650a 2020 2020 2020 2020 6672  Table.        fr
│ │ │ │ +0001a390: 6f6d 207b 506f 696e 7473 203d 3e20 302c  om {Points => 0,
│ │ │ │ +0001a3a0: 2052 616e 646f 6d20 3d3e 2030 2c20 4752   Random => 0, GR
│ │ │ │ +0001a3b0: 6576 4c65 784c 6172 6765 7374 203d 3e20  evLexLargest => 
│ │ │ │ +0001a3c0: 302c 204c 6578 536d 616c 6c65 7374 5465  0, LexSmallestTe
│ │ │ │ +0001a3d0: 726d 203d 3e0a 2020 2020 2020 2020 3235  rm =>.        25
│ │ │ │ +0001a3e0: 2c20 4c65 784c 6172 6765 7374 203d 3e20  , LexLargest => 
│ │ │ │ +0001a3f0: 302c 204c 6578 536d 616c 6c65 7374 203d  0, LexSmallest =
│ │ │ │ +0001a400: 3e20 3235 2c20 4752 6576 4c65 7853 6d61  > 25, GRevLexSma
│ │ │ │ +0001a410: 6c6c 6573 7454 6572 6d20 3d3e 2032 352c  llestTerm => 25,
│ │ │ │ +0001a420: 0a20 2020 2020 2020 2052 616e 646f 6d4e  .        RandomN
│ │ │ │ +0001a430: 6f6e 7a65 726f 203d 3e20 302c 2047 5265  onzero => 0, GRe
│ │ │ │ +0001a440: 764c 6578 536d 616c 6c65 7374 203d 3e20  vLexSmallest => 
│ │ │ │ +0001a450: 3235 7d2c 2073 7472 6174 6567 6965 7320  25}, strategies 
│ │ │ │ +0001a460: 666f 7220 6368 6f6f 7369 6e67 0a20 2020  for choosing.   
│ │ │ │ +0001a470: 2020 2020 2073 7562 6d61 7472 6963 6573       submatrices
│ │ │ │ +0001a480: 0a20 2020 2020 202a 202a 6e6f 7465 2054  .      * *note T
│ │ │ │ +0001a490: 6872 6561 6473 3a20 6973 5261 6e6b 4174  hreads: isRankAt
│ │ │ │ +0001a4a0: 4c65 6173 745f 6c70 5f70 645f 7064 5f70  Least_lp_pd_pd_p
│ │ │ │ +0001a4b0: 645f 636d 5468 7265 6164 733d 3e5f 7064  d_cmThreads=>_pd
│ │ │ │ +0001a4c0: 5f70 645f 7064 5f72 702c 203d 3e0a 2020  _pd_pd_rp, =>.  
│ │ │ │ +0001a4d0: 2020 2020 2020 2e2e 2e2c 2064 6566 6175        ..., defau
│ │ │ │ +0001a4e0: 6c74 2076 616c 7565 2031 2c20 616e 206f  lt value 1, an o
│ │ │ │ +0001a4f0: 7074 696f 6e20 666f 7220 7661 7269 6f75  ption for variou
│ │ │ │ +0001a500: 7320 6675 6e63 7469 6f6e 730a 2020 2020  s functions.    
│ │ │ │ +0001a510: 2020 2a20 5665 7262 6f73 6520 3d3e 202e    * Verbose => .
│ │ │ │ +0001a520: 2e2e 2c20 6465 6661 756c 7420 7661 6c75  .., default valu
│ │ │ │ +0001a530: 6520 6661 6c73 650a 2020 2a20 4f75 7470  e false.  * Outp
│ │ │ │ +0001a540: 7574 733a 0a20 2020 2020 202a 2061 202a  uts:.      * a *
│ │ │ │ +0001a550: 6e6f 7465 2042 6f6f 6c65 616e 2076 616c  note Boolean val
│ │ │ │ +0001a560: 7565 3a20 284d 6163 6175 6c61 7932 446f  ue: (Macaulay2Do
│ │ │ │ +0001a570: 6329 426f 6f6c 6561 6e2c 2c20 0a0a 4465  c)Boolean,, ..De
│ │ │ │ +0001a580: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ +0001a590: 3d3d 3d3d 3d0a 0a54 6869 7320 6675 6e63  =====..This func
│ │ │ │ +0001a5a0: 7469 6f6e 2074 7269 6573 2074 6f20 7175  tion tries to qu
│ │ │ │ +0001a5b0: 6963 6b6c 7920 6465 7465 726d 696e 6520  ickly determine 
│ │ │ │ +0001a5c0: 7768 6574 6865 7220 7468 6520 6d61 7472  whether the matr
│ │ │ │ +0001a5d0: 6978 2068 6173 2061 2067 6976 656e 2072  ix has a given r
│ │ │ │ +0001a5e0: 616e 6b2e 0a69 7352 616e 6b41 744c 6561  ank..isRankAtLea
│ │ │ │ +0001a5f0: 7374 2063 616c 6c73 202a 6e6f 7465 2067  st calls *note g
│ │ │ │ +0001a600: 6574 5375 626d 6174 7269 784f 6652 616e  etSubmatrixOfRan
│ │ │ │ +0001a610: 6b3a 2067 6574 5375 626d 6174 7269 784f  k: getSubmatrixO
│ │ │ │ +0001a620: 6652 616e 6b2c 2e20 2049 6620 7468 6174  fRank,.  If that
│ │ │ │ +0001a630: 0a66 756e 6374 696f 6e20 6669 6e64 7320  .function finds 
│ │ │ │ +0001a640: 6120 7375 626d 6174 7269 7820 6f66 2061  a submatrix of a
│ │ │ │ +0001a650: 2063 6572 7461 696e 2072 616e 6b2c 2074   certain rank, t
│ │ │ │ +0001a660: 6869 7320 7265 7475 726e 7320 7472 7565  his returns true
│ │ │ │ +0001a670: 2e20 2049 6620 7468 6174 0a66 756e 6374  .  If that.funct
│ │ │ │ +0001a680: 696f 6e20 6661 696c 7320 746f 2066 696e  ion fails to fin
│ │ │ │ +0001a690: 6420 6120 7375 626d 6174 7269 7820 6f66  d a submatrix of
│ │ │ │ +0001a6a0: 2061 2063 6572 7461 696e 2072 616e 6b2c   a certain rank,
│ │ │ │ +0001a6b0: 2074 6869 7320 7369 6d70 6c79 2063 616c   this simply cal
│ │ │ │ +0001a6c0: 6c73 202a 6e6f 7465 0a72 616e 6b3a 2028  ls *note.rank: (
│ │ │ │ +0001a6d0: 4d61 6361 756c 6179 3244 6f63 2972 616e  Macaulay2Doc)ran
│ │ │ │ +0001a6e0: 6b2c 2e20 2054 6f20 636f 6e74 726f 6c20  k,.  To control 
│ │ │ │ +0001a6f0: 7468 6520 6e75 6d62 6572 206f 6620 7469  the number of ti
│ │ │ │ +0001a700: 6d65 7320 6765 7453 7562 6d61 7472 6978  mes getSubmatrix
│ │ │ │ +0001a710: 4f66 5261 6e6b 0a63 6f6e 7369 6465 7273  OfRank.considers
│ │ │ │ +0001a720: 2073 7562 6d61 7472 6963 6573 2c20 7573   submatrices, us
│ │ │ │ +0001a730: 6520 7468 6520 6f70 7469 6f6e 204d 6178  e the option Max
│ │ │ │ +0001a740: 4d69 6e6f 7273 2e0a 0a2b 2d2d 2d2d 2d2d  Minors...+------
│ │ │ │ +0001a750: 2d2d 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 2d2b 0a7c 6931 203a 2052 203d  -----+.|i1 : R =
│ │ │ │ +0001a790: 2051 515b 782c 795d 3b20 2020 2020 2020   QQ[x,y];       
│ │ │ │ +0001a7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a7c0: 2020 207c 0a2b 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: 2d2b 0a7c 6932 203a 204d 203d 206d 6174  -+.|i2 : M = mat
│ │ │ │ +0001a810: 7269 787b 7b78 2c79 2c32 2c30 2c32 2a78  rix{{x,y,2,0,2*x
│ │ │ │ +0001a820: 2b79 7d2c 207b 302c 302c 312c 302c 787d  +y}, {0,0,1,0,x}
│ │ │ │ +0001a830: 2c20 7b78 2c79 2c30 2c30 2c79 7d7d 3b7c  , {x,y,0,0,y}};|
│ │ │ │ +0001a840: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0001a850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a860: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -0001a870: 203a 204d 6174 7269 7820 5220 203c 2d2d   : Matrix R  <--
│ │ │ │ -0001a880: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ -0001a890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a8a0: 2020 2020 2020 2020 207c 0a2b 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 2d2d  ----------------
│ │ │ │ -0001a8e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2072  -------+.|i3 : r
│ │ │ │ -0001a8f0: 616e 6b20 4d20 2020 2020 2020 2020 2020  ank M           
│ │ │ │ -0001a900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a920: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001a930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a870: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001a880: 2020 2020 2020 2020 2020 2020 2033 2020               3  
│ │ │ │ +0001a890: 2020 2020 3520 2020 2020 2020 2020 2020      5           
│ │ │ │ +0001a8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a8b0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0001a8c0: 203a 204d 6174 7269 7820 5220 203c 2d2d   : Matrix R  <--
│ │ │ │ +0001a8d0: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ +0001a8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a8f0: 2020 2020 2020 2020 207c 0a2b 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 2d2b 0a7c 6933 203a 2072  -------+.|i3 : r
│ │ │ │ +0001a940: 616e 6b20 4d20 2020 2020 2020 2020 2020  ank M           
│ │ │ │  0001a950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a960: 2020 207c 0a7c 6f33 203d 2032 2020 2020     |.|o3 = 2    
│ │ │ │ -0001a970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a970: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0001a980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a9a0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -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 2d2b  ---------------+
│ │ │ │ -0001a9e0: 0a7c 6934 203a 2069 7352 616e 6b41 744c  .|i4 : isRankAtL
│ │ │ │ -0001a9f0: 6561 7374 2832 2c20 4d29 2020 2020 2020  east(2, M)      
│ │ │ │ -0001aa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001aa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa50: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ -0001aa60: 203d 2074 7275 6520 2020 2020 2020 2020   = true         
│ │ │ │ +0001a9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a9b0: 2020 207c 0a7c 6f33 203d 2032 2020 2020     |.|o3 = 2    
│ │ │ │ +0001a9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a9f0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0001aa00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001aa10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001aa20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0001aa30: 0a7c 6934 203a 2069 7352 616e 6b41 744c  .|i4 : isRankAtL
│ │ │ │ +0001aa40: 6561 7374 2832 2c20 4d29 2020 2020 2020  east(2, M)      
│ │ │ │ +0001aa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001aa60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0001aa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa90: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -0001aaa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001aab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001aac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001aad0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2069  -------+.|i5 : i
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│ │ │ │ +0001aac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001aad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001aae0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
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│ │ │ │ +0001ab50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ab60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │ -0001b020: 282e 2e2e 2c54 6872 6561 6473 3d3e 2e2e  (...,Threads=>..
│ │ │ │ -0001b030: 2e29 202d 2d20 616e 206f 7074 696f 6e20  .) -- an option 
│ │ │ │ -0001b040: 666f 7220 7661 7269 6f75 7320 6675 6e63  for various func
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│ │ │ │ -0001b060: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001b070: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001b080: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001b090: 2a2a 2a2a 2a2a 2a2a 0a0a 4465 7363 7269  ********..Descri
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│ │ │ │ -0001b130: 7265 6164 732e 0a0a 5365 6520 616c 736f  reads...See also
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│ │ │ │ -0001b170: 742c 202d 2d20 6465 7465 726d 696e 6573  t, -- determines
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│ │ │ │ -0001b1b0: 2a20 2a6e 6f74 6520 6765 7453 7562 6d61  * *note getSubma
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│ │ │ │ -0001b2d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001b2e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001b2f0: 3d3d 3d3d 3d0a 0a20 202a 2022 6765 7453  =====..  * "getS
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│ │ │ │ -0001b3a0: 2066 756e 6374 696f 6e73 0a20 202a 2022   functions.  * "
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│ │ │ │ -0001b3d0: 2922 0a0a 4675 7274 6865 7220 696e 666f  )"..Further info
│ │ │ │ -0001b3e0: 726d 6174 696f 6e0a 3d3d 3d3d 3d3d 3d3d  rmation.========
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│ │ │ │ -0001b400: 2044 6566 6175 6c74 2076 616c 7565 3a20   Default value: 
│ │ │ │ -0001b410: 310a 2020 2a20 4675 6e63 7469 6f6e 3a20  1.  * Function: 
│ │ │ │ -0001b420: 2a6e 6f74 6520 6973 5261 6e6b 4174 4c65  *note isRankAtLe
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│ │ │ │ -0001b440: 7374 2c20 2d2d 2064 6574 6572 6d69 6e65  st, -- determine
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│ │ │ │ -0001b470: 6c65 6173 7420 6120 6e75 6d62 6572 0a20  least a number. 
│ │ │ │ -0001b480: 202a 204f 7074 696f 6e20 6b65 793a 202a   * Option key: *
│ │ │ │ -0001b490: 6e6f 7465 2054 6872 6561 6473 3a20 284d  note Threads: (M
│ │ │ │ -0001b4a0: 6163 6175 6c61 7932 446f 6329 5468 7265  acaulay2Doc)Thre
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│ │ │ │ -0001b4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468  ------------..Th
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│ │ │ │ -0001b5f0: 7064 5f70 645f 7064 5f63 6d54 6872 6561  pd_pd_pd_cmThrea
│ │ │ │ -0001b600: 6473 3d3e 5f70 645f 7064 5f70 645f 7270  ds=>_pd_pd_pd_rp
│ │ │ │ -0001b610: 2c20 5570 3a20 546f 700a 0a4d 6178 4d69  , Up: Top..MaxMi
│ │ │ │ -0001b620: 6e6f 7273 202d 2d20 616e 206f 7074 696f  nors -- an optio
│ │ │ │ -0001b630: 6e20 746f 2063 6f6e 7472 6f6c 2064 6570  n to control dep
│ │ │ │ -0001b640: 7468 206f 6620 7365 6172 6368 0a2a 2a2a  th of search.***
│ │ │ │ -0001b650: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001b660: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001b670: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ -0001b680: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
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│ │ │ │ -0001b6d0: 636f 6e73 6964 6572 2e20 2049 6e63 7265  consider.  Incre
│ │ │ │ -0001b6e0: 6173 696e 6720 6974 0a77 696c 6c20 6d61  asing it.will ma
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│ │ │ │ -0001b730: 7573 6566 756c 0a6f 7574 7075 7473 2e20  useful.outputs. 
│ │ │ │ -0001b740: 2054 6865 2066 756e 6374 696f 6e73 2070   The functions p
│ │ │ │ -0001b750: 726f 6a44 696d 2061 6e64 2072 6567 756c  rojDim and regul
│ │ │ │ -0001b760: 6172 496e 436f 6469 6d65 6e73 696f 6e20  arInCodimension 
│ │ │ │ -0001b770: 6361 6e20 616c 736f 2074 616b 6520 696e  can also take in
│ │ │ │ -0001b780: 206d 6f72 650a 636f 6d70 6c69 6361 7465   more.complicate
│ │ │ │ -0001b790: 6420 696e 7075 7473 2e20 2053 6565 2074  d inputs.  See t
│ │ │ │ -0001b7a0: 6865 6972 2064 6f63 756d 656e 7461 7469  heir documentati
│ │ │ │ -0001b7b0: 6f6e 2066 6f72 2064 6574 6169 6c73 2e0a  on for details..
│ │ │ │ -0001b7c0: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ -0001b7d0: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 7265  ==..  * *note re
│ │ │ │ -0001b7e0: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ -0001b7f0: 6f6e 3a20 7265 6775 6c61 7249 6e43 6f64  on: regularInCod
│ │ │ │ -0001b800: 696d 656e 7369 6f6e 2c20 2d2d 2061 7474  imension, -- att
│ │ │ │ -0001b810: 656d 7074 7320 746f 2073 686f 7720 7468  empts to show th
│ │ │ │ -0001b820: 6174 0a20 2020 2074 6865 2072 696e 6720  at.    the ring 
│ │ │ │ -0001b830: 6973 2072 6567 756c 6172 2069 6e20 636f  is regular in co
│ │ │ │ -0001b840: 6469 6d65 6e73 696f 6e20 6e0a 2020 2a20  dimension n.  * 
│ │ │ │ -0001b850: 2a6e 6f74 6520 7072 6f6a 4469 6d3a 2070  *note projDim: p
│ │ │ │ -0001b860: 726f 6a44 696d 2c20 2d2d 2066 696e 6473  rojDim, -- finds
│ │ │ │ -0001b870: 2061 6e20 7570 7065 7220 626f 756e 6420   an upper bound 
│ │ │ │ -0001b880: 666f 7220 7468 6520 7072 6f6a 6563 7469  for the projecti
│ │ │ │ -0001b890: 7665 0a20 2020 2064 696d 656e 7369 6f6e  ve.    dimension
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│ │ │ │ -0001b8b0: 202a 6e6f 7465 2069 7352 616e 6b41 744c   *note isRankAtL
│ │ │ │ -0001b8c0: 6561 7374 3a20 6973 5261 6e6b 4174 4c65  east: isRankAtLe
│ │ │ │ -0001b8d0: 6173 742c 202d 2d20 6465 7465 726d 696e  ast, -- determin
│ │ │ │ -0001b8e0: 6573 2069 6620 7468 6520 6d61 7472 6978  es if the matrix
│ │ │ │ -0001b8f0: 2068 6173 2072 616e 6b20 6174 0a20 2020   has rank at.   
│ │ │ │ -0001b900: 206c 6561 7374 2061 206e 756d 6265 720a   least a number.
│ │ │ │ -0001b910: 2020 2a20 2a6e 6f74 6520 6765 7453 7562    * *note getSub
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│ │ │ │ -0001b930: 7453 7562 6d61 7472 6978 4f66 5261 6e6b  tSubmatrixOfRank
│ │ │ │ -0001b940: 2c20 2d2d 2074 7269 6573 2074 6f20 6669  , -- tries to fi
│ │ │ │ -0001b950: 6e64 2061 2073 7562 6d61 7472 6978 0a20  nd a submatrix. 
│ │ │ │ -0001b960: 2020 206f 6620 7468 6520 6769 7665 6e20     of the given 
│ │ │ │ -0001b970: 7261 6e6b 0a0a 4675 6e63 7469 6f6e 7320  rank..Functions 
│ │ │ │ -0001b980: 7769 7468 206f 7074 696f 6e61 6c20 6172  with optional ar
│ │ │ │ -0001b990: 6775 6d65 6e74 206e 616d 6564 204d 6178  gument named Max
│ │ │ │ -0001b9a0: 4d69 6e6f 7273 3a0a 3d3d 3d3d 3d3d 3d3d  Minors:.========
│ │ │ │ -0001b9b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001b9c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001b9d0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022  =========..  * "
│ │ │ │ -0001b9e0: 6765 7453 7562 6d61 7472 6978 4f66 5261  getSubmatrixOfRa
│ │ │ │ -0001b9f0: 6e6b 282e 2e2e 2c4d 6178 4d69 6e6f 7273  nk(...,MaxMinors
│ │ │ │ -0001ba00: 3d3e 2e2e 2e29 220a 2020 2a20 2269 7352  =>...)".  * "isR
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│ │ │ │ -0001ba20: 6178 4d69 6e6f 7273 3d3e 2e2e 2e29 220a  axMinors=>...)".
│ │ │ │ -0001ba30: 2020 2a20 2270 726f 6a44 696d 282e 2e2e    * "projDim(...
│ │ │ │ -0001ba40: 2c4d 6178 4d69 6e6f 7273 3d3e 2e2e 2e29  ,MaxMinors=>...)
│ │ │ │ -0001ba50: 220a 2020 2a20 2272 6567 756c 6172 496e  ".  * "regularIn
│ │ │ │ -0001ba60: 436f 6469 6d65 6e73 696f 6e28 2e2e 2e2c  Codimension(...,
│ │ │ │ -0001ba70: 4d61 784d 696e 6f72 733d 3e2e 2e2e 2922  MaxMinors=>...)"
│ │ │ │ -0001ba80: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
│ │ │ │ -0001ba90: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
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│ │ │ │ -0001bb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bb20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bb30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bb40: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520  ----------..The 
│ │ │ │ -0001bb50: 736f 7572 6365 206f 6620 7468 6973 2064  source of this d
│ │ │ │ -0001bb60: 6f63 756d 656e 7420 6973 2069 6e0a 2f62  ocument is in./b
│ │ │ │ -0001bb70: 7569 6c64 2f72 6570 726f 6475 6369 626c  uild/reproducibl
│ │ │ │ -0001bb80: 652d 7061 7468 2f6d 6163 6175 6c61 7932  e-path/macaulay2
│ │ │ │ -0001bb90: 2d31 2e32 352e 3131 2b64 732f 4d32 2f4d  -1.25.11+ds/M2/M
│ │ │ │ -0001bba0: 6163 6175 6c61 7932 2f70 6163 6b61 6765  acaulay2/package
│ │ │ │ -0001bbb0: 732f 4661 7374 4d69 6e6f 7273 2e0a 6d32  s/FastMinors..m2
│ │ │ │ -0001bbc0: 3a32 3133 393a 302e 0a1f 0a46 696c 653a  :2139:0....File:
│ │ │ │ -0001bbd0: 2046 6173 744d 696e 6f72 732e 696e 666f   FastMinors.info
│ │ │ │ -0001bbe0: 2c20 4e6f 6465 3a20 4d69 6e44 696d 656e  , Node: MinDimen
│ │ │ │ -0001bbf0: 7369 6f6e 2c20 4e65 7874 3a20 4d6f 6475  sion, Next: Modu
│ │ │ │ -0001bc00: 6c75 732c 2050 7265 763a 204d 6178 4d69  lus, Prev: MaxMi
│ │ │ │ -0001bc10: 6e6f 7273 2c20 5570 3a20 546f 700a 0a4d  nors, Up: Top..M
│ │ │ │ -0001bc20: 696e 4469 6d65 6e73 696f 6e20 2d2d 2061  inDimension -- a
│ │ │ │ -0001bc30: 6e20 6f70 7469 6f6e 2066 6f72 2070 726f  n option for pro
│ │ │ │ -0001bc40: 6a44 696d 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a  jDim.***********
│ │ │ │ -0001bc50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001bc60: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4465 7363  **********..Desc
│ │ │ │ -0001bc70: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ -0001bc80: 3d3d 3d0a 0a54 6869 7320 6f70 7469 6f6e  ===..This option
│ │ │ │ -0001bc90: 2069 7320 7573 6564 2074 6f20 7465 6c6c   is used to tell
│ │ │ │ -0001bca0: 2074 6865 2066 756e 6374 696f 6e20 7072   the function pr
│ │ │ │ -0001bcb0: 6f6a 4469 6d20 6e6f 7420 746f 206c 6f6f  ojDim not to loo
│ │ │ │ -0001bcc0: 6b20 666f 7220 7072 6f6a 6563 7469 7665  k for projective
│ │ │ │ -0001bcd0: 0a64 696d 656e 7369 6f6e 2062 656c 6f77  .dimension below
│ │ │ │ -0001bce0: 2074 6865 206f 7074 696f 6e20 7661 6c75   the option valu
│ │ │ │ -0001bcf0: 652e 0a0a 5365 6520 616c 736f 0a3d 3d3d  e...See also.===
│ │ │ │ -0001bd00: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ -0001bd10: 2070 726f 6a44 696d 3a20 7072 6f6a 4469   projDim: projDi
│ │ │ │ -0001bd20: 6d2c 202d 2d20 6669 6e64 7320 616e 2075  m, -- finds an u
│ │ │ │ -0001bd30: 7070 6572 2062 6f75 6e64 2066 6f72 2074  pper bound for t
│ │ │ │ -0001bd40: 6865 2070 726f 6a65 6374 6976 650a 2020  he projective.  
│ │ │ │ -0001bd50: 2020 6469 6d65 6e73 696f 6e20 6f66 2061    dimension of a
│ │ │ │ -0001bd60: 206d 6f64 756c 650a 0a46 756e 6374 696f   module..Functio
│ │ │ │ -0001bd70: 6e73 2077 6974 6820 6f70 7469 6f6e 616c  ns with optional
│ │ │ │ -0001bd80: 2061 7267 756d 656e 7420 6e61 6d65 6420   argument named 
│ │ │ │ -0001bd90: 4d69 6e44 696d 656e 7369 6f6e 3a0a 3d3d  MinDimension:.==
│ │ │ │ -0001bda0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001bdb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001bdc0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001bdd0: 3d3d 0a0a 2020 2a20 2270 726f 6a44 696d  ==..  * "projDim
│ │ │ │ -0001bde0: 282e 2e2e 2c4d 696e 4469 6d65 6e73 696f  (...,MinDimensio
│ │ │ │ -0001bdf0: 6e3d 3e2e 2e2e 2922 202d 2d20 7365 6520  n=>...)" -- see 
│ │ │ │ -0001be00: 2a6e 6f74 6520 7072 6f6a 4469 6d3a 2070  *note projDim: p
│ │ │ │ -0001be10: 726f 6a44 696d 2c20 2d2d 2066 696e 6473  rojDim, -- finds
│ │ │ │ -0001be20: 2061 6e0a 2020 2020 7570 7065 7220 626f   an.    upper bo
│ │ │ │ -0001be30: 756e 6420 666f 7220 7468 6520 7072 6f6a  und for the proj
│ │ │ │ -0001be40: 6563 7469 7665 2064 696d 656e 7369 6f6e  ective dimension
│ │ │ │ -0001be50: 206f 6620 6120 6d6f 6475 6c65 0a0a 466f   of a module..Fo
│ │ │ │ -0001be60: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ -0001be70: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -0001be80: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ -0001be90: 2a6e 6f74 6520 4d69 6e44 696d 656e 7369  *note MinDimensi
│ │ │ │ -0001bea0: 6f6e 3a20 4d69 6e44 696d 656e 7369 6f6e  on: MinDimension
│ │ │ │ -0001beb0: 2c20 6973 2061 202a 6e6f 7465 2073 796d  , is a *note sym
│ │ │ │ -0001bec0: 626f 6c3a 0a28 4d61 6361 756c 6179 3244  bol:.(Macaulay2D
│ │ │ │ -0001bed0: 6f63 2953 796d 626f 6c2c 2e0a 0a2d 2d2d  oc)Symbol,...---
│ │ │ │ -0001bee0: 2d2d 2d2d 2d2d 2d2d 2d2d 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 0a0a 5468  ------------..Th
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│ │ │ │ -0001bf40: 2064 6f63 756d 656e 7420 6973 2069 6e0a   document is in.
│ │ │ │ -0001bf50: 2f62 7569 6c64 2f72 6570 726f 6475 6369  /build/reproduci
│ │ │ │ -0001bf60: 626c 652d 7061 7468 2f6d 6163 6175 6c61  ble-path/macaula
│ │ │ │ -0001bf70: 7932 2d31 2e32 352e 3131 2b64 732f 4d32  y2-1.25.11+ds/M2
│ │ │ │ -0001bf80: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61  /Macaulay2/packa
│ │ │ │ -0001bf90: 6765 732f 4661 7374 4d69 6e6f 7273 2e0a  ges/FastMinors..
│ │ │ │ -0001bfa0: 6d32 3a32 3039 323a 302e 0a1f 0a46 696c  m2:2092:0....Fil
│ │ │ │ -0001bfb0: 653a 2046 6173 744d 696e 6f72 732e 696e  e: FastMinors.in
│ │ │ │ -0001bfc0: 666f 2c20 4e6f 6465 3a20 4d6f 6475 6c75  fo, Node: Modulu
│ │ │ │ -0001bfd0: 732c 204e 6578 743a 2050 6f69 6e74 4f70  s, Next: PointOp
│ │ │ │ -0001bfe0: 7469 6f6e 732c 2050 7265 763a 204d 696e  tions, Prev: Min
│ │ │ │ -0001bff0: 4469 6d65 6e73 696f 6e2c 2055 703a 2054  Dimension, Up: T
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│ │ │ │ -0001c010: 6e20 6f70 7469 6f6e 2066 6f72 2072 6567  n option for reg
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│ │ │ │ -0001c030: 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  n.**************
│ │ │ │ -0001c040: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001c050: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ -0001c060: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d  .Description.===
│ │ │ │ -0001c070: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6973 206f  ========..This o
│ │ │ │ -0001c080: 7074 696f 6e20 6973 2075 7365 6420 746f  ption is used to
│ │ │ │ -0001c090: 2074 656c 6c20 7468 6520 6675 6e63 7469   tell the functi
│ │ │ │ -0001c0a0: 6f6e 2074 6f20 646f 2074 6865 2063 6f6d  on to do the com
│ │ │ │ -0001c0b0: 7075 7461 7469 6f6e 206d 6f64 756c 6f20  putation modulo 
│ │ │ │ -0001c0c0: 6120 7072 696d 650a 702e 0a0a 5365 6520  a prime.p...See 
│ │ │ │ -0001c0d0: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  also.========.. 
│ │ │ │ -0001c0e0: 202a 202a 6e6f 7465 2072 6567 756c 6172   * *note regular
│ │ │ │ -0001c0f0: 496e 436f 6469 6d65 6e73 696f 6e3a 2072  InCodimension: r
│ │ │ │ -0001c100: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ -0001c110: 696f 6e2c 202d 2d20 6174 7465 6d70 7473  ion, -- attempts
│ │ │ │ -0001c120: 2074 6f20 7368 6f77 2074 6861 740a 2020   to show that.  
│ │ │ │ -0001c130: 2020 7468 6520 7269 6e67 2069 7320 7265    the ring is re
│ │ │ │ -0001c140: 6775 6c61 7220 696e 2063 6f64 696d 656e  gular in codimen
│ │ │ │ -0001c150: 7369 6f6e 206e 0a0a 4675 6e63 7469 6f6e  sion n..Function
│ │ │ │ -0001c160: 7320 7769 7468 206f 7074 696f 6e61 6c20  s with optional 
│ │ │ │ -0001c170: 6172 6775 6d65 6e74 206e 616d 6564 204d  argument named M
│ │ │ │ -0001c180: 6f64 756c 7573 3a0a 3d3d 3d3d 3d3d 3d3d  odulus:.========
│ │ │ │ -0001c190: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001c1a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001c1b0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 7265  =======..  * "re
│ │ │ │ -0001c1c0: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ -0001c1d0: 6f6e 282e 2e2e 2c4d 6f64 756c 7573 3d3e  on(...,Modulus=>
│ │ │ │ -0001c1e0: 2e2e 2e29 2220 2d2d 2073 6565 202a 6e6f  ...)" -- see *no
│ │ │ │ -0001c1f0: 7465 2072 6567 756c 6172 496e 436f 6469  te regularInCodi
│ │ │ │ -0001c200: 6d65 6e73 696f 6e3a 0a20 2020 2072 6567  mension:.    reg
│ │ │ │ -0001c210: 756c 6172 496e 436f 6469 6d65 6e73 696f  ularInCodimensio
│ │ │ │ -0001c220: 6e2c 202d 2d20 6174 7465 6d70 7473 2074  n, -- attempts t
│ │ │ │ -0001c230: 6f20 7368 6f77 2074 6861 7420 7468 6520  o show that the 
│ │ │ │ -0001c240: 7269 6e67 2069 7320 7265 6775 6c61 7220  ring is regular 
│ │ │ │ -0001c250: 696e 0a20 2020 2063 6f64 696d 656e 7369  in.    codimensi
│ │ │ │ -0001c260: 6f6e 206e 0a0a 466f 7220 7468 6520 7072  on n..For the pr
│ │ │ │ -0001c270: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ -0001c280: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ -0001c290: 206f 626a 6563 7420 2a6e 6f74 6520 4d6f   object *note Mo
│ │ │ │ -0001c2a0: 6475 6c75 733a 204d 6f64 756c 7573 2c20  dulus: Modulus, 
│ │ │ │ -0001c2b0: 6973 2061 202a 6e6f 7465 2073 796d 626f  is a *note symbo
│ │ │ │ -0001c2c0: 6c3a 2028 4d61 6361 756c 6179 3244 6f63  l: (Macaulay2Doc
│ │ │ │ -0001c2d0: 2953 796d 626f 6c2c 2e0a 0a2d 2d2d 2d2d  )Symbol,...-----
│ │ │ │ -0001c2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0001b070: 282e 2e2e 2c54 6872 6561 6473 3d3e 2e2e  (...,Threads=>..
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│ │ │ │ +0001b090: 666f 7220 7661 7269 6f75 7320 6675 6e63  for various func
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│ │ │ │ +0001b0b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001b0c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001b0d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ +0001b130: 696f 6e73 2074 6f20 6d75 6c74 6974 6872  ions to multithr
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│ │ │ │ +0001b180: 7265 6164 732e 0a0a 5365 6520 616c 736f  reads...See also
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│ │ │ │ +0001b1a0: 6e6f 7465 2069 7352 616e 6b41 744c 6561  note isRankAtLea
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│ │ │ │ +0001b1f0: 6561 7374 2061 206e 756d 6265 720a 2020  east a number.  
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│ │ │ │ +0001b2f0: 7769 7468 206f 7074 696f 6e61 6c20 6172  with optional ar
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│ │ │ │ +0001b320: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0001b330: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0001b340: 3d3d 3d3d 3d0a 0a20 202a 2022 6765 7453  =====..  * "getS
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│ │ │ │ +0001b370: 220a 2020 2a20 2a6e 6f74 6520 6973 5261  ".  * *note isRa
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│ │ │ │ +0001b3a0: 2069 7352 616e 6b41 744c 6561 7374 5f6c   isRankAtLeast_l
│ │ │ │ +0001b3b0: 705f 7064 5f70 645f 7064 5f63 6d54 6872  p_pd_pd_pd_cmThr
│ │ │ │ +0001b3c0: 6561 6473 3d3e 5f70 645f 7064 5f70 645f  eads=>_pd_pd_pd_
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│ │ │ │ +0001b3f0: 2066 756e 6374 696f 6e73 0a20 202a 2022   functions.  * "
│ │ │ │ +0001b400: 7265 6375 7273 6976 654d 696e 6f72 7328  recursiveMinors(
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│ │ │ │ +0001b420: 2922 0a0a 4675 7274 6865 7220 696e 666f  )"..Further info
│ │ │ │ +0001b430: 726d 6174 696f 6e0a 3d3d 3d3d 3d3d 3d3d  rmation.========
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│ │ │ │ +0001b450: 2044 6566 6175 6c74 2076 616c 7565 3a20   Default value: 
│ │ │ │ +0001b460: 310a 2020 2a20 4675 6e63 7469 6f6e 3a20  1.  * Function: 
│ │ │ │ +0001b470: 2a6e 6f74 6520 6973 5261 6e6b 4174 4c65  *note isRankAtLe
│ │ │ │ +0001b480: 6173 743a 2069 7352 616e 6b41 744c 6561  ast: isRankAtLea
│ │ │ │ +0001b490: 7374 2c20 2d2d 2064 6574 6572 6d69 6e65  st, -- determine
│ │ │ │ +0001b4a0: 7320 6966 2074 6865 206d 6174 7269 780a  s if the matrix.
│ │ │ │ +0001b4b0: 2020 2020 6861 7320 7261 6e6b 2061 7420      has rank at 
│ │ │ │ +0001b4c0: 6c65 6173 7420 6120 6e75 6d62 6572 0a20  least a number. 
│ │ │ │ +0001b4d0: 202a 204f 7074 696f 6e20 6b65 793a 202a   * Option key: *
│ │ │ │ +0001b4e0: 6e6f 7465 2054 6872 6561 6473 3a20 284d  note Threads: (M
│ │ │ │ +0001b4f0: 6163 6175 6c61 7932 446f 6329 5468 7265  acaulay2Doc)Thre
│ │ │ │ +0001b500: 6164 732c 202d 2d20 616e 206f 7074 696f  ads, -- an optio
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│ │ │ │ +0001b540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468  ------------..Th
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│ │ │ │ +0001b650: 6473 3d3e 5f70 645f 7064 5f70 645f 7270  ds=>_pd_pd_pd_rp
│ │ │ │ +0001b660: 2c20 5570 3a20 546f 700a 0a4d 6178 4d69  , Up: Top..MaxMi
│ │ │ │ +0001b670: 6e6f 7273 202d 2d20 616e 206f 7074 696f  nors -- an optio
│ │ │ │ +0001b680: 6e20 746f 2063 6f6e 7472 6f6c 2064 6570  n to control dep
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│ │ │ │ +0001b6a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001b6b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001b6c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
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│ │ │ │ +0001b700: 7720 6d61 6e79 206d 696e 6f72 7320 7661  w many minors va
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│ │ │ │ +0001b720: 636f 6e73 6964 6572 2e20 2049 6e63 7265  consider.  Incre
│ │ │ │ +0001b730: 6173 696e 6720 6974 0a77 696c 6c20 6d61  asing it.will ma
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│ │ │ │ +0001b780: 7573 6566 756c 0a6f 7574 7075 7473 2e20  useful.outputs. 
│ │ │ │ +0001b790: 2054 6865 2066 756e 6374 696f 6e73 2070   The functions p
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│ │ │ │ +0001b7b0: 6172 496e 436f 6469 6d65 6e73 696f 6e20  arInCodimension 
│ │ │ │ +0001b7c0: 6361 6e20 616c 736f 2074 616b 6520 696e  can also take in
│ │ │ │ +0001b7d0: 206d 6f72 650a 636f 6d70 6c69 6361 7465   more.complicate
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│ │ │ │ +0001b800: 6f6e 2066 6f72 2064 6574 6169 6c73 2e0a  on for details..
│ │ │ │ +0001b810: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ +0001b820: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 7265  ==..  * *note re
│ │ │ │ +0001b830: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ +0001b840: 6f6e 3a20 7265 6775 6c61 7249 6e43 6f64  on: regularInCod
│ │ │ │ +0001b850: 696d 656e 7369 6f6e 2c20 2d2d 2061 7474  imension, -- att
│ │ │ │ +0001b860: 656d 7074 7320 746f 2073 686f 7720 7468  empts to show th
│ │ │ │ +0001b870: 6174 0a20 2020 2074 6865 2072 696e 6720  at.    the ring 
│ │ │ │ +0001b880: 6973 2072 6567 756c 6172 2069 6e20 636f  is regular in co
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│ │ │ │ +0001b8a0: 2a6e 6f74 6520 7072 6f6a 4469 6d3a 2070  *note projDim: p
│ │ │ │ +0001b8b0: 726f 6a44 696d 2c20 2d2d 2066 696e 6473  rojDim, -- finds
│ │ │ │ +0001b8c0: 2061 6e20 7570 7065 7220 626f 756e 6420   an upper bound 
│ │ │ │ +0001b8d0: 666f 7220 7468 6520 7072 6f6a 6563 7469  for the projecti
│ │ │ │ +0001b8e0: 7665 0a20 2020 2064 696d 656e 7369 6f6e  ve.    dimension
│ │ │ │ +0001b8f0: 206f 6620 6120 6d6f 6475 6c65 0a20 202a   of a module.  *
│ │ │ │ +0001b900: 202a 6e6f 7465 2069 7352 616e 6b41 744c   *note isRankAtL
│ │ │ │ +0001b910: 6561 7374 3a20 6973 5261 6e6b 4174 4c65  east: isRankAtLe
│ │ │ │ +0001b920: 6173 742c 202d 2d20 6465 7465 726d 696e  ast, -- determin
│ │ │ │ +0001b930: 6573 2069 6620 7468 6520 6d61 7472 6978  es if the matrix
│ │ │ │ +0001b940: 2068 6173 2072 616e 6b20 6174 0a20 2020   has rank at.   
│ │ │ │ +0001b950: 206c 6561 7374 2061 206e 756d 6265 720a   least a number.
│ │ │ │ +0001b960: 2020 2a20 2a6e 6f74 6520 6765 7453 7562    * *note getSub
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│ │ │ │ +0001b980: 7453 7562 6d61 7472 6978 4f66 5261 6e6b  tSubmatrixOfRank
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│ │ │ │ +0001b9b0: 2020 206f 6620 7468 6520 6769 7665 6e20     of the given 
│ │ │ │ +0001b9c0: 7261 6e6b 0a0a 4675 6e63 7469 6f6e 7320  rank..Functions 
│ │ │ │ +0001b9d0: 7769 7468 206f 7074 696f 6e61 6c20 6172  with optional ar
│ │ │ │ +0001b9e0: 6775 6d65 6e74 206e 616d 6564 204d 6178  gument named Max
│ │ │ │ +0001b9f0: 4d69 6e6f 7273 3a0a 3d3d 3d3d 3d3d 3d3d  Minors:.========
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│ │ │ │ +0001ba10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0001ba20: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022  =========..  * "
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│ │ │ │ +0001ba50: 3d3e 2e2e 2e29 220a 2020 2a20 2269 7352  =>...)".  * "isR
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│ │ │ │ +0001ba70: 6178 4d69 6e6f 7273 3d3e 2e2e 2e29 220a  axMinors=>...)".
│ │ │ │ +0001ba80: 2020 2a20 2270 726f 6a44 696d 282e 2e2e    * "projDim(...
│ │ │ │ +0001ba90: 2c4d 6178 4d69 6e6f 7273 3d3e 2e2e 2e29  ,MaxMinors=>...)
│ │ │ │ +0001baa0: 220a 2020 2a20 2272 6567 756c 6172 496e  ".  * "regularIn
│ │ │ │ +0001bab0: 436f 6469 6d65 6e73 696f 6e28 2e2e 2e2c  Codimension(...,
│ │ │ │ +0001bac0: 4d61 784d 696e 6f72 733d 3e2e 2e2e 2922  MaxMinors=>...)"
│ │ │ │ +0001bad0: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
│ │ │ │ +0001bae0: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
│ │ │ │ +0001baf0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
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│ │ │ │ +0001bb40: 2953 796d 626f 6c2c 2e0a 0a2d 2d2d 2d2d  )Symbol,...-----
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│ │ │ │ +0001bb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bb90: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520  ----------..The 
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│ │ │ │ +0001bbc0: 7569 6c64 2f72 6570 726f 6475 6369 626c  uild/reproducibl
│ │ │ │ +0001bbd0: 652d 7061 7468 2f6d 6163 6175 6c61 7932  e-path/macaulay2
│ │ │ │ +0001bbe0: 2d31 2e32 352e 3131 2b64 732f 4d32 2f4d  -1.25.11+ds/M2/M
│ │ │ │ +0001bbf0: 6163 6175 6c61 7932 2f70 6163 6b61 6765  acaulay2/package
│ │ │ │ +0001bc00: 732f 4661 7374 4d69 6e6f 7273 2e0a 6d32  s/FastMinors..m2
│ │ │ │ +0001bc10: 3a32 3133 393a 302e 0a1f 0a46 696c 653a  :2139:0....File:
│ │ │ │ +0001bc20: 2046 6173 744d 696e 6f72 732e 696e 666f   FastMinors.info
│ │ │ │ +0001bc30: 2c20 4e6f 6465 3a20 4d69 6e44 696d 656e  , Node: MinDimen
│ │ │ │ +0001bc40: 7369 6f6e 2c20 4e65 7874 3a20 4d6f 6475  sion, Next: Modu
│ │ │ │ +0001bc50: 6c75 732c 2050 7265 763a 204d 6178 4d69  lus, Prev: MaxMi
│ │ │ │ +0001bc60: 6e6f 7273 2c20 5570 3a20 546f 700a 0a4d  nors, Up: Top..M
│ │ │ │ +0001bc70: 696e 4469 6d65 6e73 696f 6e20 2d2d 2061  inDimension -- a
│ │ │ │ +0001bc80: 6e20 6f70 7469 6f6e 2066 6f72 2070 726f  n option for pro
│ │ │ │ +0001bc90: 6a44 696d 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a  jDim.***********
│ │ │ │ +0001bca0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001bcb0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4465 7363  **********..Desc
│ │ │ │ +0001bcc0: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
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│ │ │ │ +0001bce0: 2069 7320 7573 6564 2074 6f20 7465 6c6c   is used to tell
│ │ │ │ +0001bcf0: 2074 6865 2066 756e 6374 696f 6e20 7072   the function pr
│ │ │ │ +0001bd00: 6f6a 4469 6d20 6e6f 7420 746f 206c 6f6f  ojDim not to loo
│ │ │ │ +0001bd10: 6b20 666f 7220 7072 6f6a 6563 7469 7665  k for projective
│ │ │ │ +0001bd20: 0a64 696d 656e 7369 6f6e 2062 656c 6f77  .dimension below
│ │ │ │ +0001bd30: 2074 6865 206f 7074 696f 6e20 7661 6c75   the option valu
│ │ │ │ +0001bd40: 652e 0a0a 5365 6520 616c 736f 0a3d 3d3d  e...See also.===
│ │ │ │ +0001bd50: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ +0001bd60: 2070 726f 6a44 696d 3a20 7072 6f6a 4469   projDim: projDi
│ │ │ │ +0001bd70: 6d2c 202d 2d20 6669 6e64 7320 616e 2075  m, -- finds an u
│ │ │ │ +0001bd80: 7070 6572 2062 6f75 6e64 2066 6f72 2074  pper bound for t
│ │ │ │ +0001bd90: 6865 2070 726f 6a65 6374 6976 650a 2020  he projective.  
│ │ │ │ +0001bda0: 2020 6469 6d65 6e73 696f 6e20 6f66 2061    dimension of a
│ │ │ │ +0001bdb0: 206d 6f64 756c 650a 0a46 756e 6374 696f   module..Functio
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│ │ │ │ +0001bdd0: 2061 7267 756d 656e 7420 6e61 6d65 6420   argument named 
│ │ │ │ +0001bde0: 4d69 6e44 696d 656e 7369 6f6e 3a0a 3d3d  MinDimension:.==
│ │ │ │ +0001bdf0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0001be00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0001be10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0001be20: 3d3d 0a0a 2020 2a20 2270 726f 6a44 696d  ==..  * "projDim
│ │ │ │ +0001be30: 282e 2e2e 2c4d 696e 4469 6d65 6e73 696f  (...,MinDimensio
│ │ │ │ +0001be40: 6e3d 3e2e 2e2e 2922 202d 2d20 7365 6520  n=>...)" -- see 
│ │ │ │ +0001be50: 2a6e 6f74 6520 7072 6f6a 4469 6d3a 2070  *note projDim: p
│ │ │ │ +0001be60: 726f 6a44 696d 2c20 2d2d 2066 696e 6473  rojDim, -- finds
│ │ │ │ +0001be70: 2061 6e0a 2020 2020 7570 7065 7220 626f   an.    upper bo
│ │ │ │ +0001be80: 756e 6420 666f 7220 7468 6520 7072 6f6a  und for the proj
│ │ │ │ +0001be90: 6563 7469 7665 2064 696d 656e 7369 6f6e  ective dimension
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│ │ │ │ +0001beb0: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ +0001bec0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ +0001bed0: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ +0001bee0: 2a6e 6f74 6520 4d69 6e44 696d 656e 7369  *note MinDimensi
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│ │ │ │ +0001bf00: 2c20 6973 2061 202a 6e6f 7465 2073 796d  , is a *note sym
│ │ │ │ +0001bf10: 626f 6c3a 0a28 4d61 6361 756c 6179 3244  bol:.(Macaulay2D
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│ │ │ │ +0001bf30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bf40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bf50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bf60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bf70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468  ------------..Th
│ │ │ │ +0001bf80: 6520 736f 7572 6365 206f 6620 7468 6973  e source of this
│ │ │ │ +0001bf90: 2064 6f63 756d 656e 7420 6973 2069 6e0a   document is in.
│ │ │ │ +0001bfa0: 2f62 7569 6c64 2f72 6570 726f 6475 6369  /build/reproduci
│ │ │ │ +0001bfb0: 626c 652d 7061 7468 2f6d 6163 6175 6c61  ble-path/macaula
│ │ │ │ +0001bfc0: 7932 2d31 2e32 352e 3131 2b64 732f 4d32  y2-1.25.11+ds/M2
│ │ │ │ +0001bfd0: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61  /Macaulay2/packa
│ │ │ │ +0001bfe0: 6765 732f 4661 7374 4d69 6e6f 7273 2e0a  ges/FastMinors..
│ │ │ │ +0001bff0: 6d32 3a32 3039 323a 302e 0a1f 0a46 696c  m2:2092:0....Fil
│ │ │ │ +0001c000: 653a 2046 6173 744d 696e 6f72 732e 696e  e: FastMinors.in
│ │ │ │ +0001c010: 666f 2c20 4e6f 6465 3a20 4d6f 6475 6c75  fo, Node: Modulu
│ │ │ │ +0001c020: 732c 204e 6578 743a 2050 6f69 6e74 4f70  s, Next: PointOp
│ │ │ │ +0001c030: 7469 6f6e 732c 2050 7265 763a 204d 696e  tions, Prev: Min
│ │ │ │ +0001c040: 4469 6d65 6e73 696f 6e2c 2055 703a 2054  Dimension, Up: T
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│ │ │ │ +0001c060: 6e20 6f70 7469 6f6e 2066 6f72 2072 6567  n option for reg
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│ │ │ │ +0001c080: 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  n.**************
│ │ │ │ +0001c090: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001c0a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ +0001c0b0: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d  .Description.===
│ │ │ │ +0001c0c0: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6973 206f  ========..This o
│ │ │ │ +0001c0d0: 7074 696f 6e20 6973 2075 7365 6420 746f  ption is used to
│ │ │ │ +0001c0e0: 2074 656c 6c20 7468 6520 6675 6e63 7469   tell the functi
│ │ │ │ +0001c0f0: 6f6e 2074 6f20 646f 2074 6865 2063 6f6d  on to do the com
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│ │ │ │ +0001c110: 6120 7072 696d 650a 702e 0a0a 5365 6520  a prime.p...See 
│ │ │ │ +0001c120: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  also.========.. 
│ │ │ │ +0001c130: 202a 202a 6e6f 7465 2072 6567 756c 6172   * *note regular
│ │ │ │ +0001c140: 496e 436f 6469 6d65 6e73 696f 6e3a 2072  InCodimension: r
│ │ │ │ +0001c150: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ +0001c160: 696f 6e2c 202d 2d20 6174 7465 6d70 7473  ion, -- attempts
│ │ │ │ +0001c170: 2074 6f20 7368 6f77 2074 6861 740a 2020   to show that.  
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│ │ │ │  0001c8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +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  ----------------
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│ │ │ │  0001c9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0001caa0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
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│ │ │ │ -0001cb00: 2020 2020 2020 2020 486f 6d6f 6765 6e65          Homogene
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│ │ │ │ +0001cb00: 2020 2020 2020 2020 4578 7465 6e64 4669          ExtendFi
│ │ │ │ +0001cb10: 656c 6420 3d3e 2074 7275 6520 2020 2020  eld => true     
│ │ │ │  0001cb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cb40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001cb50: 2020 2020 2020 2020 4d69 6e6f 7250 6f69          MinorPoi
│ │ │ │ -0001cb60: 6e74 4174 7465 6d70 7473 203d 3e20 3520  ntAttempts => 5 
│ │ │ │ +0001cb50: 2020 2020 2020 2020 486f 6d6f 6765 6e65          Homogene
│ │ │ │ +0001cb60: 6f75 7320 3d3e 2066 616c 7365 2020 2020  ous => false    
│ │ │ │  0001cb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cb90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001cba0: 2020 2020 2020 2020 4e75 6d54 6872 6561          NumThrea
│ │ │ │ -0001cbb0: 6473 546f 5573 6520 3d3e 2031 2020 2020  dsToUse => 1    
│ │ │ │ +0001cba0: 2020 2020 2020 2020 4d69 6e6f 7250 6f69          MinorPoi
│ │ │ │ +0001cbb0: 6e74 4174 7465 6d70 7473 203d 3e20 3520  ntAttempts => 5 
│ │ │ │  0001cbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cbe0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001cbf0: 2020 2020 2020 2020 506f 696e 7443 6865          PointChe
│ │ │ │ -0001cc00: 636b 4174 7465 6d70 7473 203d 3e20 3020  ckAttempts => 0 
│ │ │ │ +0001cbf0: 2020 2020 2020 2020 4e75 6d54 6872 6561          NumThrea
│ │ │ │ +0001cc00: 6473 546f 5573 6520 3d3e 2031 2020 2020  dsToUse => 1    
│ │ │ │  0001cc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001cc40: 2020 2020 2020 2020 5265 706c 6163 656d          Replacem
│ │ │ │ -0001cc50: 656e 7420 3d3e 2042 696e 6f6d 6961 6c20  ent => Binomial 
│ │ │ │ +0001cc40: 2020 2020 2020 2020 506f 696e 7443 6865          PointChe
│ │ │ │ +0001cc50: 636b 4174 7465 6d70 7473 203d 3e20 3020  ckAttempts => 0 
│ │ │ │  0001cc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc80: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001cc90: 2020 2020 2020 2020 5374 7261 7465 6779          Strategy
│ │ │ │ -0001cca0: 203d 3e20 4465 6661 756c 7420 2020 2020   => Default     
│ │ │ │ +0001cc90: 2020 2020 2020 2020 5265 706c 6163 656d          Replacem
│ │ │ │ +0001cca0: 656e 7420 3d3e 2042 696e 6f6d 6961 6c20  ent => Binomial 
│ │ │ │  0001ccb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ccc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ccd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001cce0: 2020 2020 2020 2020 5665 7262 6f73 6520          Verbose 
│ │ │ │ -0001ccf0: 3d3e 2066 616c 7365 2020 2020 2020 2020  => false        
│ │ │ │ +0001cce0: 2020 2020 2020 2020 5374 7261 7465 6779          Strategy
│ │ │ │ +0001ccf0: 203d 3e20 4465 6661 756c 7420 2020 2020   => Default     
│ │ │ │  0001cd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001cd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cd30: 2020 2020 2020 2020 5665 7262 6f73 6520          Verbose 
│ │ │ │ +0001cd40: 3d3e 2066 616c 7365 2020 2020 2020 2020  => false        
│ │ │ │  0001cd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cd70: 2020 2020 7c0a 7c6f 3220 3a20 4f70 7469      |.|o2 : Opti
│ │ │ │ -0001cd80: 6f6e 5461 626c 6520 2020 2020 2020 2020  onTable         
│ │ │ │ +0001cd70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +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 7c0a 2b2d 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 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ce10: 2d2d 2d2d 2b0a 0a54 6865 2064 6566 6175  ----+..The defau
│ │ │ │ -0001ce20: 6c74 2073 6574 7469 6e67 2045 7874 656e  lt setting Exten
│ │ │ │ -0001ce30: 6446 6965 6c64 203d 3e20 7472 7565 206d  dField => true m
│ │ │ │ -0001ce40: 6561 6e73 2074 6861 7420 706f 696e 7473  eans that points
│ │ │ │ -0001ce50: 2077 686f 7365 2072 6573 6964 7565 2066   whose residue f
│ │ │ │ -0001ce60: 6965 6c64 0a61 7265 2066 696e 6974 6520  ield.are finite 
│ │ │ │ -0001ce70: 6578 7465 6e73 696f 6e73 206f 6620 7468  extensions of th
│ │ │ │ -0001ce80: 6520 7072 696d 6520 6669 656c 6420 6172  e prime field ar
│ │ │ │ -0001ce90: 6520 7661 6c69 642c 2061 6e64 2061 7265  e valid, and are
│ │ │ │ -0001cea0: 2075 7365 6420 746f 2073 7475 6479 2074   used to study t
│ │ │ │ -0001ceb0: 6865 0a6d 6174 7269 782e 2020 4675 7274  he.matrix.  Furt
│ │ │ │ -0001cec0: 6865 726d 6f72 652c 2077 6520 6861 7665  hermore, we have
│ │ │ │ -0001ced0: 2073 6574 2048 6f6d 6f67 656e 656f 7573   set Homogeneous
│ │ │ │ -0001cee0: 3d3e 6661 6c73 6520 6279 2064 6566 6175  =>false by defau
│ │ │ │ -0001cef0: 6c74 2077 6869 6368 206d 6561 6e73 2074  lt which means t
│ │ │ │ -0001cf00: 6865 0a6f 7269 6769 6e20 6973 2074 7265  he.origin is tre
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│ │ │ │ -0001cf20: 706f 696e 742e 2020 5365 7474 696e 6720  point.  Setting 
│ │ │ │ -0001cf30: 4578 7465 6e64 4669 656c 643d 3e66 616c  ExtendField=>fal
│ │ │ │ -0001cf40: 7365 2077 696c 6c20 736f 6d65 7469 6d65  se will sometime
│ │ │ │ -0001cf50: 730a 7370 6565 6420 7570 2063 6f6d 7075  s.speed up compu
│ │ │ │ -0001cf60: 7461 7469 6f6e 2c20 6275 7420 6361 6e20  tation, but can 
│ │ │ │ -0001cf70: 616c 736f 206d 6973 7320 736f 6d65 2069  also miss some i
│ │ │ │ -0001cf80: 6d70 6f72 7461 6e74 2073 7562 6d61 7472  mportant submatr
│ │ │ │ -0001cf90: 6963 6573 2069 6620 7468 6174 0a64 6574  ices if that.det
│ │ │ │ -0001cfa0: 6572 6d69 6e61 6e74 2028 706c 7573 2077  erminant (plus w
│ │ │ │ -0001cfb0: 6861 7420 6861 7320 616c 7265 6164 7920  hat has already 
│ │ │ │ -0001cfc0: 6265 656e 2063 6f6d 7075 7465 6429 2064  been computed) d
│ │ │ │ -0001cfd0: 6566 696e 6573 2061 2073 6368 656d 6520  efines a scheme 
│ │ │ │ -0001cfe0: 7769 7468 206e 6f20 6f72 0a72 656c 6174  with no or.relat
│ │ │ │ -0001cff0: 6976 656c 7920 6665 7720 7261 7469 6f6e  ively few ration
│ │ │ │ -0001d000: 616c 2070 6f69 6e74 732e 2020 496e 2073  al points.  In s
│ │ │ │ -0001d010: 7563 6820 6120 6361 7365 2c20 4578 7465  uch a case, Exte
│ │ │ │ -0001d020: 6e64 4669 656c 6420 3d3e 2066 616c 7365  ndField => false
│ │ │ │ -0001d030: 2077 696c 6c0a 7479 7069 6361 6c6c 7920   will.typically 
│ │ │ │ -0001d040: 7375 6273 7461 6e74 6961 6c6c 7920 736c  substantially sl
│ │ │ │ -0001d050: 6f77 2064 6f77 6e20 636f 6d70 7574 6174  ow down computat
│ │ │ │ -0001d060: 696f 6e73 2e0a 0a53 6565 2061 6c73 6f0a  ions...See also.
│ │ │ │ -0001d070: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ -0001d080: 6f74 6520 6669 6e64 414e 6f6e 5a65 726f  ote findANonZero
│ │ │ │ -0001d090: 4d69 6e6f 723a 2028 5261 6e64 6f6d 506f  Minor: (RandomPo
│ │ │ │ -0001d0a0: 696e 7473 2966 696e 6441 4e6f 6e5a 6572  ints)findANonZer
│ │ │ │ -0001d0b0: 6f4d 696e 6f72 2c20 2d2d 2066 696e 6473  oMinor, -- finds
│ │ │ │ -0001d0c0: 2061 0a20 2020 206e 6f6e 2d76 616e 6973   a.    non-vanis
│ │ │ │ -0001d0d0: 6869 6e67 206d 696e 6f72 2061 7420 736f  hing minor at so
│ │ │ │ -0001d0e0: 6d65 2072 616e 646f 6d6c 7920 6368 6f73  me randomly chos
│ │ │ │ -0001d0f0: 656e 2070 6f69 6e74 0a0a 4675 6e63 7469  en point..Functi
│ │ │ │ -0001d100: 6f6e 7320 7769 7468 206f 7074 696f 6e61  ons with optiona
│ │ │ │ -0001d110: 6c20 6172 6775 6d65 6e74 206e 616d 6564  l argument named
│ │ │ │ -0001d120: 2050 6f69 6e74 4f70 7469 6f6e 733a 0a3d   PointOptions:.=
│ │ │ │ -0001d130: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001d140: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001d150: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001d160: 3d3d 3d0a 0a20 202a 2022 6368 6f6f 7365  ===..  * "choose
│ │ │ │ -0001d170: 476f 6f64 4d69 6e6f 7273 282e 2e2e 2c50  GoodMinors(...,P
│ │ │ │ -0001d180: 6f69 6e74 4f70 7469 6f6e 733d 3e2e 2e2e  ointOptions=>...
│ │ │ │ -0001d190: 2922 0a20 202a 2022 6765 7453 7562 6d61  )".  * "getSubma
│ │ │ │ -0001d1a0: 7472 6978 4f66 5261 6e6b 282e 2e2e 2c50  trixOfRank(...,P
│ │ │ │ -0001d1b0: 6f69 6e74 4f70 7469 6f6e 733d 3e2e 2e2e  ointOptions=>...
│ │ │ │ -0001d1c0: 2922 0a20 202a 2022 6973 5261 6e6b 4174  )".  * "isRankAt
│ │ │ │ -0001d1d0: 4c65 6173 7428 2e2e 2e2c 506f 696e 744f  Least(...,PointO
│ │ │ │ -0001d1e0: 7074 696f 6e73 3d3e 2e2e 2e29 220a 2020  ptions=>...)".  
│ │ │ │ -0001d1f0: 2a20 2270 726f 6a44 696d 282e 2e2e 2c50  * "projDim(...,P
│ │ │ │ +0001cdc0: 2020 2020 7c0a 7c6f 3220 3a20 4f70 7469      |.|o2 : Opti
│ │ │ │ +0001cdd0: 6f6e 5461 626c 6520 2020 2020 2020 2020  onTable         
│ │ │ │ +0001cde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ce00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ce10: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0001ce20: 2d2d 2d2d 2d2d 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 2b0a 0a54 6865 2064 6566 6175  ----+..The defau
│ │ │ │ +0001ce70: 6c74 2073 6574 7469 6e67 2045 7874 656e  lt setting Exten
│ │ │ │ +0001ce80: 6446 6965 6c64 203d 3e20 7472 7565 206d  dField => true m
│ │ │ │ +0001ce90: 6561 6e73 2074 6861 7420 706f 696e 7473  eans that points
│ │ │ │ +0001cea0: 2077 686f 7365 2072 6573 6964 7565 2066   whose residue f
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│ │ │ │ +0001cec0: 6578 7465 6e73 696f 6e73 206f 6620 7468  extensions of th
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│ │ │ │ +0001cf10: 6865 726d 6f72 652c 2077 6520 6861 7665  hermore, we have
│ │ │ │ +0001cf20: 2073 6574 2048 6f6d 6f67 656e 656f 7573   set Homogeneous
│ │ │ │ +0001cf30: 3d3e 6661 6c73 6520 6279 2064 6566 6175  =>false by defau
│ │ │ │ +0001cf40: 6c74 2077 6869 6368 206d 6561 6e73 2074  lt which means t
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│ │ │ │ +0001cf70: 706f 696e 742e 2020 5365 7474 696e 6720  point.  Setting 
│ │ │ │ +0001cf80: 4578 7465 6e64 4669 656c 643d 3e66 616c  ExtendField=>fal
│ │ │ │ +0001cf90: 7365 2077 696c 6c20 736f 6d65 7469 6d65  se will sometime
│ │ │ │ +0001cfa0: 730a 7370 6565 6420 7570 2063 6f6d 7075  s.speed up compu
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│ │ │ │ +0001cfc0: 616c 736f 206d 6973 7320 736f 6d65 2069  also miss some i
│ │ │ │ +0001cfd0: 6d70 6f72 7461 6e74 2073 7562 6d61 7472  mportant submatr
│ │ │ │ +0001cfe0: 6963 6573 2069 6620 7468 6174 0a64 6574  ices if that.det
│ │ │ │ +0001cff0: 6572 6d69 6e61 6e74 2028 706c 7573 2077  erminant (plus w
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│ │ │ │ +0001d020: 6566 696e 6573 2061 2073 6368 656d 6520  efines a scheme 
│ │ │ │ +0001d030: 7769 7468 206e 6f20 6f72 0a72 656c 6174  with no or.relat
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│ │ │ │ +0001d050: 616c 2070 6f69 6e74 732e 2020 496e 2073  al points.  In s
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│ │ │ │ +0001d070: 6e64 4669 656c 6420 3d3e 2066 616c 7365  ndField => false
│ │ │ │ +0001d080: 2077 696c 6c0a 7479 7069 6361 6c6c 7920   will.typically 
│ │ │ │ +0001d090: 7375 6273 7461 6e74 6961 6c6c 7920 736c  substantially sl
│ │ │ │ +0001d0a0: 6f77 2064 6f77 6e20 636f 6d70 7574 6174  ow down computat
│ │ │ │ +0001d0b0: 696f 6e73 2e0a 0a53 6565 2061 6c73 6f0a  ions...See also.
│ │ │ │ +0001d0c0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ +0001d0d0: 6f74 6520 6669 6e64 414e 6f6e 5a65 726f  ote findANonZero
│ │ │ │ +0001d0e0: 4d69 6e6f 723a 2028 5261 6e64 6f6d 506f  Minor: (RandomPo
│ │ │ │ +0001d0f0: 696e 7473 2966 696e 6441 4e6f 6e5a 6572  ints)findANonZer
│ │ │ │ +0001d100: 6f4d 696e 6f72 2c20 2d2d 2066 696e 6473  oMinor, -- finds
│ │ │ │ +0001d110: 2061 0a20 2020 206e 6f6e 2d76 616e 6973   a.    non-vanis
│ │ │ │ +0001d120: 6869 6e67 206d 696e 6f72 2061 7420 736f  hing minor at so
│ │ │ │ +0001d130: 6d65 2072 616e 646f 6d6c 7920 6368 6f73  me randomly chos
│ │ │ │ +0001d140: 656e 2070 6f69 6e74 0a0a 4675 6e63 7469  en point..Functi
│ │ │ │ +0001d150: 6f6e 7320 7769 7468 206f 7074 696f 6e61  ons with optiona
│ │ │ │ +0001d160: 6c20 6172 6775 6d65 6e74 206e 616d 6564  l argument named
│ │ │ │ +0001d170: 2050 6f69 6e74 4f70 7469 6f6e 733a 0a3d   PointOptions:.=
│ │ │ │ +0001d180: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0001d190: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0001d1a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0001d1b0: 3d3d 3d0a 0a20 202a 2022 6368 6f6f 7365  ===..  * "choose
│ │ │ │ +0001d1c0: 476f 6f64 4d69 6e6f 7273 282e 2e2e 2c50  GoodMinors(...,P
│ │ │ │ +0001d1d0: 6f69 6e74 4f70 7469 6f6e 733d 3e2e 2e2e  ointOptions=>...
│ │ │ │ +0001d1e0: 2922 0a20 202a 2022 6765 7453 7562 6d61  )".  * "getSubma
│ │ │ │ +0001d1f0: 7472 6978 4f66 5261 6e6b 282e 2e2e 2c50  trixOfRank(...,P
│ │ │ │  0001d200: 6f69 6e74 4f70 7469 6f6e 733d 3e2e 2e2e  ointOptions=>...
│ │ │ │ -0001d210: 2922 0a20 202a 2022 7265 6775 6c61 7249  )".  * "regularI
│ │ │ │ -0001d220: 6e43 6f64 696d 656e 7369 6f6e 282e 2e2e  nCodimension(...
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│ │ │ │ -0001d240: 2e2e 2922 0a0a 466f 7220 7468 6520 7072  ..)"..For the pr
│ │ │ │ -0001d250: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ -0001d260: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
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│ │ │ │ -0001d2a0: 6e6f 7465 2073 796d 626f 6c3a 0a28 4d61  note symbol:.(Ma
│ │ │ │ -0001d2b0: 6361 756c 6179 3244 6f63 2953 796d 626f  caulay2Doc)Symbo
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│ │ │ │ -0001d2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d310: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365  ----..The source
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│ │ │ │ -0001d360: 3131 2b64 732f 4d32 2f4d 6163 6175 6c61  11+ds/M2/Macaula
│ │ │ │ -0001d370: 7932 2f70 6163 6b61 6765 732f 4661 7374  y2/packages/Fast
│ │ │ │ -0001d380: 4d69 6e6f 7273 2e0a 6d32 3a31 3739 343a  Minors..m2:1794:
│ │ │ │ -0001d390: 302e 0a1f 0a46 696c 653a 2046 6173 744d  0....File: FastM
│ │ │ │ -0001d3a0: 696e 6f72 732e 696e 666f 2c20 4e6f 6465  inors.info, Node
│ │ │ │ -0001d3b0: 3a20 7072 6f6a 4469 6d2c 204e 6578 743a  : projDim, Next:
│ │ │ │ -0001d3c0: 2072 6563 7572 7369 7665 4d69 6e6f 7273   recursiveMinors
│ │ │ │ -0001d3d0: 2c20 5072 6576 3a20 506f 696e 744f 7074  , Prev: PointOpt
│ │ │ │ -0001d3e0: 696f 6e73 2c20 5570 3a20 546f 700a 0a70  ions, Up: Top..p
│ │ │ │ -0001d3f0: 726f 6a44 696d 202d 2d20 6669 6e64 7320  rojDim -- finds 
│ │ │ │ -0001d400: 616e 2075 7070 6572 2062 6f75 6e64 2066  an upper bound f
│ │ │ │ -0001d410: 6f72 2074 6865 2070 726f 6a65 6374 6976  or the projectiv
│ │ │ │ -0001d420: 6520 6469 6d65 6e73 696f 6e20 6f66 2061  e dimension of a
│ │ │ │ -0001d430: 206d 6f64 756c 650a 2a2a 2a2a 2a2a 2a2a   module.********
│ │ │ │ -0001d440: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001d450: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001d460: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001d470: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001d480: 0a0a 2020 2a20 5573 6167 653a 200a 2020  ..  * Usage: .  
│ │ │ │ -0001d490: 2020 2020 2020 6e20 3d20 7072 6f6a 4469        n = projDi
│ │ │ │ -0001d4a0: 6d28 4e2c 204d 696e 4469 6d65 6e73 696f  m(N, MinDimensio
│ │ │ │ -0001d4b0: 6e3d 3e64 290a 2020 2a20 496e 7075 7473  n=>d).  * Inputs
│ │ │ │ -0001d4c0: 3a0a 2020 2020 2020 2a20 4e2c 2061 202a  :.      * N, a *
│ │ │ │ -0001d4d0: 6e6f 7465 206d 6f64 756c 653a 2028 4d61  note module: (Ma
│ │ │ │ -0001d4e0: 6361 756c 6179 3244 6f63 294d 6f64 756c  caulay2Doc)Modul
│ │ │ │ -0001d4f0: 652c 2c20 6120 6d6f 6475 6c65 206f 7665  e,, a module ove
│ │ │ │ -0001d500: 7220 6120 706f 6c79 6e6f 6d69 616c 0a20  r a polynomial. 
│ │ │ │ -0001d510: 2020 2020 2020 2072 696e 670a 2020 2020         ring.    
│ │ │ │ -0001d520: 2020 2a20 642c 2061 6e20 2a6e 6f74 6520    * d, an *note 
│ │ │ │ -0001d530: 696e 7465 6765 723a 2028 4d61 6361 756c  integer: (Macaul
│ │ │ │ -0001d540: 6179 3244 6f63 295a 5a2c 2c20 7468 6520  ay2Doc)ZZ,, the 
│ │ │ │ -0001d550: 6d69 6e69 6d75 6d20 7072 6f6a 6563 7469  minimum projecti
│ │ │ │ -0001d560: 7665 0a20 2020 2020 2020 2064 696d 656e  ve.        dimen
│ │ │ │ -0001d570: 7369 6f6e 206f 6620 7468 6520 6d6f 6475  sion of the modu
│ │ │ │ -0001d580: 6c65 0a20 202a 202a 6e6f 7465 204f 7074  le.  * *note Opt
│ │ │ │ -0001d590: 696f 6e61 6c20 696e 7075 7473 3a20 284d  ional inputs: (M
│ │ │ │ -0001d5a0: 6163 6175 6c61 7932 446f 6329 7573 696e  acaulay2Doc)usin
│ │ │ │ -0001d5b0: 6720 6675 6e63 7469 6f6e 7320 7769 7468  g functions with
│ │ │ │ -0001d5c0: 206f 7074 696f 6e61 6c20 696e 7075 7473   optional inputs
│ │ │ │ -0001d5d0: 2c3a 0a20 2020 2020 202a 204d 696e 4469  ,:.      * MinDi
│ │ │ │ -0001d5e0: 6d65 6e73 696f 6e20 3d3e 2061 202a 6e6f  mension => a *no
│ │ │ │ -0001d5f0: 7465 206e 756d 6265 723a 2028 4d61 6361  te number: (Maca
│ │ │ │ -0001d600: 756c 6179 3244 6f63 294e 756d 6265 722c  ulay2Doc)Number,
│ │ │ │ -0001d610: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ -0001d620: 302c 0a20 2020 2020 2020 2073 746f 7020  0,.        stop 
│ │ │ │ -0001d630: 6166 7465 7220 7665 7269 6679 696e 6720  after verifying 
│ │ │ │ -0001d640: 7468 6520 6d6f 6475 6c65 2068 6173 2061  the module has a
│ │ │ │ -0001d650: 7420 6d6f 7374 2061 2063 6572 7461 696e  t most a certain
│ │ │ │ -0001d660: 2070 726f 6a65 6374 6976 650a 2020 2020   projective.    
│ │ │ │ -0001d670: 2020 2020 6469 6d65 6e73 696f 6e0a 2020      dimension.  
│ │ │ │ -0001d680: 2020 2020 2a20 2a6e 6f74 6520 506f 696e      * *note Poin
│ │ │ │ -0001d690: 744f 7074 696f 6e73 3a20 506f 696e 744f  tOptions: PointO
│ │ │ │ -0001d6a0: 7074 696f 6e73 2c20 3d3e 2061 202a 6e6f  ptions, => a *no
│ │ │ │ -0001d6b0: 7465 206c 6973 743a 2028 4d61 6361 756c  te list: (Macaul
│ │ │ │ -0001d6c0: 6179 3244 6f63 294c 6973 742c 2c0a 2020  ay2Doc)List,,.  
│ │ │ │ -0001d6d0: 2020 2020 2020 6465 6661 756c 7420 7661        default va
│ │ │ │ -0001d6e0: 6c75 6520 7b53 7472 6174 6567 7920 3d3e  lue {Strategy =>
│ │ │ │ -0001d6f0: 2044 6566 6175 6c74 2c20 486f 6d6f 6765   Default, Homoge
│ │ │ │ -0001d700: 6e65 6f75 7320 3d3e 2066 616c 7365 2c20  neous => false, 
│ │ │ │ -0001d710: 5265 706c 6163 656d 656e 740a 2020 2020  Replacement.    
│ │ │ │ -0001d720: 2020 2020 3d3e 2042 696e 6f6d 6961 6c2c      => Binomial,
│ │ │ │ -0001d730: 2045 7874 656e 6446 6965 6c64 203d 3e20   ExtendField => 
│ │ │ │ -0001d740: 7472 7565 2c20 506f 696e 7443 6865 636b  true, PointCheck
│ │ │ │ -0001d750: 4174 7465 6d70 7473 203d 3e20 302c 0a20  Attempts => 0,. 
│ │ │ │ -0001d760: 2020 2020 2020 2044 6563 6f6d 706f 7369         Decomposi
│ │ │ │ -0001d770: 7469 6f6e 5374 7261 7465 6779 203d 3e20  tionStrategy => 
│ │ │ │ -0001d780: 4465 636f 6d70 6f73 652c 204e 756d 5468  Decompose, NumTh
│ │ │ │ -0001d790: 7265 6164 7354 6f55 7365 203d 3e20 312c  readsToUse => 1,
│ │ │ │ -0001d7a0: 0a20 2020 2020 2020 2044 696d 656e 7369  .        Dimensi
│ │ │ │ -0001d7b0: 6f6e 4675 6e63 7469 6f6e 203d 3e20 6469  onFunction => di
│ │ │ │ -0001d7c0: 6d2c 2056 6572 626f 7365 203d 3e20 6661  m, Verbose => fa
│ │ │ │ -0001d7d0: 6c73 657d 2c20 6f70 7469 6f6e 7320 746f  lse}, options to
│ │ │ │ -0001d7e0: 2062 6520 7061 7373 6564 2074 6f0a 2020   be passed to.  
│ │ │ │ -0001d7f0: 2020 2020 2020 7468 6520 5261 6e64 6f6d        the Random
│ │ │ │ -0001d800: 506f 696e 7473 2070 6163 6b61 6765 0a20  Points package. 
│ │ │ │ -0001d810: 2020 2020 202a 202a 6e6f 7465 204d 6178       * *note Max
│ │ │ │ -0001d820: 4d69 6e6f 7273 3a20 4d61 784d 696e 6f72  Minors: MaxMinor
│ │ │ │ -0001d830: 732c 203d 3e20 2e2e 2e2c 2064 6566 6175  s, => ..., defau
│ │ │ │ -0001d840: 6c74 2076 616c 7565 0a20 2020 2020 2020  lt value.       
│ │ │ │ -0001d850: 2046 756e 6374 696f 6e43 6c6f 7375 7265   FunctionClosure
│ │ │ │ -0001d860: 5b2e 2e2f 4661 7374 4d69 6e6f 7273 2e6d  [../FastMinors.m
│ │ │ │ -0001d870: 323a 3138 363a 3138 2d31 3836 3a34 355d  2:186:18-186:45]
│ │ │ │ -0001d880: 2c20 7573 6564 2074 6f20 636f 6e74 726f  , used to contro
│ │ │ │ -0001d890: 6c20 686f 770a 2020 2020 2020 2020 6d61  l how.        ma
│ │ │ │ -0001d8a0: 6e79 206d 696e 6f72 7320 6172 6520 636f  ny minors are co
│ │ │ │ -0001d8b0: 6d70 7574 6564 206f 6620 7468 6520 6d61  mputed of the ma
│ │ │ │ -0001d8c0: 7472 6963 6573 2069 6e20 6120 7072 6f6a  trices in a proj
│ │ │ │ -0001d8d0: 6563 7469 7665 2072 6573 6f6c 7574 696f  ective resolutio
│ │ │ │ -0001d8e0: 6e0a 2020 2020 2020 2a20 2a6e 6f74 6520  n.      * *note 
│ │ │ │ -0001d8f0: 4465 7453 7472 6174 6567 793a 2044 6574  DetStrategy: Det
│ │ │ │ -0001d900: 5374 7261 7465 6779 2c20 3d3e 202e 2e2e  Strategy, => ...
│ │ │ │ -0001d910: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ -0001d920: 436f 6661 6374 6f72 2c0a 2020 2020 2020  Cofactor,.      
│ │ │ │ -0001d930: 2020 4465 7453 7472 6174 6567 7920 6973    DetStrategy is
│ │ │ │ -0001d940: 2061 2073 7472 6174 6567 7920 666f 7220   a strategy for 
│ │ │ │ -0001d950: 616c 6c6f 7769 6e67 2074 6865 2075 7365  allowing the use
│ │ │ │ -0001d960: 7220 746f 2063 686f 6f73 6520 686f 770a  r to choose how.
│ │ │ │ -0001d970: 2020 2020 2020 2020 6465 7465 726d 696e          determin
│ │ │ │ -0001d980: 616e 7473 2028 6f72 2072 616e 6b29 2c20  ants (or rank), 
│ │ │ │ -0001d990: 6973 2063 6f6d 7075 7465 640a 2020 2020  is computed.    
│ │ │ │ -0001d9a0: 2020 2a20 2a6e 6f74 6520 5374 7261 7465    * *note Strate
│ │ │ │ -0001d9b0: 6779 3a20 5374 7261 7465 6779 4465 6661  gy: StrategyDefa
│ │ │ │ -0001d9c0: 756c 742c 203d 3e20 2e2e 2e2c 2064 6566  ult, => ..., def
│ │ │ │ -0001d9d0: 6175 6c74 2076 616c 7565 206e 6577 204f  ault value new O
│ │ │ │ -0001d9e0: 7074 696f 6e54 6162 6c65 0a20 2020 2020  ptionTable.     
│ │ │ │ -0001d9f0: 2020 2066 726f 6d20 7b50 6f69 6e74 7320     from {Points 
│ │ │ │ -0001da00: 3d3e 2030 2c20 5261 6e64 6f6d 203d 3e20  => 0, Random => 
│ │ │ │ -0001da10: 3136 2c20 4752 6576 4c65 784c 6172 6765  16, GRevLexLarge
│ │ │ │ -0001da20: 7374 203d 3e20 302c 204c 6578 536d 616c  st => 0, LexSmal
│ │ │ │ -0001da30: 6c65 7374 5465 726d 0a20 2020 2020 2020  lestTerm.       
│ │ │ │ -0001da40: 203d 3e20 3136 2c20 4c65 784c 6172 6765   => 16, LexLarge
│ │ │ │ -0001da50: 7374 203d 3e20 302c 204c 6578 536d 616c  st => 0, LexSmal
│ │ │ │ -0001da60: 6c65 7374 203d 3e20 3136 2c20 4752 6576  lest => 16, GRev
│ │ │ │ -0001da70: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ -0001da80: 3d3e 2031 362c 0a20 2020 2020 2020 2052  => 16,.        R
│ │ │ │ -0001da90: 616e 646f 6d4e 6f6e 7a65 726f 203d 3e20  andomNonzero => 
│ │ │ │ -0001daa0: 3136 2c20 4752 6576 4c65 7853 6d61 6c6c  16, GRevLexSmall
│ │ │ │ -0001dab0: 6573 7420 3d3e 2031 367d 2c20 7374 7261  est => 16}, stra
│ │ │ │ -0001dac0: 7465 6769 6573 2066 6f72 2063 686f 6f73  tegies for choos
│ │ │ │ -0001dad0: 696e 670a 2020 2020 2020 2020 7375 626d  ing.        subm
│ │ │ │ -0001dae0: 6174 7269 6365 730a 2020 2020 2020 2a20  atrices.      * 
│ │ │ │ -0001daf0: 5665 7262 6f73 6520 3d3e 202e 2e2e 2c20  Verbose => ..., 
│ │ │ │ -0001db00: 6465 6661 756c 7420 7661 6c75 6520 6661  default value fa
│ │ │ │ -0001db10: 6c73 650a 2020 2a20 4f75 7470 7574 733a  lse.  * Outputs:
│ │ │ │ -0001db20: 0a20 2020 2020 202a 206e 2c20 616e 202a  .      * n, an *
│ │ │ │ -0001db30: 6e6f 7465 2069 6e74 6567 6572 3a20 284d  note integer: (M
│ │ │ │ -0001db40: 6163 6175 6c61 7932 446f 6329 5a5a 2c2c  acaulay2Doc)ZZ,,
│ │ │ │ -0001db50: 2061 6e20 7570 7065 7220 626f 756e 6420   an upper bound 
│ │ │ │ -0001db60: 666f 7220 7468 650a 2020 2020 2020 2020  for the.        
│ │ │ │ -0001db70: 7072 6f6a 6563 7469 7665 2064 696d 656e  projective dimen
│ │ │ │ -0001db80: 7369 6f6e 206f 6620 4e0a 0a44 6573 6372  sion of N..Descr
│ │ │ │ -0001db90: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ -0001dba0: 3d3d 0a0a 5468 6520 6675 6e63 7469 6f6e  ==..The function
│ │ │ │ -0001dbb0: 2070 6469 6d20 7265 7475 726e 7320 7468   pdim returns th
│ │ │ │ -0001dbc0: 6520 6c65 6e67 7468 206f 6620 6120 7072  e length of a pr
│ │ │ │ -0001dbd0: 6f6a 6563 7469 7665 2072 6573 6f6c 7574  ojective resolut
│ │ │ │ -0001dbe0: 696f 6e2e 2049 6620 7468 6520 6d6f 6475  ion. If the modu
│ │ │ │ -0001dbf0: 6c65 0a70 6173 7365 6420 6973 206e 6f74  le.passed is not
│ │ │ │ -0001dc00: 2068 6f6d 6f67 656e 656f 7573 2c20 7468   homogeneous, th
│ │ │ │ -0001dc10: 656e 2074 6865 2070 726f 6a65 6374 6976  en the projectiv
│ │ │ │ -0001dc20: 6520 7265 736f 6c75 7469 6f6e 206d 6179  e resolution may
│ │ │ │ -0001dc30: 206e 6f74 2062 6520 6d69 6e69 6d61 6c0a   not be minimal.
│ │ │ │ -0001dc40: 616e 6420 736f 2070 6469 6d20 6361 6e20  and so pdim can 
│ │ │ │ -0001dc50: 6769 7665 2074 6865 2077 726f 6e67 2061  give the wrong a
│ │ │ │ -0001dc60: 6e73 7765 722e 2020 5468 6973 2066 756e  nswer.  This fun
│ │ │ │ -0001dc70: 6374 696f 6e20 7072 6f6a 4469 6d20 7472  ction projDim tr
│ │ │ │ -0001dc80: 6965 7320 746f 2069 6d70 726f 7665 0a74  ies to improve.t
│ │ │ │ -0001dc90: 6869 7320 626f 756e 6420 6279 2063 6f6e  his bound by con
│ │ │ │ -0001dca0: 7369 6465 7269 6e67 2069 6465 616c 7320  sidering ideals 
│ │ │ │ -0001dcb0: 6f66 2061 7070 726f 7072 6961 7465 6c79  of appropriately
│ │ │ │ -0001dcc0: 2073 697a 6564 206d 696e 6f72 7320 6f66   sized minors of
│ │ │ │ -0001dcd0: 2074 6865 0a72 6573 6f6c 7574 696f 6e20   the.resolution 
│ │ │ │ -0001dce0: 2873 7461 7274 696e 6720 6672 6f6d 2074  (starting from t
│ │ │ │ -0001dcf0: 6865 2065 6e64 206f 6620 7468 6520 7265  he end of the re
│ │ │ │ -0001dd00: 736f 6c75 7469 6f6e 2061 6e64 2077 6f72  solution and wor
│ │ │ │ -0001dd10: 6b69 6e67 2062 6163 6b77 6172 6473 292e  king backwards).
│ │ │ │ -0001dd20: 0a55 7369 6e67 2074 6865 206f 7074 696f  .Using the optio
│ │ │ │ -0001dd30: 6e20 4d69 6e44 696d 656e 7369 6f6e 2028  n MinDimension (
│ │ │ │ -0001dd40: 6465 6661 756c 7420 7661 6c75 6520 3029  default value 0)
│ │ │ │ -0001dd50: 2067 6976 6573 2061 206c 6f77 6572 2062   gives a lower b
│ │ │ │ -0001dd60: 6f75 6e64 206f 6e20 7468 650a 7072 6f6a  ound on the.proj
│ │ │ │ -0001dd70: 6563 7469 7665 2064 696d 656e 7369 6f6e  ective dimension
│ │ │ │ -0001dd80: 2c20 696e 6372 6561 7369 6e67 2069 7420  , increasing it 
│ │ │ │ -0001dd90: 6361 6e20 7468 7573 2069 6d70 726f 7665  can thus improve
│ │ │ │ -0001dda0: 2074 6865 2073 7065 6564 206f 6620 636f   the speed of co
│ │ │ │ -0001ddb0: 6d70 7574 6174 696f 6e2e 0a0a 2b2d 2d2d  mputation...+---
│ │ │ │ -0001ddc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ddd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001dde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ddf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001de00: 2d2d 2d2d 2d2b 0a7c 6931 203a 2052 203d  -----+.|i1 : R =
│ │ │ │ -0001de10: 2051 515b 782c 795d 3b20 2020 2020 2020   QQ[x,y];       
│ │ │ │ -0001de20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001de30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001de40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001de50: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -0001de60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001de70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001de80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001de90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ -0001dea0: 203a 2049 203d 2069 6465 616c 2828 785e   : I = ideal((x^
│ │ │ │ -0001deb0: 332b 7929 5e32 2c20 2878 5e32 2b79 5e32  3+y)^2, (x^2+y^2
│ │ │ │ -0001dec0: 295e 322c 2028 782b 795e 3329 5e32 2c20  )^2, (x+y^3)^2, 
│ │ │ │ -0001ded0: 2878 2a79 295e 3229 3b20 2020 2020 2020  (x*y)^2);       
│ │ │ │ -0001dee0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001def0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001df00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001df10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001df20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001df30: 207c 0a7c 6f32 203a 2049 6465 616c 206f   |.|o2 : Ideal o
│ │ │ │ -0001df40: 6620 5220 2020 2020 2020 2020 2020 2020  f R             
│ │ │ │ +0001d210: 2922 0a20 202a 2022 6973 5261 6e6b 4174  )".  * "isRankAt
│ │ │ │ +0001d220: 4c65 6173 7428 2e2e 2e2c 506f 696e 744f  Least(...,PointO
│ │ │ │ +0001d230: 7074 696f 6e73 3d3e 2e2e 2e29 220a 2020  ptions=>...)".  
│ │ │ │ +0001d240: 2a20 2270 726f 6a44 696d 282e 2e2e 2c50  * "projDim(...,P
│ │ │ │ +0001d250: 6f69 6e74 4f70 7469 6f6e 733d 3e2e 2e2e  ointOptions=>...
│ │ │ │ +0001d260: 2922 0a20 202a 2022 7265 6775 6c61 7249  )".  * "regularI
│ │ │ │ +0001d270: 6e43 6f64 696d 656e 7369 6f6e 282e 2e2e  nCodimension(...
│ │ │ │ +0001d280: 2c50 6f69 6e74 4f70 7469 6f6e 733d 3e2e  ,PointOptions=>.
│ │ │ │ +0001d290: 2e2e 2922 0a0a 466f 7220 7468 6520 7072  ..)"..For the pr
│ │ │ │ +0001d2a0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ +0001d2b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ +0001d2c0: 206f 626a 6563 7420 2a6e 6f74 6520 506f   object *note Po
│ │ │ │ +0001d2d0: 696e 744f 7074 696f 6e73 3a20 506f 696e  intOptions: Poin
│ │ │ │ +0001d2e0: 744f 7074 696f 6e73 2c20 6973 2061 202a  tOptions, is a *
│ │ │ │ +0001d2f0: 6e6f 7465 2073 796d 626f 6c3a 0a28 4d61  note symbol:.(Ma
│ │ │ │ +0001d300: 6361 756c 6179 3244 6f63 2953 796d 626f  caulay2Doc)Symbo
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│ │ │ │ +0001d330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d360: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365  ----..The source
│ │ │ │ +0001d370: 206f 6620 7468 6973 2064 6f63 756d 656e   of this documen
│ │ │ │ +0001d380: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72  t is in./build/r
│ │ │ │ +0001d390: 6570 726f 6475 6369 626c 652d 7061 7468  eproducible-path
│ │ │ │ +0001d3a0: 2f6d 6163 6175 6c61 7932 2d31 2e32 352e  /macaulay2-1.25.
│ │ │ │ +0001d3b0: 3131 2b64 732f 4d32 2f4d 6163 6175 6c61  11+ds/M2/Macaula
│ │ │ │ +0001d3c0: 7932 2f70 6163 6b61 6765 732f 4661 7374  y2/packages/Fast
│ │ │ │ +0001d3d0: 4d69 6e6f 7273 2e0a 6d32 3a31 3739 343a  Minors..m2:1794:
│ │ │ │ +0001d3e0: 302e 0a1f 0a46 696c 653a 2046 6173 744d  0....File: FastM
│ │ │ │ +0001d3f0: 696e 6f72 732e 696e 666f 2c20 4e6f 6465  inors.info, Node
│ │ │ │ +0001d400: 3a20 7072 6f6a 4469 6d2c 204e 6578 743a  : projDim, Next:
│ │ │ │ +0001d410: 2072 6563 7572 7369 7665 4d69 6e6f 7273   recursiveMinors
│ │ │ │ +0001d420: 2c20 5072 6576 3a20 506f 696e 744f 7074  , Prev: PointOpt
│ │ │ │ +0001d430: 696f 6e73 2c20 5570 3a20 546f 700a 0a70  ions, Up: Top..p
│ │ │ │ +0001d440: 726f 6a44 696d 202d 2d20 6669 6e64 7320  rojDim -- finds 
│ │ │ │ +0001d450: 616e 2075 7070 6572 2062 6f75 6e64 2066  an upper bound f
│ │ │ │ +0001d460: 6f72 2074 6865 2070 726f 6a65 6374 6976  or the projectiv
│ │ │ │ +0001d470: 6520 6469 6d65 6e73 696f 6e20 6f66 2061  e dimension of a
│ │ │ │ +0001d480: 206d 6f64 756c 650a 2a2a 2a2a 2a2a 2a2a   module.********
│ │ │ │ +0001d490: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001d4a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001d4b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001d4c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001d4d0: 0a0a 2020 2a20 5573 6167 653a 200a 2020  ..  * Usage: .  
│ │ │ │ +0001d4e0: 2020 2020 2020 6e20 3d20 7072 6f6a 4469        n = projDi
│ │ │ │ +0001d4f0: 6d28 4e2c 204d 696e 4469 6d65 6e73 696f  m(N, MinDimensio
│ │ │ │ +0001d500: 6e3d 3e64 290a 2020 2a20 496e 7075 7473  n=>d).  * Inputs
│ │ │ │ +0001d510: 3a0a 2020 2020 2020 2a20 4e2c 2061 202a  :.      * N, a *
│ │ │ │ +0001d520: 6e6f 7465 206d 6f64 756c 653a 2028 4d61  note module: (Ma
│ │ │ │ +0001d530: 6361 756c 6179 3244 6f63 294d 6f64 756c  caulay2Doc)Modul
│ │ │ │ +0001d540: 652c 2c20 6120 6d6f 6475 6c65 206f 7665  e,, a module ove
│ │ │ │ +0001d550: 7220 6120 706f 6c79 6e6f 6d69 616c 0a20  r a polynomial. 
│ │ │ │ +0001d560: 2020 2020 2020 2072 696e 670a 2020 2020         ring.    
│ │ │ │ +0001d570: 2020 2a20 642c 2061 6e20 2a6e 6f74 6520    * d, an *note 
│ │ │ │ +0001d580: 696e 7465 6765 723a 2028 4d61 6361 756c  integer: (Macaul
│ │ │ │ +0001d590: 6179 3244 6f63 295a 5a2c 2c20 7468 6520  ay2Doc)ZZ,, the 
│ │ │ │ +0001d5a0: 6d69 6e69 6d75 6d20 7072 6f6a 6563 7469  minimum projecti
│ │ │ │ +0001d5b0: 7665 0a20 2020 2020 2020 2064 696d 656e  ve.        dimen
│ │ │ │ +0001d5c0: 7369 6f6e 206f 6620 7468 6520 6d6f 6475  sion of the modu
│ │ │ │ +0001d5d0: 6c65 0a20 202a 202a 6e6f 7465 204f 7074  le.  * *note Opt
│ │ │ │ +0001d5e0: 696f 6e61 6c20 696e 7075 7473 3a20 284d  ional inputs: (M
│ │ │ │ +0001d5f0: 6163 6175 6c61 7932 446f 6329 7573 696e  acaulay2Doc)usin
│ │ │ │ +0001d600: 6720 6675 6e63 7469 6f6e 7320 7769 7468  g functions with
│ │ │ │ +0001d610: 206f 7074 696f 6e61 6c20 696e 7075 7473   optional inputs
│ │ │ │ +0001d620: 2c3a 0a20 2020 2020 202a 204d 696e 4469  ,:.      * MinDi
│ │ │ │ +0001d630: 6d65 6e73 696f 6e20 3d3e 2061 202a 6e6f  mension => a *no
│ │ │ │ +0001d640: 7465 206e 756d 6265 723a 2028 4d61 6361  te number: (Maca
│ │ │ │ +0001d650: 756c 6179 3244 6f63 294e 756d 6265 722c  ulay2Doc)Number,
│ │ │ │ +0001d660: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ +0001d670: 302c 0a20 2020 2020 2020 2073 746f 7020  0,.        stop 
│ │ │ │ +0001d680: 6166 7465 7220 7665 7269 6679 696e 6720  after verifying 
│ │ │ │ +0001d690: 7468 6520 6d6f 6475 6c65 2068 6173 2061  the module has a
│ │ │ │ +0001d6a0: 7420 6d6f 7374 2061 2063 6572 7461 696e  t most a certain
│ │ │ │ +0001d6b0: 2070 726f 6a65 6374 6976 650a 2020 2020   projective.    
│ │ │ │ +0001d6c0: 2020 2020 6469 6d65 6e73 696f 6e0a 2020      dimension.  
│ │ │ │ +0001d6d0: 2020 2020 2a20 2a6e 6f74 6520 506f 696e      * *note Poin
│ │ │ │ +0001d6e0: 744f 7074 696f 6e73 3a20 506f 696e 744f  tOptions: PointO
│ │ │ │ +0001d6f0: 7074 696f 6e73 2c20 3d3e 2061 202a 6e6f  ptions, => a *no
│ │ │ │ +0001d700: 7465 206c 6973 743a 2028 4d61 6361 756c  te list: (Macaul
│ │ │ │ +0001d710: 6179 3244 6f63 294c 6973 742c 2c0a 2020  ay2Doc)List,,.  
│ │ │ │ +0001d720: 2020 2020 2020 6465 6661 756c 7420 7661        default va
│ │ │ │ +0001d730: 6c75 6520 7b53 7472 6174 6567 7920 3d3e  lue {Strategy =>
│ │ │ │ +0001d740: 2044 6566 6175 6c74 2c20 486f 6d6f 6765   Default, Homoge
│ │ │ │ +0001d750: 6e65 6f75 7320 3d3e 2066 616c 7365 2c20  neous => false, 
│ │ │ │ +0001d760: 5265 706c 6163 656d 656e 740a 2020 2020  Replacement.    
│ │ │ │ +0001d770: 2020 2020 3d3e 2042 696e 6f6d 6961 6c2c      => Binomial,
│ │ │ │ +0001d780: 2045 7874 656e 6446 6965 6c64 203d 3e20   ExtendField => 
│ │ │ │ +0001d790: 7472 7565 2c20 506f 696e 7443 6865 636b  true, PointCheck
│ │ │ │ +0001d7a0: 4174 7465 6d70 7473 203d 3e20 302c 0a20  Attempts => 0,. 
│ │ │ │ +0001d7b0: 2020 2020 2020 2044 6563 6f6d 706f 7369         Decomposi
│ │ │ │ +0001d7c0: 7469 6f6e 5374 7261 7465 6779 203d 3e20  tionStrategy => 
│ │ │ │ +0001d7d0: 4465 636f 6d70 6f73 652c 204e 756d 5468  Decompose, NumTh
│ │ │ │ +0001d7e0: 7265 6164 7354 6f55 7365 203d 3e20 312c  readsToUse => 1,
│ │ │ │ +0001d7f0: 0a20 2020 2020 2020 2044 696d 656e 7369  .        Dimensi
│ │ │ │ +0001d800: 6f6e 4675 6e63 7469 6f6e 203d 3e20 6469  onFunction => di
│ │ │ │ +0001d810: 6d2c 2056 6572 626f 7365 203d 3e20 6661  m, Verbose => fa
│ │ │ │ +0001d820: 6c73 657d 2c20 6f70 7469 6f6e 7320 746f  lse}, options to
│ │ │ │ +0001d830: 2062 6520 7061 7373 6564 2074 6f0a 2020   be passed to.  
│ │ │ │ +0001d840: 2020 2020 2020 7468 6520 5261 6e64 6f6d        the Random
│ │ │ │ +0001d850: 506f 696e 7473 2070 6163 6b61 6765 0a20  Points package. 
│ │ │ │ +0001d860: 2020 2020 202a 202a 6e6f 7465 204d 6178       * *note Max
│ │ │ │ +0001d870: 4d69 6e6f 7273 3a20 4d61 784d 696e 6f72  Minors: MaxMinor
│ │ │ │ +0001d880: 732c 203d 3e20 2e2e 2e2c 2064 6566 6175  s, => ..., defau
│ │ │ │ +0001d890: 6c74 2076 616c 7565 0a20 2020 2020 2020  lt value.       
│ │ │ │ +0001d8a0: 2046 756e 6374 696f 6e43 6c6f 7375 7265   FunctionClosure
│ │ │ │ +0001d8b0: 5b2e 2e2f 4661 7374 4d69 6e6f 7273 2e6d  [../FastMinors.m
│ │ │ │ +0001d8c0: 323a 3138 363a 3138 2d31 3836 3a34 355d  2:186:18-186:45]
│ │ │ │ +0001d8d0: 2c20 7573 6564 2074 6f20 636f 6e74 726f  , used to contro
│ │ │ │ +0001d8e0: 6c20 686f 770a 2020 2020 2020 2020 6d61  l how.        ma
│ │ │ │ +0001d8f0: 6e79 206d 696e 6f72 7320 6172 6520 636f  ny minors are co
│ │ │ │ +0001d900: 6d70 7574 6564 206f 6620 7468 6520 6d61  mputed of the ma
│ │ │ │ +0001d910: 7472 6963 6573 2069 6e20 6120 7072 6f6a  trices in a proj
│ │ │ │ +0001d920: 6563 7469 7665 2072 6573 6f6c 7574 696f  ective resolutio
│ │ │ │ +0001d930: 6e0a 2020 2020 2020 2a20 2a6e 6f74 6520  n.      * *note 
│ │ │ │ +0001d940: 4465 7453 7472 6174 6567 793a 2044 6574  DetStrategy: Det
│ │ │ │ +0001d950: 5374 7261 7465 6779 2c20 3d3e 202e 2e2e  Strategy, => ...
│ │ │ │ +0001d960: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ +0001d970: 436f 6661 6374 6f72 2c0a 2020 2020 2020  Cofactor,.      
│ │ │ │ +0001d980: 2020 4465 7453 7472 6174 6567 7920 6973    DetStrategy is
│ │ │ │ +0001d990: 2061 2073 7472 6174 6567 7920 666f 7220   a strategy for 
│ │ │ │ +0001d9a0: 616c 6c6f 7769 6e67 2074 6865 2075 7365  allowing the use
│ │ │ │ +0001d9b0: 7220 746f 2063 686f 6f73 6520 686f 770a  r to choose how.
│ │ │ │ +0001d9c0: 2020 2020 2020 2020 6465 7465 726d 696e          determin
│ │ │ │ +0001d9d0: 616e 7473 2028 6f72 2072 616e 6b29 2c20  ants (or rank), 
│ │ │ │ +0001d9e0: 6973 2063 6f6d 7075 7465 640a 2020 2020  is computed.    
│ │ │ │ +0001d9f0: 2020 2a20 2a6e 6f74 6520 5374 7261 7465    * *note Strate
│ │ │ │ +0001da00: 6779 3a20 5374 7261 7465 6779 4465 6661  gy: StrategyDefa
│ │ │ │ +0001da10: 756c 742c 203d 3e20 2e2e 2e2c 2064 6566  ult, => ..., def
│ │ │ │ +0001da20: 6175 6c74 2076 616c 7565 206e 6577 204f  ault value new O
│ │ │ │ +0001da30: 7074 696f 6e54 6162 6c65 0a20 2020 2020  ptionTable.     
│ │ │ │ +0001da40: 2020 2066 726f 6d20 7b50 6f69 6e74 7320     from {Points 
│ │ │ │ +0001da50: 3d3e 2030 2c20 5261 6e64 6f6d 203d 3e20  => 0, Random => 
│ │ │ │ +0001da60: 3136 2c20 4752 6576 4c65 784c 6172 6765  16, GRevLexLarge
│ │ │ │ +0001da70: 7374 203d 3e20 302c 204c 6578 536d 616c  st => 0, LexSmal
│ │ │ │ +0001da80: 6c65 7374 5465 726d 0a20 2020 2020 2020  lestTerm.       
│ │ │ │ +0001da90: 203d 3e20 3136 2c20 4c65 784c 6172 6765   => 16, LexLarge
│ │ │ │ +0001daa0: 7374 203d 3e20 302c 204c 6578 536d 616c  st => 0, LexSmal
│ │ │ │ +0001dab0: 6c65 7374 203d 3e20 3136 2c20 4752 6576  lest => 16, GRev
│ │ │ │ +0001dac0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ +0001dad0: 3d3e 2031 362c 0a20 2020 2020 2020 2052  => 16,.        R
│ │ │ │ +0001dae0: 616e 646f 6d4e 6f6e 7a65 726f 203d 3e20  andomNonzero => 
│ │ │ │ +0001daf0: 3136 2c20 4752 6576 4c65 7853 6d61 6c6c  16, GRevLexSmall
│ │ │ │ +0001db00: 6573 7420 3d3e 2031 367d 2c20 7374 7261  est => 16}, stra
│ │ │ │ +0001db10: 7465 6769 6573 2066 6f72 2063 686f 6f73  tegies for choos
│ │ │ │ +0001db20: 696e 670a 2020 2020 2020 2020 7375 626d  ing.        subm
│ │ │ │ +0001db30: 6174 7269 6365 730a 2020 2020 2020 2a20  atrices.      * 
│ │ │ │ +0001db40: 5665 7262 6f73 6520 3d3e 202e 2e2e 2c20  Verbose => ..., 
│ │ │ │ +0001db50: 6465 6661 756c 7420 7661 6c75 6520 6661  default value fa
│ │ │ │ +0001db60: 6c73 650a 2020 2a20 4f75 7470 7574 733a  lse.  * Outputs:
│ │ │ │ +0001db70: 0a20 2020 2020 202a 206e 2c20 616e 202a  .      * n, an *
│ │ │ │ +0001db80: 6e6f 7465 2069 6e74 6567 6572 3a20 284d  note integer: (M
│ │ │ │ +0001db90: 6163 6175 6c61 7932 446f 6329 5a5a 2c2c  acaulay2Doc)ZZ,,
│ │ │ │ +0001dba0: 2061 6e20 7570 7065 7220 626f 756e 6420   an upper bound 
│ │ │ │ +0001dbb0: 666f 7220 7468 650a 2020 2020 2020 2020  for the.        
│ │ │ │ +0001dbc0: 7072 6f6a 6563 7469 7665 2064 696d 656e  projective dimen
│ │ │ │ +0001dbd0: 7369 6f6e 206f 6620 4e0a 0a44 6573 6372  sion of N..Descr
│ │ │ │ +0001dbe0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ +0001dbf0: 3d3d 0a0a 5468 6520 6675 6e63 7469 6f6e  ==..The function
│ │ │ │ +0001dc00: 2070 6469 6d20 7265 7475 726e 7320 7468   pdim returns th
│ │ │ │ +0001dc10: 6520 6c65 6e67 7468 206f 6620 6120 7072  e length of a pr
│ │ │ │ +0001dc20: 6f6a 6563 7469 7665 2072 6573 6f6c 7574  ojective resolut
│ │ │ │ +0001dc30: 696f 6e2e 2049 6620 7468 6520 6d6f 6475  ion. If the modu
│ │ │ │ +0001dc40: 6c65 0a70 6173 7365 6420 6973 206e 6f74  le.passed is not
│ │ │ │ +0001dc50: 2068 6f6d 6f67 656e 656f 7573 2c20 7468   homogeneous, th
│ │ │ │ +0001dc60: 656e 2074 6865 2070 726f 6a65 6374 6976  en the projectiv
│ │ │ │ +0001dc70: 6520 7265 736f 6c75 7469 6f6e 206d 6179  e resolution may
│ │ │ │ +0001dc80: 206e 6f74 2062 6520 6d69 6e69 6d61 6c0a   not be minimal.
│ │ │ │ +0001dc90: 616e 6420 736f 2070 6469 6d20 6361 6e20  and so pdim can 
│ │ │ │ +0001dca0: 6769 7665 2074 6865 2077 726f 6e67 2061  give the wrong a
│ │ │ │ +0001dcb0: 6e73 7765 722e 2020 5468 6973 2066 756e  nswer.  This fun
│ │ │ │ +0001dcc0: 6374 696f 6e20 7072 6f6a 4469 6d20 7472  ction projDim tr
│ │ │ │ +0001dcd0: 6965 7320 746f 2069 6d70 726f 7665 0a74  ies to improve.t
│ │ │ │ +0001dce0: 6869 7320 626f 756e 6420 6279 2063 6f6e  his bound by con
│ │ │ │ +0001dcf0: 7369 6465 7269 6e67 2069 6465 616c 7320  sidering ideals 
│ │ │ │ +0001dd00: 6f66 2061 7070 726f 7072 6961 7465 6c79  of appropriately
│ │ │ │ +0001dd10: 2073 697a 6564 206d 696e 6f72 7320 6f66   sized minors of
│ │ │ │ +0001dd20: 2074 6865 0a72 6573 6f6c 7574 696f 6e20   the.resolution 
│ │ │ │ +0001dd30: 2873 7461 7274 696e 6720 6672 6f6d 2074  (starting from t
│ │ │ │ +0001dd40: 6865 2065 6e64 206f 6620 7468 6520 7265  he end of the re
│ │ │ │ +0001dd50: 736f 6c75 7469 6f6e 2061 6e64 2077 6f72  solution and wor
│ │ │ │ +0001dd60: 6b69 6e67 2062 6163 6b77 6172 6473 292e  king backwards).
│ │ │ │ +0001dd70: 0a55 7369 6e67 2074 6865 206f 7074 696f  .Using the optio
│ │ │ │ +0001dd80: 6e20 4d69 6e44 696d 656e 7369 6f6e 2028  n MinDimension (
│ │ │ │ +0001dd90: 6465 6661 756c 7420 7661 6c75 6520 3029  default value 0)
│ │ │ │ +0001dda0: 2067 6976 6573 2061 206c 6f77 6572 2062   gives a lower b
│ │ │ │ +0001ddb0: 6f75 6e64 206f 6e20 7468 650a 7072 6f6a  ound on the.proj
│ │ │ │ +0001ddc0: 6563 7469 7665 2064 696d 656e 7369 6f6e  ective dimension
│ │ │ │ +0001ddd0: 2c20 696e 6372 6561 7369 6e67 2069 7420  , increasing it 
│ │ │ │ +0001dde0: 6361 6e20 7468 7573 2069 6d70 726f 7665  can thus improve
│ │ │ │ +0001ddf0: 2074 6865 2073 7065 6564 206f 6620 636f   the speed of co
│ │ │ │ +0001de00: 6d70 7574 6174 696f 6e2e 0a0a 2b2d 2d2d  mputation...+---
│ │ │ │ +0001de10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001de20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001de30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001de40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001de50: 2d2d 2d2d 2d2b 0a7c 6931 203a 2052 203d  -----+.|i1 : R =
│ │ │ │ +0001de60: 2051 515b 782c 795d 3b20 2020 2020 2020   QQ[x,y];       
│ │ │ │ +0001de70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001de80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001de90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001dea0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0001deb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001dec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ded0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001dee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +0001def0: 203a 2049 203d 2069 6465 616c 2828 785e   : I = ideal((x^
│ │ │ │ +0001df00: 332b 7929 5e32 2c20 2878 5e32 2b79 5e32  3+y)^2, (x^2+y^2
│ │ │ │ +0001df10: 295e 322c 2028 782b 795e 3329 5e32 2c20  )^2, (x+y^3)^2, 
│ │ │ │ +0001df20: 2878 2a79 295e 3229 3b20 2020 2020 2020  (x*y)^2);       
│ │ │ │ +0001df30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001df40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001df50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001df60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001df70: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -0001df80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001df90: 2d2d 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 2d2b 0a7c 6933 203a 2070  -------+.|i3 : p
│ │ │ │ -0001dfd0: 6469 6d28 6d6f 6475 6c65 2049 2920 2020  dim(module I)   
│ │ │ │ -0001dfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e010: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0001e020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001df70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001df80: 207c 0a7c 6f32 203a 2049 6465 616c 206f   |.|o2 : Ideal o
│ │ │ │ +0001df90: 6620 5220 2020 2020 2020 2020 2020 2020  f R             
│ │ │ │ +0001dfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001dfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001dfc0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0001dfd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001dfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001dff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e010: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2070  -------+.|i3 : p
│ │ │ │ +0001e020: 6469 6d28 6d6f 6475 6c65 2049 2920 2020  dim(module I)   
│ │ │ │  0001e030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e050: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001e060: 6f33 203d 2032 2020 2020 2020 2020 2020  o3 = 2          
│ │ │ │ +0001e050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e060: 2020 7c0a 7c20 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 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -0001e0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 3236 3732  | -- used 0.2672
│ │ │ │ -0001e150: 3737 7320 2863 7075 293b 2030 2e31 3538  77s (cpu); 0.158
│ │ │ │ -0001e160: 3638 3373 2028 7468 7265 6164 293b 2030  683s (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   
│ │ │ │ +0001e0a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001e0b0: 6f33 203d 2032 2020 2020 2020 2020 2020  o3 = 2          
│ │ │ │ +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: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001e100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e140: 2d2d 2d2b 0a7c 6934 203a 2074 696d 6520  ---+.|i4 : time 
│ │ │ │ +0001e150: 7072 6f6a 4469 6d28 6d6f 6475 6c65 2049  projDim(module I
│ │ │ │ +0001e160: 2c20 5374 7261 7465 6779 3d3e 5374 7261  , Strategy=>Stra
│ │ │ │ +0001e170: 7465 6779 5261 6e64 6f6d 2920 2020 2020  tegyRandom)     
│ │ │ │ +0001e180: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001e190: 7c20 2d2d 2075 7365 6420 302e 3237 3832  | -- used 0.2782
│ │ │ │ +0001e1a0: 3173 2028 6370 7529 3b20 302e 3136 3734  1s (cpu); 0.1674
│ │ │ │ +0001e1b0: 3937 7320 2874 6872 6561 6429 3b20 3073  97s (thread); 0s
│ │ │ │ +0001e1c0: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │ +0001e1d0: 2020 2020 2020 2020 207c 0a7c 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 2020 2020                  
│ │ │ │ -0001e210: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001e220: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -0001e230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e260: 2d2d 2d2d 2d2d 2d2d 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 3130 3730 3236 7320 2863 7075   0.0107026s (cpu
│ │ │ │ -0001e2d0: 293b 2030 2e30 3132 3637 3839 7320 2874  ); 0.0126789s (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
│ │ │ │ -0001e350: 203d 2031 2020 2020 2020 2020 2020 2020   = 1            
│ │ │ │ +0001e210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e220: 2020 2020 7c0a 7c6f 3420 3d20 3120 2020      |.|o4 = 1   
│ │ │ │ +0001e230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e260: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001e270: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0001e280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520  ----------+.|i5 
│ │ │ │ +0001e2c0: 3a20 7469 6d65 2070 726f 6a44 696d 286d  : time projDim(m
│ │ │ │ +0001e2d0: 6f64 756c 6520 492c 2053 7472 6174 6567  odule I, Strateg
│ │ │ │ +0001e2e0: 793d 3e53 7472 6174 6567 7952 616e 646f  y=>StrategyRando
│ │ │ │ +0001e2f0: 6d2c 204d 696e 4469 6d65 6e73 696f 6e20  m, MinDimension 
│ │ │ │ +0001e300: 3d3e 2031 297c 0a7c 202d 2d20 7573 6564  => 1)|.| -- used
│ │ │ │ +0001e310: 2030 2e30 3132 3537 3131 7320 2863 7075   0.0125711s (cpu
│ │ │ │ +0001e320: 293b 2030 2e30 3135 3038 3537 7320 2874  ); 0.0150857s (t
│ │ │ │ +0001e330: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ +0001e340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e350: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0001e360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e390: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -0001e3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e3e0: 2d2b 0a0a 5468 6520 6f70 7469 6f6e 204d  -+..The option M
│ │ │ │ -0001e3f0: 6178 4d69 6e6f 7273 2063 616e 2062 6520  axMinors can be 
│ │ │ │ -0001e400: 7573 6564 2074 6f20 636f 6e74 726f 6c20  used to control 
│ │ │ │ -0001e410: 686f 7720 6d61 6e79 206d 696e 6f72 7320  how many minors 
│ │ │ │ -0001e420: 6172 6520 636f 6d70 7574 6564 2061 740a  are computed at.
│ │ │ │ -0001e430: 6561 6368 2073 7465 702e 2049 6620 7468  each step. If th
│ │ │ │ -0001e440: 6973 2069 7320 6e6f 7420 7370 6563 6966  is is not specif
│ │ │ │ -0001e450: 6965 642c 2074 6865 206e 756d 6265 7220  ied, the number 
│ │ │ │ -0001e460: 6f66 206d 696e 6f72 7320 6973 2061 2066  of minors is a f
│ │ │ │ -0001e470: 756e 6374 696f 6e20 6f66 2074 6865 0a64  unction of the.d
│ │ │ │ -0001e480: 696d 656e 7369 6f6e 2024 6424 206f 6620  imension $d$ of 
│ │ │ │ -0001e490: 7468 6520 706f 6c79 6e6f 6d69 616c 2072  the polynomial r
│ │ │ │ -0001e4a0: 696e 6720 616e 6420 7468 6520 706f 7373  ing and the poss
│ │ │ │ -0001e4b0: 6962 6c65 206d 696e 6f72 7320 2463 242e  ible minors $c$.
│ │ │ │ -0001e4c0: 2053 7065 6369 6669 6361 6c6c 790a 6974   Specifically.it
│ │ │ │ -0001e4d0: 2069 7320 3130 202a 2064 202b 2032 202a   is 10 * d + 2 *
│ │ │ │ -0001e4e0: 206c 6f67 5f31 2e33 2863 292e 204f 7468   log_1.3(c). Oth
│ │ │ │ -0001e4f0: 6572 7769 7365 2074 6865 2075 7365 7220  erwise the user 
│ │ │ │ -0001e500: 6361 6e20 7365 7420 7468 6520 6f70 7469  can set the opti
│ │ │ │ -0001e510: 6f6e 204d 6178 4d69 6e6f 7273 0a3d 3e20  on MaxMinors.=> 
│ │ │ │ -0001e520: 5a5a 2074 6f20 7370 6563 6966 7920 7468  ZZ to specify th
│ │ │ │ -0001e530: 6174 2061 2066 6978 6564 2069 6e74 6567  at a fixed integ
│ │ │ │ -0001e540: 6572 2069 7320 7573 6564 2066 6f72 2065  er is used for e
│ │ │ │ -0001e550: 6163 6820 7374 6570 2e20 2041 6c74 6572  ach step.  Alter
│ │ │ │ -0001e560: 6e61 7469 7665 6c79 2c0a 7468 6520 7573  natively,.the us
│ │ │ │ -0001e570: 6572 2063 616e 2063 6f6e 7472 6f6c 2074  er can control t
│ │ │ │ -0001e580: 6865 206e 756d 6265 7220 6f66 206d 696e  he number of min
│ │ │ │ -0001e590: 6f72 7320 636f 6d70 7574 6564 2061 7420  ors computed at 
│ │ │ │ -0001e5a0: 6561 6368 2073 7465 7020 6279 2073 6574  each step by set
│ │ │ │ -0001e5b0: 7469 6e67 2074 6865 0a6f 7074 696f 6e20  ting the.option 
│ │ │ │ -0001e5c0: 4d61 784d 696e 6f72 7320 3d3e 204c 6973  MaxMinors => Lis
│ │ │ │ -0001e5d0: 742e 2020 496e 2074 6869 7320 6361 7365  t.  In this case
│ │ │ │ -0001e5e0: 2c20 7468 6520 6c69 7374 2073 7065 6369  , the list speci
│ │ │ │ -0001e5f0: 6669 6573 2068 6f77 206d 616e 7920 6d69  fies how many mi
│ │ │ │ -0001e600: 6e6f 7273 2074 6f0a 6265 2063 6f6d 7075  nors to.be compu
│ │ │ │ -0001e610: 7465 6420 6174 2065 6163 6820 7374 6570  ted at each step
│ │ │ │ -0001e620: 2c20 2877 6f72 6b69 6e67 2062 6163 6b77  , (working backw
│ │ │ │ -0001e630: 6172 6473 292e 2046 696e 616c 6c79 2c20  ards). Finally, 
│ │ │ │ -0001e640: 796f 7520 6361 6e20 616c 736f 2073 6574  you can also set
│ │ │ │ -0001e650: 0a4d 6178 4d69 6e6f 7273 2074 6f20 6265  .MaxMinors to be
│ │ │ │ -0001e660: 2061 2063 7573 746f 6d20 6675 6e63 7469   a custom functi
│ │ │ │ -0001e670: 6f6e 206f 6620 7468 6520 6469 6d65 6e73  on of the dimens
│ │ │ │ -0001e680: 696f 6e20 2464 2420 6f66 2074 6865 2070  ion $d$ of the p
│ │ │ │ -0001e690: 6f6c 796e 6f6d 6961 6c20 7269 6e67 0a61  olynomial ring.a
│ │ │ │ -0001e6a0: 6e64 2074 6865 206d 6178 696d 756d 206e  nd the maximum n
│ │ │ │ -0001e6b0: 756d 6265 7220 6f66 206d 696e 6f72 732e  umber of minors.
│ │ │ │ -0001e6c0: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ -0001e6d0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2070  ===..  * *note p
│ │ │ │ -0001e6e0: 6469 6d3a 2028 4d61 6361 756c 6179 3244  dim: (Macaulay2D
│ │ │ │ -0001e6f0: 6f63 2970 6469 6d2c 202d 2d20 636f 6d70  oc)pdim, -- comp
│ │ │ │ -0001e700: 7574 6520 7468 6520 7072 6f6a 6563 7469  ute the projecti
│ │ │ │ -0001e710: 7665 2064 696d 656e 7369 6f6e 0a0a 5761  ve dimension..Wa
│ │ │ │ -0001e720: 7973 2074 6f20 7573 6520 7072 6f6a 4469  ys to use projDi
│ │ │ │ -0001e730: 6d3a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  m:.=============
│ │ │ │ -0001e740: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 7072  =======..  * "pr
│ │ │ │ -0001e750: 6f6a 4469 6d28 4d6f 6475 6c65 2922 0a0a  ojDim(Module)"..
│ │ │ │ -0001e760: 466f 7220 7468 6520 7072 6f67 7261 6d6d  For the programm
│ │ │ │ -0001e770: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  er.=============
│ │ │ │ -0001e780: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563  =====..The objec
│ │ │ │ -0001e790: 7420 2a6e 6f74 6520 7072 6f6a 4469 6d3a  t *note projDim:
│ │ │ │ -0001e7a0: 2070 726f 6a44 696d 2c20 6973 2061 202a   projDim, is a *
│ │ │ │ -0001e7b0: 6e6f 7465 206d 6574 686f 6420 6675 6e63  note method func
│ │ │ │ -0001e7c0: 7469 6f6e 2077 6974 6820 6f70 7469 6f6e  tion with option
│ │ │ │ -0001e7d0: 733a 0a28 4d61 6361 756c 6179 3244 6f63  s:.(Macaulay2Doc
│ │ │ │ -0001e7e0: 294d 6574 686f 6446 756e 6374 696f 6e57  )MethodFunctionW
│ │ │ │ -0001e7f0: 6974 684f 7074 696f 6e73 2c2e 0a0a 2d2d  ithOptions,...--
│ │ │ │ -0001e800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
│ │ │ │ -0001e850: 6865 2073 6f75 7263 6520 6f66 2074 6869  he source of thi
│ │ │ │ -0001e860: 7320 646f 6375 6d65 6e74 2069 7320 696e  s document is in
│ │ │ │ -0001e870: 0a2f 6275 696c 642f 7265 7072 6f64 7563  ./build/reproduc
│ │ │ │ -0001e880: 6962 6c65 2d70 6174 682f 6d61 6361 756c  ible-path/macaul
│ │ │ │ -0001e890: 6179 322d 312e 3235 2e31 312b 6473 2f4d  ay2-1.25.11+ds/M
│ │ │ │ -0001e8a0: 322f 4d61 6361 756c 6179 322f 7061 636b  2/Macaulay2/pack
│ │ │ │ -0001e8b0: 6167 6573 2f46 6173 744d 696e 6f72 732e  ages/FastMinors.
│ │ │ │ -0001e8c0: 0a6d 323a 3230 3739 3a30 2e0a 1f0a 4669  .m2:2079:0....Fi
│ │ │ │ -0001e8d0: 6c65 3a20 4661 7374 4d69 6e6f 7273 2e69  le: FastMinors.i
│ │ │ │ -0001e8e0: 6e66 6f2c 204e 6f64 653a 2072 6563 7572  nfo, Node: recur
│ │ │ │ -0001e8f0: 7369 7665 4d69 6e6f 7273 2c20 4e65 7874  siveMinors, Next
│ │ │ │ -0001e900: 3a20 7265 6775 6c61 7249 6e43 6f64 696d  : regularInCodim
│ │ │ │ -0001e910: 656e 7369 6f6e 2c20 5072 6576 3a20 7072  ension, Prev: pr
│ │ │ │ -0001e920: 6f6a 4469 6d2c 2055 703a 2054 6f70 0a0a  ojDim, Up: Top..
│ │ │ │ -0001e930: 7265 6375 7273 6976 654d 696e 6f72 7320  recursiveMinors 
│ │ │ │ -0001e940: 2d2d 2075 7365 7320 6120 7265 6375 7273  -- uses a recurs
│ │ │ │ -0001e950: 6976 6520 636f 6661 6374 6f72 2061 6c67  ive cofactor alg
│ │ │ │ -0001e960: 6f72 6974 686d 2074 6f20 636f 6d70 7574  orithm to comput
│ │ │ │ -0001e970: 6520 7468 6520 6964 6561 6c20 6f66 206d  e the ideal of m
│ │ │ │ -0001e980: 696e 6f72 7320 6f66 2061 206d 6174 7269  inors of a matri
│ │ │ │ -0001e990: 780a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  x.**************
│ │ │ │ -0001e9a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001e9b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001e9c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001e9d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001e9e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001e9f0: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20  ***..  * Usage: 
│ │ │ │ -0001ea00: 0a20 2020 2020 2020 2049 203d 2072 6563  .        I = rec
│ │ │ │ -0001ea10: 7572 7369 7665 4d69 6e6f 7273 286e 2c20  ursiveMinors(n, 
│ │ │ │ -0001ea20: 4d2c 2054 6872 6561 6473 3d3e 742c 204d  M, Threads=>t, M
│ │ │ │ -0001ea30: 696e 6f72 7343 6163 6865 3d3e 6229 0a20  inorsCache=>b). 
│ │ │ │ -0001ea40: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ -0001ea50: 202a 206e 2c20 616e 202a 6e6f 7465 2069   * n, an *note i
│ │ │ │ -0001ea60: 6e74 6567 6572 3a20 284d 6163 6175 6c61  nteger: (Macaula
│ │ │ │ -0001ea70: 7932 446f 6329 5a5a 2c2c 2074 6865 2073  y2Doc)ZZ,, the s
│ │ │ │ -0001ea80: 697a 6520 6f66 206d 696e 6f72 7320 746f  ize of minors to
│ │ │ │ -0001ea90: 2063 6f6d 7075 7465 0a20 2020 2020 202a   compute.      *
│ │ │ │ -0001eaa0: 204d 2c20 6120 2a6e 6f74 6520 6d61 7472   M, a *note matr
│ │ │ │ -0001eab0: 6978 3a20 284d 6163 6175 6c61 7932 446f  ix: (Macaulay2Do
│ │ │ │ -0001eac0: 6329 4d61 7472 6978 2c2c 200a 2020 2020  c)Matrix,, .    
│ │ │ │ -0001ead0: 2020 2a20 742c 2061 6e20 2a6e 6f74 6520    * t, an *note 
│ │ │ │ -0001eae0: 696e 7465 6765 723a 2028 4d61 6361 756c  integer: (Macaul
│ │ │ │ -0001eaf0: 6179 3244 6f63 295a 5a2c 2c20 616e 206f  ay2Doc)ZZ,, an o
│ │ │ │ -0001eb00: 7074 696f 6e61 6c20 696e 7075 742c 2077  ptional input, w
│ │ │ │ -0001eb10: 6869 6368 0a20 2020 2020 2020 2064 6573  hich.        des
│ │ │ │ -0001eb20: 6372 6962 6573 2074 6865 206e 756d 6265  cribes the numbe
│ │ │ │ -0001eb30: 7220 6f66 2074 6872 6561 6473 2074 6f20  r of threads to 
│ │ │ │ -0001eb40: 7573 6573 0a20 2020 2020 202a 2062 2c20  uses.      * b, 
│ │ │ │ -0001eb50: 6120 2a6e 6f74 6520 426f 6f6c 6561 6e20  a *note Boolean 
│ │ │ │ -0001eb60: 7661 6c75 653a 2028 4d61 6361 756c 6179  value: (Macaulay
│ │ │ │ -0001eb70: 3244 6f63 2942 6f6f 6c65 616e 2c2c 2061  2Doc)Boolean,, a
│ │ │ │ -0001eb80: 6e20 6f70 7469 6f6e 616c 2069 6e70 7574  n optional input
│ │ │ │ -0001eb90: 2c0a 2020 2020 2020 2020 7768 6963 6820  ,.        which 
│ │ │ │ -0001eba0: 7361 7973 2077 6865 7468 6572 2074 6f20  says whether to 
│ │ │ │ -0001ebb0: 6361 6368 6520 696e 2069 6e70 7574 0a20  cache in input. 
│ │ │ │ -0001ebc0: 202a 202a 6e6f 7465 204f 7074 696f 6e61   * *note Optiona
│ │ │ │ -0001ebd0: 6c20 696e 7075 7473 3a20 284d 6163 6175  l inputs: (Macau
│ │ │ │ -0001ebe0: 6c61 7932 446f 6329 7573 696e 6720 6675  lay2Doc)using fu
│ │ │ │ -0001ebf0: 6e63 7469 6f6e 7320 7769 7468 206f 7074  nctions with opt
│ │ │ │ -0001ec00: 696f 6e61 6c20 696e 7075 7473 2c3a 0a20  ional inputs,:. 
│ │ │ │ -0001ec10: 2020 2020 202a 204d 696e 6f72 7343 6163       * MinorsCac
│ │ │ │ -0001ec20: 6865 203d 3e20 2e2e 2e2c 2064 6566 6175  he => ..., defau
│ │ │ │ -0001ec30: 6c74 2076 616c 7565 2074 7275 650a 2020  lt value true.  
│ │ │ │ -0001ec40: 2020 2020 2a20 2a6e 6f74 6520 5468 7265      * *note Thre
│ │ │ │ -0001ec50: 6164 733a 2069 7352 616e 6b41 744c 6561  ads: isRankAtLea
│ │ │ │ -0001ec60: 7374 5f6c 705f 7064 5f70 645f 7064 5f63  st_lp_pd_pd_pd_c
│ │ │ │ -0001ec70: 6d54 6872 6561 6473 3d3e 5f70 645f 7064  mThreads=>_pd_pd
│ │ │ │ -0001ec80: 5f70 645f 7270 2c20 3d3e 0a20 2020 2020  _pd_rp, =>.     
│ │ │ │ -0001ec90: 2020 202e 2e2e 2c20 6465 6661 756c 7420     ..., default 
│ │ │ │ -0001eca0: 7661 6c75 6520 302c 2061 6e20 6f70 7469  value 0, an opti
│ │ │ │ -0001ecb0: 6f6e 2066 6f72 2076 6172 696f 7573 2066  on for various f
│ │ │ │ -0001ecc0: 756e 6374 696f 6e73 0a20 2020 2020 202a  unctions.      *
│ │ │ │ -0001ecd0: 2056 6572 626f 7365 203d 3e20 2e2e 2e2c   Verbose => ...,
│ │ │ │ -0001ece0: 2064 6566 6175 6c74 2076 616c 7565 2066   default value f
│ │ │ │ -0001ecf0: 616c 7365 0a20 202a 204f 7574 7075 7473  alse.  * Outputs
│ │ │ │ -0001ed00: 3a0a 2020 2020 2020 2a20 492c 2061 6e20  :.      * I, an 
│ │ │ │ -0001ed10: 2a6e 6f74 6520 6964 6561 6c3a 2028 4d61  *note ideal: (Ma
│ │ │ │ -0001ed20: 6361 756c 6179 3244 6f63 2949 6465 616c  caulay2Doc)Ideal
│ │ │ │ -0001ed30: 2c2c 2074 6865 2069 6465 616c 206f 6620  ,, the ideal of 
│ │ │ │ -0001ed40: 6d69 6e6f 7273 206f 6620 4d0a 0a44 6573  minors of M..Des
│ │ │ │ -0001ed50: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ -0001ed60: 3d3d 3d3d 0a0a 4769 7665 6e20 6120 6d61  ====..Given a ma
│ │ │ │ -0001ed70: 7472 6978 2024 4d24 2c20 7468 6973 2063  trix $M$, this c
│ │ │ │ -0001ed80: 6f6d 7075 7465 7320 7468 6520 6964 6561  omputes the idea
│ │ │ │ -0001ed90: 6c20 6f66 2064 6574 6572 6d69 6e61 6e74  l of determinant
│ │ │ │ -0001eda0: 7320 6f66 2073 697a 6520 246e 205c 7469  s of size $n \ti
│ │ │ │ -0001edb0: 6d65 730a 6e24 2073 7562 6d61 7472 6963  mes.n$ submatric
│ │ │ │ -0001edc0: 6573 2e20 5468 6520 7265 6375 7273 6976  es. The recursiv
│ │ │ │ -0001edd0: 654d 696e 6f72 7320 6675 6e63 7469 6f6e  eMinors function
│ │ │ │ -0001ede0: 2075 7365 7320 6120 7265 6375 7273 6976   uses a recursiv
│ │ │ │ -0001edf0: 6520 7374 7261 7465 6779 2c20 6b65 6570  e strategy, keep
│ │ │ │ -0001ee00: 696e 670a 7472 6163 6b20 6f66 2074 6865  ing.track of the
│ │ │ │ -0001ee10: 2073 6d61 6c6c 6572 206d 696e 6f72 7320   smaller minors 
│ │ │ │ -0001ee20: 636f 6d70 7574 6564 2073 6f20 6661 722c  computed so far,
│ │ │ │ -0001ee30: 2075 6e6c 696b 6520 7468 6520 6275 696c   unlike the buil
│ │ │ │ -0001ee40: 742d 696e 2043 6f66 6163 746f 720a 7374  t-in Cofactor.st
│ │ │ │ -0001ee50: 7261 7465 6779 2066 6f72 206d 696e 6f72  rategy for minor
│ │ │ │ -0001ee60: 730a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  s..+------------
│ │ │ │ -0001ee70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ee80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ee90: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
│ │ │ │ -0001eea0: 2052 203d 2051 515b 782c 795d 3b20 2020   R = QQ[x,y];   
│ │ │ │ -0001eeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001eec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001eed0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -0001eee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001eef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ef00: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a  ---------+.|i2 :
│ │ │ │ -0001ef10: 204d 203d 2072 616e 646f 6d28 525e 7b35   M = random(R^{5
│ │ │ │ -0001ef20: 2c35 2c35 2c35 2c35 2c35 7d2c 2052 5e37  ,5,5,5,5,5}, R^7
│ │ │ │ -0001ef30: 293b 2020 2020 2020 2020 2020 2020 2020  );              
│ │ │ │ -0001ef40: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0001ef50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ef60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ef70: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0001ef80: 2020 2020 2020 2020 2036 2020 2020 2020           6      
│ │ │ │ -0001ef90: 3720 2020 2020 2020 2020 2020 2020 2020  7               
│ │ │ │ +0001e390: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ +0001e3a0: 203d 2031 2020 2020 2020 2020 2020 2020   = 1            
│ │ │ │ +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 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001e3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e430: 2d2b 0a0a 5468 6520 6f70 7469 6f6e 204d  -+..The option M
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│ │ │ │ +0001e450: 7573 6564 2074 6f20 636f 6e74 726f 6c20  used to control 
│ │ │ │ +0001e460: 686f 7720 6d61 6e79 206d 696e 6f72 7320  how many minors 
│ │ │ │ +0001e470: 6172 6520 636f 6d70 7574 6564 2061 740a  are computed at.
│ │ │ │ +0001e480: 6561 6368 2073 7465 702e 2049 6620 7468  each step. If th
│ │ │ │ +0001e490: 6973 2069 7320 6e6f 7420 7370 6563 6966  is is not specif
│ │ │ │ +0001e4a0: 6965 642c 2074 6865 206e 756d 6265 7220  ied, the number 
│ │ │ │ +0001e4b0: 6f66 206d 696e 6f72 7320 6973 2061 2066  of minors is a f
│ │ │ │ +0001e4c0: 756e 6374 696f 6e20 6f66 2074 6865 0a64  unction of the.d
│ │ │ │ +0001e4d0: 696d 656e 7369 6f6e 2024 6424 206f 6620  imension $d$ of 
│ │ │ │ +0001e4e0: 7468 6520 706f 6c79 6e6f 6d69 616c 2072  the polynomial r
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│ │ │ │ +0001e500: 6962 6c65 206d 696e 6f72 7320 2463 242e  ible minors $c$.
│ │ │ │ +0001e510: 2053 7065 6369 6669 6361 6c6c 790a 6974   Specifically.it
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│ │ │ │ +0001e530: 206c 6f67 5f31 2e33 2863 292e 204f 7468   log_1.3(c). Oth
│ │ │ │ +0001e540: 6572 7769 7365 2074 6865 2075 7365 7220  erwise the user 
│ │ │ │ +0001e550: 6361 6e20 7365 7420 7468 6520 6f70 7469  can set the opti
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│ │ │ │ +0001e570: 5a5a 2074 6f20 7370 6563 6966 7920 7468  ZZ to specify th
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│ │ │ │ +0001e5d0: 6865 206e 756d 6265 7220 6f66 206d 696e  he number of min
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│ │ │ │ +0001e5f0: 6561 6368 2073 7465 7020 6279 2073 6574  each step by set
│ │ │ │ +0001e600: 7469 6e67 2074 6865 0a6f 7074 696f 6e20  ting the.option 
│ │ │ │ +0001e610: 4d61 784d 696e 6f72 7320 3d3e 204c 6973  MaxMinors => Lis
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│ │ │ │ +0001e650: 6e6f 7273 2074 6f0a 6265 2063 6f6d 7075  nors to.be compu
│ │ │ │ +0001e660: 7465 6420 6174 2065 6163 6820 7374 6570  ted at each step
│ │ │ │ +0001e670: 2c20 2877 6f72 6b69 6e67 2062 6163 6b77  , (working backw
│ │ │ │ +0001e680: 6172 6473 292e 2046 696e 616c 6c79 2c20  ards). Finally, 
│ │ │ │ +0001e690: 796f 7520 6361 6e20 616c 736f 2073 6574  you can also set
│ │ │ │ +0001e6a0: 0a4d 6178 4d69 6e6f 7273 2074 6f20 6265  .MaxMinors to be
│ │ │ │ +0001e6b0: 2061 2063 7573 746f 6d20 6675 6e63 7469   a custom functi
│ │ │ │ +0001e6c0: 6f6e 206f 6620 7468 6520 6469 6d65 6e73  on of the dimens
│ │ │ │ +0001e6d0: 696f 6e20 2464 2420 6f66 2074 6865 2070  ion $d$ of the p
│ │ │ │ +0001e6e0: 6f6c 796e 6f6d 6961 6c20 7269 6e67 0a61  olynomial ring.a
│ │ │ │ +0001e6f0: 6e64 2074 6865 206d 6178 696d 756d 206e  nd the maximum n
│ │ │ │ +0001e700: 756d 6265 7220 6f66 206d 696e 6f72 732e  umber of minors.
│ │ │ │ +0001e710: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ +0001e720: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2070  ===..  * *note p
│ │ │ │ +0001e730: 6469 6d3a 2028 4d61 6361 756c 6179 3244  dim: (Macaulay2D
│ │ │ │ +0001e740: 6f63 2970 6469 6d2c 202d 2d20 636f 6d70  oc)pdim, -- comp
│ │ │ │ +0001e750: 7574 6520 7468 6520 7072 6f6a 6563 7469  ute the projecti
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│ │ │ │ +0001e790: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 7072  =======..  * "pr
│ │ │ │ +0001e7a0: 6f6a 4469 6d28 4d6f 6475 6c65 2922 0a0a  ojDim(Module)"..
│ │ │ │ +0001e7b0: 466f 7220 7468 6520 7072 6f67 7261 6d6d  For the programm
│ │ │ │ +0001e7c0: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  er.=============
│ │ │ │ +0001e7d0: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563  =====..The objec
│ │ │ │ +0001e7e0: 7420 2a6e 6f74 6520 7072 6f6a 4469 6d3a  t *note projDim:
│ │ │ │ +0001e7f0: 2070 726f 6a44 696d 2c20 6973 2061 202a   projDim, is a *
│ │ │ │ +0001e800: 6e6f 7465 206d 6574 686f 6420 6675 6e63  note method func
│ │ │ │ +0001e810: 7469 6f6e 2077 6974 6820 6f70 7469 6f6e  tion with option
│ │ │ │ +0001e820: 733a 0a28 4d61 6361 756c 6179 3244 6f63  s:.(Macaulay2Doc
│ │ │ │ +0001e830: 294d 6574 686f 6446 756e 6374 696f 6e57  )MethodFunctionW
│ │ │ │ +0001e840: 6974 684f 7074 696f 6e73 2c2e 0a0a 2d2d  ithOptions,...--
│ │ │ │ +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 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
│ │ │ │ +0001e8a0: 6865 2073 6f75 7263 6520 6f66 2074 6869  he source of thi
│ │ │ │ +0001e8b0: 7320 646f 6375 6d65 6e74 2069 7320 696e  s document is in
│ │ │ │ +0001e8c0: 0a2f 6275 696c 642f 7265 7072 6f64 7563  ./build/reproduc
│ │ │ │ +0001e8d0: 6962 6c65 2d70 6174 682f 6d61 6361 756c  ible-path/macaul
│ │ │ │ +0001e8e0: 6179 322d 312e 3235 2e31 312b 6473 2f4d  ay2-1.25.11+ds/M
│ │ │ │ +0001e8f0: 322f 4d61 6361 756c 6179 322f 7061 636b  2/Macaulay2/pack
│ │ │ │ +0001e900: 6167 6573 2f46 6173 744d 696e 6f72 732e  ages/FastMinors.
│ │ │ │ +0001e910: 0a6d 323a 3230 3739 3a30 2e0a 1f0a 4669  .m2:2079:0....Fi
│ │ │ │ +0001e920: 6c65 3a20 4661 7374 4d69 6e6f 7273 2e69  le: FastMinors.i
│ │ │ │ +0001e930: 6e66 6f2c 204e 6f64 653a 2072 6563 7572  nfo, Node: recur
│ │ │ │ +0001e940: 7369 7665 4d69 6e6f 7273 2c20 4e65 7874  siveMinors, Next
│ │ │ │ +0001e950: 3a20 7265 6775 6c61 7249 6e43 6f64 696d  : regularInCodim
│ │ │ │ +0001e960: 656e 7369 6f6e 2c20 5072 6576 3a20 7072  ension, Prev: pr
│ │ │ │ +0001e970: 6f6a 4469 6d2c 2055 703a 2054 6f70 0a0a  ojDim, Up: Top..
│ │ │ │ +0001e980: 7265 6375 7273 6976 654d 696e 6f72 7320  recursiveMinors 
│ │ │ │ +0001e990: 2d2d 2075 7365 7320 6120 7265 6375 7273  -- uses a recurs
│ │ │ │ +0001e9a0: 6976 6520 636f 6661 6374 6f72 2061 6c67  ive cofactor alg
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│ │ │ │ +0001e9c0: 6520 7468 6520 6964 6561 6c20 6f66 206d  e the ideal of m
│ │ │ │ +0001e9d0: 696e 6f72 7320 6f66 2061 206d 6174 7269  inors of a matri
│ │ │ │ +0001e9e0: 780a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  x.**************
│ │ │ │ +0001e9f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001ea00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001ea10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001ea20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001ea30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001ea40: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20  ***..  * Usage: 
│ │ │ │ +0001ea50: 0a20 2020 2020 2020 2049 203d 2072 6563  .        I = rec
│ │ │ │ +0001ea60: 7572 7369 7665 4d69 6e6f 7273 286e 2c20  ursiveMinors(n, 
│ │ │ │ +0001ea70: 4d2c 2054 6872 6561 6473 3d3e 742c 204d  M, Threads=>t, M
│ │ │ │ +0001ea80: 696e 6f72 7343 6163 6865 3d3e 6229 0a20  inorsCache=>b). 
│ │ │ │ +0001ea90: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ +0001eaa0: 202a 206e 2c20 616e 202a 6e6f 7465 2069   * n, an *note i
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│ │ │ │ +0001eac0: 7932 446f 6329 5a5a 2c2c 2074 6865 2073  y2Doc)ZZ,, the s
│ │ │ │ +0001ead0: 697a 6520 6f66 206d 696e 6f72 7320 746f  ize of minors to
│ │ │ │ +0001eae0: 2063 6f6d 7075 7465 0a20 2020 2020 202a   compute.      *
│ │ │ │ +0001eaf0: 204d 2c20 6120 2a6e 6f74 6520 6d61 7472   M, a *note matr
│ │ │ │ +0001eb00: 6978 3a20 284d 6163 6175 6c61 7932 446f  ix: (Macaulay2Do
│ │ │ │ +0001eb10: 6329 4d61 7472 6978 2c2c 200a 2020 2020  c)Matrix,, .    
│ │ │ │ +0001eb20: 2020 2a20 742c 2061 6e20 2a6e 6f74 6520    * t, an *note 
│ │ │ │ +0001eb30: 696e 7465 6765 723a 2028 4d61 6361 756c  integer: (Macaul
│ │ │ │ +0001eb40: 6179 3244 6f63 295a 5a2c 2c20 616e 206f  ay2Doc)ZZ,, an o
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│ │ │ │ +0001eb70: 6372 6962 6573 2074 6865 206e 756d 6265  cribes the numbe
│ │ │ │ +0001eb80: 7220 6f66 2074 6872 6561 6473 2074 6f20  r of threads to 
│ │ │ │ +0001eb90: 7573 6573 0a20 2020 2020 202a 2062 2c20  uses.      * b, 
│ │ │ │ +0001eba0: 6120 2a6e 6f74 6520 426f 6f6c 6561 6e20  a *note Boolean 
│ │ │ │ +0001ebb0: 7661 6c75 653a 2028 4d61 6361 756c 6179  value: (Macaulay
│ │ │ │ +0001ebc0: 3244 6f63 2942 6f6f 6c65 616e 2c2c 2061  2Doc)Boolean,, a
│ │ │ │ +0001ebd0: 6e20 6f70 7469 6f6e 616c 2069 6e70 7574  n optional input
│ │ │ │ +0001ebe0: 2c0a 2020 2020 2020 2020 7768 6963 6820  ,.        which 
│ │ │ │ +0001ebf0: 7361 7973 2077 6865 7468 6572 2074 6f20  says whether to 
│ │ │ │ +0001ec00: 6361 6368 6520 696e 2069 6e70 7574 0a20  cache in input. 
│ │ │ │ +0001ec10: 202a 202a 6e6f 7465 204f 7074 696f 6e61   * *note Optiona
│ │ │ │ +0001ec20: 6c20 696e 7075 7473 3a20 284d 6163 6175  l inputs: (Macau
│ │ │ │ +0001ec30: 6c61 7932 446f 6329 7573 696e 6720 6675  lay2Doc)using fu
│ │ │ │ +0001ec40: 6e63 7469 6f6e 7320 7769 7468 206f 7074  nctions with opt
│ │ │ │ +0001ec50: 696f 6e61 6c20 696e 7075 7473 2c3a 0a20  ional inputs,:. 
│ │ │ │ +0001ec60: 2020 2020 202a 204d 696e 6f72 7343 6163       * MinorsCac
│ │ │ │ +0001ec70: 6865 203d 3e20 2e2e 2e2c 2064 6566 6175  he => ..., defau
│ │ │ │ +0001ec80: 6c74 2076 616c 7565 2074 7275 650a 2020  lt value true.  
│ │ │ │ +0001ec90: 2020 2020 2a20 2a6e 6f74 6520 5468 7265      * *note Thre
│ │ │ │ +0001eca0: 6164 733a 2069 7352 616e 6b41 744c 6561  ads: isRankAtLea
│ │ │ │ +0001ecb0: 7374 5f6c 705f 7064 5f70 645f 7064 5f63  st_lp_pd_pd_pd_c
│ │ │ │ +0001ecc0: 6d54 6872 6561 6473 3d3e 5f70 645f 7064  mThreads=>_pd_pd
│ │ │ │ +0001ecd0: 5f70 645f 7270 2c20 3d3e 0a20 2020 2020  _pd_rp, =>.     
│ │ │ │ +0001ece0: 2020 202e 2e2e 2c20 6465 6661 756c 7420     ..., default 
│ │ │ │ +0001ecf0: 7661 6c75 6520 302c 2061 6e20 6f70 7469  value 0, an opti
│ │ │ │ +0001ed00: 6f6e 2066 6f72 2076 6172 696f 7573 2066  on for various f
│ │ │ │ +0001ed10: 756e 6374 696f 6e73 0a20 2020 2020 202a  unctions.      *
│ │ │ │ +0001ed20: 2056 6572 626f 7365 203d 3e20 2e2e 2e2c   Verbose => ...,
│ │ │ │ +0001ed30: 2064 6566 6175 6c74 2076 616c 7565 2066   default value f
│ │ │ │ +0001ed40: 616c 7365 0a20 202a 204f 7574 7075 7473  alse.  * Outputs
│ │ │ │ +0001ed50: 3a0a 2020 2020 2020 2a20 492c 2061 6e20  :.      * I, an 
│ │ │ │ +0001ed60: 2a6e 6f74 6520 6964 6561 6c3a 2028 4d61  *note ideal: (Ma
│ │ │ │ +0001ed70: 6361 756c 6179 3244 6f63 2949 6465 616c  caulay2Doc)Ideal
│ │ │ │ +0001ed80: 2c2c 2074 6865 2069 6465 616c 206f 6620  ,, the ideal of 
│ │ │ │ +0001ed90: 6d69 6e6f 7273 206f 6620 4d0a 0a44 6573  minors of M..Des
│ │ │ │ +0001eda0: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ +0001edb0: 3d3d 3d3d 0a0a 4769 7665 6e20 6120 6d61  ====..Given a ma
│ │ │ │ +0001edc0: 7472 6978 2024 4d24 2c20 7468 6973 2063  trix $M$, this c
│ │ │ │ +0001edd0: 6f6d 7075 7465 7320 7468 6520 6964 6561  omputes the idea
│ │ │ │ +0001ede0: 6c20 6f66 2064 6574 6572 6d69 6e61 6e74  l of determinant
│ │ │ │ +0001edf0: 7320 6f66 2073 697a 6520 246e 205c 7469  s of size $n \ti
│ │ │ │ +0001ee00: 6d65 730a 6e24 2073 7562 6d61 7472 6963  mes.n$ submatric
│ │ │ │ +0001ee10: 6573 2e20 5468 6520 7265 6375 7273 6976  es. The recursiv
│ │ │ │ +0001ee20: 654d 696e 6f72 7320 6675 6e63 7469 6f6e  eMinors function
│ │ │ │ +0001ee30: 2075 7365 7320 6120 7265 6375 7273 6976   uses a recursiv
│ │ │ │ +0001ee40: 6520 7374 7261 7465 6779 2c20 6b65 6570  e strategy, keep
│ │ │ │ +0001ee50: 696e 670a 7472 6163 6b20 6f66 2074 6865  ing.track of the
│ │ │ │ +0001ee60: 2073 6d61 6c6c 6572 206d 696e 6f72 7320   smaller minors 
│ │ │ │ +0001ee70: 636f 6d70 7574 6564 2073 6f20 6661 722c  computed so far,
│ │ │ │ +0001ee80: 2075 6e6c 696b 6520 7468 6520 6275 696c   unlike the buil
│ │ │ │ +0001ee90: 742d 696e 2043 6f66 6163 746f 720a 7374  t-in Cofactor.st
│ │ │ │ +0001eea0: 7261 7465 6779 2066 6f72 206d 696e 6f72  rategy for minor
│ │ │ │ +0001eeb0: 730a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  s..+------------
│ │ │ │ +0001eec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001eed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001eee0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
│ │ │ │ +0001eef0: 2052 203d 2051 515b 782c 795d 3b20 2020   R = QQ[x,y];   
│ │ │ │ +0001ef00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ef10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ef20: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0001ef30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ef40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ef50: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a  ---------+.|i2 :
│ │ │ │ +0001ef60: 204d 203d 2072 616e 646f 6d28 525e 7b35   M = random(R^{5
│ │ │ │ +0001ef70: 2c35 2c35 2c35 2c35 2c35 7d2c 2052 5e37  ,5,5,5,5,5}, R^7
│ │ │ │ +0001ef80: 293b 2020 2020 2020 2020 2020 2020 2020  );              
│ │ │ │ +0001ef90: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001efa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001efb0: 207c 0a7c 6f32 203a 204d 6174 7269 7820   |.|o2 : Matrix 
│ │ │ │ -0001efc0: 5220 203c 2d2d 2052 2020 2020 2020 2020  R  <-- R        
│ │ │ │ -0001efd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001efe0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -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 2e35 3134 3935 3273 2028  used 0.514952s (
│ │ │ │ -0001f070: 6370 7529 3b20 302e 3436 3135 3434 7320  cpu); 0.461544s 
│ │ │ │ -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                  
│ │ │ │ +0001efb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001efc0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001efd0: 2020 2020 2020 2020 2036 2020 2020 2020           6      
│ │ │ │ +0001efe0: 3720 2020 2020 2020 2020 2020 2020 2020  7               
│ │ │ │ +0001eff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f000: 207c 0a7c 6f32 203a 204d 6174 7269 7820   |.|o2 : Matrix 
│ │ │ │ +0001f010: 5220 203c 2d2d 2052 2020 2020 2020 2020  R  <-- R        
│ │ │ │ +0001f020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f030: 2020 2020 2020 2020 207c 0a2b 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: 2d2b 0a7c 6933 203a 2074 696d 6520 4932  -+.|i3 : time I2
│ │ │ │ +0001f080: 203d 2072 6563 7572 7369 7665 4d69 6e6f   = recursiveMino
│ │ │ │ +0001f090: 7273 2834 2c20 4d2c 2054 6872 6561 6473  rs(4, M, Threads
│ │ │ │ +0001f0a0: 3d3e 3029 3b20 2020 207c 0a7c 202d 2d20  =>0);    |.| -- 
│ │ │ │ +0001f0b0: 7573 6564 2030 2e35 3638 3931 3873 2028  used 0.568918s (
│ │ │ │ +0001f0c0: 6370 7529 3b20 302e 3530 3735 3331 7320  cpu); 0.507531s 
│ │ │ │ +0001f0d0: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ +0001f0e0: 297c 0a7c 2020 2020 2020 2020 2020 2020  )|.|            
│ │ │ │  0001f0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f100: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -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: 3738 3831 7320 2863 7075 293b 2031 2e32  7881s (cpu); 1.2
│ │ │ │ -0001f190: 3736 3837 7320 2874 6872 6561 6429 3b20  7687s (thread); 
│ │ │ │ -0001f1a0: 3073 2028 6763 2920 207c 0a7c 2020 2020  0s (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             
│ │ │ │ +0001f100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f110: 2020 2020 2020 2020 207c 0a7c 6f33 203a           |.|o3 :
│ │ │ │ +0001f120: 2049 6465 616c 206f 6620 5220 2020 2020   Ideal of R     
│ │ │ │ +0001f130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f150: 207c 0a2b 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 2d2b 0a7c 6934 203a  ---------+.|i4 :
│ │ │ │ +0001f190: 2074 696d 6520 4931 203d 206d 696e 6f72   time I1 = minor
│ │ │ │ +0001f1a0: 7328 342c 204d 2c20 5374 7261 7465 6779  s(4, M, Strategy
│ │ │ │ +0001f1b0: 3d3e 436f 6661 6374 6f72 293b 2020 2020  =>Cofactor);    
│ │ │ │ +0001f1c0: 207c 0a7c 202d 2d20 7573 6564 2031 2e33   |.| -- used 1.3
│ │ │ │ +0001f1d0: 3731 3235 7320 2863 7075 293b 2031 2e32  7125s (cpu); 1.2
│ │ │ │ +0001f1e0: 3534 3831 7320 2874 6872 6561 6429 3b20  5481s (thread); 
│ │ │ │ +0001f1f0: 3073 2028 6763 2920 207c 0a7c 2020 2020  0s (gc)  |.|    
│ │ │ │  0001f200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f210: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -0001f220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f250: 2d2b 0a7c 6935 203a 2049 3120 3d3d 2049  -+.|i5 : I1 == I
│ │ │ │ -0001f260: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -0001f270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f280: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0001f290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f2c0: 207c 0a7c 6f35 203d 2074 7275 6520 2020   |.|o5 = true   
│ │ │ │ -0001f2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f230: 207c 0a7c 6f34 203a 2049 6465 616c 206f   |.|o4 : Ideal o
│ │ │ │ +0001f240: 6620 5220 2020 2020 2020 2020 2020 2020  f R             
│ │ │ │ +0001f250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f260: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001f270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f2a0: 2d2b 0a7c 6935 203a 2049 3120 3d3d 2049  -+.|i5 : I1 == I
│ │ │ │ +0001f2b0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +0001f2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f2d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0001f2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f2f0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -0001f300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f330: 2d2b 0a0a 5365 6520 616c 736f 0a3d 3d3d  -+..See also.===
│ │ │ │ -0001f340: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ -0001f350: 206d 696e 6f72 733a 2028 4d61 6361 756c   minors: (Macaul
│ │ │ │ -0001f360: 6179 3244 6f63 296d 696e 6f72 735f 6c70  ay2Doc)minors_lp
│ │ │ │ -0001f370: 5a5a 5f63 6d4d 6174 7269 785f 7270 2c20  ZZ_cmMatrix_rp, 
│ │ │ │ -0001f380: 2d2d 2069 6465 616c 2067 656e 6572 6174  -- ideal generat
│ │ │ │ -0001f390: 6564 2062 790a 2020 2020 6d69 6e6f 7273  ed by.    minors
│ │ │ │ -0001f3a0: 0a0a 5761 7973 2074 6f20 7573 6520 7265  ..Ways to use re
│ │ │ │ -0001f3b0: 6375 7273 6976 654d 696e 6f72 733a 0a3d  cursiveMinors:.=
│ │ │ │ -0001f3c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001f3d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ -0001f3e0: 2022 7265 6375 7273 6976 654d 696e 6f72   "recursiveMinor
│ │ │ │ -0001f3f0: 7328 5a5a 2c4d 6174 7269 7829 220a 0a46  s(ZZ,Matrix)"..F
│ │ │ │ -0001f400: 6f72 2074 6865 2070 726f 6772 616d 6d65  or the programme
│ │ │ │ -0001f410: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r.==============
│ │ │ │ -0001f420: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ -0001f430: 202a 6e6f 7465 2072 6563 7572 7369 7665   *note recursive
│ │ │ │ -0001f440: 4d69 6e6f 7273 3a20 7265 6375 7273 6976  Minors: recursiv
│ │ │ │ -0001f450: 654d 696e 6f72 732c 2069 7320 6120 2a6e  eMinors, is a *n
│ │ │ │ -0001f460: 6f74 6520 6d65 7468 6f64 2066 756e 6374  ote method funct
│ │ │ │ -0001f470: 696f 6e0a 7769 7468 206f 7074 696f 6e73  ion.with options
│ │ │ │ -0001f480: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -0001f490: 4d65 7468 6f64 4675 6e63 7469 6f6e 5769  MethodFunctionWi
│ │ │ │ -0001f4a0: 7468 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d  thOptions,...---
│ │ │ │ -0001f4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468  ------------..Th
│ │ │ │ -0001f500: 6520 736f 7572 6365 206f 6620 7468 6973  e source of this
│ │ │ │ -0001f510: 2064 6f63 756d 656e 7420 6973 2069 6e0a   document is in.
│ │ │ │ -0001f520: 2f62 7569 6c64 2f72 6570 726f 6475 6369  /build/reproduci
│ │ │ │ -0001f530: 626c 652d 7061 7468 2f6d 6163 6175 6c61  ble-path/macaula
│ │ │ │ -0001f540: 7932 2d31 2e32 352e 3131 2b64 732f 4d32  y2-1.25.11+ds/M2
│ │ │ │ -0001f550: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61  /Macaulay2/packa
│ │ │ │ -0001f560: 6765 732f 4661 7374 4d69 6e6f 7273 2e0a  ges/FastMinors..
│ │ │ │ -0001f570: 6d32 3a32 3033 303a 302e 0a1f 0a46 696c  m2:2030:0....Fil
│ │ │ │ -0001f580: 653a 2046 6173 744d 696e 6f72 732e 696e  e: FastMinors.in
│ │ │ │ -0001f590: 666f 2c20 4e6f 6465 3a20 7265 6775 6c61  fo, Node: regula
│ │ │ │ -0001f5a0: 7249 6e43 6f64 696d 656e 7369 6f6e 2c20  rInCodimension, 
│ │ │ │ -0001f5b0: 4e65 7874 3a20 5265 6775 6c61 7249 6e43  Next: RegularInC
│ │ │ │ -0001f5c0: 6f64 696d 656e 7369 6f6e 5475 746f 7269  odimensionTutori
│ │ │ │ -0001f5d0: 616c 2c20 5072 6576 3a20 7265 6375 7273  al, Prev: recurs
│ │ │ │ -0001f5e0: 6976 654d 696e 6f72 732c 2055 703a 2054  iveMinors, Up: T
│ │ │ │ -0001f5f0: 6f70 0a0a 7265 6775 6c61 7249 6e43 6f64  op..regularInCod
│ │ │ │ -0001f600: 696d 656e 7369 6f6e 202d 2d20 6174 7465  imension -- atte
│ │ │ │ -0001f610: 6d70 7473 2074 6f20 7368 6f77 2074 6861  mpts to show tha
│ │ │ │ -0001f620: 7420 7468 6520 7269 6e67 2069 7320 7265  t the ring is re
│ │ │ │ -0001f630: 6775 6c61 7220 696e 2063 6f64 696d 656e  gular in codimen
│ │ │ │ -0001f640: 7369 6f6e 206e 0a2a 2a2a 2a2a 2a2a 2a2a  sion n.*********
│ │ │ │ -0001f650: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001f660: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001f670: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001f680: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001f690: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055  *********..  * U
│ │ │ │ -0001f6a0: 7361 6765 3a20 0a20 2020 2020 2020 2072  sage: .        r
│ │ │ │ -0001f6b0: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ -0001f6c0: 696f 6e28 6e2c 2052 290a 2020 2a20 496e  ion(n, R).  * In
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│ │ │ │ -0001fc90: 7420 7661 6c75 6520 436f 6661 6374 6f72  t value Cofactor
│ │ │ │ -0001fca0: 2c0a 2020 2020 2020 2020 4465 7453 7472  ,.        DetStr
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│ │ │ │ -0001fda0: 2045 7874 656e 6446 6965 6c64 203d 3e0a   ExtendField =>.
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│ │ │ │ -0001fe00: 2020 204e 756d 5468 7265 6164 7354 6f55     NumThreadsToU
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│ │ │ │ -0001fe30: 6d2c 2056 6572 626f 7365 203d 3e20 6661  m, Verbose => fa
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│ │ │ │ -0001fe50: 7469 6f6e 7320 746f 2070 6173 7320 746f  tions to pass to
│ │ │ │ -0001fe60: 2066 756e 6374 696f 6e73 2069 6e20 7468   functions in th
│ │ │ │ -0001fe70: 6520 7061 636b 6167 6520 5261 6e64 6f6d  e package Random
│ │ │ │ -0001fe80: 506f 696e 7473 0a20 2020 2020 202a 202a  Points.      * *
│ │ │ │ -0001fe90: 6e6f 7465 2053 7472 6174 6567 793a 2053  note Strategy: S
│ │ │ │ -0001fea0: 7472 6174 6567 7944 6566 6175 6c74 2c20  trategyDefault, 
│ │ │ │ -0001feb0: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ -0001fec0: 7661 6c75 6520 6e65 7720 4f70 7469 6f6e  value new Option
│ │ │ │ -0001fed0: 5461 626c 650a 2020 2020 2020 2020 6672  Table.        fr
│ │ │ │ -0001fee0: 6f6d 207b 506f 696e 7473 203d 3e20 302c  om {Points => 0,
│ │ │ │ -0001fef0: 2052 616e 646f 6d20 3d3e 2031 362c 2047   Random => 16, G
│ │ │ │ -0001ff00: 5265 764c 6578 4c61 7267 6573 7420 3d3e  RevLexLargest =>
│ │ │ │ -0001ff10: 2030 2c20 4c65 7853 6d61 6c6c 6573 7454   0, LexSmallestT
│ │ │ │ -0001ff20: 6572 6d0a 2020 2020 2020 2020 3d3e 2031  erm.        => 1
│ │ │ │ -0001ff30: 362c 204c 6578 4c61 7267 6573 7420 3d3e  6, LexLargest =>
│ │ │ │ -0001ff40: 2030 2c20 4c65 7853 6d61 6c6c 6573 7420   0, LexSmallest 
│ │ │ │ -0001ff50: 3d3e 2031 362c 2047 5265 764c 6578 536d  => 16, GRevLexSm
│ │ │ │ -0001ff60: 616c 6c65 7374 5465 726d 203d 3e20 3136  allestTerm => 16
│ │ │ │ -0001ff70: 2c0a 2020 2020 2020 2020 5261 6e64 6f6d  ,.        Random
│ │ │ │ -0001ff80: 4e6f 6e7a 6572 6f20 3d3e 2031 362c 2047  Nonzero => 16, G
│ │ │ │ -0001ff90: 5265 764c 6578 536d 616c 6c65 7374 203d  RevLexSmallest =
│ │ │ │ -0001ffa0: 3e20 3136 7d2c 2073 7472 6174 6567 6965  > 16}, strategie
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│ │ │ │ -0001ffd0: 6573 0a20 2020 2020 202a 2056 6572 626f  es.      * Verbo
│ │ │ │ -0001ffe0: 7365 203d 3e20 2e2e 2e2c 2064 6566 6175  se => ..., defau
│ │ │ │ -0001fff0: 6c74 2076 616c 7565 2066 616c 7365 0a20  lt value false. 
│ │ │ │ -00020000: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ -00020010: 2020 2a20 7472 7565 2c20 6966 2074 6865    * true, if the
│ │ │ │ -00020020: 2072 696e 6720 6973 2072 6567 756c 6172   ring is regular
│ │ │ │ -00020030: 2069 6e20 636f 6469 6d65 6e73 696f 6e20   in codimension 
│ │ │ │ -00020040: 6e2c 2066 616c 7365 2069 6620 6974 2064  n, false if it d
│ │ │ │ -00020050: 6574 6572 6d69 6e65 730a 2020 2020 2020  etermines.      
│ │ │ │ -00020060: 2020 6974 2069 7320 6e6f 742c 2061 6e64    it is not, and
│ │ │ │ -00020070: 206e 756c 6c20 6966 206e 6f20 6465 7465   null if no dete
│ │ │ │ -00020080: 726d 696e 6174 696f 6e20 6973 206d 6164  rmination is mad
│ │ │ │ -00020090: 650a 0a44 6573 6372 6970 7469 6f6e 0a3d  e..Description.=
│ │ │ │ -000200a0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6973  ==========..This
│ │ │ │ -000200b0: 2066 756e 6374 696f 6e20 7265 7475 726e   function return
│ │ │ │ -000200c0: 7320 7472 7565 2069 6620 5220 6973 2072  s true if R is r
│ │ │ │ -000200d0: 6567 756c 6172 2069 6e20 636f 6469 6d65  egular in codime
│ │ │ │ -000200e0: 6e73 696f 6e20 6e2c 2066 616c 7365 2069  nsion n, false i
│ │ │ │ -000200f0: 6620 6974 2069 730a 6e6f 742c 2061 6e64  f it is.not, and
│ │ │ │ -00020100: 206e 756c 6c20 6966 2069 7420 6469 6420   null if it did 
│ │ │ │ -00020110: 6e6f 7420 6d61 6b65 2061 2064 6574 6572  not make a deter
│ │ │ │ -00020120: 6d69 6e61 7469 6f6e 2e20 4974 2063 6f6e  mination. It con
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│ │ │ │ -00020150: 206a 6163 6f62 6961 6e20 6d61 7472 6978   jacobian matrix
│ │ │ │ -00020160: 2074 6f20 7472 7920 746f 2076 6572 6966   to try to verif
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│ │ │ │ -000201a0: 6973 2066 7265 7175 656e 746c 7920 6d75  is frequently mu
│ │ │ │ -000201b0: 6368 2066 6173 7465 7220 6174 2067 6976  ch faster at giv
│ │ │ │ -000201c0: 696e 6720 616e 2061 6666 6972 6d61 7469  ing an affirmati
│ │ │ │ -000201d0: 7665 2061 6e73 7765 720a 7468 616e 2063  ve answer.than c
│ │ │ │ -000201e0: 6f6d 7075 7469 6e67 2074 6865 2064 696d  omputing the dim
│ │ │ │ -000201f0: 656e 7369 6f6e 206f 6620 7468 6520 6964  ension of the id
│ │ │ │ -00020200: 6561 6c20 6f66 2061 6c6c 206d 696e 6f72  eal of all minor
│ │ │ │ -00020210: 7320 6f66 2074 6865 204a 6163 6f62 6961  s of the Jacobia
│ │ │ │ -00020220: 6e2e 2057 650a 6265 6769 6e20 7769 7468  n. We.begin with
│ │ │ │ -00020230: 2061 2073 696d 706c 6520 6578 616d 706c   a simple exampl
│ │ │ │ -00020240: 6520 7768 6963 6820 6973 2052 312c 2062  e which is R1, b
│ │ │ │ -00020250: 7574 206e 6f74 2052 322e 0a0a 2b2d 2d2d  ut not R2...+---
│ │ │ │ -00020260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020280: 2d2b 0a7c 6931 203a 2052 203d 2051 515b  -+.|i1 : R = QQ[
│ │ │ │ -00020290: 782c 2079 2c20 7a5d 2f69 6465 616c 2878  x, y, z]/ideal(x
│ │ │ │ -000202a0: 2a79 2d7a 5e32 293b 7c0a 2b2d 2d2d 2d2d  *y-z^2);|.+-----
│ │ │ │ +0001f2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f310: 207c 0a7c 6f35 203d 2074 7275 6520 2020   |.|o5 = true   
│ │ │ │ +0001f320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f340: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001f350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f380: 2d2b 0a0a 5365 6520 616c 736f 0a3d 3d3d  -+..See also.===
│ │ │ │ +0001f390: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ +0001f3a0: 206d 696e 6f72 733a 2028 4d61 6361 756c   minors: (Macaul
│ │ │ │ +0001f3b0: 6179 3244 6f63 296d 696e 6f72 735f 6c70  ay2Doc)minors_lp
│ │ │ │ +0001f3c0: 5a5a 5f63 6d4d 6174 7269 785f 7270 2c20  ZZ_cmMatrix_rp, 
│ │ │ │ +0001f3d0: 2d2d 2069 6465 616c 2067 656e 6572 6174  -- ideal generat
│ │ │ │ +0001f3e0: 6564 2062 790a 2020 2020 6d69 6e6f 7273  ed by.    minors
│ │ │ │ +0001f3f0: 0a0a 5761 7973 2074 6f20 7573 6520 7265  ..Ways to use re
│ │ │ │ +0001f400: 6375 7273 6976 654d 696e 6f72 733a 0a3d  cursiveMinors:.=
│ │ │ │ +0001f410: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0001f420: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ +0001f430: 2022 7265 6375 7273 6976 654d 696e 6f72   "recursiveMinor
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│ │ │ │ +0001fce0: 7420 7661 6c75 6520 436f 6661 6374 6f72  t value Cofactor
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│ │ │ │ +0001fe80: 6d2c 2056 6572 626f 7365 203d 3e20 6661  m, Verbose => fa
│ │ │ │ +0001fe90: 6c73 657d 2c0a 2020 2020 2020 2020 6f70  lse},.        op
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│ │ │ │ +0001feb0: 2066 756e 6374 696f 6e73 2069 6e20 7468   functions in th
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│ │ │ │ +0001fee0: 6e6f 7465 2053 7472 6174 6567 793a 2053  note Strategy: S
│ │ │ │ +0001fef0: 7472 6174 6567 7944 6566 6175 6c74 2c20  trategyDefault, 
│ │ │ │ +0001ff00: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ +0001ff10: 7661 6c75 6520 6e65 7720 4f70 7469 6f6e  value new Option
│ │ │ │ +0001ff20: 5461 626c 650a 2020 2020 2020 2020 6672  Table.        fr
│ │ │ │ +0001ff30: 6f6d 207b 506f 696e 7473 203d 3e20 302c  om {Points => 0,
│ │ │ │ +0001ff40: 2052 616e 646f 6d20 3d3e 2031 362c 2047   Random => 16, G
│ │ │ │ +0001ff50: 5265 764c 6578 4c61 7267 6573 7420 3d3e  RevLexLargest =>
│ │ │ │ +0001ff60: 2030 2c20 4c65 7853 6d61 6c6c 6573 7454   0, LexSmallestT
│ │ │ │ +0001ff70: 6572 6d0a 2020 2020 2020 2020 3d3e 2031  erm.        => 1
│ │ │ │ +0001ff80: 362c 204c 6578 4c61 7267 6573 7420 3d3e  6, LexLargest =>
│ │ │ │ +0001ff90: 2030 2c20 4c65 7853 6d61 6c6c 6573 7420   0, LexSmallest 
│ │ │ │ +0001ffa0: 3d3e 2031 362c 2047 5265 764c 6578 536d  => 16, GRevLexSm
│ │ │ │ +0001ffb0: 616c 6c65 7374 5465 726d 203d 3e20 3136  allestTerm => 16
│ │ │ │ +0001ffc0: 2c0a 2020 2020 2020 2020 5261 6e64 6f6d  ,.        Random
│ │ │ │ +0001ffd0: 4e6f 6e7a 6572 6f20 3d3e 2031 362c 2047  Nonzero => 16, G
│ │ │ │ +0001ffe0: 5265 764c 6578 536d 616c 6c65 7374 203d  RevLexSmallest =
│ │ │ │ +0001fff0: 3e20 3136 7d2c 2073 7472 6174 6567 6965  > 16}, strategie
│ │ │ │ +00020000: 7320 666f 7220 6368 6f6f 7369 6e67 0a20  s for choosing. 
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│ │ │ │ +00020020: 6573 0a20 2020 2020 202a 2056 6572 626f  es.      * Verbo
│ │ │ │ +00020030: 7365 203d 3e20 2e2e 2e2c 2064 6566 6175  se => ..., defau
│ │ │ │ +00020040: 6c74 2076 616c 7565 2066 616c 7365 0a20  lt value false. 
│ │ │ │ +00020050: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ +00020060: 2020 2a20 7472 7565 2c20 6966 2074 6865    * true, if the
│ │ │ │ +00020070: 2072 696e 6720 6973 2072 6567 756c 6172   ring is regular
│ │ │ │ +00020080: 2069 6e20 636f 6469 6d65 6e73 696f 6e20   in codimension 
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│ │ │ │ +000200a0: 6574 6572 6d69 6e65 730a 2020 2020 2020  etermines.      
│ │ │ │ +000200b0: 2020 6974 2069 7320 6e6f 742c 2061 6e64    it is not, and
│ │ │ │ +000200c0: 206e 756c 6c20 6966 206e 6f20 6465 7465   null if no dete
│ │ │ │ +000200d0: 726d 696e 6174 696f 6e20 6973 206d 6164  rmination is mad
│ │ │ │ +000200e0: 650a 0a44 6573 6372 6970 7469 6f6e 0a3d  e..Description.=
│ │ │ │ +000200f0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6973  ==========..This
│ │ │ │ +00020100: 2066 756e 6374 696f 6e20 7265 7475 726e   function return
│ │ │ │ +00020110: 7320 7472 7565 2069 6620 5220 6973 2072  s true if R is r
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│ │ │ │ +00020160: 6e6f 7420 6d61 6b65 2061 2064 6574 6572  not make a deter
│ │ │ │ +00020170: 6d69 6e61 7469 6f6e 2e20 4974 2063 6f6e  mination. It con
│ │ │ │ +00020180: 7369 6465 7273 2069 6e74 6572 6573 7469  siders interesti
│ │ │ │ +00020190: 6e67 0a6d 696e 6f72 7320 6f66 2074 6865  ng.minors of the
│ │ │ │ +000201a0: 206a 6163 6f62 6961 6e20 6d61 7472 6978   jacobian matrix
│ │ │ │ +000201b0: 2074 6f20 7472 7920 746f 2076 6572 6966   to try to verif
│ │ │ │ +000201c0: 7920 7468 6174 2074 6865 2072 696e 6720  y that the ring 
│ │ │ │ +000201d0: 6973 2072 6567 756c 6172 2069 6e0a 636f  is regular in.co
│ │ │ │ +000201e0: 6469 6d65 6e73 696f 6e20 6e2e 2049 7420  dimension n. It 
│ │ │ │ +000201f0: 6973 2066 7265 7175 656e 746c 7920 6d75  is frequently mu
│ │ │ │ +00020200: 6368 2066 6173 7465 7220 6174 2067 6976  ch faster at giv
│ │ │ │ +00020210: 696e 6720 616e 2061 6666 6972 6d61 7469  ing an affirmati
│ │ │ │ +00020220: 7665 2061 6e73 7765 720a 7468 616e 2063  ve answer.than c
│ │ │ │ +00020230: 6f6d 7075 7469 6e67 2074 6865 2064 696d  omputing the dim
│ │ │ │ +00020240: 656e 7369 6f6e 206f 6620 7468 6520 6964  ension of the id
│ │ │ │ +00020250: 6561 6c20 6f66 2061 6c6c 206d 696e 6f72  eal of all minor
│ │ │ │ +00020260: 7320 6f66 2074 6865 204a 6163 6f62 6961  s of the Jacobia
│ │ │ │ +00020270: 6e2e 2057 650a 6265 6769 6e20 7769 7468  n. We.begin with
│ │ │ │ +00020280: 2061 2073 696d 706c 6520 6578 616d 706c   a simple exampl
│ │ │ │ +00020290: 6520 7768 6963 6820 6973 2052 312c 2062  e which is R1, b
│ │ │ │ +000202a0: 7574 206e 6f74 2052 322e 0a0a 2b2d 2d2d  ut not R2...+---
│ │ │ │  000202b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000202c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000202d0: 0a7c 6932 203a 2072 6567 756c 6172 496e  .|i2 : regularIn
│ │ │ │ -000202e0: 436f 6469 6d65 6e73 696f 6e28 312c 2052  Codimension(1, R
│ │ │ │ -000202f0: 2920 2020 2020 7c0a 7c20 2020 2020 2020  )     |.|       
│ │ │ │ -00020300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020310: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00020320: 6f32 203d 2074 7275 6520 2020 2020 2020  o2 = true       
│ │ │ │ -00020330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020340: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00020350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │ -00020370: 203a 2072 6567 756c 6172 496e 436f 6469   : regularInCodi
│ │ │ │ -00020380: 6d65 6e73 696f 6e28 322c 2052 2920 2020  mension(2, R)   
│ │ │ │ -00020390: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +000202c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000202d0: 2d2b 0a7c 6931 203a 2052 203d 2051 515b  -+.|i1 : R = QQ[
│ │ │ │ +000202e0: 782c 2079 2c20 7a5d 2f69 6465 616c 2878  x, y, z]/ideal(x
│ │ │ │ +000202f0: 2a79 2d7a 5e32 293b 7c0a 2b2d 2d2d 2d2d  *y-z^2);|.+-----
│ │ │ │ +00020300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00020320: 0a7c 6932 203a 2072 6567 756c 6172 496e  .|i2 : regularIn
│ │ │ │ +00020330: 436f 6469 6d65 6e73 696f 6e28 312c 2052  Codimension(1, R
│ │ │ │ +00020340: 2920 2020 2020 7c0a 7c20 2020 2020 2020  )     |.|       
│ │ │ │ +00020350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020360: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00020370: 6f32 203d 2074 7275 6520 2020 2020 2020  o2 = true       
│ │ │ │ +00020380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020390: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  000203a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000203b0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4e65 7874  ---------+..Next
│ │ │ │ -000203c0: 2077 6520 636f 6e73 6964 6572 2061 206d   we consider a m
│ │ │ │ -000203d0: 6f72 6520 696e 7465 7265 7374 696e 6720  ore interesting 
│ │ │ │ -000203e0: 6578 616d 706c 6520 7468 6174 2069 7320  example that is 
│ │ │ │ -000203f0: 5231 2062 7574 206e 6f74 2052 322c 2061  R1 but not R2, a
│ │ │ │ -00020400: 6e64 0a68 6967 686c 6967 6874 2074 6865  nd.highlight the
│ │ │ │ -00020410: 2073 7065 6564 2064 6966 6665 7265 6e63   speed differenc
│ │ │ │ -00020420: 6573 2e20 204e 6f74 6520 7468 6174 2072  es.  Note that r
│ │ │ │ -00020430: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ -00020440: 696f 6e28 322c 2052 2920 7265 7475 726e  ion(2, R) return
│ │ │ │ -00020450: 730a 6e6f 7468 696e 672c 2061 7320 7468  s.nothing, as th
│ │ │ │ -00020460: 6520 6675 6e63 7469 6f6e 2064 6964 206e  e function did n
│ │ │ │ -00020470: 6f74 2064 6574 6572 6d69 6e65 2077 6865  ot determine whe
│ │ │ │ -00020480: 7468 6572 2074 6865 2072 696e 6720 7761  ther the ring wa
│ │ │ │ -00020490: 7320 7265 6775 6c61 7220 696e 0a63 6f64  s regular in.cod
│ │ │ │ -000204a0: 696d 656e 7369 6f6e 206e 2e0a 0a2b 2d2d  imension n...+--
│ │ │ │ -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 2d2d  ----------------
│ │ │ │ -000204e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000204f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ -00020500: 203a 2054 203d 205a 5a2f 3130 315b 7831   : T = ZZ/101[x1
│ │ │ │ -00020510: 2c78 322c 7833 2c78 342c 7835 2c78 362c  ,x2,x3,x4,x5,x6,
│ │ │ │ -00020520: 7837 5d3b 2020 2020 2020 2020 2020 2020  x7];            
│ │ │ │ -00020530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020540: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -00020550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935  -----------+.|i5
│ │ │ │ -000205a0: 203a 2049 203d 2020 6964 6561 6c28 7835   : I =  ideal(x5
│ │ │ │ -000205b0: 2a78 362d 7834 2a78 372c 7831 2a78 362d  *x6-x4*x7,x1*x6-
│ │ │ │ -000205c0: 7832 2a78 372c 7835 5e32 2d78 312a 7837  x2*x7,x5^2-x1*x7
│ │ │ │ -000205d0: 2c78 342a 7835 2d78 322a 7837 2c78 345e  ,x4*x5-x2*x7,x4^
│ │ │ │ -000205e0: 322d 7832 2a78 362c 7831 2a7c 0a7c 2020  2-x2*x6,x1*|.|  
│ │ │ │ -000205f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020630: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ -00020640: 203a 2049 6465 616c 206f 6620 5420 2020   : Ideal of T   
│ │ │ │ +000203b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │ +000203c0: 203a 2072 6567 756c 6172 496e 436f 6469   : regularInCodi
│ │ │ │ +000203d0: 6d65 6e73 696f 6e28 322c 2052 2920 2020  mension(2, R)   
│ │ │ │ +000203e0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +000203f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020400: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4e65 7874  ---------+..Next
│ │ │ │ +00020410: 2077 6520 636f 6e73 6964 6572 2061 206d   we consider a m
│ │ │ │ +00020420: 6f72 6520 696e 7465 7265 7374 696e 6720  ore interesting 
│ │ │ │ +00020430: 6578 616d 706c 6520 7468 6174 2069 7320  example that is 
│ │ │ │ +00020440: 5231 2062 7574 206e 6f74 2052 322c 2061  R1 but not R2, a
│ │ │ │ +00020450: 6e64 0a68 6967 686c 6967 6874 2074 6865  nd.highlight the
│ │ │ │ +00020460: 2073 7065 6564 2064 6966 6665 7265 6e63   speed differenc
│ │ │ │ +00020470: 6573 2e20 204e 6f74 6520 7468 6174 2072  es.  Note that r
│ │ │ │ +00020480: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ +00020490: 696f 6e28 322c 2052 2920 7265 7475 726e  ion(2, R) return
│ │ │ │ +000204a0: 730a 6e6f 7468 696e 672c 2061 7320 7468  s.nothing, as th
│ │ │ │ +000204b0: 6520 6675 6e63 7469 6f6e 2064 6964 206e  e function did n
│ │ │ │ +000204c0: 6f74 2064 6574 6572 6d69 6e65 2077 6865  ot determine whe
│ │ │ │ +000204d0: 7468 6572 2074 6865 2072 696e 6720 7761  ther the ring wa
│ │ │ │ +000204e0: 7320 7265 6775 6c61 7220 696e 0a63 6f64  s regular in.cod
│ │ │ │ +000204f0: 696d 656e 7369 6f6e 206e 2e0a 0a2b 2d2d  imension n...+--
│ │ │ │ +00020500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ +00020550: 203a 2054 203d 205a 5a2f 3130 315b 7831   : T = ZZ/101[x1
│ │ │ │ +00020560: 2c78 322c 7833 2c78 342c 7835 2c78 362c  ,x2,x3,x4,x5,x6,
│ │ │ │ +00020570: 7837 5d3b 2020 2020 2020 2020 2020 2020  x7];            
│ │ │ │ +00020580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020590: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000205a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000205b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000205c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000205d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000205e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935  -----------+.|i5
│ │ │ │ +000205f0: 203a 2049 203d 2020 6964 6561 6c28 7835   : I =  ideal(x5
│ │ │ │ +00020600: 2a78 362d 7834 2a78 372c 7831 2a78 362d  *x6-x4*x7,x1*x6-
│ │ │ │ +00020610: 7832 2a78 372c 7835 5e32 2d78 312a 7837  x2*x7,x5^2-x1*x7
│ │ │ │ +00020620: 2c78 342a 7835 2d78 322a 7837 2c78 345e  ,x4*x5-x2*x7,x4^
│ │ │ │ +00020630: 322d 7832 2a78 362c 7831 2a7c 0a7c 2020  2-x2*x6,x1*|.|  
│ │ │ │ +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                  
│ │ │ │ -00020680: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ -00020690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000206a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000206b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000206c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000206d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 7834  -----------|.|x4
│ │ │ │ -000206e0: 2d78 322a 7835 2c78 322a 7833 5e33 2a78  -x2*x5,x2*x3^3*x
│ │ │ │ -000206f0: 352b 332a 7832 2a78 335e 322a 7837 2b38  5+3*x2*x3^2*x7+8
│ │ │ │ -00020700: 2a78 325e 322a 7835 2b33 2a78 332a 7834  *x2^2*x5+3*x3*x4
│ │ │ │ -00020710: 2a78 372d 382a 7834 2a78 372b 7836 2a78  *x7-8*x4*x7+x6*x
│ │ │ │ -00020720: 372c 7831 2a78 335e 332a 207c 0a7c 2d2d  7,x1*x3^3* |.|--
│ │ │ │ -00020730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 7835  -----------|.|x5
│ │ │ │ -00020780: 2b33 2a78 312a 7833 5e32 2a78 372b 382a  +3*x1*x3^2*x7+8*
│ │ │ │ -00020790: 7831 2a78 322a 7835 2b33 2a78 332a 7835  x1*x2*x5+3*x3*x5
│ │ │ │ -000207a0: 2a78 372d 382a 7835 2a78 372b 7837 5e32  *x7-8*x5*x7+x7^2
│ │ │ │ -000207b0: 2c78 322a 7833 5e33 2a78 342b 332a 7832  ,x2*x3^3*x4+3*x2
│ │ │ │ -000207c0: 2a78 335e 322a 7836 2b38 2a7c 0a7c 2d2d  *x3^2*x6+8*|.|--
│ │ │ │ -000207d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000207e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000207f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 7832  -----------|.|x2
│ │ │ │ -00020820: 5e32 2a78 342b 332a 7833 2a78 342a 7836  ^2*x4+3*x3*x4*x6
│ │ │ │ -00020830: 2d38 2a78 342a 7836 2b78 365e 322c 7832  -8*x4*x6+x6^2,x2
│ │ │ │ -00020840: 5e32 2a78 335e 332b 332a 7832 2a78 335e  ^2*x3^3+3*x2*x3^
│ │ │ │ -00020850: 322a 7834 2b38 2a78 325e 332b 332a 7832  2*x4+8*x2^3+3*x2
│ │ │ │ -00020860: 2a78 332a 7836 2d38 2a78 327c 0a7c 2d2d  *x3*x6-8*x2|.|--
│ │ │ │ -00020870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000208a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000208b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2a78  -----------|.|*x
│ │ │ │ -000208c0: 362b 7834 2a78 362c 7831 2a78 322a 7833  6+x4*x6,x1*x2*x3
│ │ │ │ -000208d0: 5e33 2b33 2a78 322a 7833 5e32 2a78 352b  ^3+3*x2*x3^2*x5+
│ │ │ │ -000208e0: 382a 7831 2a78 325e 322b 332a 7832 2a78  8*x1*x2^2+3*x2*x
│ │ │ │ -000208f0: 332a 7837 2d38 2a78 322a 7837 2b78 342a  3*x7-8*x2*x7+x4*
│ │ │ │ -00020900: 7837 2c78 315e 322a 7833 5e7c 0a7c 2d2d  x7,x1^2*x3^|.|--
│ │ │ │ -00020910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 2d7c 0a7c 332b  -----------|.|3+
│ │ │ │ -00020960: 332a 7831 2a78 335e 322a 7835 2b38 2a78  3*x1*x3^2*x5+8*x
│ │ │ │ -00020970: 315e 322a 7832 2b33 2a78 312a 7833 2a78  1^2*x2+3*x1*x3*x
│ │ │ │ -00020980: 372d 382a 7831 2a78 372b 7835 2a78 3729  7-8*x1*x7+x5*x7)
│ │ │ │ -00020990: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ -000209a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -000209b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000209c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000209d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000209e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000209f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936  -----------+.|i6
│ │ │ │ -00020a00: 203a 2053 203d 2054 2f49 3b20 2020 2020   : S = T/I;     
│ │ │ │ -00020a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020a40: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -00020a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00020a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937  -----------+.|i7
│ │ │ │ -00020aa0: 203a 2064 696d 2053 2020 2020 2020 2020   : dim S        
│ │ │ │ -00020ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020ae0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00020af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020680: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ +00020690: 203a 2049 6465 616c 206f 6620 5420 2020   : Ideal of T   
│ │ │ │ +000206a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000206b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000206c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000206d0: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ +000206e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000206f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 7834  -----------|.|x4
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│ │ │ │ +00020770: 372c 7831 2a78 335e 332a 207c 0a7c 2d2d  7,x1*x3^3* |.|--
│ │ │ │ +00020780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +000207b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000207c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 7835  -----------|.|x5
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│ │ │ │ +00020820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00020840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00020870: 5e32 2a78 342b 332a 7833 2a78 342a 7836  ^2*x4+3*x3*x4*x6
│ │ │ │ +00020880: 2d38 2a78 342a 7836 2b78 365e 322c 7832  -8*x4*x6+x6^2,x2
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│ │ │ │ +000208b0: 2a78 332a 7836 2d38 2a78 327c 0a7c 2d2d  *x3*x6-8*x2|.|--
│ │ │ │ +000208c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000208d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000208e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000208f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2a78  -----------|.|*x
│ │ │ │ +00020910: 362b 7834 2a78 362c 7831 2a78 322a 7833  6+x4*x6,x1*x2*x3
│ │ │ │ +00020920: 5e33 2b33 2a78 322a 7833 5e32 2a78 352b  ^3+3*x2*x3^2*x5+
│ │ │ │ +00020930: 382a 7831 2a78 325e 322b 332a 7832 2a78  8*x1*x2^2+3*x2*x
│ │ │ │ +00020940: 332a 7837 2d38 2a78 322a 7837 2b78 342a  3*x7-8*x2*x7+x4*
│ │ │ │ +00020950: 7837 2c78 315e 322a 7833 5e7c 0a7c 2d2d  x7,x1^2*x3^|.|--
│ │ │ │ +00020960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000209a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 332b  -----------|.|3+
│ │ │ │ +000209b0: 332a 7831 2a78 335e 322a 7835 2b38 2a78  3*x1*x3^2*x5+8*x
│ │ │ │ +000209c0: 315e 322a 7832 2b33 2a78 312a 7833 2a78  1^2*x2+3*x1*x3*x
│ │ │ │ +000209d0: 372d 382a 7831 2a78 372b 7835 2a78 3729  7-8*x1*x7+x5*x7)
│ │ │ │ +000209e0: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ +000209f0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00020a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936  -----------+.|i6
│ │ │ │ +00020a50: 203a 2053 203d 2054 2f49 3b20 2020 2020   : S = T/I;     
│ │ │ │ +00020a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020a90: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00020aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00020ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937  -----------+.|i7
│ │ │ │ +00020af0: 203a 2064 696d 2053 2020 2020 2020 2020   : dim S        
│ │ │ │  00020b00: 2020 2020 2020 2020 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 207c 0a7c 6f37             |.|o7
│ │ │ │ -00020b40: 203d 2033 2020 2020 2020 2020 2020 2020   = 3            
│ │ │ │ +00020b30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00020b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020b80: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -00020b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938  -----------+.|i8
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│ │ │ │ -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 3534 3339 3273  - used 0.654392s
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│ │ │ │ -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                  
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│ │ │ │ -00020cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020cc0: 2020 2020 2020 2020 2020 207c 0a7c 6f38             |.|o8
│ │ │ │ -00020cd0: 203d 2074 7275 6520 2020 2020 2020 2020   = true         
│ │ │ │ +00020b80: 2020 2020 2020 2020 2020 207c 0a7c 6f37             |.|o7
│ │ │ │ +00020b90: 203d 2033 2020 2020 2020 2020 2020 2020   = 3            
│ │ │ │ +00020ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020bd0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
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│ │ │ │ +00020c70: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
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│ │ │ │ +00020cc0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
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│ │ │ │  00020ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00020d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020d10: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
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│ │ │ │ -00020d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00020d90: 5329 2020 2020 2020 2020 2020 2020 2020  S)              
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│ │ │ │ -00020db0: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
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│ │ │ │ -00020e00: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
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│ │ │ │ -00020e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5468  -----------+..Th
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│ │ │ │ -00020ff0: 6f66 2064 696d 2073 696e 6775 6c61 724c  of dim singularL
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│ │ │ │ -00021030: 616d 652e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  ame...+---------
│ │ │ │ -00021040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021080: 2d2d 2d2d 2b0a 7c69 3130 203a 2052 203d  ----+.|i10 : R =
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│ │ │ │ -000210a0: 6964 6561 6c28 675e 332b 685e 332b 312c  ideal(g^3+h^3+1,
│ │ │ │ -000210b0: 662a 675e 332b 662a 685e 332b 662c 632a  f*g^3+f*h^3+f,c*
│ │ │ │ -000210c0: 675e 332b 632a 685e 332b 632c 665e 322a  g^3+c*h^3+c,f^2*
│ │ │ │ -000210d0: 675e 332b 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  g^3+|.|---------
│ │ │ │ -000210e0: 2d2d 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 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021120: 2d2d 2d2d 7c0a 7c66 5e32 2a68 5e33 2b66  ----|.|f^2*h^3+f
│ │ │ │ -00021130: 5e32 2c63 2a66 2a67 5e33 2b63 2a66 2a68  ^2,c*f*g^3+c*f*h
│ │ │ │ -00021140: 5e33 2b63 2a66 2c63 5e32 2a67 5e33 2b63  ^3+c*f,c^2*g^3+c
│ │ │ │ -00021150: 5e32 2a68 5e33 2b63 5e32 2c66 5e33 2a67  ^2*h^3+c^2,f^3*g
│ │ │ │ -00021160: 5e33 2b66 5e33 2a68 5e33 2b66 5e33 2c63  ^3+f^3*h^3+f^3,c
│ │ │ │ -00021170: 2a66 5e32 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  *f^2|.|---------
│ │ │ │ -00021180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000211a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000211b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000211c0: 2d2d 2d2d 7c0a 7c2a 675e 332b 632a 665e  ----|.|*g^3+c*f^
│ │ │ │ -000211d0: 322a 685e 332b 632a 665e 322c 635e 322a  2*h^3+c*f^2,c^2*
│ │ │ │ -000211e0: 662a 675e 332b 635e 322a 662a 685e 332b  f*g^3+c^2*f*h^3+
│ │ │ │ -000211f0: 635e 322a 662c 635e 332d 665e 322d 632c  c^2*f,c^3-f^2-c,
│ │ │ │ -00021200: 635e 332a 682d 665e 322a 682d 632a 682c  c^3*h-f^2*h-c*h,
│ │ │ │ -00021210: 635e 332a 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  c^3*|.|---------
│ │ │ │ -00021220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021260: 2d2d 2d2d 7c0a 7c67 2d66 5e32 2a67 2d63  ----|.|g-f^2*g-c
│ │ │ │ -00021270: 2a67 2c63 5e33 2a68 5e32 2d66 5e32 2a68  *g,c^3*h^2-f^2*h
│ │ │ │ -00021280: 5e32 2d63 2a68 5e32 2c63 5e33 2a67 2a68  ^2-c*h^2,c^3*g*h
│ │ │ │ -00021290: 2d66 5e32 2a67 2a68 2d63 2a67 2a68 2c63  -f^2*g*h-c*g*h,c
│ │ │ │ -000212a0: 5e33 2a67 5e32 2d66 5e32 2a67 5e32 2d63  ^3*g^2-f^2*g^2-c
│ │ │ │ -000212b0: 2a67 5e32 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  *g^2|.|---------
│ │ │ │ -000212c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000212d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000212e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000212f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021300: 2d2d 2d2d 7c0a 7c2c 635e 332a 685e 332d  ----|.|,c^3*h^3-
│ │ │ │ -00021310: 665e 322a 685e 332d 632a 685e 332c 635e  f^2*h^3-c*h^3,c^
│ │ │ │ -00021320: 332a 672a 685e 322d 665e 322a 672a 685e  3*g*h^2-f^2*g*h^
│ │ │ │ -00021330: 322d 632a 672a 685e 322c 635e 332a 675e  2-c*g*h^2,c^3*g^
│ │ │ │ -00021340: 322a 682d 665e 322a 675e 322a 682d 632a  2*h-f^2*g^2*h-c*
│ │ │ │ -00021350: 675e 322a 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  g^2*|.|---------
│ │ │ │ -00021360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000213a0: 2d2d 2d2d 7c0a 7c68 2c63 5e33 2a67 5e33  ----|.|h,c^3*g^3
│ │ │ │ -000213b0: 2b66 5e32 2a68 5e33 2b63 2a68 5e33 2b66  +f^2*h^3+c*h^3+f
│ │ │ │ -000213c0: 5e32 2b63 293b 2020 2020 2020 2020 2020  ^2+c);          
│ │ │ │ -000213d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000213e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000213f0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00021400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021440: 2d2d 2d2d 2b0a 7c69 3131 203a 2064 696d  ----+.|i11 : dim
│ │ │ │ -00021450: 2852 2920 2020 2020 2020 2020 2020 2020  (R)             
│ │ │ │ -00021460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021490: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000214a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020d10: 2020 2020 2020 2020 2020 207c 0a7c 6f38             |.|o8
│ │ │ │ +00020d20: 203d 2074 7275 6520 2020 2020 2020 2020   = true         
│ │ │ │ +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: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00020d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939  -----------+.|i9
│ │ │ │ +00020dc0: 203a 2074 696d 6520 7265 6775 6c61 7249   : time regularI
│ │ │ │ +00020dd0: 6e43 6f64 696d 656e 7369 6f6e 2832 2c20  nCodimension(2, 
│ │ │ │ +00020de0: 5329 2020 2020 2020 2020 2020 2020 2020  S)              
│ │ │ │ +00020df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020e00: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ +00020e10: 2d20 7573 6564 2037 2e30 3730 3338 7320  - used 7.07038s 
│ │ │ │ +00020e20: 2863 7075 293b 2035 2e33 3538 3334 7320  (cpu); 5.35834s 
│ │ │ │ +00020e30: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ +00020e40: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +00020e50: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00020e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5468  -----------+..Th
│ │ │ │ +00020eb0: 6572 6520 6172 6520 6e75 6d65 726f 7573  ere are numerous
│ │ │ │ +00020ec0: 2065 7861 6d70 6c65 7320 7768 6572 6520   examples where 
│ │ │ │ +00020ed0: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ +00020ee0: 7369 6f6e 2069 7320 7365 7665 7261 6c20  sion is several 
│ │ │ │ +00020ef0: 6f72 6465 7273 206f 660a 6d61 676e 6974  orders of.magnit
│ │ │ │ +00020f00: 7564 6520 6661 7374 6572 2074 6861 7420  ude faster that 
│ │ │ │ +00020f10: 6361 6c6c 7320 6f66 2064 696d 2073 696e  calls of dim sin
│ │ │ │ +00020f20: 6775 6c61 724c 6f63 7573 2e0a 0a54 6865  gularLocus...The
│ │ │ │ +00020f30: 2066 6f6c 6c6f 7769 6e67 2069 7320 6120   following is a 
│ │ │ │ +00020f40: 2870 7275 6e65 6429 2061 6666 696e 6520  (pruned) affine 
│ │ │ │ +00020f50: 6368 6172 7420 6f6e 2061 6e20 4162 656c  chart on an Abel
│ │ │ │ +00020f60: 6961 6e20 7375 7266 6163 6520 6f62 7461  ian surface obta
│ │ │ │ +00020f70: 696e 6564 2061 7320 610a 7072 6f64 7563  ined as a.produc
│ │ │ │ +00020f80: 7420 6f66 2074 776f 2065 6c6c 6970 7469  t of two ellipti
│ │ │ │ +00020f90: 6320 6375 7276 6573 2e20 2049 7420 6973  c curves.  It is
│ │ │ │ +00020fa0: 206e 6f6e 7369 6e67 756c 6172 2c20 6173   nonsingular, as
│ │ │ │ +00020fb0: 206f 7572 2066 756e 6374 696f 6e20 7665   our function ve
│ │ │ │ +00020fc0: 7269 6669 6573 2e0a 4966 206f 6e65 2064  rifies..If one d
│ │ │ │ +00020fd0: 6f65 7320 6e6f 7420 7072 756e 6520 6974  oes not prune it
│ │ │ │ +00020fe0: 2c20 7468 656e 2074 6865 2064 696d 2073  , then the dim s
│ │ │ │ +00020ff0: 696e 6775 6c61 724c 6f63 7573 2063 616c  ingularLocus cal
│ │ │ │ +00021000: 6c20 7461 6b65 7320 616e 2065 6e6f 726d  l takes an enorm
│ │ │ │ +00021010: 6f75 730a 616d 6f75 6e74 206f 6620 7469  ous.amount of ti
│ │ │ │ +00021020: 6d65 2c20 6f74 6865 7277 6973 6520 7468  me, otherwise th
│ │ │ │ +00021030: 6520 7275 6e6e 696e 6720 7469 6d65 7320  e running times 
│ │ │ │ +00021040: 6f66 2064 696d 2073 696e 6775 6c61 724c  of dim singularL
│ │ │ │ +00021050: 6f63 7573 2061 6e64 206f 7572 0a66 756e  ocus and our.fun
│ │ │ │ +00021060: 6374 696f 6e20 6172 6520 6672 6571 7565  ction are freque
│ │ │ │ +00021070: 6e74 6c79 2061 626f 7574 2074 6865 2073  ntly about the s
│ │ │ │ +00021080: 616d 652e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  ame...+---------
│ │ │ │ +00021090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000210a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000210b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000210c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000210d0: 2d2d 2d2d 2b0a 7c69 3130 203a 2052 203d  ----+.|i10 : R =
│ │ │ │ +000210e0: 2051 515b 632c 2066 2c20 672c 2068 5d2f   QQ[c, f, g, h]/
│ │ │ │ +000210f0: 6964 6561 6c28 675e 332b 685e 332b 312c  ideal(g^3+h^3+1,
│ │ │ │ +00021100: 662a 675e 332b 662a 685e 332b 662c 632a  f*g^3+f*h^3+f,c*
│ │ │ │ +00021110: 675e 332b 632a 685e 332b 632c 665e 322a  g^3+c*h^3+c,f^2*
│ │ │ │ +00021120: 675e 332b 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  g^3+|.|---------
│ │ │ │ +00021130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021170: 2d2d 2d2d 7c0a 7c66 5e32 2a68 5e33 2b66  ----|.|f^2*h^3+f
│ │ │ │ +00021180: 5e32 2c63 2a66 2a67 5e33 2b63 2a66 2a68  ^2,c*f*g^3+c*f*h
│ │ │ │ +00021190: 5e33 2b63 2a66 2c63 5e32 2a67 5e33 2b63  ^3+c*f,c^2*g^3+c
│ │ │ │ +000211a0: 5e32 2a68 5e33 2b63 5e32 2c66 5e33 2a67  ^2*h^3+c^2,f^3*g
│ │ │ │ +000211b0: 5e33 2b66 5e33 2a68 5e33 2b66 5e33 2c63  ^3+f^3*h^3+f^3,c
│ │ │ │ +000211c0: 2a66 5e32 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  *f^2|.|---------
│ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021210: 2d2d 2d2d 7c0a 7c2a 675e 332b 632a 665e  ----|.|*g^3+c*f^
│ │ │ │ +00021220: 322a 685e 332b 632a 665e 322c 635e 322a  2*h^3+c*f^2,c^2*
│ │ │ │ +00021230: 662a 675e 332b 635e 322a 662a 685e 332b  f*g^3+c^2*f*h^3+
│ │ │ │ +00021240: 635e 322a 662c 635e 332d 665e 322d 632c  c^2*f,c^3-f^2-c,
│ │ │ │ +00021250: 635e 332a 682d 665e 322a 682d 632a 682c  c^3*h-f^2*h-c*h,
│ │ │ │ +00021260: 635e 332a 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  c^3*|.|---------
│ │ │ │ +00021270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000212a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000212b0: 2d2d 2d2d 7c0a 7c67 2d66 5e32 2a67 2d63  ----|.|g-f^2*g-c
│ │ │ │ +000212c0: 2a67 2c63 5e33 2a68 5e32 2d66 5e32 2a68  *g,c^3*h^2-f^2*h
│ │ │ │ +000212d0: 5e32 2d63 2a68 5e32 2c63 5e33 2a67 2a68  ^2-c*h^2,c^3*g*h
│ │ │ │ +000212e0: 2d66 5e32 2a67 2a68 2d63 2a67 2a68 2c63  -f^2*g*h-c*g*h,c
│ │ │ │ +000212f0: 5e33 2a67 5e32 2d66 5e32 2a67 5e32 2d63  ^3*g^2-f^2*g^2-c
│ │ │ │ +00021300: 2a67 5e32 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  *g^2|.|---------
│ │ │ │ +00021310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021350: 2d2d 2d2d 7c0a 7c2c 635e 332a 685e 332d  ----|.|,c^3*h^3-
│ │ │ │ +00021360: 665e 322a 685e 332d 632a 685e 332c 635e  f^2*h^3-c*h^3,c^
│ │ │ │ +00021370: 332a 672a 685e 322d 665e 322a 672a 685e  3*g*h^2-f^2*g*h^
│ │ │ │ +00021380: 322d 632a 672a 685e 322c 635e 332a 675e  2-c*g*h^2,c^3*g^
│ │ │ │ +00021390: 322a 682d 665e 322a 675e 322a 682d 632a  2*h-f^2*g^2*h-c*
│ │ │ │ +000213a0: 675e 322a 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d  g^2*|.|---------
│ │ │ │ +000213b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000213c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000213d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000213e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000213f0: 2d2d 2d2d 7c0a 7c68 2c63 5e33 2a67 5e33  ----|.|h,c^3*g^3
│ │ │ │ +00021400: 2b66 5e32 2a68 5e33 2b63 2a68 5e33 2b66  +f^2*h^3+c*h^3+f
│ │ │ │ +00021410: 5e32 2b63 293b 2020 2020 2020 2020 2020  ^2+c);          
│ │ │ │ +00021420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021440: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00021450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021490: 2d2d 2d2d 2b0a 7c69 3131 203a 2064 696d  ----+.|i11 : dim
│ │ │ │ +000214a0: 2852 2920 2020 2020 2020 2020 2020 2020  (R)             
│ │ │ │  000214b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000214c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000214d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000214e0: 2020 2020 7c0a 7c6f 3131 203d 2032 2020      |.|o11 = 2  
│ │ │ │ +000214e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  000214f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021530: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00021540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 3031 3939 3938 3873 2028 6370 7529  0.0199988s (cpu)
│ │ │ │ -000215f0: 3b20 302e 3031 3938 3039 3673 2028 7468  ; 0.0198096s (th
│ │ │ │ -00021600: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │ +00021530: 2020 2020 7c0a 7c6f 3131 203d 2032 2020      |.|o11 = 2  
│ │ │ │ +00021540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021580: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00021590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000215a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000215b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000215c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000215d0: 2d2d 2d2d 2b0a 7c69 3132 203a 2074 696d  ----+.|i12 : tim
│ │ │ │ +000215e0: 6520 2864 696d 2073 696e 6775 6c61 724c  e (dim singularL
│ │ │ │ +000215f0: 6f63 7573 2028 5229 2920 2020 2020 2020  ocus (R))       
│ │ │ │ +00021600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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                  
│ │ │ │ +00021620: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ +00021630: 302e 3032 3339 3434 3473 2028 6370 7529  0.0239444s (cpu)
│ │ │ │ +00021640: 3b20 302e 3032 3131 3433 3673 2028 7468  ; 0.0211436s (th
│ │ │ │ +00021650: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │  00021660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021670: 2020 2020 7c0a 7c6f 3132 203d 202d 3120      |.|o12 = -1 
│ │ │ │ +00021670: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00021680: 2020 2020 2020 2020 2020 2020 2020 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 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -000216d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000216e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000216f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 3835 7320 2863 7075 293b  0.182885s (cpu);
│ │ │ │ -00021780: 2030 2e31 3335 3134 3773 2028 7468 7265   0.135147s (thre
│ │ │ │ -00021790: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │ +000216c0: 2020 2020 7c0a 7c6f 3132 203d 202d 3120      |.|o12 = -1 
│ │ │ │ +000216d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000216e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000216f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021710: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00021720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021760: 2d2d 2d2d 2b0a 7c69 3133 203a 2074 696d  ----+.|i13 : tim
│ │ │ │ +00021770: 6520 7265 6775 6c61 7249 6e43 6f64 696d  e regularInCodim
│ │ │ │ +00021780: 656e 7369 6f6e 2832 2c20 5229 2020 2020  ension(2, R)    
│ │ │ │ +00021790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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                  
│ │ │ │ +000217b0: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ +000217c0: 302e 3139 3830 3539 7320 2863 7075 293b  0.198059s (cpu);
│ │ │ │ +000217d0: 2030 2e31 3530 3630 3873 2028 7468 7265   0.150608s (thre
│ │ │ │ +000217e0: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │  000217f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021800: 2020 2020 7c0a 7c6f 3133 203d 2074 7275      |.|o13 = tru
│ │ │ │ -00021810: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ +00021800: 2020 2020 7c0a 7c20 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021850: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00021860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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 3931 3934 3973 2028 6370 7529 3b20  0.91949s (cpu); 
│ │ │ │ -00021910: 302e 3537 3233 3332 7320 2874 6872 6561  0.572332s (threa
│ │ │ │ -00021920: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +00021850: 2020 2020 7c0a 7c6f 3133 203d 2074 7275      |.|o13 = tru
│ │ │ │ +00021860: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ +00021870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000218a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000218b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000218c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000218d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000218e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000218f0: 2d2d 2d2d 2b0a 7c69 3134 203a 2074 696d  ----+.|i14 : tim
│ │ │ │ +00021900: 6520 7265 6775 6c61 7249 6e43 6f64 696d  e regularInCodim
│ │ │ │ +00021910: 656e 7369 6f6e 2832 2c20 5229 2020 2020  ension(2, R)    
│ │ │ │ +00021920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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                  
│ │ │ │ +00021940: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ +00021950: 312e 3035 3439 3273 2028 6370 7529 3b20  1.05492s (cpu); 
│ │ │ │ +00021960: 302e 3637 3234 3734 7320 2874 6872 6561  0.672474s (threa
│ │ │ │ +00021970: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │  00021980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021990: 2020 2020 7c0a 7c6f 3134 203d 2074 7275      |.|o14 = tru
│ │ │ │ -000219a0: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ +00021990: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000219a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000219b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000219c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000219d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000219e0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -000219f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 3234 3130 3373 2028 6370 7529 3b20  1.24103s (cpu); 
│ │ │ │ -00021aa0: 302e 3836 3633 3033 7320 2874 6872 6561  0.866303s (threa
│ │ │ │ -00021ab0: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +000219e0: 2020 2020 7c0a 7c6f 3134 203d 2074 7275      |.|o14 = tru
│ │ │ │ +000219f0: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ +00021a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021a30: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00021a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021a80: 2d2d 2d2d 2b0a 7c69 3135 203a 2074 696d  ----+.|i15 : tim
│ │ │ │ +00021a90: 6520 7265 6775 6c61 7249 6e43 6f64 696d  e regularInCodim
│ │ │ │ +00021aa0: 656e 7369 6f6e 2832 2c20 5229 2020 2020  ension(2, R)    
│ │ │ │ +00021ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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                  
│ │ │ │ +00021ad0: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ +00021ae0: 312e 3436 3939 3873 2028 6370 7529 3b20  1.46998s (cpu); 
│ │ │ │ +00021af0: 312e 3030 3836 3873 2028 7468 7265 6164  1.00868s (thread
│ │ │ │ +00021b00: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │  00021b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021b20: 2020 2020 7c0a 7c6f 3135 203d 2074 7275      |.|o15 = tru
│ │ │ │ -00021b30: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ +00021b20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00021b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021b70: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00021b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021bc0: 2d2d 2d2d 2b0a 0a54 6865 2066 756e 6374  ----+..The funct
│ │ │ │ -00021bd0: 696f 6e20 776f 726b 7320 6279 2063 686f  ion works by cho
│ │ │ │ -00021be0: 6f73 696e 6720 696e 7465 7265 7374 696e  osing interestin
│ │ │ │ -00021bf0: 6720 6c6f 6f6b 696e 6720 7375 626d 6174  g looking submat
│ │ │ │ -00021c00: 7269 6365 732c 2063 6f6d 7075 7469 6e67  rices, computing
│ │ │ │ -00021c10: 2074 6865 6972 0a64 6574 6572 6d69 6e61   their.determina
│ │ │ │ -00021c20: 6e74 732c 2061 6e64 2070 6572 696f 6469  nts, and periodi
│ │ │ │ -00021c30: 6361 6c6c 7920 2862 6173 6564 206f 6e20  cally (based on 
│ │ │ │ -00021c40: 6120 6c6f 6761 7269 7468 6d69 6320 6772  a logarithmic gr
│ │ │ │ -00021c50: 6f77 7468 2073 6574 7469 6e67 292c 0a63  owth setting),.c
│ │ │ │ -00021c60: 6f6d 7075 7469 6e67 2074 6865 2064 696d  omputing the dim
│ │ │ │ -00021c70: 656e 7369 6f6e 206f 6620 6120 7375 6269  ension of a subi
│ │ │ │ -00021c80: 6465 616c 206f 6620 7468 6520 4a61 636f  deal of the Jaco
│ │ │ │ -00021c90: 6269 616e 2e20 5468 6520 6f70 7469 6f6e  bian. The option
│ │ │ │ -00021ca0: 2056 6572 626f 7365 2063 616e 0a62 6520   Verbose can.be 
│ │ │ │ -00021cb0: 7573 6564 2074 6f20 7365 6520 7468 6973  used to see this
│ │ │ │ -00021cc0: 2069 6e20 6163 7469 6f6e 2e0a 0a2b 2d2d   in action...+--
│ │ │ │ -00021cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00021d20: 3620 3a20 7469 6d65 2072 6567 756c 6172  6 : time regular
│ │ │ │ -00021d30: 496e 436f 6469 6d65 6e73 696f 6e28 322c  InCodimension(2,
│ │ │ │ -00021d40: 2053 2c20 5665 7262 6f73 653d 3e74 7275   S, Verbose=>tru
│ │ │ │ -00021d50: 6529 2020 2020 2020 2020 2020 2020 2020  e)              
│ │ │ │ -00021d60: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │ -00021d70: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ -00021d80: 6f6e 3a20 7269 6e67 2064 696d 656e 7369  on: ring dimensi
│ │ │ │ -00021d90: 6f6e 203d 332c 2074 6865 7265 2061 7265  on =3, there are
│ │ │ │ -00021da0: 2031 3733 3235 2070 6f73 7369 626c 6520   17325 possible 
│ │ │ │ -00021db0: 3420 6279 2034 206d 696e 6f7c 0a7c 7265  4 by 4 mino|.|re
│ │ │ │ +00021b70: 2020 2020 7c0a 7c6f 3135 203d 2074 7275      |.|o15 = tru
│ │ │ │ +00021b80: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ +00021b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021bc0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00021bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021c10: 2d2d 2d2d 2b0a 0a54 6865 2066 756e 6374  ----+..The funct
│ │ │ │ +00021c20: 696f 6e20 776f 726b 7320 6279 2063 686f  ion works by cho
│ │ │ │ +00021c30: 6f73 696e 6720 696e 7465 7265 7374 696e  osing interestin
│ │ │ │ +00021c40: 6720 6c6f 6f6b 696e 6720 7375 626d 6174  g looking submat
│ │ │ │ +00021c50: 7269 6365 732c 2063 6f6d 7075 7469 6e67  rices, computing
│ │ │ │ +00021c60: 2074 6865 6972 0a64 6574 6572 6d69 6e61   their.determina
│ │ │ │ +00021c70: 6e74 732c 2061 6e64 2070 6572 696f 6469  nts, and periodi
│ │ │ │ +00021c80: 6361 6c6c 7920 2862 6173 6564 206f 6e20  cally (based on 
│ │ │ │ +00021c90: 6120 6c6f 6761 7269 7468 6d69 6320 6772  a logarithmic gr
│ │ │ │ +00021ca0: 6f77 7468 2073 6574 7469 6e67 292c 0a63  owth setting),.c
│ │ │ │ +00021cb0: 6f6d 7075 7469 6e67 2074 6865 2064 696d  omputing the dim
│ │ │ │ +00021cc0: 656e 7369 6f6e 206f 6620 6120 7375 6269  ension of a subi
│ │ │ │ +00021cd0: 6465 616c 206f 6620 7468 6520 4a61 636f  deal of the Jaco
│ │ │ │ +00021ce0: 6269 616e 2e20 5468 6520 6f70 7469 6f6e  bian. The option
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│ │ │ │ +00021d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
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│ │ │ │ +00021d90: 2053 2c20 5665 7262 6f73 653d 3e74 7275   S, Verbose=>tru
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│ │ │ │ +00021db0: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
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│ │ │ │ -00021e20: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ +00021df0: 2031 3733 3235 2070 6f73 7369 626c 6520   17325 possible 
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│ │ │ │ +00021e10: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
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│ │ │ │  00021e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021e50: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ +00021e70: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │  00021ea0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00021ec0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ +00021ec0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │  00021ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021ef0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │  00021f10: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │  00021f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021f40: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ +00021f60: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │  00021f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021f90: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │  00021fb0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00021fc0: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
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│ │ │ │  00021fe0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ +00022000: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │  00022020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022030: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00022050: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00022050: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │  00022070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022080: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │  000220a0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │  000220c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000220d0: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
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│ │ │ │ -00022110: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
│ │ │ │ -00022120: 6d61 7472 6963 6573 2063 6f7c 0a7c 7265  matrices co|.|re
│ │ │ │ +000220d0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ +000220f0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00022100: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │ +00022110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022120: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
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│ │ │ │ -00022150: 6173 7420 6661 696c 6564 2c20 636f 6d70  ast failed, comp
│ │ │ │ -00022160: 7574 696e 6720 636f 6469 6d2e 2020 2020  uting codim.    
│ │ │ │ -00022170: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
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│ │ │ │ +00022150: 6162 6f75 7420 746f 2063 6f6d 7075 7465  about to compute
│ │ │ │ +00022160: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
│ │ │ │ +00022170: 6d61 7472 6963 6573 2063 6f7c 0a7c 7265  matrices co|.|re
│ │ │ │  00022180: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ -00022190: 6f6e 3a20 2070 6172 7469 616c 2073 696e  on:  partial sin
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│ │ │ │ -000221b0: 6e73 696f 6e20 636f 6d70 7574 6564 2c20  nsion computed, 
│ │ │ │ -000221c0: 3d20 3220 2020 2020 2020 207c 0a7c 696e  = 2        |.|in
│ │ │ │ -000221d0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -000221e0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ -00022200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022210: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ +000221a0: 6173 7420 6661 696c 6564 2c20 636f 6d70  ast failed, comp
│ │ │ │ +000221b0: 7574 696e 6720 636f 6469 6d2e 2020 2020  uting codim.    
│ │ │ │ +000221c0: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
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│ │ │ │  00022220: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00022230: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00022240: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │ +00022230: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │  00022250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022260: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
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│ │ │ │ -000222a0: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
│ │ │ │ -000222b0: 6d61 7472 6963 6573 2063 6f7c 0a7c 7265  matrices co|.|re
│ │ │ │ +00022260: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ +00022280: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +000222a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000222b0: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
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│ │ │ │ -000222e0: 6173 7420 6661 696c 6564 2c20 636f 6d70  ast failed, comp
│ │ │ │ -000222f0: 7574 696e 6720 636f 6469 6d2e 2020 2020  uting codim.    
│ │ │ │ -00022300: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
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│ │ │ │ +000222f0: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
│ │ │ │ +00022300: 6d61 7472 6963 6573 2063 6f7c 0a7c 7265  matrices co|.|re
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│ │ │ │ -00022390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000223a0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -000223c0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +000223c0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │  000223f0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │  00022410: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │  00022440: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00022460: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00022460: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │  00022480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022490: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
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│ │ │ │ -000224c0: 6162 6f75 7420 746f 2063 6f6d 7075 7465  about to compute
│ │ │ │ -000224d0: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
│ │ │ │ -000224e0: 6d61 7472 6963 6573 2063 6f7c 0a7c 7265  matrices co|.|re
│ │ │ │ +00022490: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │ +000224a0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ +000224b0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +000224d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000224e0: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
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│ │ │ │ -00022500: 6f6e 3a20 2069 7343 6f64 696d 4174 4c65  on:  isCodimAtLe
│ │ │ │ -00022510: 6173 7420 6661 696c 6564 2c20 636f 6d70  ast failed, comp
│ │ │ │ -00022520: 7574 696e 6720 636f 6469 6d2e 2020 2020  uting codim.    
│ │ │ │ -00022530: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │ +00022500: 6f6e 3a20 204c 6f6f 7020 7374 6570 2c20  on:  Loop step, 
│ │ │ │ +00022510: 6162 6f75 7420 746f 2063 6f6d 7075 7465  about to compute
│ │ │ │ +00022520: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
│ │ │ │ +00022530: 6d61 7472 6963 6573 2063 6f7c 0a7c 7265  matrices co|.|re
│ │ │ │  00022540: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ -00022550: 6f6e 3a20 2070 6172 7469 616c 2073 696e  on:  partial sin
│ │ │ │ -00022560: 6775 6c61 7220 6c6f 6375 7320 6469 6d65  gular locus dime
│ │ │ │ -00022570: 6e73 696f 6e20 636f 6d70 7574 6564 2c20  nsion computed, 
│ │ │ │ -00022580: 3d20 3220 2020 2020 2020 207c 0a7c 696e  = 2        |.|in
│ │ │ │ -00022590: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -000225a0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -000225b0: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │ -000225c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000225d0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │ +00022550: 6f6e 3a20 2069 7343 6f64 696d 4174 4c65  on:  isCodimAtLe
│ │ │ │ +00022560: 6173 7420 6661 696c 6564 2c20 636f 6d70  ast failed, comp
│ │ │ │ +00022570: 7574 696e 6720 636f 6469 6d2e 2020 2020  uting codim.    
│ │ │ │ +00022580: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │ +00022590: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ +000225a0: 6f6e 3a20 2070 6172 7469 616c 2073 696e  on:  partial sin
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│ │ │ │ +000225c0: 6e73 696f 6e20 636f 6d70 7574 6564 2c20  nsion computed, 
│ │ │ │ +000225d0: 3d20 3220 2020 2020 2020 207c 0a7c 696e  = 2        |.|in
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│ │ │ │ -000225f0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ +000225f0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00022600: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  00022610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022620: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00022630: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00022640: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00022650: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ +00022650: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │  00022660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022670: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00022680: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00022690: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -000226a0: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │ +00022690: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +000226a0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │  000226b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000226c0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  000226d0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -000226e0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -000226f0: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │ +000226e0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +000226f0: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00022700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022710: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00022720: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00022730: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │  00022740: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │  00022750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022760: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │ -00022770: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
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│ │ │ │ -00022790: 6162 6f75 7420 746f 2063 6f6d 7075 7465  about to compute
│ │ │ │ -000227a0: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
│ │ │ │ -000227b0: 6d61 7472 6963 6573 2063 6f7c 0a7c 7265  matrices co|.|re
│ │ │ │ +00022760: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │ +00022770: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ +00022780: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +000227a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000227b0: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
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│ │ │ │ -000227e0: 6173 7420 6661 696c 6564 2c20 636f 6d70  ast failed, comp
│ │ │ │ -000227f0: 7574 696e 6720 636f 6469 6d2e 2020 2020  uting codim.    
│ │ │ │ -00022800: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │ +000227d0: 6f6e 3a20 204c 6f6f 7020 7374 6570 2c20  on:  Loop step, 
│ │ │ │ +000227e0: 6162 6f75 7420 746f 2063 6f6d 7075 7465  about to compute
│ │ │ │ +000227f0: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
│ │ │ │ +00022800: 6d61 7472 6963 6573 2063 6f7c 0a7c 7265  matrices co|.|re
│ │ │ │  00022810: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
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│ │ │ │ -00022830: 6775 6c61 7220 6c6f 6375 7320 6469 6d65  gular locus dime
│ │ │ │ -00022840: 6e73 696f 6e20 636f 6d70 7574 6564 2c20  nsion computed, 
│ │ │ │ -00022850: 3d20 3120 2020 2020 2020 207c 0a7c 696e  = 1        |.|in
│ │ │ │ -00022860: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00022870: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00022880: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │ -00022890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000228a0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │ +00022820: 6f6e 3a20 2069 7343 6f64 696d 4174 4c65  on:  isCodimAtLe
│ │ │ │ +00022830: 6173 7420 6661 696c 6564 2c20 636f 6d70  ast failed, comp
│ │ │ │ +00022840: 7574 696e 6720 636f 6469 6d2e 2020 2020  uting codim.    
│ │ │ │ +00022850: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │ +00022860: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
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│ │ │ │ -000228c0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +000228c0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +000228d0: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │  000228e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000228f0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │  00022910: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │  00022930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022940: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00022950: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00022960: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00022970: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │ +00022960: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00022970: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00022980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022990: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -000229b0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -000229c0: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │ +000229b0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +000229c0: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │  000229d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000229e0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │  00022a00: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │  00022a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022a30: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00022a50: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00022a60: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ +00022a50: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00022a60: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  00022a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022a80: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │ -00022a90: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
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│ │ │ │ -00022ab0: 6162 6f75 7420 746f 2063 6f6d 7075 7465  about to compute
│ │ │ │ -00022ac0: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
│ │ │ │ -00022ad0: 6d61 7472 6963 6573 2063 6f7c 0a7c 7265  matrices co|.|re
│ │ │ │ +00022a80: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │ +00022a90: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ +00022aa0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00022ab0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ +00022ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022ad0: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
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│ │ │ │ -00022b00: 6173 7420 6661 696c 6564 2c20 636f 6d70  ast failed, comp
│ │ │ │ -00022b10: 7574 696e 6720 636f 6469 6d2e 2020 2020  uting codim.    
│ │ │ │ -00022b20: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │ +00022af0: 6f6e 3a20 204c 6f6f 7020 7374 6570 2c20  on:  Loop step, 
│ │ │ │ +00022b00: 6162 6f75 7420 746f 2063 6f6d 7075 7465  about to compute
│ │ │ │ +00022b10: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
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│ │ │ │  00022b30: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
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│ │ │ │ -00022b90: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ -00022bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022bc0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ +00022b60: 7574 696e 6720 636f 6469 6d2e 2020 2020  uting codim.    
│ │ │ │ +00022b70: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
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│ │ │ │ -00022be0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00022bf0: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │ +00022be0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00022bf0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │  00022c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022c10: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00022c20: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00022c30: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │  00022c40: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │  00022c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022c60: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00022c70: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00022c80: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00022c90: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │ +00022c80: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00022c90: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │  00022ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022cb0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00022cd0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00022ce0: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │ +00022cd0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00022ce0: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00022cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022d00: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00022d20: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00022d30: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │ +00022d20: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00022d30: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │  00022d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022d50: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │  00022d70: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │  00022d80: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  00022d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022da0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00022dc0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00022dc0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00022dd0: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  00022de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022df0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00022e10: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00022e20: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ +00022e10: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00022e20: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │  00022e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022e40: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │ -00022e50: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ -00022e60: 6f6e 3a20 204c 6f6f 7020 7374 6570 2c20  on:  Loop step, 
│ │ │ │ -00022e70: 6162 6f75 7420 746f 2063 6f6d 7075 7465  about to compute
│ │ │ │ -00022e80: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
│ │ │ │ -00022e90: 6d61 7472 6963 6573 2063 6f7c 0a7c 7265  matrices co|.|re
│ │ │ │ +00022e40: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │ +00022e50: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ +00022e60: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00022e70: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ +00022e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022e90: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │  00022ea0: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ -00022eb0: 6f6e 3a20 2069 7343 6f64 696d 4174 4c65  on:  isCodimAtLe
│ │ │ │ -00022ec0: 6173 7420 6661 696c 6564 2c20 636f 6d70  ast failed, comp
│ │ │ │ -00022ed0: 7574 696e 6720 636f 6469 6d2e 2020 2020  uting codim.    
│ │ │ │ -00022ee0: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │ +00022eb0: 6f6e 3a20 204c 6f6f 7020 7374 6570 2c20  on:  Loop step, 
│ │ │ │ +00022ec0: 6162 6f75 7420 746f 2063 6f6d 7075 7465  about to compute
│ │ │ │ +00022ed0: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
│ │ │ │ +00022ee0: 6d61 7472 6963 6573 2063 6f7c 0a7c 7265  matrices co|.|re
│ │ │ │  00022ef0: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ -00022f00: 6f6e 3a20 2070 6172 7469 616c 2073 696e  on:  partial sin
│ │ │ │ -00022f10: 6775 6c61 7220 6c6f 6375 7320 6469 6d65  gular locus dime
│ │ │ │ -00022f20: 6e73 696f 6e20 636f 6d70 7574 6564 2c20  nsion computed, 
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│ │ │ │ -000237c0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00023890: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ +00023980: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
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│ │ │ │ +000239f0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00023a00: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
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│ │ │ │  00023a20: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ +00023bd0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ +00024050: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
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│ │ │ │ +000240a0: 7574 696e 6720 636f 6469 6d2e 2020 2020  uting codim.    
│ │ │ │ +000240b0: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
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│ │ │ │ +00024120: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │  00024150: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00024180: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │ +00024170: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │  000241c0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +000241d0: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
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│ │ │ │ -00024210: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00024210: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │  00024230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024240: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ +00024260: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │  00024290: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -000242b0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -000242c0: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │ +000242b0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │  000242e0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │  00024300: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00024310: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  00024320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024330: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │  00024350: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00024360: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
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│ │ │ │  00024380: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │  00024c60: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ -00024cb0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ +00024cb0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00024d50: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │  00024da0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00024df0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ -00024e40: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ -000254f0: 7574 696e 6720 636f 6469 6d2e 2020 2020  uting codim.    
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│ │ │ │ +000256b0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00025700: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ +00025750: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ -000257a0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +000257a0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +000257b0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
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│ │ │ │ -000257f0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ +000257f0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00025800: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
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│ │ │ │  00025820: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00025840: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00025850: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │ +00025840: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ +00025890: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +000258a0: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
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│ │ │ │  000258c0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │  000259d0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │  000259e0: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  000259f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00025a20: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │  00025a30: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  00025a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00025a70: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00025a80: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  00025a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025aa0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00025ac0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00025ad0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ +00025ac0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │  00025ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025af0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00025b00: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00025b10: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00025b10: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00025b20: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │  00025b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025b40: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00025b50: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00025b60: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00025b60: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00025b70: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00025b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025b90: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00025bb0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00025bc0: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │ +00025bb0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │  00025bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025be0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00025c00: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00025c00: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00025c10: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  00025c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025c30: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00025c50: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00025c50: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │  00025c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025c80: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00025ca0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00025ca0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00025cb0: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00025cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025cd0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00025cf0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ +00025cf0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00025d00: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │  00025d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025d20: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │  00025d40: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00025d50: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ +00025d50: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │  00025d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025d70: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00025d90: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00025da0: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │ +00025d90: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00025da0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │  00025db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025dc0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00025dd0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00025de0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00025df0: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │ +00025de0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00025df0: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  00025e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025e10: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00025e30: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00025e40: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │ +00025e30: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00025e40: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │  00025e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025e60: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00025e80: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00025e80: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00025e90: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  00025ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00025ed0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00025ed0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │  00025f00: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00025f20: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ +00025f20: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00025f30: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00025f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025f50: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00025f70: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00025f70: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │  00025fa0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00025fc0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00025fc0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00025fd0: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00025fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025ff0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ +00026020: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
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│ │ │ │ -00026060: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00026070: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ +00026060: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │  00026090: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -000260b0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -000260c0: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │ +000260b0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +000260c0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │  000260d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000260e0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  000260f0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00026100: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │  00026110: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  00026120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026130: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00026140: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00026150: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00026160: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ +00026150: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00026160: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  00026170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026180: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00026190: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -000261a0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -000261b0: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │ +000261a0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +000261b0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │  000261c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000261d0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  000261e0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -000261f0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00026200: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │ +000261f0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00026200: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │  00026210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026220: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00026230: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00026240: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00026250: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │ +00026240: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00026250: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00026260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026270: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00026280: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00026290: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -000262a0: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │ +000262a0: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │  000262b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000262c0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  000262d0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  000262e0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │  000262f0: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │  00026300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026310: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │ -00026320: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ -00026330: 6f6e 3a20 204c 6f6f 7020 7374 6570 2c20  on:  Loop step, 
│ │ │ │ -00026340: 6162 6f75 7420 746f 2063 6f6d 7075 7465  about to compute
│ │ │ │ -00026350: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
│ │ │ │ -00026360: 6d61 7472 6963 6573 2063 6f7c 0a7c 7265  matrices co|.|re
│ │ │ │ +00026310: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ +00026330: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00026340: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │ +00026350: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00026ba0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ -00026bf0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00026bf0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ -00026c40: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00026c40: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00026c90: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ -00026ce0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00026ce0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
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│ │ │ │ -00026d30: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +00026d30: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00026d80: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00026d90: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
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│ │ │ │ -00026dd0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +00026dd0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00026de0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
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│ │ │ │ -00026e30: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
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│ │ │ │ +000273c0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ +000276b0: 7574 696e 6720 636f 6469 6d2e 2020 2020  uting codim.    
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│ │ │ │ +00027820: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │ -000278d0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
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│ │ │ │  000278e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000278f0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00027900: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00027910: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00027920: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │ +00027920: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │  00027930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027940: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00027950: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00027960: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00027970: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │ +00027960: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00027970: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │  00027980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027990: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  000279a0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -000279b0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -000279c0: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │ +000279b0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +000279c0: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │  000279d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000279e0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  000279f0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00027a00: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00027a10: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │ +00027a00: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00027a10: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  00027a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027a30: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00027a40: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00027a50: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │  00027a60: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │  00027a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027a80: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00027a90: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00027aa0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00027ab0: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │ +00027ab0: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
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│ │ │ │  00027ae0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00027af0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00027b00: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │ +00027af0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00027b00: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │  00027b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027b20: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00027b30: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00027b40: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00027b50: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ +00027b50: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │  00027b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027b70: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00027b80: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00027b90: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00027ba0: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │ +00027b90: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00027ba0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │  00027bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027bc0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00027bd0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00027be0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │  00027bf0: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │  00027c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027c10: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00027c20: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00027c30: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00027c40: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ +00027c30: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00027c40: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │  00027c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027c60: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00027c70: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00027c80: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00027c90: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │ +00027c80: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00027c90: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │  00027ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027cb0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00027cc0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00027cd0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00027ce0: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │ +00027cd0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00027ce0: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00027cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027d00: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00027d10: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00027d20: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00027d30: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │ +00027d20: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00027d30: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │  00027d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027d50: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00027d60: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00027d70: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00027d80: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │ +00027d70: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00027d80: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00027d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027da0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00027db0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00027dc0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │  00027dd0: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │  00027de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027df0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00027e00: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00027e10: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00027e20: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │ +00027e10: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00027e20: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │  00027e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027e40: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00027e50: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00027e60: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00027e70: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │ +00027e70: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  00027e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027e90: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00027ea0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00027eb0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │  00027ec0: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00027ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027ee0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00027ef0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00027f00: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00027f10: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │ +00027f00: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00027f10: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00027f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027f30: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00027f40: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00027f50: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00027f60: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │ +00027f50: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00027f60: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │  00027f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027f80: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00027f90: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00027fa0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00027fb0: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │ +00027fa0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00027fb0: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │  00027fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027fd0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00027fe0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00027ff0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00028000: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │ +00027ff0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00028000: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00028010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028020: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028030: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00028040: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │  00028050: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │  00028060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028070: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028080: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028090: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -000280a0: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │ +00028090: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +000280a0: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │  000280b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000280c0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  000280d0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -000280e0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
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│ │ │ │ +000280e0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
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│ │ │ │  00028100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028110: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028120: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028130: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00028140: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │ +00028130: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00028140: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00028150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028160: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028170: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00028180: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │  00028190: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │  000281a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000281b0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  000281c0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -000281d0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -000281e0: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │ +000281d0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +000281e0: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │  000281f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028200: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028210: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028220: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00028230: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │ +00028220: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00028230: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00028240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028250: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028260: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00028270: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │  00028280: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │  00028290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000282a0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  000282b0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  000282c0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -000282d0: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │ +000282d0: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
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│ │ │ │  000282f0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028300: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028310: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00028320: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │ +00028310: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00028320: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │  00028330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028340: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028350: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028360: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00028370: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │ +00028360: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00028370: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │  00028380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028390: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  000283a0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  000283b0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -000283c0: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │ +000283c0: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  000283d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000283e0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  000283f0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00028400: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │  00028410: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00028420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028430: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028440: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028450: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00028460: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ +00028450: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00028460: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00028470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028480: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028490: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -000284a0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -000284b0: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │ +000284a0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +000284b0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │  000284c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000284d0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  000284e0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -000284f0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00028500: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │ +000284f0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00028500: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │  00028510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028520: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028530: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028540: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00028550: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │ +00028540: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00028550: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │  00028560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028570: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028580: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00028590: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │  000285a0: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │  000285b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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: 3536 3836 3573 2028 6370 7529 3b20 342e  56865s (cpu); 4.
│ │ │ │ -00028650: 3934 3932 3873 2028 7468 7265 6164 293b  94928s (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
│ │ │ │ -000286d0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -000286e0: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │ +00028630: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00028640: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │ +00028650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028660: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │ +00028670: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ +00028680: 723a 2043 6820 2d2d 2075 7365 6420 372e  r: Ch -- used 7.
│ │ │ │ +00028690: 3634 3136 3173 2028 6370 7529 3b20 352e  64161s (cpu); 5.
│ │ │ │ +000286a0: 3634 3535 3373 2028 7468 7265 6164 293b  64553s (thread);
│ │ │ │ +000286b0: 2030 7320 2867 6329 2020 207c 0a7c 6f6f   0s (gc)   |.|oo
│ │ │ │ +000286c0: 7369 6e67 2047 5265 764c 6578 536d 616c  sing GRevLexSmal
│ │ │ │ +000286d0: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │ +000286e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000286f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028700: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028710: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028720: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00028730: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │ +00028720: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00028730: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00028740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028750: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028760: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028770: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00028780: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │ +00028770: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00028780: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │  00028790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000287a0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  000287b0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  000287c0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │  000287d0: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  000287e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000287f0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028800: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028810: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00028820: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │ +00028810: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00028820: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  00028830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028840: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028850: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028860: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00028870: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ +00028860: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00028870: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │  00028880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028890: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  000288a0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -000288b0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -000288c0: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │ +000288b0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +000288c0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │  000288d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000288e0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  000288f0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028900: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00028910: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │ +00028900: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00028910: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │  00028920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028930: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028940: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00028950: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │  00028960: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  00028970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028980: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028990: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -000289a0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -000289b0: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │ +000289a0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +000289b0: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  000289c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000289d0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -000289f0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00028a00: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │ +000289f0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00028a00: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │  00028a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028a20: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028a30: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028a40: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00028a50: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │ +00028a40: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00028a50: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00028a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028a70: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028a80: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00028a90: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00028aa0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ +00028aa0: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │  00028ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028ac0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028ad0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028ae0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00028af0: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │ +00028ae0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00028af0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │  00028b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028b10: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028b20: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028b30: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00028b40: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ +00028b30: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00028b40: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │  00028b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028b60: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028b70: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00028b80: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00028b90: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │ +00028b90: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │  00028ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028bb0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028bc0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028bd0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00028be0: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │ +00028bd0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00028be0: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │  00028bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028c00: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028c10: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028c20: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00028c30: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │ +00028c20: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00028c30: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │  00028c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028c50: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028c60: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028c70: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00028c80: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
│ │ │ │ +00028c70: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00028c80: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00028c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028ca0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028cb0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00028cc0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00028cd0: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │ +00028cd0: 6d61 6c6c 6573 7454 6572 6d20 2020 2020  mallestTerm     
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│ │ │ │  00028cf0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00028d10: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00028d20: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ +00028d10: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00028d20: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │  00028d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028d40: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028d50: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00028d60: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00028d70: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │ +00028d70: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │  00028d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028d90: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028da0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00028db0: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │  00028dc0: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │  00028dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028de0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
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│ │ │ │ -00028e00: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00028e10: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │ +00028e00: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00028e10: 4c65 7853 6d61 6c6c 6573 7420 2020 2020  LexSmallest     
│ │ │ │  00028e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028e30: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │ -00028e40: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ -00028e50: 6f6e 3a20 204c 6f6f 7020 7374 6570 2c20  on:  Loop step, 
│ │ │ │ -00028e60: 6162 6f75 7420 746f 2063 6f6d 7075 7465  about to compute
│ │ │ │ -00028e70: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
│ │ │ │ -00028e80: 6d61 7472 6963 6573 2063 6f7c 0a7c 7265  matrices co|.|re
│ │ │ │ +00028e30: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │ +00028e40: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ +00028e50: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00028e60: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │ +00028e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028e80: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │  00028e90: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ -00028ea0: 6f6e 3a20 2069 7343 6f64 696d 4174 4c65  on:  isCodimAtLe
│ │ │ │ -00028eb0: 6173 7420 6661 696c 6564 2c20 636f 6d70  ast failed, comp
│ │ │ │ -00028ec0: 7574 696e 6720 636f 6469 6d2e 2020 2020  uting codim.    
│ │ │ │ -00028ed0: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │ +00028ea0: 6f6e 3a20 204c 6f6f 7020 7374 6570 2c20  on:  Loop step, 
│ │ │ │ +00028eb0: 6162 6f75 7420 746f 2063 6f6d 7075 7465  about to compute
│ │ │ │ +00028ec0: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
│ │ │ │ +00028ed0: 6d61 7472 6963 6573 2063 6f7c 0a7c 7265  matrices co|.|re
│ │ │ │  00028ee0: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ -00028ef0: 6f6e 3a20 2070 6172 7469 616c 2073 696e  on:  partial sin
│ │ │ │ -00028f00: 6775 6c61 7220 6c6f 6375 7320 6469 6d65  gular locus dime
│ │ │ │ -00028f10: 6e73 696f 6e20 636f 6d70 7574 6564 2c20  nsion computed, 
│ │ │ │ -00028f20: 3d20 3120 2020 2020 2020 207c 0a7c 696e  = 1        |.|in
│ │ │ │ -00028f30: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028f40: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00028f50: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ -00028f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028f70: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │ +00028ef0: 6f6e 3a20 2069 7343 6f64 696d 4174 4c65  on:  isCodimAtLe
│ │ │ │ +00028f00: 6173 7420 6661 696c 6564 2c20 636f 6d70  ast failed, comp
│ │ │ │ +00028f10: 7574 696e 6720 636f 6469 6d2e 2020 2020  uting codim.    
│ │ │ │ +00028f20: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │ +00028f30: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ +00028f40: 6f6e 3a20 2070 6172 7469 616c 2073 696e  on:  partial sin
│ │ │ │ +00028f50: 6775 6c61 7220 6c6f 6375 7320 6469 6d65  gular locus dime
│ │ │ │ +00028f60: 6e73 696f 6e20 636f 6d70 7574 6564 2c20  nsion computed, 
│ │ │ │ +00028f70: 3d20 3120 2020 2020 2020 207c 0a7c 696e  = 1        |.|in
│ │ │ │  00028f80: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028f90: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -00028fa0: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │ +00028f90: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00028fa0: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │  00028fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028fc0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00028fd0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00028fe0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00028ff0: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │ +00028fe0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00028ff0: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │  00029000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029010: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00029020: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00029030: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │  00029040: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  00029050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029060: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00029070: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  00029080: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │  00029090: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  000290a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000290b0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  000290c0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -000290d0: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ -000290e0: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │ +000290d0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +000290e0: 6f6d 4e6f 6e5a 6572 6f20 2020 2020 2020  omNonZero       
│ │ │ │  000290f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029100: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00029110: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00029120: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ -00029130: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │ +00029120: 723a 2043 686f 6f73 696e 6720 4c65 7853  r: Choosing LexS
│ │ │ │ +00029130: 6d61 6c6c 6573 7420 2020 2020 2020 2020  mallest         
│ │ │ │  00029140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029150: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  00029160: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ -00029170: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ -00029180: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │ +00029170: 723a 2043 686f 6f73 696e 6720 4752 6576  r: Choosing GRev
│ │ │ │ +00029180: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20  LexSmallestTerm 
│ │ │ │  00029190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000291a0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │  000291b0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │  000291c0: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │  000291d0: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │  000291e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000291f0: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │ -00029200: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ -00029210: 6f6e 3a20 204c 6f6f 7020 7374 6570 2c20  on:  Loop step, 
│ │ │ │ -00029220: 6162 6f75 7420 746f 2063 6f6d 7075 7465  about to compute
│ │ │ │ -00029230: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
│ │ │ │ -00029240: 6d61 7472 6963 6573 2063 6f7c 0a7c 7265  matrices co|.|re
│ │ │ │ +000291f0: 2020 2020 2020 2020 2020 207c 0a7c 696e             |.|in
│ │ │ │ +00029200: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f  ternalChooseMino
│ │ │ │ +00029210: 723a 2043 686f 6f73 696e 6720 5261 6e64  r: Choosing Rand
│ │ │ │ +00029220: 6f6d 2020 2020 2020 2020 2020 2020 2020  om              
│ │ │ │ +00029230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029240: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │  00029250: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ -00029260: 6f6e 3a20 2069 7343 6f64 696d 4174 4c65  on:  isCodimAtLe
│ │ │ │ -00029270: 6173 7420 6661 696c 6564 2c20 636f 6d70  ast failed, comp
│ │ │ │ -00029280: 7574 696e 6720 636f 6469 6d2e 2020 2020  uting codim.    
│ │ │ │ -00029290: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │ +00029260: 6f6e 3a20 204c 6f6f 7020 7374 6570 2c20  on:  Loop step, 
│ │ │ │ +00029270: 6162 6f75 7420 746f 2063 6f6d 7075 7465  about to compute
│ │ │ │ +00029280: 2064 696d 656e 7369 6f6e 2e20 2053 7562   dimension.  Sub
│ │ │ │ +00029290: 6d61 7472 6963 6573 2063 6f7c 0a7c 7265  matrices co|.|re
│ │ │ │  000292a0: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ -000292b0: 6f6e 3a20 2070 6172 7469 616c 2073 696e  on:  partial sin
│ │ │ │ -000292c0: 6775 6c61 7220 6c6f 6375 7320 6469 6d65  gular locus dime
│ │ │ │ -000292d0: 6e73 696f 6e20 636f 6d70 7574 6564 2c20  nsion computed, 
│ │ │ │ -000292e0: 3d20 3120 2020 2020 2020 207c 0a7c 7265  = 1        |.|re
│ │ │ │ +000292b0: 6f6e 3a20 2069 7343 6f64 696d 4174 4c65  on:  isCodimAtLe
│ │ │ │ +000292c0: 6173 7420 6661 696c 6564 2c20 636f 6d70  ast failed, comp
│ │ │ │ +000292d0: 7574 696e 6720 636f 6469 6d2e 2020 2020  uting codim.    
│ │ │ │ +000292e0: 2020 2020 2020 2020 2020 207c 0a7c 7265             |.|re
│ │ │ │  000292f0: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ -00029300: 6f6e 3a20 204c 6f6f 7020 636f 6d70 6c65  on:  Loop comple
│ │ │ │ -00029310: 7465 642c 2073 7562 6d61 7472 6963 6573  ted, submatrices
│ │ │ │ -00029320: 2063 6f6e 7369 6465 7265 6420 3d20 3332   considered = 32
│ │ │ │ -00029330: 382c 2061 6e64 2063 6f6d 707c 0a7c 2d2d  8, and comp|.|--
│ │ │ │ -00029340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 7273  -----------|.|rs
│ │ │ │ -00029390: 2c20 7765 2077 696c 6c20 636f 6d70 7574  , we will comput
│ │ │ │ -000293a0: 6520 7570 2074 6f20 3332 372e 3539 3920  e up to 327.599 
│ │ │ │ -000293b0: 6f66 2074 6865 6d2e 2020 2020 2020 2020  of them.        
│ │ │ │ -000293c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000293d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000293e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000293f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029300: 6f6e 3a20 2070 6172 7469 616c 2073 696e  on:  partial sin
│ │ │ │ +00029310: 6775 6c61 7220 6c6f 6375 7320 6469 6d65  gular locus dime
│ │ │ │ +00029320: 6e73 696f 6e20 636f 6d70 7574 6564 2c20  nsion computed, 
│ │ │ │ +00029330: 3d20 3120 2020 2020 2020 207c 0a7c 7265  = 1        |.|re
│ │ │ │ +00029340: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ +00029350: 6f6e 3a20 204c 6f6f 7020 636f 6d70 6c65  on:  Loop comple
│ │ │ │ +00029360: 7465 642c 2073 7562 6d61 7472 6963 6573  ted, submatrices
│ │ │ │ +00029370: 2063 6f6e 7369 6465 7265 6420 3d20 3332   considered = 32
│ │ │ │ +00029380: 382c 2061 6e64 2063 6f6d 707c 0a7c 2d2d  8, and comp|.|--
│ │ │ │ +00029390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000293a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000293b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000293c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000293d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 7273  -----------|.|rs
│ │ │ │ +000293e0: 2c20 7765 2077 696c 6c20 636f 6d70 7574  , we will comput
│ │ │ │ +000293f0: 6520 7570 2074 6f20 3332 372e 3539 3920  e up to 327.599 
│ │ │ │ +00029400: 6f66 2074 6865 6d2e 2020 2020 2020 2020  of them.        
│ │ │ │  00029410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029420: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00029430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029470: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ @@ -10601,22 +10601,22 @@
│ │ │ │  00029680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000296a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000296b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000296c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000296d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000296e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000296f0: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ -00029700: 6964 6572 6564 3a20 392c 2061 6e64 2063  idered: 9, and c
│ │ │ │ -00029710: 6f6d 7075 7465 6420 3d20 3920 2020 2020  omputed = 9     
│ │ │ │ +000296f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00029700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029740: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00029750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029740: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ +00029750: 6964 6572 6564 3a20 392c 2061 6e64 2063  idered: 9, and c
│ │ │ │ +00029760: 6f6d 7075 7465 6420 3d20 3920 2020 2020  omputed = 9     
│ │ │ │  00029770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029790: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000297a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000297b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000297c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000297d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -10626,22 +10626,22 @@
│ │ │ │  00029810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029830: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00029840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029880: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ -00029890: 6964 6572 6564 3a20 3131 2c20 616e 6420  idered: 11, and 
│ │ │ │ -000298a0: 636f 6d70 7574 6564 203d 2031 3020 2020  computed = 10   
│ │ │ │ +00029880: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00029890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000298a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000298b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000298c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000298d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000298e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000298f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000298d0: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ +000298e0: 6964 6572 6564 3a20 3131 2c20 616e 6420  idered: 11, and 
│ │ │ │ +000298f0: 636f 6d70 7574 6564 203d 2031 3020 2020  computed = 10   
│ │ │ │  00029900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029920: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00029930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -10661,22 +10661,22 @@
│ │ │ │  00029a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029a60: 2020 2020 2020 2020 2020 207c 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029ab0: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ -00029ac0: 6964 6572 6564 3a20 3135 2c20 616e 6420  idered: 15, and 
│ │ │ │ -00029ad0: 636f 6d70 7574 6564 203d 2031 3320 2020  computed = 13   
│ │ │ │ +00029ab0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00029ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029b00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00029b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029b00: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ +00029b10: 6964 6572 6564 3a20 3135 2c20 616e 6420  idered: 15, and 
│ │ │ │ +00029b20: 636f 6d70 7574 6564 203d 2031 3320 2020  computed = 13   
│ │ │ │  00029b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029b50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -10706,22 +10706,22 @@
│ │ │ │  00029d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029d30: 2020 2020 2020 2020 2020 207c 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029d80: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ -00029d90: 6964 6572 6564 3a20 3231 2c20 616e 6420  idered: 21, and 
│ │ │ │ -00029da0: 636f 6d70 7574 6564 203d 2031 3820 2020  computed = 18   
│ │ │ │ +00029d80: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00029d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029dd0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00029de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029dd0: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ +00029de0: 6964 6572 6564 3a20 3231 2c20 616e 6420  idered: 21, and 
│ │ │ │ +00029df0: 636f 6d70 7574 6564 203d 2031 3820 2020  computed = 18   
│ │ │ │  00029e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029e20: 2020 2020 2020 2020 2020 207c 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -10756,22 +10756,22 @@
│ │ │ │  0002a030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a050: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002a060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a0a0: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ -0002a0b0: 6964 6572 6564 3a20 3238 2c20 616e 6420  idered: 28, and 
│ │ │ │ -0002a0c0: 636f 6d70 7574 6564 203d 2032 3320 2020  computed = 23   
│ │ │ │ +0002a0a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002a0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a0f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002a100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a0f0: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ +0002a100: 6964 6572 6564 3a20 3238 2c20 616e 6420  idered: 28, and 
│ │ │ │ +0002a110: 636f 6d70 7574 6564 203d 2032 3320 2020  computed = 23   
│ │ │ │  0002a120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a140: 2020 2020 2020 2020 2020 207c 0a7c 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 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -10816,22 +10816,22 @@
│ │ │ │  0002a3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a410: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002a420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a460: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ -0002a470: 6964 6572 6564 3a20 3337 2c20 616e 6420  idered: 37, and 
│ │ │ │ -0002a480: 636f 6d70 7574 6564 203d 2033 3020 2020  computed = 30   
│ │ │ │ +0002a460: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002a470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a4b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002a4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a4b0: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ +0002a4c0: 6964 6572 6564 3a20 3337 2c20 616e 6420  idered: 37, and 
│ │ │ │ +0002a4d0: 636f 6d70 7574 6564 203d 2033 3020 2020  computed = 30   
│ │ │ │  0002a4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a500: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002a510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -10891,22 +10891,22 @@
│ │ │ │  0002a8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a8c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002a8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a910: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ -0002a920: 6964 6572 6564 3a20 3439 2c20 616e 6420  idered: 49, and 
│ │ │ │ -0002a930: 636f 6d70 7574 6564 203d 2033 3620 2020  computed = 36   
│ │ │ │ +0002a910: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002a920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a960: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002a970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a960: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ +0002a970: 6964 6572 6564 3a20 3439 2c20 616e 6420  idered: 49, and 
│ │ │ │ +0002a980: 636f 6d70 7574 6564 203d 2033 3620 2020  computed = 36   
│ │ │ │  0002a990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a9b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002a9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -10981,22 +10981,22 @@
│ │ │ │  0002ae40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ae50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ae60: 2020 2020 2020 2020 2020 207c 0a7c 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 2020                  
│ │ │ │  0002aea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aeb0: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ -0002aec0: 6964 6572 6564 3a20 3634 2c20 616e 6420  idered: 64, and 
│ │ │ │ -0002aed0: 636f 6d70 7574 6564 203d 2034 3420 2020  computed = 44   
│ │ │ │ +0002aeb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002aec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002aed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002af00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002af10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002af20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002af00: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ +0002af10: 6964 6572 6564 3a20 3634 2c20 616e 6420  idered: 64, and 
│ │ │ │ +0002af20: 636f 6d70 7574 6564 203d 2034 3420 2020  computed = 44   
│ │ │ │  0002af30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002af40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002af50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002af60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002af70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002af80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002af90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -11096,22 +11096,22 @@
│ │ │ │  0002b570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b590: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002b5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b5e0: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ -0002b5f0: 6964 6572 6564 3a20 3834 2c20 616e 6420  idered: 84, and 
│ │ │ │ -0002b600: 636f 6d70 7574 6564 203d 2035 3620 2020  computed = 56   
│ │ │ │ +0002b5e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +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 2020                  
│ │ │ │  0002b620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b630: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002b640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b630: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ +0002b640: 6964 6572 6564 3a20 3834 2c20 616e 6420  idered: 84, and 
│ │ │ │ +0002b650: 636f 6d70 7574 6564 203d 2035 3620 2020  computed = 56   
│ │ │ │  0002b660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b680: 2020 2020 2020 2020 2020 207c 0a7c 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 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -11241,22 +11241,22 @@
│ │ │ │  0002be80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002be90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bea0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002beb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bef0: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ -0002bf00: 6964 6572 6564 3a20 3131 302c 2061 6e64  idered: 110, and
│ │ │ │ -0002bf10: 2063 6f6d 7075 7465 6420 3d20 3639 2020   computed = 69  
│ │ │ │ +0002bef0: 2020 2020 2020 2020 2020 207c 0a7c 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 2020 2020 2020                  
│ │ │ │ -0002bf40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002bf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bf40: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ +0002bf50: 6964 6572 6564 3a20 3131 302c 2061 6e64  idered: 110, and
│ │ │ │ +0002bf60: 2063 6f6d 7075 7465 6420 3d20 3639 2020   computed = 69  
│ │ │ │  0002bf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bf90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002bfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -11426,22 +11426,22 @@
│ │ │ │  0002ca10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ca20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ca30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  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 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ca80: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ -0002ca90: 6964 6572 6564 3a20 3134 342c 2061 6e64  idered: 144, and
│ │ │ │ -0002caa0: 2063 6f6d 7075 7465 6420 3d20 3833 2020   computed = 83  
│ │ │ │ +0002ca80: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002ca90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002caa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cad0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002cae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002caf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cad0: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ +0002cae0: 6964 6572 6564 3a20 3134 342c 2061 6e64  idered: 144, and
│ │ │ │ +0002caf0: 2063 6f6d 7075 7465 6420 3d20 3833 2020   computed = 83  
│ │ │ │  0002cb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cb20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002cb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -11661,22 +11661,22 @@
│ │ │ │  0002d8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d8e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002d8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d930: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ -0002d940: 6964 6572 6564 3a20 3138 382c 2061 6e64  idered: 188, and
│ │ │ │ -0002d950: 2063 6f6d 7075 7465 6420 3d20 3130 3520   computed = 105 
│ │ │ │ +0002d930: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002d940: 2020 2020 2020 2020 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 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002d990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d980: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ +0002d990: 6964 6572 6564 3a20 3138 382c 2061 6e64  idered: 188, and
│ │ │ │ +0002d9a0: 2063 6f6d 7075 7465 6420 3d20 3130 3520   computed = 105 
│ │ │ │  0002d9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d9d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002d9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002da00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002da10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -11961,22 +11961,22 @@
│ │ │ │  0002eb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002eb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002eba0: 2020 2020 2020 2020 2020 207c 0a7c 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 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ebf0: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ -0002ec00: 6964 6572 6564 3a20 3234 352c 2061 6e64  idered: 245, and
│ │ │ │ -0002ec10: 2063 6f6d 7075 7465 6420 3d20 3133 3520   computed = 135 
│ │ │ │ +0002ebf0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002ec00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ec10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ec20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ec30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ec40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002ec50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ec60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ec40: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ +0002ec50: 6964 6572 6564 3a20 3234 352c 2061 6e64  idered: 245, and
│ │ │ │ +0002ec60: 2063 6f6d 7075 7465 6420 3d20 3133 3520   computed = 135 
│ │ │ │  0002ec70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ec80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ec90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002eca0: 2020 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                  
│ │ │ │ @@ -12351,22 +12351,22 @@
│ │ │ │  000303e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000303f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030400: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00030410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030450: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ -00030460: 6964 6572 6564 3a20 3331 392c 2061 6e64  idered: 319, and
│ │ │ │ -00030470: 2063 6f6d 7075 7465 6420 3d20 3137 3520   computed = 175 
│ │ │ │ +00030450: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00030460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000304a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000304b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000304c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000304a0: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ +000304b0: 6964 6572 6564 3a20 3331 392c 2061 6e64  idered: 319, and
│ │ │ │ +000304c0: 2063 6f6d 7075 7465 6420 3d20 3137 3520   computed = 175 
│ │ │ │  000304d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000304e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000304f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00030500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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                  
│ │ │ │ @@ -12411,357 +12411,357 @@
│ │ │ │  000307a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000307b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000307c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000307d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 207c 0a7c 6e73             |.|ns
│ │ │ │ -00030820: 6964 6572 6564 3a20 3332 382c 2061 6e64  idered: 328, and
│ │ │ │ -00030830: 2063 6f6d 7075 7465 6420 3d20 3138 3020   computed = 180 
│ │ │ │ +00030810: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00030820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030860: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00030870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030860: 2020 2020 2020 2020 2020 207c 0a7c 6e73             |.|ns
│ │ │ │ +00030870: 6964 6572 6564 3a20 3332 382c 2061 6e64  idered: 328, and
│ │ │ │ +00030880: 2063 6f6d 7075 7465 6420 3d20 3138 3020   computed = 180 
│ │ │ │  00030890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000308a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000308b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000308c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000308d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000308e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000308f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030900: 2020 2020 2020 2020 2020 207c 0a7c 7574             |.|ut
│ │ │ │ -00030910: 6564 203d 2031 3830 2e20 2073 696e 6775  ed = 180.  singu
│ │ │ │ -00030920: 6c61 7220 6c6f 6375 7320 6469 6d65 6e73  lar locus dimens
│ │ │ │ -00030930: 696f 6e20 6170 7065 6172 7320 746f 2062  ion appears to b
│ │ │ │ -00030940: 6520 3d20 3120 2020 2020 2020 2020 2020  e = 1           
│ │ │ │ -00030950: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -00030960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000309a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5468  -----------+..Th
│ │ │ │ -000309b0: 6520 6d61 7869 6d75 6d20 6e75 6d62 6572  e maximum number
│ │ │ │ -000309c0: 206f 6620 6d69 6e6f 7273 2063 6f6e 7369   of minors consi
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│ │ │ │ -00030a00: 2020 416c 7465 726e 6174 6976 656c 792c    Alternatively,
│ │ │ │ -00030a10: 2069 7420 6361 6e20 6265 2063 6f6e 7472   it can be contr
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│ │ │ │ -00030a30: 7072 6563 6973 6520 7761 7920 6279 0a70  precise way by.p
│ │ │ │ -00030a40: 6173 7369 6e67 2061 2066 756e 6374 696f  assing a functio
│ │ │ │ -00030a50: 6e20 746f 2074 6865 206f 7074 696f 6e20  n to the option 
│ │ │ │ -00030a60: 4d61 784d 696e 6f72 732e 2020 5468 6973  MaxMinors.  This
│ │ │ │ -00030a70: 2066 756e 6374 696f 6e20 7368 6f75 6c64   function should
│ │ │ │ -00030a80: 2068 6176 6520 7477 6f0a 696e 7075 7473   have two.inputs
│ │ │ │ -00030a90: 3b20 7468 6520 6669 7273 7420 6973 206d  ; the first is m
│ │ │ │ -00030aa0: 696e 696d 756d 206e 756d 6265 7220 6f66  inimum number of
│ │ │ │ -00030ab0: 206d 696e 6f72 7320 6e65 6564 6564 2074   minors needed t
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│ │ │ │ -00030ad0: 6865 7220 7468 650a 7269 6e67 2069 7320  her the.ring is 
│ │ │ │ -00030ae0: 7265 6775 6c61 7220 696e 2063 6f64 696d  regular in codim
│ │ │ │ -00030af0: 656e 7369 6f6e 206e 2c20 616e 6420 7468  ension n, and th
│ │ │ │ -00030b00: 6520 7365 636f 6e64 2069 7320 7468 6520  e second is the 
│ │ │ │ -00030b10: 746f 7461 6c20 6e75 6d62 6572 206f 6620  total number of 
│ │ │ │ -00030b20: 6d69 6e6f 7273 0a61 7661 696c 6162 6c65  minors.available
│ │ │ │ -00030b30: 2069 6e20 7468 6520 4a61 636f 6269 616e   in the Jacobian
│ │ │ │ -00030b40: 2e20 5468 6520 6675 6e63 7469 6f6e 2072  . The function r
│ │ │ │ -00030b50: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ -00030b60: 696f 6e20 646f 6573 206e 6f74 2072 6563  ion does not rec
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│ │ │ │ -00030b90: 7320 6f72 2069 7320 6f6e 6c79 2061 6e20  s or is only an 
│ │ │ │ -00030ba0: 7570 7065 7220 626f 756e 6420 6f6e 2074  upper bound on t
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│ │ │ │ -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 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 3732 3033   -- used 1.27203
│ │ │ │ -00030c80: 7320 2863 7075 293b 2030 2e39 3538 3532  s (cpu); 0.95852
│ │ │ │ -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
│ │ │ │ -00030d00: 6520 3420 6279 2034 206d 696e 6f7c 0a7c  e 4 by 4 mino|.|
│ │ │ │ +00030900: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00030910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +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 207c 0a7c 7574             |.|ut
│ │ │ │ +00030960: 6564 203d 2031 3830 2e20 2073 696e 6775  ed = 180.  singu
│ │ │ │ +00030970: 6c61 7220 6c6f 6375 7320 6469 6d65 6e73  lar locus dimens
│ │ │ │ +00030980: 696f 6e20 6170 7065 6172 7320 746f 2062  ion appears to b
│ │ │ │ +00030990: 6520 3d20 3120 2020 2020 2020 2020 2020  e = 1           
│ │ │ │ +000309a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000309b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000309c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000309d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000309e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000309f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5468  -----------+..Th
│ │ │ │ +00030a00: 6520 6d61 7869 6d75 6d20 6e75 6d62 6572  e maximum number
│ │ │ │ +00030a10: 206f 6620 6d69 6e6f 7273 2063 6f6e 7369   of minors consi
│ │ │ │ +00030a20: 6465 7265 6420 6361 6e20 6265 2063 6f6e  dered can be con
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│ │ │ │ +00030a50: 2020 416c 7465 726e 6174 6976 656c 792c    Alternatively,
│ │ │ │ +00030a60: 2069 7420 6361 6e20 6265 2063 6f6e 7472   it can be contr
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│ │ │ │ +00030a80: 7072 6563 6973 6520 7761 7920 6279 0a70  precise way by.p
│ │ │ │ +00030a90: 6173 7369 6e67 2061 2066 756e 6374 696f  assing a functio
│ │ │ │ +00030aa0: 6e20 746f 2074 6865 206f 7074 696f 6e20  n to the option 
│ │ │ │ +00030ab0: 4d61 784d 696e 6f72 732e 2020 5468 6973  MaxMinors.  This
│ │ │ │ +00030ac0: 2066 756e 6374 696f 6e20 7368 6f75 6c64   function should
│ │ │ │ +00030ad0: 2068 6176 6520 7477 6f0a 696e 7075 7473   have two.inputs
│ │ │ │ +00030ae0: 3b20 7468 6520 6669 7273 7420 6973 206d  ; the first is m
│ │ │ │ +00030af0: 696e 696d 756d 206e 756d 6265 7220 6f66  inimum number of
│ │ │ │ +00030b00: 206d 696e 6f72 7320 6e65 6564 6564 2074   minors needed t
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│ │ │ │ +00030b20: 6865 7220 7468 650a 7269 6e67 2069 7320  her the.ring is 
│ │ │ │ +00030b30: 7265 6775 6c61 7220 696e 2063 6f64 696d  regular in codim
│ │ │ │ +00030b40: 656e 7369 6f6e 206e 2c20 616e 6420 7468  ension n, and th
│ │ │ │ +00030b50: 6520 7365 636f 6e64 2069 7320 7468 6520  e second is the 
│ │ │ │ +00030b60: 746f 7461 6c20 6e75 6d62 6572 206f 6620  total number of 
│ │ │ │ +00030b70: 6d69 6e6f 7273 0a61 7661 696c 6162 6c65  minors.available
│ │ │ │ +00030b80: 2069 6e20 7468 6520 4a61 636f 6269 616e   in the Jacobian
│ │ │ │ +00030b90: 2e20 5468 6520 6675 6e63 7469 6f6e 2072  . The function r
│ │ │ │ +00030ba0: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ +00030bb0: 696f 6e20 646f 6573 206e 6f74 2072 6563  ion does not rec
│ │ │ │ +00030bc0: 6f6d 7075 7465 0a64 6574 6572 6d69 6e61  ompute.determina
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│ │ │ │ +00030be0: 7320 6f72 2069 7320 6f6e 6c79 2061 6e20  s or is only an 
│ │ │ │ +00030bf0: 7570 7065 7220 626f 756e 6420 6f6e 2074  upper bound on t
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│ │ │ │ +00030c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00030c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00030c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00030c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00030c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00030c70: 6931 3720 3a20 7469 6d65 2072 6567 756c  i17 : time regul
│ │ │ │ +00030c80: 6172 496e 436f 6469 6d65 6e73 696f 6e28  arInCodimension(
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│ │ │ │ +00030ca0: 7275 652c 204d 6178 4d69 6e6f 7273 3d3e  rue, MaxMinors=>
│ │ │ │ +00030cb0: 3330 2920 2020 2020 2020 2020 207c 0a7c  30)          |.|
│ │ │ │ +00030cc0: 202d 2d20 7573 6564 2031 2e36 3137 3831   -- used 1.61781
│ │ │ │ +00030cd0: 7320 2863 7075 293b 2031 2e32 3235 3873  s (cpu); 1.2258s
│ │ │ │ +00030ce0: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ +00030cf0: 6329 2020 2020 2020 2020 2020 2020 2020  c)              
│ │ │ │ +00030d00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00030d10: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
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│ │ │ │ -00030d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030d50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ +00030d20: 7369 6f6e 3a20 7269 6e67 2064 696d 656e  sion: ring dimen
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│ │ │ │  00030d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030da0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00030db0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
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│ │ │ │  00030de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030df0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00030e00: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
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│ │ │ │  00030e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030e40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00030e50: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
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│ │ │ │  00030e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030e90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00030ea0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
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│ │ │ │ +00030eb0: 6e6f 723a 2043 686f 6f73 696e 6720 4752  nor: Choosing GR
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│ │ │ │  00030ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030ee0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00030ef0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
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│ │ │ │  00030f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030f30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00030f40: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
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│ │ │ │ +00030f50: 6e6f 723a 2043 686f 6f73 696e 6720 5261  nor: Choosing Ra
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│ │ │ │  00030f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030f80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00030f90: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -00030fa0: 6e6f 723a 2043 686f 6f73 696e 6720 5261  nor: Choosing Ra
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│ │ │ │  00030fe0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
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│ │ │ │ +00031000: 6e64 6f6d 2020 2020 2020 2020 2020 2020  ndom            
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│ │ │ │ +00031060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031070: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ -000310c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ -00031140: 6e64 6f6d 2020 2020 2020 2020 2020 2020  ndom            
│ │ │ │ -00031150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031160: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000310e0: 7369 6f6e 3a20 2069 7343 6f64 696d 4174  sion:  isCodimAt
│ │ │ │ +000310f0: 4c65 6173 7420 6661 696c 6564 2c20 636f  Least failed, co
│ │ │ │ +00031100: 6d70 7574 696e 6720 636f 6469 6d2e 2020  mputing codim.  
│ │ │ │ +00031110: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031120: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ +00031130: 7369 6f6e 3a20 2070 6172 7469 616c 2073  sion:  partial s
│ │ │ │ +00031140: 696e 6775 6c61 7220 6c6f 6375 7320 6469  ingular locus di
│ │ │ │ +00031150: 6d65 6e73 696f 6e20 636f 6d70 7574 6564  mension computed
│ │ │ │ +00031160: 2c20 3d20 3220 2020 2020 2020 207c 0a7c  , = 2        |.|
│ │ │ │  00031170: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -00031180: 6e6f 723a 2043 686f 6f73 696e 6720 4752  nor: Choosing GR
│ │ │ │ -00031190: 6576 4c65 7853 6d61 6c6c 6573 7420 2020  evLexSmallest   
│ │ │ │ +00031180: 6e6f 723a 2043 686f 6f73 696e 6720 5261  nor: Choosing Ra
│ │ │ │ +00031190: 6e64 6f6d 2020 2020 2020 2020 2020 2020  ndom            
│ │ │ │  000311a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000311b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000311c0: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -000311d0: 7369 6f6e 3a20 204c 6f6f 7020 7374 6570  sion:  Loop step
│ │ │ │ -000311e0: 2c20 6162 6f75 7420 746f 2063 6f6d 7075  , about to compu
│ │ │ │ -000311f0: 7465 2064 696d 656e 7369 6f6e 2e20 2053  te dimension.  S
│ │ │ │ -00031200: 7562 6d61 7472 6963 6573 2063 6f7c 0a7c  ubmatrices co|.|
│ │ │ │ +000311c0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ +000311d0: 6e6f 723a 2043 686f 6f73 696e 6720 4752  nor: Choosing GR
│ │ │ │ +000311e0: 6576 4c65 7853 6d61 6c6c 6573 7420 2020  evLexSmallest   
│ │ │ │ +000311f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031200: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00031210: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -00031220: 7369 6f6e 3a20 2069 7343 6f64 696d 4174  sion:  isCodimAt
│ │ │ │ -00031230: 4c65 6173 7420 6661 696c 6564 2c20 636f  Least failed, co
│ │ │ │ -00031240: 6d70 7574 696e 6720 636f 6469 6d2e 2020  mputing codim.  
│ │ │ │ -00031250: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031220: 7369 6f6e 3a20 204c 6f6f 7020 7374 6570  sion:  Loop step
│ │ │ │ +00031230: 2c20 6162 6f75 7420 746f 2063 6f6d 7075  , about to compu
│ │ │ │ +00031240: 7465 2064 696d 656e 7369 6f6e 2e20 2053  te dimension.  S
│ │ │ │ +00031250: 7562 6d61 7472 6963 6573 2063 6f7c 0a7c  ubmatrices co|.|
│ │ │ │  00031260: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -00031270: 7369 6f6e 3a20 2070 6172 7469 616c 2073  sion:  partial s
│ │ │ │ -00031280: 696e 6775 6c61 7220 6c6f 6375 7320 6469  ingular locus di
│ │ │ │ -00031290: 6d65 6e73 696f 6e20 636f 6d70 7574 6564  mension computed
│ │ │ │ -000312a0: 2c20 3d20 3220 2020 2020 2020 207c 0a7c  , = 2        |.|
│ │ │ │ -000312b0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -000312c0: 6e6f 723a 2043 686f 6f73 696e 6720 4c65  nor: Choosing Le
│ │ │ │ -000312d0: 7853 6d61 6c6c 6573 7454 6572 6d20 2020  xSmallestTerm   
│ │ │ │ -000312e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000312f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031270: 7369 6f6e 3a20 2069 7343 6f64 696d 4174  sion:  isCodimAt
│ │ │ │ +00031280: 4c65 6173 7420 6661 696c 6564 2c20 636f  Least failed, co
│ │ │ │ +00031290: 6d70 7574 696e 6720 636f 6469 6d2e 2020  mputing codim.  
│ │ │ │ +000312a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000312b0: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ +000312c0: 7369 6f6e 3a20 2070 6172 7469 616c 2073  sion:  partial s
│ │ │ │ +000312d0: 696e 6775 6c61 7220 6c6f 6375 7320 6469  ingular locus di
│ │ │ │ +000312e0: 6d65 6e73 696f 6e20 636f 6d70 7574 6564  mension computed
│ │ │ │ +000312f0: 2c20 3d20 3220 2020 2020 2020 207c 0a7c  , = 2        |.|
│ │ │ │  00031300: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │  00031310: 6e6f 723a 2043 686f 6f73 696e 6720 4c65  nor: Choosing Le
│ │ │ │ -00031320: 7853 6d61 6c6c 6573 7420 2020 2020 2020  xSmallest       
│ │ │ │ +00031320: 7853 6d61 6c6c 6573 7454 6572 6d20 2020  xSmallestTerm   
│ │ │ │  00031330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031340: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00031350: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │  00031360: 6e6f 723a 2043 686f 6f73 696e 6720 4c65  nor: Choosing Le
│ │ │ │  00031370: 7853 6d61 6c6c 6573 7420 2020 2020 2020  xSmallest       
│ │ │ │  00031380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031390: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000313a0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -000313b0: 6e6f 723a 2043 686f 6f73 696e 6720 5261  nor: Choosing Ra
│ │ │ │ -000313c0: 6e64 6f6d 4e6f 6e5a 6572 6f20 2020 2020  ndomNonZero     
│ │ │ │ +000313b0: 6e6f 723a 2043 686f 6f73 696e 6720 4c65  nor: Choosing Le
│ │ │ │ +000313c0: 7853 6d61 6c6c 6573 7420 2020 2020 2020  xSmallest       
│ │ │ │  000313d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000313e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000313f0: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -00031400: 7369 6f6e 3a20 204c 6f6f 7020 7374 6570  sion:  Loop step
│ │ │ │ -00031410: 2c20 6162 6f75 7420 746f 2063 6f6d 7075  , about to compu
│ │ │ │ -00031420: 7465 2064 696d 656e 7369 6f6e 2e20 2053  te dimension.  S
│ │ │ │ -00031430: 7562 6d61 7472 6963 6573 2063 6f7c 0a7c  ubmatrices co|.|
│ │ │ │ +000313f0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ +00031400: 6e6f 723a 2043 686f 6f73 696e 6720 5261  nor: Choosing Ra
│ │ │ │ +00031410: 6e64 6f6d 4e6f 6e5a 6572 6f20 2020 2020  ndomNonZero     
│ │ │ │ +00031420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031430: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00031440: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -00031450: 7369 6f6e 3a20 2069 7343 6f64 696d 4174  sion:  isCodimAt
│ │ │ │ -00031460: 4c65 6173 7420 6661 696c 6564 2c20 636f  Least failed, co
│ │ │ │ -00031470: 6d70 7574 696e 6720 636f 6469 6d2e 2020  mputing codim.  
│ │ │ │ -00031480: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031450: 7369 6f6e 3a20 204c 6f6f 7020 7374 6570  sion:  Loop step
│ │ │ │ +00031460: 2c20 6162 6f75 7420 746f 2063 6f6d 7075  , about to compu
│ │ │ │ +00031470: 7465 2064 696d 656e 7369 6f6e 2e20 2053  te dimension.  S
│ │ │ │ +00031480: 7562 6d61 7472 6963 6573 2063 6f7c 0a7c  ubmatrices co|.|
│ │ │ │  00031490: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -000314a0: 7369 6f6e 3a20 2070 6172 7469 616c 2073  sion:  partial s
│ │ │ │ -000314b0: 696e 6775 6c61 7220 6c6f 6375 7320 6469  ingular locus di
│ │ │ │ -000314c0: 6d65 6e73 696f 6e20 636f 6d70 7574 6564  mension computed
│ │ │ │ -000314d0: 2c20 3d20 3120 2020 2020 2020 207c 0a7c  , = 1        |.|
│ │ │ │ -000314e0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -000314f0: 6e6f 723a 2043 686f 6f73 696e 6720 4752  nor: Choosing GR
│ │ │ │ -00031500: 6576 4c65 7853 6d61 6c6c 6573 7454 6572  evLexSmallestTer
│ │ │ │ -00031510: 6d20 2020 2020 2020 2020 2020 2020 2020  m               
│ │ │ │ -00031520: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000314a0: 7369 6f6e 3a20 2069 7343 6f64 696d 4174  sion:  isCodimAt
│ │ │ │ +000314b0: 4c65 6173 7420 6661 696c 6564 2c20 636f  Least failed, co
│ │ │ │ +000314c0: 6d70 7574 696e 6720 636f 6469 6d2e 2020  mputing codim.  
│ │ │ │ +000314d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000314e0: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ +000314f0: 7369 6f6e 3a20 2070 6172 7469 616c 2073  sion:  partial s
│ │ │ │ +00031500: 696e 6775 6c61 7220 6c6f 6375 7320 6469  ingular locus di
│ │ │ │ +00031510: 6d65 6e73 696f 6e20 636f 6d70 7574 6564  mension computed
│ │ │ │ +00031520: 2c20 3d20 3120 2020 2020 2020 207c 0a7c  , = 1        |.|
│ │ │ │  00031530: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -00031540: 6e6f 723a 2043 686f 6f73 696e 6720 4c65  nor: Choosing Le
│ │ │ │ -00031550: 7853 6d61 6c6c 6573 7454 6572 6d20 2020  xSmallestTerm   
│ │ │ │ -00031560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031540: 6e6f 723a 2043 686f 6f73 696e 6720 4752  nor: Choosing GR
│ │ │ │ +00031550: 6576 4c65 7853 6d61 6c6c 6573 7454 6572  evLexSmallestTer
│ │ │ │ +00031560: 6d20 2020 2020 2020 2020 2020 2020 2020  m               
│ │ │ │  00031570: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00031580: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -00031590: 6e6f 723a 2043 686f 6f73 696e 6720 5261  nor: Choosing Ra
│ │ │ │ -000315a0: 6e64 6f6d 4e6f 6e5a 6572 6f20 2020 2020  ndomNonZero     
│ │ │ │ +00031590: 6e6f 723a 2043 686f 6f73 696e 6720 4c65  nor: Choosing Le
│ │ │ │ +000315a0: 7853 6d61 6c6c 6573 7454 6572 6d20 2020  xSmallestTerm   
│ │ │ │  000315b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000315c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000315d0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -000315e0: 6e6f 723a 2043 686f 6f73 696e 6720 4c65  nor: Choosing Le
│ │ │ │ -000315f0: 7853 6d61 6c6c 6573 7454 6572 6d20 2020  xSmallestTerm   
│ │ │ │ +000315e0: 6e6f 723a 2043 686f 6f73 696e 6720 5261  nor: Choosing Ra
│ │ │ │ +000315f0: 6e64 6f6d 4e6f 6e5a 6572 6f20 2020 2020  ndomNonZero     
│ │ │ │  00031600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031610: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00031620: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │  00031630: 6e6f 723a 2043 686f 6f73 696e 6720 4c65  nor: Choosing Le
│ │ │ │ -00031640: 7853 6d61 6c6c 6573 7420 2020 2020 2020  xSmallest       
│ │ │ │ +00031640: 7853 6d61 6c6c 6573 7454 6572 6d20 2020  xSmallestTerm   
│ │ │ │  00031650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031660: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00031670: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -00031680: 6e6f 723a 2043 686f 6f73 696e 6720 5261  nor: Choosing Ra
│ │ │ │ -00031690: 6e64 6f6d 4e6f 6e5a 6572 6f20 2020 2020  ndomNonZero     
│ │ │ │ +00031680: 6e6f 723a 2043 686f 6f73 696e 6720 4c65  nor: Choosing Le
│ │ │ │ +00031690: 7853 6d61 6c6c 6573 7420 2020 2020 2020  xSmallest       
│ │ │ │  000316a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000316b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000316c0: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -000316d0: 7369 6f6e 3a20 204c 6f6f 7020 7374 6570  sion:  Loop step
│ │ │ │ -000316e0: 2c20 6162 6f75 7420 746f 2063 6f6d 7075  , about to compu
│ │ │ │ -000316f0: 7465 2064 696d 656e 7369 6f6e 2e20 2053  te dimension.  S
│ │ │ │ -00031700: 7562 6d61 7472 6963 6573 2063 6f7c 0a7c  ubmatrices co|.|
│ │ │ │ +000316c0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ +000316d0: 6e6f 723a 2043 686f 6f73 696e 6720 5261  nor: Choosing Ra
│ │ │ │ +000316e0: 6e64 6f6d 4e6f 6e5a 6572 6f20 2020 2020  ndomNonZero     
│ │ │ │ +000316f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031700: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00031710: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -00031720: 7369 6f6e 3a20 2069 7343 6f64 696d 4174  sion:  isCodimAt
│ │ │ │ -00031730: 4c65 6173 7420 6661 696c 6564 2c20 636f  Least failed, co
│ │ │ │ -00031740: 6d70 7574 696e 6720 636f 6469 6d2e 2020  mputing codim.  
│ │ │ │ -00031750: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031720: 7369 6f6e 3a20 204c 6f6f 7020 7374 6570  sion:  Loop step
│ │ │ │ +00031730: 2c20 6162 6f75 7420 746f 2063 6f6d 7075  , about to compu
│ │ │ │ +00031740: 7465 2064 696d 656e 7369 6f6e 2e20 2053  te dimension.  S
│ │ │ │ +00031750: 7562 6d61 7472 6963 6573 2063 6f7c 0a7c  ubmatrices co|.|
│ │ │ │  00031760: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -00031770: 7369 6f6e 3a20 2070 6172 7469 616c 2073  sion:  partial s
│ │ │ │ -00031780: 696e 6775 6c61 7220 6c6f 6375 7320 6469  ingular locus di
│ │ │ │ -00031790: 6d65 6e73 696f 6e20 636f 6d70 7574 6564  mension computed
│ │ │ │ -000317a0: 2c20 3d20 3120 2020 2020 2020 207c 0a7c  , = 1        |.|
│ │ │ │ -000317b0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -000317c0: 6e6f 723a 2043 686f 6f73 696e 6720 5261  nor: Choosing Ra
│ │ │ │ -000317d0: 6e64 6f6d 4e6f 6e5a 6572 6f20 2020 2020  ndomNonZero     
│ │ │ │ -000317e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000317f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031770: 7369 6f6e 3a20 2069 7343 6f64 696d 4174  sion:  isCodimAt
│ │ │ │ +00031780: 4c65 6173 7420 6661 696c 6564 2c20 636f  Least failed, co
│ │ │ │ +00031790: 6d70 7574 696e 6720 636f 6469 6d2e 2020  mputing codim.  
│ │ │ │ +000317a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000317b0: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ +000317c0: 7369 6f6e 3a20 2070 6172 7469 616c 2073  sion:  partial s
│ │ │ │ +000317d0: 696e 6775 6c61 7220 6c6f 6375 7320 6469  ingular locus di
│ │ │ │ +000317e0: 6d65 6e73 696f 6e20 636f 6d70 7574 6564  mension computed
│ │ │ │ +000317f0: 2c20 3d20 3120 2020 2020 2020 207c 0a7c  , = 1        |.|
│ │ │ │  00031800: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -00031810: 6e6f 723a 2043 686f 6f73 696e 6720 4c65  nor: Choosing Le
│ │ │ │ -00031820: 7853 6d61 6c6c 6573 7420 2020 2020 2020  xSmallest       
│ │ │ │ +00031810: 6e6f 723a 2043 686f 6f73 696e 6720 5261  nor: Choosing Ra
│ │ │ │ +00031820: 6e64 6f6d 4e6f 6e5a 6572 6f20 2020 2020  ndomNonZero     
│ │ │ │  00031830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031840: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00031850: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -00031860: 6e6f 723a 2043 686f 6f73 696e 6720 4752  nor: Choosing GR
│ │ │ │ -00031870: 6576 4c65 7853 6d61 6c6c 6573 7420 2020  evLexSmallest   
│ │ │ │ +00031860: 6e6f 723a 2043 686f 6f73 696e 6720 4c65  nor: Choosing Le
│ │ │ │ +00031870: 7853 6d61 6c6c 6573 7420 2020 2020 2020  xSmallest       
│ │ │ │  00031880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031890: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000318a0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │  000318b0: 6e6f 723a 2043 686f 6f73 696e 6720 4752  nor: Choosing GR
│ │ │ │ -000318c0: 6576 4c65 7853 6d61 6c6c 6573 7454 6572  evLexSmallestTer
│ │ │ │ -000318d0: 6d20 2020 2020 2020 2020 2020 2020 2020  m               
│ │ │ │ +000318c0: 6576 4c65 7853 6d61 6c6c 6573 7420 2020  evLexSmallest   
│ │ │ │ +000318d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000318e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000318f0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │  00031900: 6e6f 723a 2043 686f 6f73 696e 6720 4752  nor: Choosing GR
│ │ │ │  00031910: 6576 4c65 7853 6d61 6c6c 6573 7454 6572  evLexSmallestTer
│ │ │ │  00031920: 6d20 2020 2020 2020 2020 2020 2020 2020  m               
│ │ │ │  00031930: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00031940: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │  00031950: 6e6f 723a 2043 686f 6f73 696e 6720 4752  nor: Choosing GR
│ │ │ │ -00031960: 6576 4c65 7853 6d61 6c6c 6573 7420 2020  evLexSmallest   
│ │ │ │ -00031970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031960: 6576 4c65 7853 6d61 6c6c 6573 7454 6572  evLexSmallestTer
│ │ │ │ +00031970: 6d20 2020 2020 2020 2020 2020 2020 2020  m               
│ │ │ │  00031980: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00031990: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │  000319a0: 6e6f 723a 2043 686f 6f73 696e 6720 4752  nor: Choosing GR
│ │ │ │  000319b0: 6576 4c65 7853 6d61 6c6c 6573 7420 2020  evLexSmallest   
│ │ │ │  000319c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000319d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000319e0: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -000319f0: 7369 6f6e 3a20 204c 6f6f 7020 7374 6570  sion:  Loop step
│ │ │ │ -00031a00: 2c20 6162 6f75 7420 746f 2063 6f6d 7075  , about to compu
│ │ │ │ -00031a10: 7465 2064 696d 656e 7369 6f6e 2e20 2053  te dimension.  S
│ │ │ │ -00031a20: 7562 6d61 7472 6963 6573 2063 6f7c 0a7c  ubmatrices co|.|
│ │ │ │ +000319e0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ +000319f0: 6e6f 723a 2043 686f 6f73 696e 6720 4752  nor: Choosing GR
│ │ │ │ +00031a00: 6576 4c65 7853 6d61 6c6c 6573 7420 2020  evLexSmallest   
│ │ │ │ +00031a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031a20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00031a30: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -00031a40: 7369 6f6e 3a20 2069 7343 6f64 696d 4174  sion:  isCodimAt
│ │ │ │ -00031a50: 4c65 6173 7420 6661 696c 6564 2c20 636f  Least failed, co
│ │ │ │ -00031a60: 6d70 7574 696e 6720 636f 6469 6d2e 2020  mputing codim.  
│ │ │ │ -00031a70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031a40: 7369 6f6e 3a20 204c 6f6f 7020 7374 6570  sion:  Loop step
│ │ │ │ +00031a50: 2c20 6162 6f75 7420 746f 2063 6f6d 7075  , about to compu
│ │ │ │ +00031a60: 7465 2064 696d 656e 7369 6f6e 2e20 2053  te dimension.  S
│ │ │ │ +00031a70: 7562 6d61 7472 6963 6573 2063 6f7c 0a7c  ubmatrices co|.|
│ │ │ │  00031a80: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -00031a90: 7369 6f6e 3a20 2070 6172 7469 616c 2073  sion:  partial s
│ │ │ │ -00031aa0: 696e 6775 6c61 7220 6c6f 6375 7320 6469  ingular locus di
│ │ │ │ -00031ab0: 6d65 6e73 696f 6e20 636f 6d70 7574 6564  mension computed
│ │ │ │ -00031ac0: 2c20 3d20 3120 2020 2020 2020 207c 0a7c  , = 1        |.|
│ │ │ │ -00031ad0: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ -00031ae0: 6e6f 723a 2043 686f 6f73 696e 6720 4752  nor: Choosing GR
│ │ │ │ -00031af0: 6576 4c65 7853 6d61 6c6c 6573 7420 2020  evLexSmallest   
│ │ │ │ -00031b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031b10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031a90: 7369 6f6e 3a20 2069 7343 6f64 696d 4174  sion:  isCodimAt
│ │ │ │ +00031aa0: 4c65 6173 7420 6661 696c 6564 2c20 636f  Least failed, co
│ │ │ │ +00031ab0: 6d70 7574 696e 6720 636f 6469 6d2e 2020  mputing codim.  
│ │ │ │ +00031ac0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031ad0: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ +00031ae0: 7369 6f6e 3a20 2070 6172 7469 616c 2073  sion:  partial s
│ │ │ │ +00031af0: 696e 6775 6c61 7220 6c6f 6375 7320 6469  ingular locus di
│ │ │ │ +00031b00: 6d65 6e73 696f 6e20 636f 6d70 7574 6564  mension computed
│ │ │ │ +00031b10: 2c20 3d20 3120 2020 2020 2020 207c 0a7c  , = 1        |.|
│ │ │ │  00031b20: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │  00031b30: 6e6f 723a 2043 686f 6f73 696e 6720 4752  nor: Choosing GR
│ │ │ │ -00031b40: 6576 4c65 7853 6d61 6c6c 6573 7454 6572  evLexSmallestTer
│ │ │ │ -00031b50: 6d20 2020 2020 2020 2020 2020 2020 2020  m               
│ │ │ │ +00031b40: 6576 4c65 7853 6d61 6c6c 6573 7420 2020  evLexSmallest   
│ │ │ │ +00031b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031b60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00031b70: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -00031b80: 7369 6f6e 3a20 204c 6f6f 7020 7374 6570  sion:  Loop step
│ │ │ │ -00031b90: 2c20 6162 6f75 7420 746f 2063 6f6d 7075  , about to compu
│ │ │ │ -00031ba0: 7465 2064 696d 656e 7369 6f6e 2e20 2053  te dimension.  S
│ │ │ │ -00031bb0: 7562 6d61 7472 6963 6573 2063 6f7c 0a7c  ubmatrices co|.|
│ │ │ │ +00031b70: 696e 7465 726e 616c 4368 6f6f 7365 4d69  internalChooseMi
│ │ │ │ +00031b80: 6e6f 723a 2043 686f 6f73 696e 6720 4752  nor: Choosing GR
│ │ │ │ +00031b90: 6576 4c65 7853 6d61 6c6c 6573 7454 6572  evLexSmallestTer
│ │ │ │ +00031ba0: 6d20 2020 2020 2020 2020 2020 2020 2020  m               
│ │ │ │ +00031bb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00031bc0: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -00031bd0: 7369 6f6e 3a20 2069 7343 6f64 696d 4174  sion:  isCodimAt
│ │ │ │ -00031be0: 4c65 6173 7420 6661 696c 6564 2c20 636f  Least failed, co
│ │ │ │ -00031bf0: 6d70 7574 696e 6720 636f 6469 6d2e 2020  mputing codim.  
│ │ │ │ -00031c00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00031bd0: 7369 6f6e 3a20 204c 6f6f 7020 7374 6570  sion:  Loop step
│ │ │ │ +00031be0: 2c20 6162 6f75 7420 746f 2063 6f6d 7075  , about to compu
│ │ │ │ +00031bf0: 7465 2064 696d 656e 7369 6f6e 2e20 2053  te dimension.  S
│ │ │ │ +00031c00: 7562 6d61 7472 6963 6573 2063 6f7c 0a7c  ubmatrices co|.|
│ │ │ │  00031c10: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -00031c20: 7369 6f6e 3a20 2070 6172 7469 616c 2073  sion:  partial s
│ │ │ │ -00031c30: 696e 6775 6c61 7220 6c6f 6375 7320 6469  ingular locus di
│ │ │ │ -00031c40: 6d65 6e73 696f 6e20 636f 6d70 7574 6564  mension computed
│ │ │ │ -00031c50: 2c20 3d20 3120 2020 2020 2020 207c 0a7c  , = 1        |.|
│ │ │ │ +00031c20: 7369 6f6e 3a20 2069 7343 6f64 696d 4174  sion:  isCodimAt
│ │ │ │ +00031c30: 4c65 6173 7420 6661 696c 6564 2c20 636f  Least failed, co
│ │ │ │ +00031c40: 6d70 7574 696e 6720 636f 6469 6d2e 2020  mputing codim.  
│ │ │ │ +00031c50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00031c60: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ -00031c70: 7369 6f6e 3a20 204c 6f6f 7020 636f 6d70  sion:  Loop comp
│ │ │ │ -00031c80: 6c65 7465 642c 2073 7562 6d61 7472 6963  leted, submatric
│ │ │ │ -00031c90: 6573 2063 6f6e 7369 6465 7265 6420 3d20  es considered = 
│ │ │ │ -00031ca0: 3330 2c20 616e 6420 636f 6d70 757c 0a7c  30, and compu|.|
│ │ │ │ -00031cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ -00031d00: 7273 2c20 7765 2077 696c 6c20 636f 6d70  rs, we will comp
│ │ │ │ -00031d10: 7574 6520 7570 2074 6f20 3330 206f 6620  ute up to 30 of 
│ │ │ │ -00031d20: 7468 656d 2e20 2020 2020 2020 2020 2020  them.           
│ │ │ │ -00031d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031d40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00031d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031c70: 7369 6f6e 3a20 2070 6172 7469 616c 2073  sion:  partial s
│ │ │ │ +00031c80: 696e 6775 6c61 7220 6c6f 6375 7320 6469  ingular locus di
│ │ │ │ +00031c90: 6d65 6e73 696f 6e20 636f 6d70 7574 6564  mension computed
│ │ │ │ +00031ca0: 2c20 3d20 3120 2020 2020 2020 207c 0a7c  , = 1        |.|
│ │ │ │ +00031cb0: 7265 6775 6c61 7249 6e43 6f64 696d 656e  regularInCodimen
│ │ │ │ +00031cc0: 7369 6f6e 3a20 204c 6f6f 7020 636f 6d70  sion:  Loop comp
│ │ │ │ +00031cd0: 6c65 7465 642c 2073 7562 6d61 7472 6963  leted, submatric
│ │ │ │ +00031ce0: 6573 2063 6f6e 7369 6465 7265 6420 3d20  es considered = 
│ │ │ │ +00031cf0: 3330 2c20 616e 6420 636f 6d70 757c 0a7c  30, and compu|.|
│ │ │ │ +00031d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00031d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00031d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00031d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00031d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ +00031d50: 7273 2c20 7765 2077 696c 6c20 636f 6d70  rs, we will comp
│ │ │ │ +00031d60: 7574 6520 7570 2074 6f20 3330 206f 6620  ute up to 30 of 
│ │ │ │ +00031d70: 7468 656d 2e20 2020 2020 2020 2020 2020  them.           
│ │ │ │  00031d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031d90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00031da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031de0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ @@ -12801,21 +12801,21 @@
│ │ │ │  00032000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032010: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00032020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032060: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00032070: 6e73 6964 6572 6564 3a20 392c 2061 6e64  nsidered: 9, and
│ │ │ │ -00032080: 2063 6f6d 7075 7465 6420 3d20 3720 2020   computed = 7   
│ │ │ │ +00032070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000320a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000320b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000320c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000320d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000320c0: 6e73 6964 6572 6564 3a20 392c 2061 6e64  nsidered: 9, and
│ │ │ │ +000320d0: 2063 6f6d 7075 7465 6420 3d20 3720 2020   computed = 7   
│ │ │ │  000320e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000320f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032100: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00032110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -12826,21 +12826,21 @@
│ │ │ │  00032190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000321a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000321b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000321c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000321d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000321e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000321f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00032200: 6e73 6964 6572 6564 3a20 3131 2c20 616e  nsidered: 11, an
│ │ │ │ -00032210: 6420 636f 6d70 7574 6564 203d 2039 2020  d computed = 9  
│ │ │ │ +00032200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032240: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00032250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032250: 6e73 6964 6572 6564 3a20 3131 2c20 616e  nsidered: 11, an
│ │ │ │ +00032260: 6420 636f 6d70 7574 6564 203d 2039 2020  d computed = 9  
│ │ │ │  00032270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032290: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000322a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000322b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000322c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000322d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -12861,21 +12861,21 @@
│ │ │ │  000323c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000323d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000323e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000323f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032420: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00032430: 6e73 6964 6572 6564 3a20 3135 2c20 616e  nsidered: 15, an
│ │ │ │ -00032440: 6420 636f 6d70 7574 6564 203d 2031 3120  d computed = 11 
│ │ │ │ +00032430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032470: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00032480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032480: 6e73 6964 6572 6564 3a20 3135 2c20 616e  nsidered: 15, an
│ │ │ │ +00032490: 6420 636f 6d70 7574 6564 203d 2031 3120  d computed = 11 
│ │ │ │  000324a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000324b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000324c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000324d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000324e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000324f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -12906,21 +12906,21 @@
│ │ │ │  00032690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000326a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000326b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000326c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000326d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000326e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000326f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00032700: 6e73 6964 6572 6564 3a20 3231 2c20 616e  nsidered: 21, an
│ │ │ │ -00032710: 6420 636f 6d70 7574 6564 203d 2031 3620  d computed = 16 
│ │ │ │ +00032700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032740: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00032750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032750: 6e73 6964 6572 6564 3a20 3231 2c20 616e  nsidered: 21, an
│ │ │ │ +00032760: 6420 636f 6d70 7574 6564 203d 2031 3620  d computed = 16 
│ │ │ │  00032770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032790: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000327a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000327b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000327c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000327d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -12956,21 +12956,21 @@
│ │ │ │  000329b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000329c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000329d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000329e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000329f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032a10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00032a20: 6e73 6964 6572 6564 3a20 3238 2c20 616e  nsidered: 28, an
│ │ │ │ -00032a30: 6420 636f 6d70 7574 6564 203d 2032 3220  d computed = 22 
│ │ │ │ +00032a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032a60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00032a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032a70: 6e73 6964 6572 6564 3a20 3238 2c20 616e  nsidered: 28, an
│ │ │ │ +00032a80: 6420 636f 6d70 7574 6564 203d 2032 3220  d computed = 22 
│ │ │ │  00032a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032ab0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00032ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -12981,4040 +12981,4045 @@
│ │ │ │  00032b40: 2020 2020 2020 2020 2020 2020 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 207c 0a7c               |.|
│ │ │ │ -00032bb0: 6e73 6964 6572 6564 3a20 3330 2c20 616e  nsidered: 30, an
│ │ │ │ -00032bc0: 6420 636f 6d70 7574 6564 203d 2032 3320  d computed = 23 
│ │ │ │ +00032bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +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 207c 0a7c               |.|
│ │ │ │ -00032c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032c00: 6e73 6964 6572 6564 3a20 3330 2c20 616e  nsidered: 30, an
│ │ │ │ +00032c10: 6420 636f 6d70 7574 6564 203d 2032 3320  d computed = 23 
│ │ │ │  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 207c 0a7c               |.|
│ │ │ │  00032c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 207c 0a7c               |.|
│ │ │ │ -00032ca0: 7465 6420 3d20 3233 2e20 2073 696e 6775  ted = 23.  singu
│ │ │ │ -00032cb0: 6c61 7220 6c6f 6375 7320 6469 6d65 6e73  lar locus dimens
│ │ │ │ -00032cc0: 696f 6e20 6170 7065 6172 7320 746f 2062  ion appears to b
│ │ │ │ -00032cd0: 6520 3d20 3120 2020 2020 2020 2020 2020  e = 1           
│ │ │ │ -00032ce0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00032cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ -00032d40: 5468 6973 2066 756e 6374 696f 6e20 6861  This function ha
│ │ │ │ -00032d50: 7320 6d61 6e79 206f 7074 696f 6e73 2077  s many options w
│ │ │ │ -00032d60: 6869 6368 2061 6c6c 6f77 2079 6f75 2074  hich allow you t
│ │ │ │ -00032d70: 6f20 6669 6e65 2074 756e 6520 7468 6520  o fine tune the 
│ │ │ │ -00032d80: 7374 7261 7465 6779 2075 7365 640a 746f  strategy used.to
│ │ │ │ -00032d90: 2066 696e 6420 696e 7465 7265 7374 696e   find interestin
│ │ │ │ -00032da0: 6720 6d69 6e6f 7273 2e20 596f 7520 6361  g minors. You ca
│ │ │ │ -00032db0: 6e20 7061 7373 2069 7420 6120 4861 7368  n pass it a Hash
│ │ │ │ -00032dc0: 5461 626c 6520 7370 6563 6966 7969 6e67  Table specifying
│ │ │ │ -00032dd0: 2074 6865 2073 7472 6174 6567 790a 7669   the strategy.vi
│ │ │ │ -00032de0: 6120 7468 6520 6f70 7469 6f6e 2053 7472  a the option Str
│ │ │ │ -00032df0: 6174 6567 792e 2020 5365 6520 2a6e 6f74  ategy.  See *not
│ │ │ │ -00032e00: 6520 4c65 7853 6d61 6c6c 6573 743a 2053  e LexSmallest: S
│ │ │ │ -00032e10: 7472 6174 6567 7944 6566 6175 6c74 2c20  trategyDefault, 
│ │ │ │ -00032e20: 666f 7220 686f 7720 746f 0a63 6f6e 7374  for how to.const
│ │ │ │ -00032e30: 7275 6374 2074 6869 7320 4861 7368 5461  ruct this HashTa
│ │ │ │ -00032e40: 626c 652e 2054 6865 2064 6566 6175 6c74  ble. The default
│ │ │ │ -00032e50: 2073 7472 6174 6567 7920 6973 2053 7472   strategy is Str
│ │ │ │ -00032e60: 6174 6567 7944 6566 6175 6c74 2c20 7768  ategyDefault, wh
│ │ │ │ -00032e70: 6963 6820 7365 656d 730a 746f 2077 6f72  ich seems.to wor
│ │ │ │ -00032e80: 6b20 7765 6c6c 206f 6e20 7468 6520 6578  k well on the ex
│ │ │ │ -00032e90: 616d 706c 6573 2077 6520 6861 7665 2065  amples we have e
│ │ │ │ -00032ea0: 7870 6c6f 7265 642e 2020 486f 7765 7665  xplored.  Howeve
│ │ │ │ -00032eb0: 722c 2063 6175 7469 6f6e 206d 7573 7420  r, caution must 
│ │ │ │ -00032ec0: 6265 0a65 7865 7263 6973 6564 2c20 6265  be.exercised, be
│ │ │ │ -00032ed0: 6361 7573 652c 2065 7665 6e20 696e 2074  cause, even in t
│ │ │ │ -00032ee0: 6865 2065 7861 6d70 6c65 7320 6162 6f76  he examples abov
│ │ │ │ -00032ef0: 652c 2063 6572 7461 696e 2073 7472 6174  e, certain strat
│ │ │ │ -00032f00: 6567 6965 7320 776f 726b 2077 656c 6c0a  egies work well.
│ │ │ │ -00032f10: 7768 696c 6520 6f74 6865 7273 2064 6f20  while others do 
│ │ │ │ -00032f20: 6e6f 742e 2020 496e 2074 6865 2041 6265  not.  In the Abe
│ │ │ │ -00032f30: 6c69 616e 2073 7572 6661 6365 2065 7861  lian surface exa
│ │ │ │ -00032f40: 6d70 6c65 2c20 4c65 7853 6d61 6c6c 6573  mple, LexSmalles
│ │ │ │ -00032f50: 7420 776f 726b 7320 7665 7279 0a77 656c  t works very.wel
│ │ │ │ -00032f60: 6c2c 2077 6869 6c65 204c 6578 536d 616c  l, while LexSmal
│ │ │ │ -00032f70: 6c65 7374 5465 726d 2064 6f65 7320 6e6f  lestTerm does no
│ │ │ │ -00032f80: 7420 6576 656e 2074 7970 6963 616c 6c79  t even typically
│ │ │ │ -00032f90: 2063 6f72 7265 6374 6c79 2069 6465 6e74   correctly ident
│ │ │ │ -00032fa0: 6966 7920 7468 6520 7269 6e67 0a61 7320  ify the ring.as 
│ │ │ │ -00032fb0: 6e6f 6e73 696e 6775 6c61 7220 2874 6869  nonsingular (thi
│ │ │ │ -00032fc0: 7320 6973 2062 6563 6175 7365 2074 6865  s is because the
│ │ │ │ -00032fd0: 7265 2061 7265 2061 2073 6d61 6c6c 206e  re are a small n
│ │ │ │ -00032fe0: 756d 6265 7220 6f66 2065 6e74 7269 6573  umber of entries
│ │ │ │ -00032ff0: 2077 6974 680a 6e6f 6e7a 6572 6f20 636f   with.nonzero co
│ │ │ │ -00033000: 6e73 7461 6e74 2074 6572 6d73 2c20 7768  nstant terms, wh
│ │ │ │ -00033010: 6963 6820 6172 6520 7365 6c65 6374 6564  ich are selected
│ │ │ │ -00033020: 2072 6570 6561 7465 646c 7929 2e20 486f   repeatedly). Ho
│ │ │ │ -00033030: 7765 7665 722c 2069 6e20 6f75 7220 6669  wever, in our fi
│ │ │ │ -00033040: 7273 740a 6578 616d 706c 652c 2074 6865  rst.example, the
│ │ │ │ -00033050: 204c 6578 536d 616c 6c65 7374 5465 726d   LexSmallestTerm
│ │ │ │ -00033060: 2069 7320 6d75 6368 2066 6173 7465 722c   is much faster,
│ │ │ │ -00033070: 2061 6e64 2052 616e 646f 6d20 646f 6573   and Random does
│ │ │ │ -00033080: 206e 6f74 2070 6572 666f 726d 2077 656c   not perform wel
│ │ │ │ -00033090: 6c0a 6174 2061 6c6c 2e0a 0a2b 2d2d 2d2d  l.at all...+----
│ │ │ │ -000330a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 2b0a 7c69  ------------+.|i
│ │ │ │ -000330e0: 3138 203a 2053 7472 6174 6567 7943 7572  18 : StrategyCur
│ │ │ │ -000330f0: 7265 6e74 2352 616e 646f 6d20 3d20 303b  rent#Random = 0;
│ │ │ │ -00033100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033110: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00033120: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00033130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033160: 2d2d 2b0a 7c69 3139 203a 2053 7472 6174  --+.|i19 : Strat
│ │ │ │ -00033170: 6567 7943 7572 7265 6e74 234c 6578 536d  egyCurrent#LexSm
│ │ │ │ -00033180: 616c 6c65 7374 203d 2031 3030 3b20 2020  allest = 100;   
│ │ │ │ -00033190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000331a0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -000331b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000331c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000331d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000331e0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3230 203a  --------+.|i20 :
│ │ │ │ -000331f0: 2053 7472 6174 6567 7943 7572 7265 6e74   StrategyCurrent
│ │ │ │ -00033200: 234c 6578 536d 616c 6c65 7374 5465 726d  #LexSmallestTerm
│ │ │ │ -00033210: 203d 2030 3b20 2020 2020 2020 2020 2020   = 0;           
│ │ │ │ -00033220: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -00033230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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: 3033 3036 7320 2863 7075 293b 2030 2e32  0306s (cpu); 0.2
│ │ │ │ -000332d0: 3137 3137 3673 2028 7468 7265 6164 293b  17176s (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            
│ │ │ │ +00032ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032ce0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00032cf0: 7465 6420 3d20 3233 2e20 2073 696e 6775  ted = 23.  singu
│ │ │ │ +00032d00: 6c61 7220 6c6f 6375 7320 6469 6d65 6e73  lar locus dimens
│ │ │ │ +00032d10: 696f 6e20 6170 7065 6172 7320 746f 2062  ion appears to b
│ │ │ │ +00032d20: 6520 3d20 3120 2020 2020 2020 2020 2020  e = 1           
│ │ │ │ +00032d30: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00032d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00032d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00032d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00032d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00032d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ +00032d90: 5468 6973 2066 756e 6374 696f 6e20 6861  This function ha
│ │ │ │ +00032da0: 7320 6d61 6e79 206f 7074 696f 6e73 2077  s many options w
│ │ │ │ +00032db0: 6869 6368 2061 6c6c 6f77 2079 6f75 2074  hich allow you t
│ │ │ │ +00032dc0: 6f20 6669 6e65 2074 756e 6520 7468 6520  o fine tune the 
│ │ │ │ +00032dd0: 7374 7261 7465 6779 2075 7365 640a 746f  strategy used.to
│ │ │ │ +00032de0: 2066 696e 6420 696e 7465 7265 7374 696e   find interestin
│ │ │ │ +00032df0: 6720 6d69 6e6f 7273 2e20 596f 7520 6361  g minors. You ca
│ │ │ │ +00032e00: 6e20 7061 7373 2069 7420 6120 4861 7368  n pass it a Hash
│ │ │ │ +00032e10: 5461 626c 6520 7370 6563 6966 7969 6e67  Table specifying
│ │ │ │ +00032e20: 2074 6865 2073 7472 6174 6567 790a 7669   the strategy.vi
│ │ │ │ +00032e30: 6120 7468 6520 6f70 7469 6f6e 2053 7472  a the option Str
│ │ │ │ +00032e40: 6174 6567 792e 2020 5365 6520 2a6e 6f74  ategy.  See *not
│ │ │ │ +00032e50: 6520 4c65 7853 6d61 6c6c 6573 743a 2053  e LexSmallest: S
│ │ │ │ +00032e60: 7472 6174 6567 7944 6566 6175 6c74 2c20  trategyDefault, 
│ │ │ │ +00032e70: 666f 7220 686f 7720 746f 0a63 6f6e 7374  for how to.const
│ │ │ │ +00032e80: 7275 6374 2074 6869 7320 4861 7368 5461  ruct this HashTa
│ │ │ │ +00032e90: 626c 652e 2054 6865 2064 6566 6175 6c74  ble. The default
│ │ │ │ +00032ea0: 2073 7472 6174 6567 7920 6973 2053 7472   strategy is Str
│ │ │ │ +00032eb0: 6174 6567 7944 6566 6175 6c74 2c20 7768  ategyDefault, wh
│ │ │ │ +00032ec0: 6963 6820 7365 656d 730a 746f 2077 6f72  ich seems.to wor
│ │ │ │ +00032ed0: 6b20 7765 6c6c 206f 6e20 7468 6520 6578  k well on the ex
│ │ │ │ +00032ee0: 616d 706c 6573 2077 6520 6861 7665 2065  amples we have e
│ │ │ │ +00032ef0: 7870 6c6f 7265 642e 2020 486f 7765 7665  xplored.  Howeve
│ │ │ │ +00032f00: 722c 2063 6175 7469 6f6e 206d 7573 7420  r, caution must 
│ │ │ │ +00032f10: 6265 0a65 7865 7263 6973 6564 2c20 6265  be.exercised, be
│ │ │ │ +00032f20: 6361 7573 652c 2065 7665 6e20 696e 2074  cause, even in t
│ │ │ │ +00032f30: 6865 2065 7861 6d70 6c65 7320 6162 6f76  he examples abov
│ │ │ │ +00032f40: 652c 2063 6572 7461 696e 2073 7472 6174  e, certain strat
│ │ │ │ +00032f50: 6567 6965 7320 776f 726b 2077 656c 6c0a  egies work well.
│ │ │ │ +00032f60: 7768 696c 6520 6f74 6865 7273 2064 6f20  while others do 
│ │ │ │ +00032f70: 6e6f 742e 2020 496e 2074 6865 2041 6265  not.  In the Abe
│ │ │ │ +00032f80: 6c69 616e 2073 7572 6661 6365 2065 7861  lian surface exa
│ │ │ │ +00032f90: 6d70 6c65 2c20 4c65 7853 6d61 6c6c 6573  mple, LexSmalles
│ │ │ │ +00032fa0: 7420 776f 726b 7320 7665 7279 0a77 656c  t works very.wel
│ │ │ │ +00032fb0: 6c2c 2077 6869 6c65 204c 6578 536d 616c  l, while LexSmal
│ │ │ │ +00032fc0: 6c65 7374 5465 726d 2064 6f65 7320 6e6f  lestTerm does no
│ │ │ │ +00032fd0: 7420 6576 656e 2074 7970 6963 616c 6c79  t even typically
│ │ │ │ +00032fe0: 2063 6f72 7265 6374 6c79 2069 6465 6e74   correctly ident
│ │ │ │ +00032ff0: 6966 7920 7468 6520 7269 6e67 0a61 7320  ify the ring.as 
│ │ │ │ +00033000: 6e6f 6e73 696e 6775 6c61 7220 2874 6869  nonsingular (thi
│ │ │ │ +00033010: 7320 6973 2062 6563 6175 7365 2074 6865  s is because the
│ │ │ │ +00033020: 7265 2061 7265 2061 2073 6d61 6c6c 206e  re are a small n
│ │ │ │ +00033030: 756d 6265 7220 6f66 2065 6e74 7269 6573  umber of entries
│ │ │ │ +00033040: 2077 6974 680a 6e6f 6e7a 6572 6f20 636f   with.nonzero co
│ │ │ │ +00033050: 6e73 7461 6e74 2074 6572 6d73 2c20 7768  nstant terms, wh
│ │ │ │ +00033060: 6963 6820 6172 6520 7365 6c65 6374 6564  ich are selected
│ │ │ │ +00033070: 2072 6570 6561 7465 646c 7929 2e20 486f   repeatedly). Ho
│ │ │ │ +00033080: 7765 7665 722c 2069 6e20 6f75 7220 6669  wever, in our fi
│ │ │ │ +00033090: 7273 740a 6578 616d 706c 652c 2074 6865  rst.example, the
│ │ │ │ +000330a0: 204c 6578 536d 616c 6c65 7374 5465 726d   LexSmallestTerm
│ │ │ │ +000330b0: 2069 7320 6d75 6368 2066 6173 7465 722c   is much faster,
│ │ │ │ +000330c0: 2061 6e64 2052 616e 646f 6d20 646f 6573   and Random does
│ │ │ │ +000330d0: 206e 6f74 2070 6572 666f 726d 2077 656c   not perform wel
│ │ │ │ +000330e0: 6c0a 6174 2061 6c6c 2e0a 0a2b 2d2d 2d2d  l.at all...+----
│ │ │ │ +000330f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00033130: 3138 203a 2053 7472 6174 6567 7943 7572  18 : StrategyCur
│ │ │ │ +00033140: 7265 6e74 2352 616e 646f 6d20 3d20 303b  rent#Random = 0;
│ │ │ │ +00033150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033160: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00033170: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00033180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000331a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000331b0: 2d2d 2b0a 7c69 3139 203a 2053 7472 6174  --+.|i19 : Strat
│ │ │ │ +000331c0: 6567 7943 7572 7265 6e74 234c 6578 536d  egyCurrent#LexSm
│ │ │ │ +000331d0: 616c 6c65 7374 203d 2031 3030 3b20 2020  allest = 100;   
│ │ │ │ +000331e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000331f0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00033200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033230: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3230 203a  --------+.|i20 :
│ │ │ │ +00033240: 2053 7472 6174 6567 7943 7572 7265 6e74   StrategyCurrent
│ │ │ │ +00033250: 234c 6578 536d 616c 6c65 7374 5465 726d  #LexSmallestTerm
│ │ │ │ +00033260: 203d 2030 3b20 2020 2020 2020 2020 2020   = 0;           
│ │ │ │ +00033270: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00033280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000332a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000332b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000332c0: 7c69 3231 203a 2074 696d 6520 7265 6775  |i21 : time regu
│ │ │ │ +000332d0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ +000332e0: 2832 2c20 522c 2053 7472 6174 6567 793d  (2, R, Strategy=
│ │ │ │ +000332f0: 3e53 7472 6174 6567 7943 7572 7265 6e74  >StrategyCurrent
│ │ │ │ +00033300: 297c 0a7c 202d 2d20 7573 6564 2030 2e33  )|.| -- used 0.3
│ │ │ │ +00033310: 3737 3732 3573 2028 6370 7529 3b20 302e  77725s (cpu); 0.
│ │ │ │ +00033320: 3234 3435 3036 7320 2874 6872 6561 6429  244506s (thread)
│ │ │ │ +00033330: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ +00033340: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00033350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033370: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -00033380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 3131  |.| -- used 0.11
│ │ │ │ -00033410: 3336 3538 7320 2863 7075 293b 2030 2e30  3658s (cpu); 0.0
│ │ │ │ -00033420: 3739 3238 3831 7320 2874 6872 6561 6429  792881s (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             
│ │ │ │ +00033370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033380: 2020 2020 2020 207c 0a7c 6f32 3120 3d20         |.|o21 = 
│ │ │ │ +00033390: 7472 7565 2020 2020 2020 2020 2020 2020  true            
│ │ │ │ +000333a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000333b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000333c0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +000333d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000333e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000333f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00033410: 6932 3220 3a20 7469 6d65 2072 6567 756c  i22 : time regul
│ │ │ │ +00033420: 6172 496e 436f 6469 6d65 6e73 696f 6e28  arInCodimension(
│ │ │ │ +00033430: 322c 2052 2c20 5374 7261 7465 6779 3d3e  2, R, Strategy=>
│ │ │ │ +00033440: 5374 7261 7465 6779 4375 7272 656e 7429  StrategyCurrent)
│ │ │ │ +00033450: 7c0a 7c20 2d2d 2075 7365 6420 302e 3134  |.| -- used 0.14
│ │ │ │ +00033460: 3639 3837 7320 2863 7075 293b 2030 2e30  6987s (cpu); 0.0
│ │ │ │ +00033470: 3833 3439 3273 2028 7468 7265 6164 293b  83492s (thread);
│ │ │ │ +00033480: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │ +00033490: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  000334a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000334b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000334c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -000334d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 3636  .| -- used 0.366
│ │ │ │ -00033560: 3034 3473 2028 6370 7529 3b20 302e 3237  044s (cpu); 0.27
│ │ │ │ -00033570: 3237 3233 7320 2874 6872 6561 6429 3b20  2723s (thread); 
│ │ │ │ -00033580: 3073 2028 6763 2920 2020 2020 2020 2020  0s (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              
│ │ │ │ +000334c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000334d0: 2020 2020 2020 7c0a 7c6f 3232 203d 2074        |.|o22 = t
│ │ │ │ +000334e0: 7275 6520 2020 2020 2020 2020 2020 2020  rue             
│ │ │ │ +000334f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033510: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00033520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00033560: 3233 203a 2074 696d 6520 7265 6775 6c61  23 : time regula
│ │ │ │ +00033570: 7249 6e43 6f64 696d 656e 7369 6f6e 2831  rInCodimension(1
│ │ │ │ +00033580: 2c20 532c 2053 7472 6174 6567 793d 3e53  , S, Strategy=>S
│ │ │ │ +00033590: 7472 6174 6567 7943 7572 7265 6e74 297c  trategyCurrent)|
│ │ │ │ +000335a0: 0a7c 202d 2d20 7573 6564 2030 2e34 3532  .| -- used 0.452
│ │ │ │ +000335b0: 3634 3173 2028 6370 7529 3b20 302e 3331  641s (cpu); 0.31
│ │ │ │ +000335c0: 3835 3633 7320 2874 6872 6561 6429 3b20  8563s (thread); 
│ │ │ │ +000335d0: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │ +000335e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000335f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033610: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -00033620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 3733 3338  | -- used 1.7338
│ │ │ │ -000336b0: 3173 2028 6370 7529 3b20 312e 3235 3132  1s (cpu); 1.2512
│ │ │ │ -000336c0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ -000336d0: 6763 2920 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               
│ │ │ │ +00033610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033620: 2020 2020 207c 0a7c 6f32 3320 3d20 7472       |.|o23 = tr
│ │ │ │ +00033630: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ +00033640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033660: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00033670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000336a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +000336b0: 3420 3a20 7469 6d65 2072 6567 756c 6172  4 : time regular
│ │ │ │ +000336c0: 496e 436f 6469 6d65 6e73 696f 6e28 312c  InCodimension(1,
│ │ │ │ +000336d0: 2053 2c20 5374 7261 7465 6779 3d3e 5374   S, Strategy=>St
│ │ │ │ +000336e0: 7261 7465 6779 4375 7272 656e 7429 7c0a  rategyCurrent)|.
│ │ │ │ +000336f0: 7c20 2d2d 2075 7365 6420 312e 3934 3534  | -- used 1.9454
│ │ │ │ +00033700: 3373 2028 6370 7529 3b20 312e 3430 3730  3s (cpu); 1.4070
│ │ │ │ +00033710: 3473 2028 7468 7265 6164 293b 2030 7320  4s (thread); 0s 
│ │ │ │ +00033720: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │ +00033730: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00033740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033760: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -00033770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000337a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3235  ----------+.|i25
│ │ │ │ -000337b0: 203a 2053 7472 6174 6567 7943 7572 7265   : StrategyCurre
│ │ │ │ -000337c0: 6e74 234c 6578 536d 616c 6c65 7374 203d  nt#LexSmallest =
│ │ │ │ -000337d0: 2030 3b20 2020 2020 2020 2020 2020 2020   0;             
│ │ │ │ -000337e0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -000337f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033830: 2b0a 7c69 3236 203a 2053 7472 6174 6567  +.|i26 : Strateg
│ │ │ │ -00033840: 7943 7572 7265 6e74 234c 6578 536d 616c  yCurrent#LexSmal
│ │ │ │ -00033850: 6c65 7374 5465 726d 203d 2031 3030 3b20  lestTerm = 100; 
│ │ │ │ -00033860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033870: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -00033880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 2e33 3239 3632 7320 2863  used 2.32962s (c
│ │ │ │ -00033910: 7075 293b 2031 2e36 3434 3538 7320 2874  pu); 1.64458s (t
│ │ │ │ -00033920: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 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: 3338 3734 3173 2028 6370 7529 3b20 312e  38741s (cpu); 1.
│ │ │ │ -000339e0: 3631 3339 3673 2028 7468 7265 6164 293b  61396s (thread);
│ │ │ │ -000339f0: 2030 7320 2867 6329 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           
│ │ │ │ +00033760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033770: 2020 2020 7c0a 7c6f 3234 203d 2074 7275      |.|o24 = tru
│ │ │ │ +00033780: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ +00033790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000337a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000337b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000337c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000337d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000337e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000337f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3235  ----------+.|i25
│ │ │ │ +00033800: 203a 2053 7472 6174 6567 7943 7572 7265   : StrategyCurre
│ │ │ │ +00033810: 6e74 234c 6578 536d 616c 6c65 7374 203d  nt#LexSmallest =
│ │ │ │ +00033820: 2030 3b20 2020 2020 2020 2020 2020 2020   0;             
│ │ │ │ +00033830: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00033840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033880: 2b0a 7c69 3236 203a 2053 7472 6174 6567  +.|i26 : Strateg
│ │ │ │ +00033890: 7943 7572 7265 6e74 234c 6578 536d 616c  yCurrent#LexSmal
│ │ │ │ +000338a0: 6c65 7374 5465 726d 203d 2031 3030 3b20  lestTerm = 100; 
│ │ │ │ +000338b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000338c0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +000338d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000338e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000338f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033900: 2d2d 2d2d 2d2d 2b0a 7c69 3237 203a 2074  ------+.|i27 : t
│ │ │ │ +00033910: 696d 6520 7265 6775 6c61 7249 6e43 6f64  ime regularInCod
│ │ │ │ +00033920: 696d 656e 7369 6f6e 2832 2c20 522c 2053  imension(2, R, S
│ │ │ │ +00033930: 7472 6174 6567 793d 3e53 7472 6174 6567  trategy=>Strateg
│ │ │ │ +00033940: 7943 7572 7265 6e74 297c 0a7c 202d 2d20  yCurrent)|.| -- 
│ │ │ │ +00033950: 7573 6564 2032 2e35 3832 3833 7320 2863  used 2.58283s (c
│ │ │ │ +00033960: 7075 293b 2031 2e38 3031 3973 2028 7468  pu); 1.8019s (th
│ │ │ │ +00033970: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │ +00033980: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00033990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000339a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000339b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000339c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +000339d0: 0a7c 6932 3820 3a20 7469 6d65 2072 6567  .|i28 : time reg
│ │ │ │ +000339e0: 756c 6172 496e 436f 6469 6d65 6e73 696f  ularInCodimensio
│ │ │ │ +000339f0: 6e28 322c 2052 2c20 5374 7261 7465 6779  n(2, R, Strategy
│ │ │ │ +00033a00: 3d3e 5374 7261 7465 6779 4375 7272 656e  =>StrategyCurren
│ │ │ │ +00033a10: 7429 7c0a 7c20 2d2d 2075 7365 6420 322e  t)|.| -- used 2.
│ │ │ │ +00033a20: 3732 3731 3773 2028 6370 7529 3b20 312e  72717s (cpu); 1.
│ │ │ │ +00033a30: 3831 3639 7320 2874 6872 6561 6429 3b20  8169s (thread); 
│ │ │ │ +00033a40: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │ +00033a50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00033a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033a80: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -00033a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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: 3632 3935 3673 2028 6370 7529 3b20 302e  62956s (cpu); 0.
│ │ │ │ -00033b30: 3337 3031 3633 7320 2874 6872 6561 6429  370163s (thread)
│ │ │ │ -00033b40: 3b20 3073 2028 6763 2920 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            
│ │ │ │ +00033a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033a90: 2020 2020 2020 2020 7c0a 7c6f 3238 203d          |.|o28 =
│ │ │ │ +00033aa0: 2074 7275 6520 2020 2020 2020 2020 2020   true           
│ │ │ │ +00033ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033ad0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00033ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00033b20: 7c69 3239 203a 2074 696d 6520 7265 6775  |i29 : time regu
│ │ │ │ +00033b30: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ +00033b40: 2831 2c20 532c 2053 7472 6174 6567 793d  (1, S, Strategy=
│ │ │ │ +00033b50: 3e53 7472 6174 6567 7943 7572 7265 6e74  >StrategyCurrent
│ │ │ │ +00033b60: 297c 0a7c 202d 2d20 7573 6564 2030 2e34  )|.| -- used 0.4
│ │ │ │ +00033b70: 3737 3232 3773 2028 6370 7529 3b20 302e  77227s (cpu); 0.
│ │ │ │ +00033b80: 3335 3636 3933 7320 2874 6872 6561 6429  356693s (thread)
│ │ │ │ +00033b90: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ +00033ba0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00033bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033bd0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -00033be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 3737  |.| -- used 0.77
│ │ │ │ -00033c70: 3032 3032 7320 2863 7075 293b 2030 2e36  0202s (cpu); 0.6
│ │ │ │ -00033c80: 3034 3237 3273 2028 7468 7265 6164 293b  04272s (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             
│ │ │ │ +00033bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033be0: 2020 2020 2020 207c 0a7c 6f32 3920 3d20         |.|o29 = 
│ │ │ │ +00033bf0: 7472 7565 2020 2020 2020 2020 2020 2020  true            
│ │ │ │ +00033c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033c20: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00033c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00033c70: 6933 3020 3a20 7469 6d65 2072 6567 756c  i30 : time regul
│ │ │ │ +00033c80: 6172 496e 436f 6469 6d65 6e73 696f 6e28  arInCodimension(
│ │ │ │ +00033c90: 312c 2053 2c20 5374 7261 7465 6779 3d3e  1, S, Strategy=>
│ │ │ │ +00033ca0: 5374 7261 7465 6779 4375 7272 656e 7429  StrategyCurrent)
│ │ │ │ +00033cb0: 7c0a 7c20 2d2d 2075 7365 6420 302e 3938  |.| -- used 0.98
│ │ │ │ +00033cc0: 3830 3831 7320 2863 7075 293b 2030 2e37  8081s (cpu); 0.7
│ │ │ │ +00033cd0: 3932 3636 3373 2028 7468 7265 6164 293b  92663s (thread);
│ │ │ │ +00033ce0: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │ +00033cf0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00033d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033d20: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -00033d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 2031 2e30 3738  .| -- used 1.078
│ │ │ │ -00033dc0: 3732 7320 2863 7075 293b 2030 2e38 3731  72s (cpu); 0.871
│ │ │ │ -00033dd0: 3233 3573 2028 7468 7265 6164 293b 2030  235s (thread); 0
│ │ │ │ -00033de0: 7320 2867 6329 2020 2020 2020 2020 2020  s (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              
│ │ │ │ +00033d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033d30: 2020 2020 2020 7c0a 7c6f 3330 203d 2074        |.|o30 = t
│ │ │ │ +00033d40: 7275 6520 2020 2020 2020 2020 2020 2020  rue             
│ │ │ │ +00033d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033d70: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00033d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00033dc0: 3331 203a 2074 696d 6520 7265 6775 6c61  31 : time regula
│ │ │ │ +00033dd0: 7249 6e43 6f64 696d 656e 7369 6f6e 2831  rInCodimension(1
│ │ │ │ +00033de0: 2c20 532c 2053 7472 6174 6567 793d 3e53  , S, Strategy=>S
│ │ │ │ +00033df0: 7472 6174 6567 7952 616e 646f 6d29 207c  trategyRandom) |
│ │ │ │ +00033e00: 0a7c 202d 2d20 7573 6564 2031 2e32 3234  .| -- used 1.224
│ │ │ │ +00033e10: 3634 7320 2863 7075 293b 2030 2e39 3637  64s (cpu); 0.967
│ │ │ │ +00033e20: 3739 3473 2028 7468 7265 6164 293b 2030  794s (thread); 0
│ │ │ │ +00033e30: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │ +00033e40: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00033e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033e70: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -00033e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 3835 3933  | -- used 1.8593
│ │ │ │ -00033f10: 3673 2028 6370 7529 3b20 312e 3438 3037  6s (cpu); 1.4807
│ │ │ │ -00033f20: 3373 2028 7468 7265 6164 293b 2030 7320  3s (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               
│ │ │ │ +00033e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033e80: 2020 2020 207c 0a7c 6f33 3120 3d20 7472       |.|o31 = tr
│ │ │ │ +00033e90: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ +00033ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033ec0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00033ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │ +00033f10: 3220 3a20 7469 6d65 2072 6567 756c 6172  2 : time regular
│ │ │ │ +00033f20: 496e 436f 6469 6d65 6e73 696f 6e28 312c  InCodimension(1,
│ │ │ │ +00033f30: 2053 2c20 5374 7261 7465 6779 3d3e 5374   S, Strategy=>St
│ │ │ │ +00033f40: 7261 7465 6779 5261 6e64 6f6d 2920 7c0a  rategyRandom) |.
│ │ │ │ +00033f50: 7c20 2d2d 2075 7365 6420 312e 3938 3036  | -- used 1.9806
│ │ │ │ +00033f60: 3373 2028 6370 7529 3b20 312e 3439 3932  3s (cpu); 1.4992
│ │ │ │ +00033f70: 3473 2028 7468 7265 6164 293b 2030 7320  4s (thread); 0s 
│ │ │ │ +00033f80: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │ +00033f90: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00033fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033fc0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -00033fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034000: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a54 6865  ----------+..The
│ │ │ │ -00034010: 206d 696e 696d 756d 206e 756d 6265 7220   minimum number 
│ │ │ │ -00034020: 6f66 206d 696e 6f72 7320 636f 6d70 7574  of minors comput
│ │ │ │ -00034030: 6564 2062 6566 6f72 6520 6368 6563 6b69  ed before checki
│ │ │ │ -00034040: 6e67 2074 6865 2063 6f64 696d 656e 7369  ng the codimensi
│ │ │ │ -00034050: 6f6e 2063 616e 2061 6c73 6f0a 6265 2063  on can also.be c
│ │ │ │ -00034060: 6f6e 7472 6f6c 6c65 6420 6279 2061 6e20  ontrolled by an 
│ │ │ │ -00034070: 6f70 7469 6f6e 204d 696e 4d69 6e6f 7273  option MinMinors
│ │ │ │ -00034080: 4675 6e63 7469 6f6e 2e20 2054 6869 7320  Function.  This 
│ │ │ │ -00034090: 6973 2073 686f 756c 6420 6265 2061 2066  is should be a f
│ │ │ │ -000340a0: 756e 6374 696f 6e20 6f66 0a61 2073 696e  unction of.a sin
│ │ │ │ -000340b0: 676c 6520 7661 7269 6162 6c65 2c20 7468  gle variable, th
│ │ │ │ -000340c0: 6520 6e75 6d62 6572 206f 6620 6d69 6e6f  e number of mino
│ │ │ │ -000340d0: 7273 2063 6f6d 7075 7465 642e 2020 4669  rs computed.  Fi
│ │ │ │ -000340e0: 6e61 6c6c 792c 2076 6961 2074 6865 206f  nally, via the o
│ │ │ │ -000340f0: 7074 696f 6e0a 436f 6469 6d43 6865 636b  ption.CodimCheck
│ │ │ │ -00034100: 4675 6e63 7469 6f6e 2c20 796f 7520 6361  Function, you ca
│ │ │ │ -00034110: 6e20 7061 7373 2074 6865 2072 6567 756c  n pass the regul
│ │ │ │ -00034120: 6172 496e 436f 6469 6d65 6e73 696f 6e20  arInCodimension 
│ │ │ │ -00034130: 6120 6675 6e63 7469 6f6e 2077 6869 6368  a function which
│ │ │ │ -00034140: 0a63 6f6e 7472 6f6c 7320 686f 7720 6672  .controls how fr
│ │ │ │ -00034150: 6571 7565 6e74 6c79 2074 6865 2063 6f64  equently the cod
│ │ │ │ -00034160: 696d 656e 7369 6f6e 206f 6620 7468 6520  imension of the 
│ │ │ │ -00034170: 7061 7274 6961 6c20 4a61 636f 6269 616e  partial Jacobian
│ │ │ │ -00034180: 2069 6465 616c 2069 730a 636f 6d70 7574   ideal is.comput
│ │ │ │ -00034190: 6564 2e20 2042 7920 6465 6661 756c 7420  ed.  By default 
│ │ │ │ -000341a0: 7468 6973 2069 7320 7468 6520 666c 6f6f  this is the floo
│ │ │ │ -000341b0: 7220 6f66 2031 2e33 5e6b 2e20 4669 6e61  r of 1.3^k. Fina
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│ │ │ │ -000341e0: 3d3e 2070 2077 696c 6c20 646f 2074 6865  => p will do the
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│ │ │ │ -00034270: 696d 4174 4c65 6173 742e 0a59 6f75 2063  imAtLeast..You c
│ │ │ │ -00034280: 616e 2074 7572 6e20 6f66 6620 696e 7465  an turn off inte
│ │ │ │ -00034290: 726e 616c 2063 616c 6c73 2074 6f20 636f  rnal calls to co
│ │ │ │ -000342a0: 6469 6d2f 6469 6d2c 2061 6e64 206f 6e6c  dim/dim, and onl
│ │ │ │ -000342b0: 7920 7573 6520 6973 436f 6469 6d41 744c  y use isCodimAtL
│ │ │ │ -000342c0: 6561 7374 2062 790a 7365 7474 696e 6720  east by.setting 
│ │ │ │ -000342d0: 5573 654f 6e6c 7946 6173 7443 6f64 696d  UseOnlyFastCodim
│ │ │ │ -000342e0: 203d 3e20 7472 7565 2e0a 0a53 6565 2061   => true...See a
│ │ │ │ -000342f0: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  lso.========..  
│ │ │ │ -00034300: 2a20 2a6e 6f74 6520 6973 436f 6469 6d41  * *note isCodimA
│ │ │ │ -00034310: 744c 6561 7374 3a20 6973 436f 6469 6d41  tLeast: isCodimA
│ │ │ │ -00034320: 744c 6561 7374 2c20 2d2d 2072 6574 7572  tLeast, -- retur
│ │ │ │ -00034330: 6e73 2074 7275 6520 6966 2077 6520 6361  ns true if we ca
│ │ │ │ -00034340: 6e20 7175 6963 6b6c 7920 7365 650a 2020  n quickly see.  
│ │ │ │ -00034350: 2020 7768 6574 6865 7220 7468 6520 636f    whether the co
│ │ │ │ -00034360: 6469 6d20 6973 2061 7420 6c65 6173 7420  dim is at least 
│ │ │ │ -00034370: 6120 6769 7665 6e20 6e75 6d62 6572 0a0a  a given number..
│ │ │ │ -00034380: 5761 7973 2074 6f20 7573 6520 7265 6775  Ways to use regu
│ │ │ │ -00034390: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ -000343a0: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ -000343b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000343c0: 3d3d 3d0a 0a20 202a 2022 7265 6775 6c61  ===..  * "regula
│ │ │ │ -000343d0: 7249 6e43 6f64 696d 656e 7369 6f6e 285a  rInCodimension(Z
│ │ │ │ -000343e0: 5a2c 5269 6e67 2922 0a0a 466f 7220 7468  Z,Ring)"..For th
│ │ │ │ -000343f0: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ -00034400: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -00034410: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ -00034420: 6520 7265 6775 6c61 7249 6e43 6f64 696d  e regularInCodim
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│ │ │ │ +000346e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000346f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00034700: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00034710: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00034720: 2a2a 2a2a 2a2a 0a0a 4465 7363 7269 7074  ******..Descript
│ │ │ │ +00034730: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ion.===========.
│ │ │ │ +00034740: 0a49 6e20 7468 6973 2074 7574 6f72 6961  .In this tutoria
│ │ │ │ +00034750: 6c20 7765 2065 7870 6c6f 7265 2074 6865  l we explore the
│ │ │ │ +00034760: 2064 6966 6665 7265 6e74 206f 7074 696f   different optio
│ │ │ │ +00034770: 6e73 206f 6620 5265 6775 6c61 7249 6e43  ns of RegularInC
│ │ │ │ +00034780: 6f64 696d 656e 7369 6f6e 2028 616e 640a  odimension (and.
│ │ │ │ +00034790: 7265 6c61 7465 6420 6675 6e63 7469 6f6e  related function
│ │ │ │ +000347a0: 7329 206f 6e20 736f 6d65 2063 6f6e 6520  s) on some cone 
│ │ │ │ +000347b0: 7369 6e67 756c 6172 6974 6965 732e 2020  singularities.  
│ │ │ │ +000347c0: 466f 7220 7468 6520 6d6f 7374 2070 6172  For the most par
│ │ │ │ +000347d0: 7420 7765 2077 696c 6c20 6e6f 740a 7461  t we will not.ta
│ │ │ │ +000347e0: 6c6b 2061 626f 7574 2074 6865 2053 7472  lk about the Str
│ │ │ │ +000347f0: 6174 6567 7920 6f70 7469 6f6e 2c20 7765  ategy option, we
│ │ │ │ +00034800: 2068 6176 6520 6120 7365 7061 7261 7465   have a separate
│ │ │ │ +00034810: 2074 7574 6f72 6961 6c20 666f 7220 7468   tutorial for th
│ │ │ │ +00034820: 6174 202a 6e6f 7465 0a46 6173 744d 696e  at *note.FastMin
│ │ │ │ +00034830: 6f72 7353 7472 6174 6567 7954 7574 6f72  orsStrategyTutor
│ │ │ │ +00034840: 6961 6c3a 2046 6173 744d 696e 6f72 7353  ial: FastMinorsS
│ │ │ │ +00034850: 7472 6174 6567 7954 7574 6f72 6961 6c2c  trategyTutorial,
│ │ │ │ +00034860: 2e0a 0a57 6520 6265 6769 6e20 7769 7468  ...We begin with
│ │ │ │ +00034870: 2074 6865 2066 6f6c 6c6f 7769 6e67 2069   the following i
│ │ │ │ +00034880: 6465 616c 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  deal...+--------
│ │ │ │ +00034890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000348a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000348b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000348c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000348d0: 2d2d 2d2d 2d2b 0a7c 6931 203a 2053 203d  -----+.|i1 : S =
│ │ │ │ +000348e0: 205a 5a2f 3130 335b 785f 312e 2e78 5f39   ZZ/103[x_1..x_9
│ │ │ │ +000348f0: 5d3b 2020 2020 2020 2020 2020 2020 2020  ];              
│ │ │ │ +00034900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034920: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00034930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034970: 2d2d 2d2d 2d2b 0a7c 6932 203a 204a 203d  -----+.|i2 : J =
│ │ │ │ +00034980: 2069 6465 616c 2878 5f36 2a78 5f38 2d78   ideal(x_6*x_8-x
│ │ │ │ +00034990: 5f35 2a78 5f39 2c78 5f33 2a78 5f38 2d78  _5*x_9,x_3*x_8-x
│ │ │ │ +000349a0: 5f32 2a78 5f39 2c78 5f36 2a78 5f37 2d78  _2*x_9,x_6*x_7-x
│ │ │ │ +000349b0: 5f34 2a78 5f39 2c78 5f35 2a78 5f37 2d78  _4*x_9,x_5*x_7-x
│ │ │ │ +000349c0: 5f34 2a78 5f7c 0a7c 2020 2020 2020 2020  _4*x_|.|        
│ │ │ │ +000349d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000349e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000349f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034a10: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ -00034a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034a60: 2d2d 2d2d 2d7c 0a7c 382c 785f 332a 785f  -----|.|8,x_3*x_
│ │ │ │ -00034a70: 372d 785f 312a 785f 392c 785f 322a 785f  7-x_1*x_9,x_2*x_
│ │ │ │ -00034a80: 372d 785f 312a 785f 382c 785f 332a 785f  7-x_1*x_8,x_3*x_
│ │ │ │ -00034a90: 352d 785f 322a 785f 362c 785f 332a 785f  5-x_2*x_6,x_3*x_
│ │ │ │ -00034aa0: 342d 785f 312a 785f 362c 785f 322a 785f  4-x_1*x_6,x_2*x_
│ │ │ │ -00034ab0: 342d 785f 317c 0a7c 2d2d 2d2d 2d2d 2d2d  4-x_1|.|--------
│ │ │ │ -00034ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034b00: 2d2d 2d2d 2d7c 0a7c 2a78 5f35 2c78 5f33  -----|.|*x_5,x_3
│ │ │ │ -00034b10: 5e33 2d78 5f36 5e33 2d78 5f39 5e33 2c78  ^3-x_6^3-x_9^3,x
│ │ │ │ -00034b20: 5f32 2a78 5f33 5e32 2d78 5f35 2a78 5f36  _2*x_3^2-x_5*x_6
│ │ │ │ -00034b30: 5e32 2d78 5f38 2a78 5f39 5e32 2c78 5f31  ^2-x_8*x_9^2,x_1
│ │ │ │ -00034b40: 2a78 5f33 5e32 2d78 5f34 2a78 5f36 5e32  *x_3^2-x_4*x_6^2
│ │ │ │ -00034b50: 2d78 5f37 2a7c 0a7c 2d2d 2d2d 2d2d 2d2d  -x_7*|.|--------
│ │ │ │ -00034b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034ba0: 2d2d 2d2d 2d7c 0a7c 785f 395e 322c 785f  -----|.|x_9^2,x_
│ │ │ │ -00034bb0: 325e 322a 785f 332d 785f 355e 322a 785f  2^2*x_3-x_5^2*x_
│ │ │ │ -00034bc0: 362d 785f 385e 322a 785f 392c 785f 312a  6-x_8^2*x_9,x_1*
│ │ │ │ -00034bd0: 785f 322a 785f 332d 785f 342a 785f 352a  x_2*x_3-x_4*x_5*
│ │ │ │ -00034be0: 785f 362d 785f 372a 785f 382a 785f 392c  x_6-x_7*x_8*x_9,
│ │ │ │ -00034bf0: 785f 315e 327c 0a7c 2d2d 2d2d 2d2d 2d2d  x_1^2|.|--------
│ │ │ │ -00034c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034c40: 2d2d 2d2d 2d7c 0a7c 2a78 5f33 2d78 5f34  -----|.|*x_3-x_4
│ │ │ │ -00034c50: 5e32 2a78 5f36 2d78 5f37 5e32 2a78 5f39  ^2*x_6-x_7^2*x_9
│ │ │ │ -00034c60: 2c78 5f32 5e33 2d78 5f35 5e33 2d78 5f38  ,x_2^3-x_5^3-x_8
│ │ │ │ -00034c70: 5e33 2c78 5f31 2a78 5f32 5e32 2d78 5f34  ^3,x_1*x_2^2-x_4
│ │ │ │ -00034c80: 2a78 5f35 5e32 2d78 5f37 2a78 5f38 5e32  *x_5^2-x_7*x_8^2
│ │ │ │ -00034c90: 2c78 5f31 5e7c 0a7c 2d2d 2d2d 2d2d 2d2d  ,x_1^|.|--------
│ │ │ │ -00034ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034ce0: 2d2d 2d2d 2d7c 0a7c 322a 785f 322d 785f  -----|.|2*x_2-x_
│ │ │ │ -00034cf0: 345e 322a 785f 352d 785f 375e 322a 785f  4^2*x_5-x_7^2*x_
│ │ │ │ -00034d00: 382c 785f 315e 332d 785f 345e 332d 785f  8,x_1^3-x_4^3-x_
│ │ │ │ -00034d10: 375e 3329 3b20 2020 2020 2020 2020 2020  7^3);           
│ │ │ │ -00034d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034d30: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00034d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034d80: 2d2d 2d2d 2d2b 0a7c 6933 203a 2064 696d  -----+.|i3 : dim
│ │ │ │ -00034d90: 2028 532f 4a29 2020 2020 2020 2020 2020   (S/J)          
│ │ │ │ -00034da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034dd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00034de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034a10: 2020 2020 207c 0a7c 6f32 203a 2049 6465       |.|o2 : Ide
│ │ │ │ +00034a20: 616c 206f 6620 5320 2020 2020 2020 2020  al of S         
│ │ │ │ +00034a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034a60: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +00034a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034ab0: 2d2d 2d2d 2d7c 0a7c 382c 785f 332a 785f  -----|.|8,x_3*x_
│ │ │ │ +00034ac0: 372d 785f 312a 785f 392c 785f 322a 785f  7-x_1*x_9,x_2*x_
│ │ │ │ +00034ad0: 372d 785f 312a 785f 382c 785f 332a 785f  7-x_1*x_8,x_3*x_
│ │ │ │ +00034ae0: 352d 785f 322a 785f 362c 785f 332a 785f  5-x_2*x_6,x_3*x_
│ │ │ │ +00034af0: 342d 785f 312a 785f 362c 785f 322a 785f  4-x_1*x_6,x_2*x_
│ │ │ │ +00034b00: 342d 785f 317c 0a7c 2d2d 2d2d 2d2d 2d2d  4-x_1|.|--------
│ │ │ │ +00034b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034b50: 2d2d 2d2d 2d7c 0a7c 2a78 5f35 2c78 5f33  -----|.|*x_5,x_3
│ │ │ │ +00034b60: 5e33 2d78 5f36 5e33 2d78 5f39 5e33 2c78  ^3-x_6^3-x_9^3,x
│ │ │ │ +00034b70: 5f32 2a78 5f33 5e32 2d78 5f35 2a78 5f36  _2*x_3^2-x_5*x_6
│ │ │ │ +00034b80: 5e32 2d78 5f38 2a78 5f39 5e32 2c78 5f31  ^2-x_8*x_9^2,x_1
│ │ │ │ +00034b90: 2a78 5f33 5e32 2d78 5f34 2a78 5f36 5e32  *x_3^2-x_4*x_6^2
│ │ │ │ +00034ba0: 2d78 5f37 2a7c 0a7c 2d2d 2d2d 2d2d 2d2d  -x_7*|.|--------
│ │ │ │ +00034bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034bf0: 2d2d 2d2d 2d7c 0a7c 785f 395e 322c 785f  -----|.|x_9^2,x_
│ │ │ │ +00034c00: 325e 322a 785f 332d 785f 355e 322a 785f  2^2*x_3-x_5^2*x_
│ │ │ │ +00034c10: 362d 785f 385e 322a 785f 392c 785f 312a  6-x_8^2*x_9,x_1*
│ │ │ │ +00034c20: 785f 322a 785f 332d 785f 342a 785f 352a  x_2*x_3-x_4*x_5*
│ │ │ │ +00034c30: 785f 362d 785f 372a 785f 382a 785f 392c  x_6-x_7*x_8*x_9,
│ │ │ │ +00034c40: 785f 315e 327c 0a7c 2d2d 2d2d 2d2d 2d2d  x_1^2|.|--------
│ │ │ │ +00034c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034c90: 2d2d 2d2d 2d7c 0a7c 2a78 5f33 2d78 5f34  -----|.|*x_3-x_4
│ │ │ │ +00034ca0: 5e32 2a78 5f36 2d78 5f37 5e32 2a78 5f39  ^2*x_6-x_7^2*x_9
│ │ │ │ +00034cb0: 2c78 5f32 5e33 2d78 5f35 5e33 2d78 5f38  ,x_2^3-x_5^3-x_8
│ │ │ │ +00034cc0: 5e33 2c78 5f31 2a78 5f32 5e32 2d78 5f34  ^3,x_1*x_2^2-x_4
│ │ │ │ +00034cd0: 2a78 5f35 5e32 2d78 5f37 2a78 5f38 5e32  *x_5^2-x_7*x_8^2
│ │ │ │ +00034ce0: 2c78 5f31 5e7c 0a7c 2d2d 2d2d 2d2d 2d2d  ,x_1^|.|--------
│ │ │ │ +00034cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034d30: 2d2d 2d2d 2d7c 0a7c 322a 785f 322d 785f  -----|.|2*x_2-x_
│ │ │ │ +00034d40: 345e 322a 785f 352d 785f 375e 322a 785f  4^2*x_5-x_7^2*x_
│ │ │ │ +00034d50: 382c 785f 315e 332d 785f 345e 332d 785f  8,x_1^3-x_4^3-x_
│ │ │ │ +00034d60: 375e 3329 3b20 2020 2020 2020 2020 2020  7^3);           
│ │ │ │ +00034d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034d80: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00034d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034dd0: 2d2d 2d2d 2d2b 0a7c 6933 203a 2064 696d  -----+.|i3 : dim
│ │ │ │ +00034de0: 2028 532f 4a29 2020 2020 2020 2020 2020   (S/J)          
│ │ │ │  00034df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034e20: 2020 2020 207c 0a7c 6f33 203d 2034 2020       |.|o3 = 4  
│ │ │ │ +00034e20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00034e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034e70: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00034e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00034fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00034fe0: 532f 4a29 2020 2020 2020 2020 2020 2020  S/J)            
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│ │ │ │ +00034e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034ec0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00034ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034f10: 2d2d 2d2d 2d2b 0a0a 4974 2069 7320 7468  -----+..It is th
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│ │ │ │ +00034ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  00035080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035090: 2020 2020 2020 2020 2020 207c 0a2b 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  ----------------
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│ │ │ │ -00035150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00035160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -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  ----------------
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│ │ │ │ +00035190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000351a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000351b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +000354b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000354c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000354d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000354e0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │ +000354f0: 7469 6d65 2072 6567 756c 6172 496e 436f  time regularInCo
│ │ │ │ +00035500: 6469 6d65 6e73 696f 6e28 312c 2053 2f4a  dimension(1, S/J
│ │ │ │ +00035510: 2c20 5665 7262 6f73 653d 3e74 7275 6529  , Verbose=>true)
│ │ │ │ +00035520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035530: 2020 2020 2020 2020 7c0a 7c72 6567 756c          |.|regul
│ │ │ │ +00035540: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00035550: 2072 696e 6720 6469 6d65 6e73 696f 6e20   ring dimension 
│ │ │ │ +00035560: 3d34 2c20 7468 6572 6520 6172 6520 3134  =4, there are 14
│ │ │ │ +00035570: 3635 3132 3820 706f 7373 6962 6c65 2035  65128 possible 5
│ │ │ │ +00035580: 2062 7920 3520 6d69 7c0a 7c72 6567 756c   by 5 mi|.|regul
│ │ │ │ +00035590: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +000355a0: 2041 626f 7574 2074 6f20 656e 7465 7220   About to enter 
│ │ │ │ +000355b0: 6c6f 6f70 2020 2020 2020 2020 2020 2020  loop            
│ │ │ │ +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 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +00035600: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │  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 2052 616e 646f 6d20  Choosing Random 
│ │ │ │ +00035650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +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 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +000356a0: 536d 616c 6c65 7374 2020 2020 2020 2020  Smallest        
│ │ │ │  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 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +00035740: 6c65 7374 2020 2020 2020 2020 2020 2020  lest            
│ │ │ │ +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 5465 726d 2020 2020  SmallestTerm    
│ │ │ │ +000357a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000357b0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +000357c0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +000357d0: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +000357e0: 536d 616c 6c65 7374 2020 2020 2020 2020  Smallest        
│ │ │ │ +000357f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035800: 2020 2020 2020 2020 7c0a 7c72 6567 756c          |.|regul
│ │ │ │ +00035810: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00035820: 2020 4c6f 6f70 2073 7465 702c 2061 626f    Loop step, abo
│ │ │ │ +00035830: 7574 2074 6f20 636f 6d70 7574 6520 6469  ut to compute di
│ │ │ │ +00035840: 6d65 6e73 696f 6e2e 2020 5375 626d 6174  mension.  Submat
│ │ │ │ +00035850: 7269 6365 7320 636f 7c0a 7c72 6567 756c  rices co|.|regul
│ │ │ │ +00035860: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00035870: 2020 6973 436f 6469 6d41 744c 6561 7374    isCodimAtLeast
│ │ │ │ +00035880: 2066 6169 6c65 642c 2063 6f6d 7075 7469   failed, computi
│ │ │ │ +00035890: 6e67 2063 6f64 696d 2e20 2020 2020 2020  ng codim.       
│ │ │ │ +000358a0: 2020 2020 2020 2020 7c0a 7c72 6567 756c          |.|regul
│ │ │ │ +000358b0: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +000358c0: 2020 7061 7274 6961 6c20 7369 6e67 756c    partial singul
│ │ │ │ +000358d0: 6172 206c 6f63 7573 2064 696d 656e 7369  ar locus dimensi
│ │ │ │ +000358e0: 6f6e 2063 6f6d 7075 7465 642c 203d 2033  on computed, = 3
│ │ │ │ +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 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │  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 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +00035970: 6c65 7374 2020 2020 2020 2020 2020 2020  lest            
│ │ │ │  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   
│ │ │ │ -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        
│ │ │ │ +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 2052 616e 646f 6d4e  Choosing RandomN
│ │ │ │ +000359c0: 6f6e 5a65 726f 2020 2020 2020 2020 2020  onZero          
│ │ │ │ +000359d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000359e0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +000359f0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00035a00: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +00035a10: 536d 616c 6c65 7374 2020 2020 2020 2020  Smallest        
│ │ │ │ +00035a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035a30: 2020 2020 2020 2020 7c0a 7c72 6567 756c          |.|regul
│ │ │ │ +00035a40: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00035a50: 2020 4c6f 6f70 2073 7465 702c 2061 626f    Loop step, abo
│ │ │ │ +00035a60: 7574 2074 6f20 636f 6d70 7574 6520 6469  ut to compute di
│ │ │ │ +00035a70: 6d65 6e73 696f 6e2e 2020 5375 626d 6174  mension.  Submat
│ │ │ │ +00035a80: 7269 6365 7320 636f 7c0a 7c72 6567 756c  rices co|.|regul
│ │ │ │ +00035a90: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00035aa0: 2020 6973 436f 6469 6d41 744c 6561 7374    isCodimAtLeast
│ │ │ │ +00035ab0: 2066 6169 6c65 642c 2063 6f6d 7075 7469   failed, computi
│ │ │ │ +00035ac0: 6e67 2063 6f64 696d 2e20 2020 2020 2020  ng codim.       
│ │ │ │ +00035ad0: 2020 2020 2020 2020 7c0a 7c72 6567 756c          |.|regul
│ │ │ │ +00035ae0: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00035af0: 2020 7061 7274 6961 6c20 7369 6e67 756c    partial singul
│ │ │ │ +00035b00: 6172 206c 6f63 7573 2064 696d 656e 7369  ar locus dimensi
│ │ │ │ +00035b10: 6f6e 2063 6f6d 7075 7465 642c 203d 2033  on computed, = 3
│ │ │ │ +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 2020 2020 2020 2020 2020 2020  lest            
│ │ │ │ +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 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +00035ba0: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │  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 
│ │ │ │ -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 
│ │ │ │ +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 2052 616e 646f 6d20  Choosing Random 
│ │ │ │ +00035bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035c10: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00035c20: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00035c30: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +00035c40: 536d 616c 6c65 7374 5465 726d 2020 2020  SmallestTerm    
│ │ │ │ +00035c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035c60: 2020 2020 2020 2020 7c0a 7c72 6567 756c          |.|regul
│ │ │ │ +00035c70: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00035c80: 2020 4c6f 6f70 2073 7465 702c 2061 626f    Loop step, abo
│ │ │ │ +00035c90: 7574 2074 6f20 636f 6d70 7574 6520 6469  ut to compute di
│ │ │ │ +00035ca0: 6d65 6e73 696f 6e2e 2020 5375 626d 6174  mension.  Submat
│ │ │ │ +00035cb0: 7269 6365 7320 636f 7c0a 7c72 6567 756c  rices co|.|regul
│ │ │ │ +00035cc0: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00035cd0: 2020 6973 436f 6469 6d41 744c 6561 7374    isCodimAtLeast
│ │ │ │ +00035ce0: 2066 6169 6c65 642c 2063 6f6d 7075 7469   failed, computi
│ │ │ │ +00035cf0: 6e67 2063 6f64 696d 2e20 2020 2020 2020  ng codim.       
│ │ │ │ +00035d00: 2020 2020 2020 2020 7c0a 7c72 6567 756c          |.|regul
│ │ │ │ +00035d10: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00035d20: 2020 7061 7274 6961 6c20 7369 6e67 756c    partial singul
│ │ │ │ +00035d30: 6172 206c 6f63 7573 2064 696d 656e 7369  ar locus dimensi
│ │ │ │ +00035d40: 6f6e 2063 6f6d 7075 7465 642c 203d 2033  on computed, = 3
│ │ │ │ +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 2052 616e 646f 6d4e  Choosing RandomN
│ │ │ │ +00035e20: 6f6e 5a65 726f 2020 2020 2020 2020 2020  onZero          
│ │ │ │ +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 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +00035e70: 536d 616c 6c65 7374 2020 2020 2020 2020  Smallest        
│ │ │ │ +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 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +00035ec0: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │  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 
│ │ │ │ -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        
│ │ │ │ +00035ee0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00035ef0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00035f00: 4368 6f6f 7369 6e67 2052 616e 646f 6d20  Choosing Random 
│ │ │ │ +00035f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035f30: 2020 2020 2020 2020 7c0a 7c72 6567 756c          |.|regul
│ │ │ │ +00035f40: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00035f50: 2020 4c6f 6f70 2073 7465 702c 2061 626f    Loop step, abo
│ │ │ │ +00035f60: 7574 2074 6f20 636f 6d70 7574 6520 6469  ut to compute di
│ │ │ │ +00035f70: 6d65 6e73 696f 6e2e 2020 5375 626d 6174  mension.  Submat
│ │ │ │ +00035f80: 7269 6365 7320 636f 7c0a 7c72 6567 756c  rices co|.|regul
│ │ │ │ +00035f90: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00035fa0: 2020 6973 436f 6469 6d41 744c 6561 7374    isCodimAtLeast
│ │ │ │ +00035fb0: 2066 6169 6c65 642c 2063 6f6d 7075 7469   failed, computi
│ │ │ │ +00035fc0: 6e67 2063 6f64 696d 2e20 2020 2020 2020  ng codim.       
│ │ │ │ +00035fd0: 2020 2020 2020 2020 7c0a 7c72 6567 756c          |.|regul
│ │ │ │ +00035fe0: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00035ff0: 2020 7061 7274 6961 6c20 7369 6e67 756c    partial singul
│ │ │ │ +00036000: 6172 206c 6f63 7573 2064 696d 656e 7369  ar locus dimensi
│ │ │ │ +00036010: 6f6e 2063 6f6d 7075 7465 642c 203d 2033  on computed, = 3
│ │ │ │ +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 2052 616e 646f 6d4e  Choosing RandomN
│ │ │ │ +00036050: 6f6e 5a65 726f 2020 2020 2020 2020 2020  onZero          
│ │ │ │ +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 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +000360a0: 536d 616c 6c65 7374 5465 726d 2020 2020  SmallestTerm    
│ │ │ │  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 2052 616e 646f 6d20  Choosing Random 
│ │ │ │ +000360f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +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 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +00036140: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │ +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 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +00036190: 536d 616c 6c65 7374 2020 2020 2020 2020  Smallest        
│ │ │ │  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 2020 2020 2020 2020 2020 2020  lest            
│ │ │ │ +000361f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036200: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00036210: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00036220: 4368 6f6f 7369 6e67 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +00036230: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │ +00036240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036250: 2020 2020 2020 2020 7c0a 7c72 6567 756c          |.|regul
│ │ │ │ +00036260: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00036270: 2020 4c6f 6f70 2073 7465 702c 2061 626f    Loop step, abo
│ │ │ │ +00036280: 7574 2074 6f20 636f 6d70 7574 6520 6469  ut to compute di
│ │ │ │ +00036290: 6d65 6e73 696f 6e2e 2020 5375 626d 6174  mension.  Submat
│ │ │ │ +000362a0: 7269 6365 7320 636f 7c0a 7c72 6567 756c  rices co|.|regul
│ │ │ │ +000362b0: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +000362c0: 2020 6973 436f 6469 6d41 744c 6561 7374    isCodimAtLeast
│ │ │ │ +000362d0: 2066 6169 6c65 642c 2063 6f6d 7075 7469   failed, computi
│ │ │ │ +000362e0: 6e67 2063 6f64 696d 2e20 2020 2020 2020  ng codim.       
│ │ │ │ +000362f0: 2020 2020 2020 2020 7c0a 7c72 6567 756c          |.|regul
│ │ │ │ +00036300: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00036310: 2020 7061 7274 6961 6c20 7369 6e67 756c    partial singul
│ │ │ │ +00036320: 6172 206c 6f63 7573 2064 696d 656e 7369  ar locus dimensi
│ │ │ │ +00036330: 6f6e 2063 6f6d 7075 7465 642c 203d 2033  on computed, = 3
│ │ │ │ +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 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +00036370: 536d 616c 6c65 7374 2020 2020 2020 2020  Smallest        
│ │ │ │  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 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +000363c0: 6c65 7374 2020 2020 2020 2020 2020 2020  lest            
│ │ │ │  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 6d4e  Choosing RandomN
│ │ │ │ +00036410: 6f6e 5a65 726f 2020 2020 2020 2020 2020  onZero          
│ │ │ │  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 6d20  Choosing Random 
│ │ │ │ +00036460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 2052 616e 646f 6d4e  Choosing RandomN
│ │ │ │ +000364b0: 6f6e 5a65 726f 2020 2020 2020 2020 2020  onZero          
│ │ │ │ +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 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │  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   
│ │ │ │ -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        
│ │ │ │ +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 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +00036550: 6c65 7374 2020 2020 2020 2020 2020 2020  lest            
│ │ │ │ +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 7c69 6e74 6572          |.|inter
│ │ │ │ +000365d0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +000365e0: 4368 6f6f 7369 6e67 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +000365f0: 536d 616c 6c65 7374 5465 726d 2020 2020  SmallestTerm    
│ │ │ │ +00036600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036610: 2020 2020 2020 2020 7c0a 7c72 6567 756c          |.|regul
│ │ │ │ +00036620: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00036630: 2020 4c6f 6f70 2073 7465 702c 2061 626f    Loop step, abo
│ │ │ │ +00036640: 7574 2074 6f20 636f 6d70 7574 6520 6469  ut to compute di
│ │ │ │ +00036650: 6d65 6e73 696f 6e2e 2020 5375 626d 6174  mension.  Submat
│ │ │ │ +00036660: 7269 6365 7320 636f 7c0a 7c72 6567 756c  rices co|.|regul
│ │ │ │ +00036670: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00036680: 2020 6973 436f 6469 6d41 744c 6561 7374    isCodimAtLeast
│ │ │ │ +00036690: 2066 6169 6c65 642c 2063 6f6d 7075 7469   failed, computi
│ │ │ │ +000366a0: 6e67 2063 6f64 696d 2e20 2020 2020 2020  ng codim.       
│ │ │ │ +000366b0: 2020 2020 2020 2020 7c0a 7c72 6567 756c          |.|regul
│ │ │ │ +000366c0: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +000366d0: 2020 7061 7274 6961 6c20 7369 6e67 756c    partial singul
│ │ │ │ +000366e0: 6172 206c 6f63 7573 2064 696d 656e 7369  ar locus dimensi
│ │ │ │ +000366f0: 6f6e 2063 6f6d 7075 7465 642c 203d 2033  on computed, = 3
│ │ │ │ +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 2020 2020 2020 2020 2020 2020  lest            
│ │ │ │ +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 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +000367d0: 6c65 7374 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │  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 6d20  Choosing Random 
│ │ │ │ +00036820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 2052 616e 646f 6d4e  Choosing RandomN
│ │ │ │ +00036870: 6f6e 5a65 726f 2020 2020 2020 2020 2020  onZero          
│ │ │ │ +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 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +000368c0: 536d 616c 6c65 7374 2020 2020 2020 2020  Smallest        
│ │ │ │  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 2052 616e 646f 6d20  Choosing Random 
│ │ │ │ +00036910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +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 2047 5265 764c 6578  Choosing GRevLex
│ │ │ │ +00036960: 536d 616c 6c65 7374 5465 726d 2020 2020  SmallestTerm    
│ │ │ │  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 2020 2020 2020 2020 2020 2020  lest            
│ │ │ │ +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 5465 726d 2020 2020 2020 2020  lestTerm        
│ │ │ │  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 204c 6578 536d 616c  Choosing LexSmal
│ │ │ │ +00036a50: 6c65 7374 2020 2020 2020 2020 2020 2020  lest            
│ │ │ │  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   |.|            
│ │ │ │ -00036c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036a70: 2020 2020 2020 2020 7c0a 7c69 6e74 6572          |.|inter
│ │ │ │ +00036a80: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20  nalChooseMinor: 
│ │ │ │ +00036a90: 4368 6f6f 7369 6e67 2052 616e 646f 6d20  Choosing Random 
│ │ │ │ +00036aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036ac0: 2020 2020 2020 2020 7c0a 7c72 6567 756c          |.|regul
│ │ │ │ +00036ad0: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00036ae0: 2020 4c6f 6f70 2073 7465 702c 2061 626f    Loop step, abo
│ │ │ │ +00036af0: 7574 2074 6f20 636f 6d70 7574 6520 6469  ut to compute di
│ │ │ │ +00036b00: 6d65 6e73 696f 6e2e 2020 5375 626d 6174  mension.  Submat
│ │ │ │ +00036b10: 7269 6365 7320 636f 7c0a 7c72 6567 756c  rices co|.|regul
│ │ │ │ +00036b20: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00036b30: 2020 7369 6e67 756c 6172 4c6f 6375 7320    singularLocus 
│ │ │ │ +00036b40: 6469 6d65 6e73 696f 6e20 7665 7269 6669  dimension verifi
│ │ │ │ +00036b50: 6564 2062 7920 6973 436f 6469 6d41 744c  ed by isCodimAtL
│ │ │ │ +00036b60: 6561 7374 2020 2020 7c0a 7c72 6567 756c  east    |.|regul
│ │ │ │ +00036b70: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00036b80: 2020 7061 7274 6961 6c20 7369 6e67 756c    partial singul
│ │ │ │ +00036b90: 6172 206c 6f63 7573 2064 696d 656e 7369  ar locus dimensi
│ │ │ │ +00036ba0: 6f6e 2063 6f6d 7075 7465 642c 203d 2032  on computed, = 2
│ │ │ │ +00036bb0: 2020 2020 2020 2020 7c0a 7c72 6567 756c          |.|regul
│ │ │ │ +00036bc0: 6172 496e 436f 6469 6d65 6e73 696f 6e3a  arInCodimension:
│ │ │ │ +00036bd0: 2020 4c6f 6f70 2063 6f6d 706c 6574 6564    Loop completed
│ │ │ │ +00036be0: 2c20 7375 626d 6174 7269 6365 7320 636f  , submatrices co
│ │ │ │ +00036bf0: 6e73 6964 6572 6564 203d 2034 392c 2061  nsidered = 49, a
│ │ │ │ +00036c00: 6e64 2063 6f6d 7075 7c0a 7c64 203d 2033  nd compu|.|d = 3
│ │ │ │ +00036c10: 392e 2020 7369 6e67 756c 6172 206c 6f63  9.  singular loc
│ │ │ │ +00036c20: 7573 2064 696d 656e 7369 6f6e 2061 7070  us dimension app
│ │ │ │ +00036c30: 6561 7273 2074 6f20 6265 203d 2032 2020  ears to be = 2  
│ │ │ │  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 7c20 2020 2020          |.|     
│ │ │ │ +00036c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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   |.|------------
│ │ │ │ -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.
│ │ │ │ -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   |.|            
│ │ │ │ -00036d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036ca0: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20          |.|o6 = 
│ │ │ │ +00036cb0: 7472 7565 2020 2020 2020 2020 2020 2020  true            
│ │ │ │ +00036cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036cf0: 2020 2020 2020 2020 7c0a 7c2d 2d2d 2d2d          |.|-----
│ │ │ │ +00036d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00036d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00036d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00036d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00036d40: 2d2d 2d2d 2d2d 2d2d 7c0a 7c6e 6f72 732c  --------|.|nors,
│ │ │ │ +00036d50: 2077 6520 7769 6c6c 2063 6f6d 7075 7465   we will compute
│ │ │ │ +00036d60: 2075 7020 746f 2034 3532 2e39 3038 206f   up to 452.908 o
│ │ │ │ +00036d70: 6620 7468 656d 2e20 2020 2020 2020 2020  f them.         
│ │ │ │  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 7c20 2020 2020          |.|     
│ │ │ │ +00036fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +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   |.|            
│ │ │ │ -00037030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037010: 2020 2020 2020 2020 7c0a 7c6e 7369 6465          |.|nside
│ │ │ │ +00037020: 7265 643a 2037 2c20 616e 6420 636f 6d70  red: 7, and comp
│ │ │ │ +00037030: 7574 6564 203d 2037 2020 2020 2020 2020  uted = 7        
│ │ │ │  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 7c20 2020 2020          |.|     
│ │ │ │ +00037200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +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   |.|            
│ │ │ │ -00037260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037240: 2020 2020 2020 2020 7c0a 7c6e 7369 6465          |.|nside
│ │ │ │ +00037250: 7265 643a 2031 312c 2061 6e64 2063 6f6d  red: 11, and com
│ │ │ │ +00037260: 7075 7465 6420 3d20 3130 2020 2020 2020  puted = 10      
│ │ │ │  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 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000372a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000372b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000372c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000372d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000372e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000372f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000372e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000372f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037340: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037330: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 2020                  
│ │ │ │ -00037430: 207c 0a7c 6e73 6964 6572 6564 3a20 3135   |.|nsidered: 15
│ │ │ │ -00037440: 2c20 616e 6420 636f 6d70 7574 6564 203d  , and computed =
│ │ │ │ -00037450: 2031 3320 2020 2020 2020 2020 2020 2020   13             
│ │ │ │ +00037420: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037480: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00037490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037470: 2020 2020 2020 2020 7c0a 7c6e 7369 6465          |.|nside
│ │ │ │ +00037480: 7265 643a 2031 352c 2061 6e64 2063 6f6d  red: 15, and com
│ │ │ │ +00037490: 7075 7465 6420 3d20 3133 2020 2020 2020  puted = 13      
│ │ │ │  000374a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000374b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000374c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000374d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000374c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000374d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000374e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000374f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037520: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037510: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037570: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037560: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000375a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000375b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000375c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000375b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000375c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000375d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000375e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000375f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037610: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037600: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037660: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037650: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000376a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000376b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000376a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000376b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000376c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000376d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000376e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000376f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037700: 207c 0a7c 6e73 6964 6572 6564 3a20 3231   |.|nsidered: 21
│ │ │ │ -00037710: 2c20 616e 6420 636f 6d70 7574 6564 203d  , and computed =
│ │ │ │ -00037720: 2031 3820 2020 2020 2020 2020 2020 2020   18             
│ │ │ │ +000376f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037750: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00037760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037740: 2020 2020 2020 2020 7c0a 7c6e 7369 6465          |.|nside
│ │ │ │ +00037750: 7265 643a 2032 312c 2061 6e64 2063 6f6d  red: 21, and com
│ │ │ │ +00037760: 7075 7465 6420 3d20 3138 2020 2020 2020  puted = 18      
│ │ │ │  00037770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000377a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037790: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000377a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000377e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000377f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037840: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037830: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037890: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037880: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000378a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000378b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000378c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000378d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000378e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000378d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000378e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000378f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037930: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037920: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037980: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037970: 2020 2020 2020 2020 7c0a 7c20 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 2020 2020                  
│ │ │ │ -000379c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000379d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000379c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000379d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000379e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000379f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037a20: 207c 0a7c 6e73 6964 6572 6564 3a20 3238   |.|nsidered: 28
│ │ │ │ -00037a30: 2c20 616e 6420 636f 6d70 7574 6564 203d  , and computed =
│ │ │ │ -00037a40: 2032 3420 2020 2020 2020 2020 2020 2020   24             
│ │ │ │ +00037a10: 2020 2020 2020 2020 7c0a 7c20 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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037a70: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00037a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037a60: 2020 2020 2020 2020 7c0a 7c6e 7369 6465          |.|nside
│ │ │ │ +00037a70: 7265 643a 2032 382c 2061 6e64 2063 6f6d  red: 28, and com
│ │ │ │ +00037a80: 7075 7465 6420 3d20 3234 2020 2020 2020  puted = 24      
│ │ │ │  00037a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037ac0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037ab0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037b10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037b00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037b60: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037b50: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037bb0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037ba0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037c00: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037bf0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037c50: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037c40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037ca0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037c90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037cf0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037ce0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037d40: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037d30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037d90: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037d80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037de0: 207c 0a7c 6e73 6964 6572 6564 3a20 3337   |.|nsidered: 37
│ │ │ │ -00037df0: 2c20 616e 6420 636f 6d70 7574 6564 203d  , and computed =
│ │ │ │ -00037e00: 2033 3020 2020 2020 2020 2020 2020 2020   30             
│ │ │ │ +00037dd0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037e30: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00037e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037e20: 2020 2020 2020 2020 7c0a 7c6e 7369 6465          |.|nside
│ │ │ │ +00037e30: 7265 643a 2033 372c 2061 6e64 2063 6f6d  red: 37, and com
│ │ │ │ +00037e40: 7075 7465 6420 3d20 3330 2020 2020 2020  puted = 30      
│ │ │ │  00037e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037e80: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037e70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037ed0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037ec0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037f20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037f10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037f70: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037f60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037fc0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00037fb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038010: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00038000: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00038010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038060: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00038050: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00038060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000380a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000380b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000380a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000380b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000380c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000380d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000380e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000380f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038100: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000380f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00038100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038150: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00038140: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00038150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000381a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00038190: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000381a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000381b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000381c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000381d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000381e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000381f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000381e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000381f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038240: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00038230: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00038240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038280: 2020 2020 2020 2020 2020 2020 2020 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 7c20 2020 2020          |.|     
│ │ │ │ +00038290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000382a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +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   |.|            
│ │ │ │ -000382f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000382d0: 2020 2020 2020 2020 7c0a 7c6e 7369 6465          |.|nside
│ │ │ │ +000382e0: 7265 643a 2034 392c 2061 6e64 2063 6f6d  red: 49, and com
│ │ │ │ +000382f0: 7075 7465 6420 3d20 3339 2020 2020 2020  puted = 39      
│ │ │ │  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: 2e34 3334 3538 7320 2863 7075 293b 2031  .43458s (cpu); 1
│ │ │ │ -000383a0: 2e30 3136 3036 7320 2874 6872 6561 6429  .01606s (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   |.+------------
│ │ │ │ -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...+--------
│ │ │ │ -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 3735 3839 3973 2028 6370 7529   0.175899s (cpu)
│ │ │ │ -00038570: 3b20 302e 3132 3435 3273 2028 7468 7265  ; 0.12452s (thre
│ │ │ │ -00038580: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 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    
│ │ │ │ +00038370: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00038380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000383a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000383b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00038ba0: 696f 6e3a 2020 4c6f 6f70 2063 6f6d 706c  ion:  Loop compl
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│ │ │ │ +00038be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00038c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00038c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00038c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c6e  ------------|.|n
│ │ │ │ +00038c30: 6f72 732c 2077 6520 7769 6c6c 2063 6f6d  ors, we will com
│ │ │ │ +00038c40: 7075 7465 2075 7020 746f 2031 3020 6f66  pute up to 10 of
│ │ │ │ +00038c50: 2074 6865 6d2e 2020 2020 2020 2020 2020   them.          
│ │ │ │  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                  
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│ │ │ │ -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                  
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│ │ │ │ -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                  
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│ │ │ │ +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                  
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│ │ │ │ -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                  
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│ │ │ │ -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 7c20              |.| 
│ │ │ │ +00038eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +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       |.|        
│ │ │ │ -00038f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038ef0: 2020 2020 2020 2020 2020 2020 7c0a 7c6e              |.|n
│ │ │ │ +00038f00: 7369 6465 7265 643a 2037 2c20 616e 6420  sidered: 7, and 
│ │ │ │ +00038f10: 636f 6d70 7574 6564 203d 2037 2020 2020  computed = 7    
│ │ │ │  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                  
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│ │ │ │ -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                  
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│ │ │ │ -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 7c20              |.| 
│ │ │ │ +00039090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000390a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +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       |.|        
│ │ │ │ -000390f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000390d0: 2020 2020 2020 2020 2020 2020 7c0a 7c6e              |.|n
│ │ │ │ +000390e0: 7369 6465 7265 643a 2031 302c 2061 6e64  sidered: 10, and
│ │ │ │ +000390f0: 2063 6f6d 7075 7465 6420 3d20 3130 2020   computed = 10  
│ │ │ │  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       |.+--------
│ │ │ │ -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:..+-----------
│ │ │ │ -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: 3135 3730 3773 2028 6370 7529 3b20 302e  15707s (cpu); 0.
│ │ │ │ -00039450: 3130 3637 3233 7320 2874 6872 6561 6429  106723s (thread)
│ │ │ │ -00039460: 3b20 3073 2028 6763 2920 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  
│ │ │ │ -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       
│ │ │ │ +00039170: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00039180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000391a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000391b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000391c0: 2020 2020 2020 2020 2020 2020 7c0a 7c74              |.|t
│ │ │ │ +000391d0: 6564 203d 2031 302e 2020 7369 6e67 756c  ed = 10.  singul
│ │ │ │ +000391e0: 6172 206c 6f63 7573 2064 696d 656e 7369  ar locus dimensi
│ │ │ │ +000391f0: 6f6e 2061 7070 6561 7273 2074 6f20 6265  on appears to be
│ │ │ │ +00039200: 203d 2033 2020 2020 2020 2020 2020 2020   = 3            
│ │ │ │ +00039210: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00039220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a54  ------------+..T
│ │ │ │ +00039270: 6865 7265 2061 7265 206f 7468 6572 2066  here are other f
│ │ │ │ +00039280: 696e 6572 2077 6179 7320 746f 2063 6f6e  iner ways to con
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│ │ │ │ +000392a0: 7273 206f 7074 696f 6e2c 2062 7574 2074  rs option, but t
│ │ │ │ +000392b0: 6865 7920 7769 6c6c 206e 6f74 0a62 6520  hey will not.be 
│ │ │ │ +000392c0: 6469 7363 7573 7365 6420 696e 2074 6869  discussed in thi
│ │ │ │ +000392d0: 7320 7475 746f 7269 616c 2e20 2053 6565  s tutorial.  See
│ │ │ │ +000392e0: 202a 6e6f 7465 2072 6567 756c 6172 496e   *note regularIn
│ │ │ │ +000392f0: 436f 6469 6d65 6e73 696f 6e3a 0a72 6567  Codimension:.reg
│ │ │ │ +00039300: 756c 6172 496e 436f 6469 6d65 6e73 696f  ularInCodimensio
│ │ │ │ +00039310: 6e2c 2e0a 0a53 656c 6563 7469 6e67 2073  n,...Selecting s
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│ │ │ │ +00039340: 616c 736f 2073 6565 206f 7574 7075 7420  also see output 
│ │ │ │ +00039350: 6c69 6b65 3a20 6060 4368 6f6f 7369 6e67  like: ``Choosing
│ │ │ │ +00039360: 0a4c 6578 536d 616c 6c65 7374 2727 206f  .LexSmallest'' o
│ │ │ │ +00039370: 7220 6060 4368 6f6f 7369 6e67 2052 616e  r ``Choosing Ran
│ │ │ │ +00039380: 646f 6d27 272e 2020 5468 6973 2069 7320  dom''.  This is 
│ │ │ │ +00039390: 7361 7969 6e67 2068 6f77 2077 6520 6172  saying how we ar
│ │ │ │ +000393a0: 6520 7365 6c65 6374 696e 6720 610a 6769  e selecting a.gi
│ │ │ │ +000393b0: 7665 6e20 7375 626d 6174 7269 782e 2020  ven submatrix.  
│ │ │ │ +000393c0: 466f 7220 696e 7374 616e 6365 2c20 7765  For instance, we
│ │ │ │ +000393d0: 2063 616e 2072 756e 3a0a 0a2b 2d2d 2d2d   can run:..+----
│ │ │ │ +000393e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000393f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039420: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a  ---------+.|i8 :
│ │ │ │ +00039430: 2074 696d 6520 7265 6775 6c61 7249 6e43   time regularInC
│ │ │ │ +00039440: 6f64 696d 656e 7369 6f6e 2831 2c20 532f  odimension(1, S/
│ │ │ │ +00039450: 4a2c 204d 6178 4d69 6e6f 7273 3d3e 3130  J, MaxMinors=>10
│ │ │ │ +00039460: 2c20 5374 7261 7465 6779 3d3e 5374 7261  , Strategy=>Stra
│ │ │ │ +00039470: 7465 6779 5261 6e64 6f7c 0a7c 202d 2d20  tegyRando|.| -- 
│ │ │ │ +00039480: 7573 6564 2030 2e31 3731 3331 3273 2028  used 0.171312s (
│ │ │ │ +00039490: 6370 7529 3b20 302e 3131 3736 3738 7320  cpu); 0.117678s 
│ │ │ │ +000394a0: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ +000394b0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +000394c0: 2020 2020 2020 2020 207c 0a7c 7265 6775           |.|regu
│ │ │ │ +000394d0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ +000394e0: 3a20 7269 6e67 2064 696d 656e 7369 6f6e  : ring dimension
│ │ │ │ +000394f0: 203d 342c 2074 6865 7265 2061 7265 2031   =4, there are 1
│ │ │ │ +00039500: 3436 3531 3238 2070 6f73 7369 626c 6520  465128 possible 
│ │ │ │ +00039510: 3520 6279 2035 206d 697c 0a7c 7265 6775  5 by 5 mi|.|regu
│ │ │ │ +00039520: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ +00039530: 3a20 4162 6f75 7420 746f 2065 6e74 6572  : About to enter
│ │ │ │ +00039540: 206c 6f6f 7020 2020 2020 2020 2020 2020   loop           
│ │ │ │  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       
│ │ │ │ -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       
│ │ │ │ +00039740: 2020 2020 2020 2020 207c 0a7c 696e 7465           |.|inte
│ │ │ │ +00039750: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ +00039760: 2043 686f 6f73 696e 6720 5261 6e64 6f6d   Choosing Random
│ │ │ │ +00039770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039790: 2020 2020 2020 2020 207c 0a7c 7265 6775           |.|regu
│ │ │ │ +000397a0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ +000397b0: 3a20 204c 6f6f 7020 7374 6570 2c20 6162  :  Loop step, ab
│ │ │ │ +000397c0: 6f75 7420 746f 2063 6f6d 7075 7465 2064  out to compute d
│ │ │ │ +000397d0: 696d 656e 7369 6f6e 2e20 2053 7562 6d61  imension.  Subma
│ │ │ │ +000397e0: 7472 6963 6573 2063 6f7c 0a7c 7265 6775  trices co|.|regu
│ │ │ │ +000397f0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ +00039800: 3a20 2069 7343 6f64 696d 4174 4c65 6173  :  isCodimAtLeas
│ │ │ │ +00039810: 7420 6661 696c 6564 2c20 636f 6d70 7574  t failed, comput
│ │ │ │ +00039820: 696e 6720 636f 6469 6d2e 2020 2020 2020  ing codim.      
│ │ │ │ +00039830: 2020 2020 2020 2020 207c 0a7c 7265 6775           |.|regu
│ │ │ │ +00039840: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ +00039850: 3a20 2070 6172 7469 616c 2073 696e 6775  :  partial singu
│ │ │ │ +00039860: 6c61 7220 6c6f 6375 7320 6469 6d65 6e73  lar locus dimens
│ │ │ │ +00039870: 696f 6e20 636f 6d70 7574 6564 2c20 3d20  ion computed, = 
│ │ │ │ +00039880: 3320 2020 2020 2020 207c 0a7c 696e 7465  3        |.|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|.|-----------
│ │ │ │ -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)          
│ │ │ │ -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    |.|           
│ │ │ │ +00039920: 2020 2020 2020 2020 207c 0a7c 696e 7465           |.|inte
│ │ │ │ +00039930: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a  rnalChooseMinor:
│ │ │ │ +00039940: 2043 686f 6f73 696e 6720 5261 6e64 6f6d   Choosing Random
│ │ │ │ +00039950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039970: 2020 2020 2020 2020 207c 0a7c 7265 6775           |.|regu
│ │ │ │ +00039980: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ +00039990: 3a20 204c 6f6f 7020 7374 6570 2c20 6162  :  Loop step, ab
│ │ │ │ +000399a0: 6f75 7420 746f 2063 6f6d 7075 7465 2064  out to compute d
│ │ │ │ +000399b0: 696d 656e 7369 6f6e 2e20 2053 7562 6d61  imension.  Subma
│ │ │ │ +000399c0: 7472 6963 6573 2063 6f7c 0a7c 7265 6775  trices co|.|regu
│ │ │ │ +000399d0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ +000399e0: 3a20 2069 7343 6f64 696d 4174 4c65 6173  :  isCodimAtLeas
│ │ │ │ +000399f0: 7420 6661 696c 6564 2c20 636f 6d70 7574  t failed, comput
│ │ │ │ +00039a00: 696e 6720 636f 6469 6d2e 2020 2020 2020  ing codim.      
│ │ │ │ +00039a10: 2020 2020 2020 2020 207c 0a7c 7265 6775           |.|regu
│ │ │ │ +00039a20: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ +00039a30: 3a20 2070 6172 7469 616c 2073 696e 6775  :  partial singu
│ │ │ │ +00039a40: 6c61 7220 6c6f 6375 7320 6469 6d65 6e73  lar locus dimens
│ │ │ │ +00039a50: 696f 6e20 636f 6d70 7574 6564 2c20 3d20  ion computed, = 
│ │ │ │ +00039a60: 3320 2020 2020 2020 207c 0a7c 7265 6775  3        |.|regu
│ │ │ │ +00039a70: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e  larInCodimension
│ │ │ │ +00039a80: 3a20 204c 6f6f 7020 636f 6d70 6c65 7465  :  Loop complete
│ │ │ │ +00039a90: 642c 2073 7562 6d61 7472 6963 6573 2063  d, submatrices c
│ │ │ │ +00039aa0: 6f6e 7369 6465 7265 6420 3d20 3130 2c20  onsidered = 10, 
│ │ │ │ +00039ab0: 616e 6420 636f 6d70 757c 0a7c 2d2d 2d2d  and compu|.|----
│ │ │ │ +00039ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039b00: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 6d2c 2056  ---------|.|m, V
│ │ │ │ +00039b10: 6572 626f 7365 3d3e 7472 7565 2920 2020  erbose=>true)   
│ │ │ │  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 2020 2020           |.|    
│ │ │ │ +00039b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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    |.|           
│ │ │ │ -00039bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039ba0: 2020 2020 2020 2020 207c 0a7c 6e6f 7273           |.|nors
│ │ │ │ +00039bb0: 2c20 7765 2077 696c 6c20 636f 6d70 7574  , we will comput
│ │ │ │ +00039bc0: 6520 7570 2074 6f20 3130 206f 6620 7468  e up to 10 of th
│ │ │ │ +00039bd0: 656d 2e20 2020 2020 2020 2020 2020 2020  em.             
│ │ │ │  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 2020 2020           |.|    
│ │ │ │ +00039e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +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    |.|           
│ │ │ │ -00039e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039e70: 2020 2020 2020 2020 207c 0a7c 6e73 6964           |.|nsid
│ │ │ │ +00039e80: 6572 6564 3a20 372c 2061 6e64 2063 6f6d  ered: 7, and com
│ │ │ │ +00039e90: 7075 7465 6420 3d20 3720 2020 2020 2020  puted = 7       
│ │ │ │  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    |.|           
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│ │ │ │ +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    |.|           
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│ │ │ │ +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    |.|           
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│ │ │ │ +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                  
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│ │ │ │ -0003a030: 3d20 3130 2020 2020 2020 2020 2020 2020  = 10            
│ │ │ │ +0003a000: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003a010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +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    |.|           
│ │ │ │ -0003a070: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  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                  
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│ │ │ │ -0003a140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a150: 2020 7c0a 2b2d 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
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│ │ │ │ -0003a1c0: 2061 7265 2063 686f 7365 6e2e 2020 5765   are chosen.  We
│ │ │ │ -0003a1d0: 2064 6973 6375 7373 2073 7472 6174 6567   discuss strateg
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│ │ │ │ -0003a1f0: 0a73 7562 6d61 7472 6963 6573 206d 7563  .submatrices muc
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│ │ │ │ -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
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│ │ │ │ -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,
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│ │ │ │ -0003a300: 7075 7465 6420 3d20 3727 272e 2020 5765  puted = 7''.  We
│ │ │ │ -0003a310: 206f 6e6c 7920 636f 6d70 7574 650a 6d69   only compute.mi
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│ │ │ │ -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
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│ │ │ │ -0003a580: 2d2d 2075 7365 6420 302e 3632 3737 3739  -- used 0.627779
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│ │ │ │ -0003a5a0: 3873 2028 7468 7265 6164 293b 2030 7320  8s (thread); 0s 
│ │ │ │ -0003a5b0: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │ -0003a5c0: 2020 2020 2020 2020 2020 2020 7c0a 7c72              |.|r
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│ │ │ │ -0003a620: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ -0003a630: 696f 6e3a 2041 626f 7574 2074 6f20 656e  ion: About to en
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│ │ │ │ -0003a650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a660: 2020 2020 2020 2020 2020 2020 7c0a 7c69              |.|i
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│ │ │ │ +0003a100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a130: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0003a160: 6c6f 6375 7320 6469 6d65 6e73 696f 6e20  locus dimension 
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│ │ │ │ +0003a1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a1e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 616e 6420  ---------+..and 
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│ │ │ │ +0003a220: 7374 7261 7465 6769 6573 2066 6f72 2063  strategies for c
│ │ │ │ +0003a230: 686f 6f73 696e 670a 7375 626d 6174 7269  hoosing.submatri
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│ │ │ │ +0003a330: 2063 6f6e 7369 6465 7265 643a 2037 2c20   considered: 7, 
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│ │ │ │ +0003a3a0: 206d 696e 6f72 732c 2074 6865 7265 2063   minors, there c
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│ │ │ │ +0003a3f0: 6d69 6e6f 7273 2076 7320 636f 6e73 6964  minors vs consid
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│ │ │ │ +0003a430: 6572 696f 6469 6361 6c6c 7920 7765 2063  eriodically we c
│ │ │ │ +0003a440: 6f6d 7075 7465 2074 6865 2063 6f64 696d  ompute the codim
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│ │ │ │ +0003a490: 5468 6572 6520 6172 6520 7477 6f20 6f70  There are two op
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│ │ │ │ +0003a540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0003a660: 206d 697c 0a7c 7265 6775 6c61 7249 6e43   mi|.|regularInC
│ │ │ │ +0003a670: 6f64 696d 656e 7369 6f6e 3a20 4162 6f75  odimension: Abou
│ │ │ │ +0003a680: 7420 746f 2065 6e74 6572 206c 6f6f 7020  t to enter loop 
│ │ │ │ +0003a690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 4e6f 6e5a 6572  ing RandomNonZer
│ │ │ │ +0003a6e0: 6f20 2020 2020 2020 2020 2020 2020 2020  o               
│ │ │ │  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 5261 6e64 6f6d 2020 2020 2020  ing Random      
│ │ │ │ +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    
│ │ │ │ -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
│ │ │ │ +0003a750: 2020 207c 0a7c 696e 7465 726e 616c 4368     |.|internalCh
│ │ │ │ +0003a760: 6f6f 7365 4d69 6e6f 723a 2043 686f 6f73  ooseMinor: Choos
│ │ │ │ +0003a770: 696e 6720 4c65 7853 6d61 6c6c 6573 7420  ing LexSmallest 
│ │ │ │ +0003a780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a7a0: 2020 207c 0a7c 7265 6775 6c61 7249 6e43     |.|regularInC
│ │ │ │ +0003a7b0: 6f64 696d 656e 7369 6f6e 3a20 204c 6f6f  odimension:  Loo
│ │ │ │ +0003a7c0: 7020 7374 6570 2c20 6162 6f75 7420 746f  p step, about to
│ │ │ │ +0003a7d0: 2063 6f6d 7075 7465 2064 696d 656e 7369   compute dimensi
│ │ │ │ +0003a7e0: 6f6e 2e20 2053 7562 6d61 7472 6963 6573  on.  Submatrices
│ │ │ │ +0003a7f0: 2063 6f7c 0a7c 7265 6775 6c61 7249 6e43   co|.|regularInC
│ │ │ │ +0003a800: 6f64 696d 656e 7369 6f6e 3a20 2069 7343  odimension:  isC
│ │ │ │ +0003a810: 6f64 696d 4174 4c65 6173 7420 6661 696c  odimAtLeast fail
│ │ │ │ +0003a820: 6564 2c20 636f 6d70 7574 696e 6720 636f  ed, computing co
│ │ │ │ +0003a830: 6469 6d2e 2020 2020 2020 2020 2020 2020  dim.            
│ │ │ │ +0003a840: 2020 207c 0a7c 7265 6775 6c61 7249 6e43     |.|regularInC
│ │ │ │ +0003a850: 6f64 696d 656e 7369 6f6e 3a20 2070 6172  odimension:  par
│ │ │ │ +0003a860: 7469 616c 2073 696e 6775 6c61 7220 6c6f  tial singular lo
│ │ │ │ +0003a870: 6375 7320 6469 6d65 6e73 696f 6e20 636f  cus dimension co
│ │ │ │ +0003a880: 6d70 7574 6564 2c20 3d20 3320 2020 2020  mputed, = 3     
│ │ │ │ +0003a890: 2020 207c 0a7c 696e 7465 726e 616c 4368     |.|internalCh
│ │ │ │ +0003a8a0: 6f6f 7365 4d69 6e6f 723a 2043 686f 6f73  ooseMinor: Choos
│ │ │ │ +0003a8b0: 696e 6720 4c65 7853 6d61 6c6c 6573 7454  ing LexSmallestT
│ │ │ │ +0003a8c0: 6572 6d20 2020 2020 2020 2020 2020 2020  erm             
│ │ │ │  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 7454 6572 6d20 2020 2020 2020 2020  estTerm         
│ │ │ │  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
│ │ │ │ -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    
│ │ │ │ +0003a930: 2020 207c 0a7c 696e 7465 726e 616c 4368     |.|internalCh
│ │ │ │ +0003a940: 6f6f 7365 4d69 6e6f 723a 2043 686f 6f73  ooseMinor: Choos
│ │ │ │ +0003a950: 696e 6720 4752 6576 4c65 7853 6d61 6c6c  ing GRevLexSmall
│ │ │ │ +0003a960: 6573 7420 2020 2020 2020 2020 2020 2020  est             
│ │ │ │ +0003a970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a980: 2020 207c 0a7c 7265 6775 6c61 7249 6e43     |.|regularInC
│ │ │ │ +0003a990: 6f64 696d 656e 7369 6f6e 3a20 204c 6f6f  odimension:  Loo
│ │ │ │ +0003a9a0: 7020 7374 6570 2c20 6162 6f75 7420 746f  p step, about to
│ │ │ │ +0003a9b0: 2063 6f6d 7075 7465 2064 696d 656e 7369   compute dimensi
│ │ │ │ +0003a9c0: 6f6e 2e20 2053 7562 6d61 7472 6963 6573  on.  Submatrices
│ │ │ │ +0003a9d0: 2063 6f7c 0a7c 7265 6775 6c61 7249 6e43   co|.|regularInC
│ │ │ │ +0003a9e0: 6f64 696d 656e 7369 6f6e 3a20 2069 7343  odimension:  isC
│ │ │ │ +0003a9f0: 6f64 696d 4174 4c65 6173 7420 6661 696c  odimAtLeast fail
│ │ │ │ +0003aa00: 6564 2c20 636f 6d70 7574 696e 6720 636f  ed, computing co
│ │ │ │ +0003aa10: 6469 6d2e 2020 2020 2020 2020 2020 2020  dim.            
│ │ │ │ +0003aa20: 2020 207c 0a7c 7265 6775 6c61 7249 6e43     |.|regularInC
│ │ │ │ +0003aa30: 6f64 696d 656e 7369 6f6e 3a20 2070 6172  odimension:  par
│ │ │ │ +0003aa40: 7469 616c 2073 696e 6775 6c61 7220 6c6f  tial singular lo
│ │ │ │ +0003aa50: 6375 7320 6469 6d65 6e73 696f 6e20 636f  cus dimension co
│ │ │ │ +0003aa60: 6d70 7574 6564 2c20 3d20 3320 2020 2020  mputed, = 3     
│ │ │ │ +0003aa70: 2020 207c 0a7c 696e 7465 726e 616c 4368     |.|internalCh
│ │ │ │ +0003aa80: 6f6f 7365 4d69 6e6f 723a 2043 686f 6f73  ooseMinor: Choos
│ │ │ │ +0003aa90: 696e 6720 4752 6576 4c65 7853 6d61 6c6c  ing GRevLexSmall
│ │ │ │ +0003aaa0: 6573 7454 6572 6d20 2020 2020 2020 2020  estTerm         
│ │ │ │  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    
│ │ │ │ -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
│ │ │ │ +0003aac0: 2020 207c 0a7c 696e 7465 726e 616c 4368     |.|internalCh
│ │ │ │ +0003aad0: 6f6f 7365 4d69 6e6f 723a 2043 686f 6f73  ooseMinor: Choos
│ │ │ │ +0003aae0: 696e 6720 4c65 7853 6d61 6c6c 6573 7454  ing LexSmallestT
│ │ │ │ +0003aaf0: 6572 6d20 2020 2020 2020 2020 2020 2020  erm             
│ │ │ │ +0003ab00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ab10: 2020 207c 0a7c 7265 6775 6c61 7249 6e43     |.|regularInC
│ │ │ │ +0003ab20: 6f64 696d 656e 7369 6f6e 3a20 204c 6f6f  odimension:  Loo
│ │ │ │ +0003ab30: 7020 7374 6570 2c20 6162 6f75 7420 746f  p step, about to
│ │ │ │ +0003ab40: 2063 6f6d 7075 7465 2064 696d 656e 7369   compute dimensi
│ │ │ │ +0003ab50: 6f6e 2e20 2053 7562 6d61 7472 6963 6573  on.  Submatrices
│ │ │ │ +0003ab60: 2063 6f7c 0a7c 7265 6775 6c61 7249 6e43   co|.|regularInC
│ │ │ │ +0003ab70: 6f64 696d 656e 7369 6f6e 3a20 2069 7343  odimension:  isC
│ │ │ │ +0003ab80: 6f64 696d 4174 4c65 6173 7420 6661 696c  odimAtLeast fail
│ │ │ │ +0003ab90: 6564 2c20 636f 6d70 7574 696e 6720 636f  ed, computing co
│ │ │ │ +0003aba0: 6469 6d2e 2020 2020 2020 2020 2020 2020  dim.            
│ │ │ │ +0003abb0: 2020 207c 0a7c 7265 6775 6c61 7249 6e43     |.|regularInC
│ │ │ │ +0003abc0: 6f64 696d 656e 7369 6f6e 3a20 2070 6172  odimension:  par
│ │ │ │ +0003abd0: 7469 616c 2073 696e 6775 6c61 7220 6c6f  tial singular lo
│ │ │ │ +0003abe0: 6375 7320 6469 6d65 6e73 696f 6e20 636f  cus dimension co
│ │ │ │ +0003abf0: 6d70 7574 6564 2c20 3d20 3320 2020 2020  mputed, = 3     
│ │ │ │ +0003ac00: 2020 207c 0a7c 696e 7465 726e 616c 4368     |.|internalCh
│ │ │ │ +0003ac10: 6f6f 7365 4d69 6e6f 723a 2043 686f 6f73  ooseMinor: Choos
│ │ │ │ +0003ac20: 696e 6720 4c65 7853 6d61 6c6c 6573 7454  ing LexSmallestT
│ │ │ │ +0003ac30: 6572 6d20 2020 2020 2020 2020 2020 2020  erm             
│ │ │ │  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|.|-
│ │ │ │ -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  )               
│ │ │ │ -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              |.| 
│ │ │ │ -0003ae40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ac50: 2020 207c 0a7c 696e 7465 726e 616c 4368     |.|internalCh
│ │ │ │ +0003ac60: 6f6f 7365 4d69 6e6f 723a 2043 686f 6f73  ooseMinor: Choos
│ │ │ │ +0003ac70: 696e 6720 4752 6576 4c65 7853 6d61 6c6c  ing GRevLexSmall
│ │ │ │ +0003ac80: 6573 7454 6572 6d20 2020 2020 2020 2020  estTerm         
│ │ │ │ +0003ac90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003aca0: 2020 207c 0a7c 7265 6775 6c61 7249 6e43     |.|regularInC
│ │ │ │ +0003acb0: 6f64 696d 656e 7369 6f6e 3a20 204c 6f6f  odimension:  Loo
│ │ │ │ +0003acc0: 7020 7374 6570 2c20 6162 6f75 7420 746f  p step, about to
│ │ │ │ +0003acd0: 2063 6f6d 7075 7465 2064 696d 656e 7369   compute dimensi
│ │ │ │ +0003ace0: 6f6e 2e20 2053 7562 6d61 7472 6963 6573  on.  Submatrices
│ │ │ │ +0003acf0: 2063 6f7c 0a7c 7265 6775 6c61 7249 6e43   co|.|regularInC
│ │ │ │ +0003ad00: 6f64 696d 656e 7369 6f6e 3a20 2069 7343  odimension:  isC
│ │ │ │ +0003ad10: 6f64 696d 4174 4c65 6173 7420 6661 696c  odimAtLeast fail
│ │ │ │ +0003ad20: 6564 2c20 636f 6d70 7574 696e 6720 636f  ed, computing co
│ │ │ │ +0003ad30: 6469 6d2e 2020 2020 2020 2020 2020 2020  dim.            
│ │ │ │ +0003ad40: 2020 207c 0a7c 7265 6775 6c61 7249 6e43     |.|regularInC
│ │ │ │ +0003ad50: 6f64 696d 656e 7369 6f6e 3a20 2070 6172  odimension:  par
│ │ │ │ +0003ad60: 7469 616c 2073 696e 6775 6c61 7220 6c6f  tial singular lo
│ │ │ │ +0003ad70: 6375 7320 6469 6d65 6e73 696f 6e20 636f  cus dimension co
│ │ │ │ +0003ad80: 6d70 7574 6564 2c20 3d20 3320 2020 2020  mputed, = 3     
│ │ │ │ +0003ad90: 2020 207c 0a7c 7265 6775 6c61 7249 6e43     |.|regularInC
│ │ │ │ +0003ada0: 6f64 696d 656e 7369 6f6e 3a20 204c 6f6f  odimension:  Loo
│ │ │ │ +0003adb0: 7020 636f 6d70 6c65 7465 642c 2073 7562  p completed, sub
│ │ │ │ +0003adc0: 6d61 7472 6963 6573 2063 6f6e 7369 6465  matrices conside
│ │ │ │ +0003add0: 7265 6420 3d20 3130 2c20 616e 6420 636f  red = 10, and co
│ │ │ │ +0003ade0: 6d70 757c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d  mpu|.|----------
│ │ │ │ +0003adf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ae00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ae10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ae20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ae30: 2d2d 2d7c 0a7c 3e33 2c20 5665 7262 6f73  ---|.|>3, Verbos
│ │ │ │ +0003ae40: 653d 3e74 7275 6529 2020 2020 2020 2020  e=>true)        
│ │ │ │  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 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003ae90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003aea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +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              |.| 
│ │ │ │ -0003aee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003aed0: 2020 207c 0a7c 6e6f 7273 2c20 7765 2077     |.|nors, we w
│ │ │ │ +0003aee0: 696c 6c20 636f 6d70 7574 6520 7570 2074  ill compute up t
│ │ │ │ +0003aef0: 6f20 3130 206f 6620 7468 656d 2e20 2020  o 10 of them.   
│ │ │ │  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 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003b020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +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 7c20              |.| 
│ │ │ │ -0003b070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b060: 2020 207c 0a7c 6e73 6964 6572 6564 3a20     |.|nsidered: 
│ │ │ │ +0003b070: 332c 2061 6e64 2063 6f6d 7075 7465 6420  3, and computed 
│ │ │ │ +0003b080: 3d20 3320 2020 2020 2020 2020 2020 2020  = 3             
│ │ │ │  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 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003b200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +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              |.| 
│ │ │ │ -0003b250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b240: 2020 207c 0a7c 6e73 6964 6572 6564 3a20     |.|nsidered: 
│ │ │ │ +0003b250: 362c 2061 6e64 2063 6f6d 7075 7465 6420  6, and computed 
│ │ │ │ +0003b260: 3d20 3620 2020 2020 2020 2020 2020 2020  = 6             
│ │ │ │  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 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003b390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +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              |.| 
│ │ │ │ -0003b3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b3d0: 2020 207c 0a7c 6e73 6964 6572 6564 3a20     |.|nsidered: 
│ │ │ │ +0003b3e0: 382c 2061 6e64 2063 6f6d 7075 7465 6420  8, and computed 
│ │ │ │ +0003b3f0: 3d20 3820 2020 2020 2020 2020 2020 2020  = 8             
│ │ │ │  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 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 2020 2020 2020 2020                  
│ │ │ │  0003b550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b560: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003b570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b560: 2020 207c 0a7c 6e73 6964 6572 6564 3a20     |.|nsidered: 
│ │ │ │ +0003b570: 3130 2c20 616e 6420 636f 6d70 7574 6564  10, and computed
│ │ │ │ +0003b580: 203d 2031 3020 2020 2020 2020 2020 2020   = 10           
│ │ │ │  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              |.+-
│ │ │ │ -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...+----
│ │ │ │ -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 2e37 3339 3834 3473 2028  used 0.739844s (
│ │ │ │ -0003bac0: 6370 7529 3b20 302e 3530 3430 3835 7320  cpu); 0.504085s 
│ │ │ │ -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
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│ │ │ │ -0003bb30: 3436 3531 3238 2070 6f73 7369 626c 6520  465128 possible 
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│ │ │ │ -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           
│ │ │ │ -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   
│ │ │ │ +0003b600: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003b610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b650: 2020 207c 0a7c 7465 6420 3d20 3130 2e20     |.|ted = 10. 
│ │ │ │ +0003b660: 2073 696e 6775 6c61 7220 6c6f 6375 7320   singular locus 
│ │ │ │ +0003b670: 6469 6d65 6e73 696f 6e20 6170 7065 6172  dimension appear
│ │ │ │ +0003b680: 7320 746f 2062 6520 3d20 3320 2020 2020  s to be = 3     
│ │ │ │ +0003b690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b6a0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0003b6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003b6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003b6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003b6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003b6f0: 2d2d 2d2b 0a0a 4d69 6e4d 696e 6f72 7346  ---+..MinMinorsF
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│ │ │ │ +0003b710: 204d 696e 4d69 6e6f 7273 4675 6e63 7469   MinMinorsFuncti
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│ │ │ │ +0003b750: 6d69 6e6f 7273 206e 6565 6465 6420 746f  minors needed to
│ │ │ │ +0003b760: 2076 6572 6966 7920 7468 6174 2073 6f6d   verify that som
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│ │ │ │ +0003b780: 7220 696e 2063 6f64 696d 656e 7369 6f6e  r in codimension
│ │ │ │ +0003b790: 2024 6e24 0a28 7768 6963 6820 6973 2061   $n$.(which is a
│ │ │ │ +0003b7a0: 6c77 6179 7320 246e 2b31 2429 2074 6f20  lways $n+1$) to 
│ │ │ │ +0003b7b0: 7468 6520 6e75 6d62 6572 206f 6620 6d69  the number of mi
│ │ │ │ +0003b7c0: 6e6f 7273 2074 6f20 636f 6d70 7574 6520  nors to compute 
│ │ │ │ +0003b7d0: 6265 666f 7265 2063 6f6d 7075 7469 6e67  before computing
│ │ │ │ +0003b7e0: 2074 6865 0a64 696d 656e 7369 6f6e 206f   the.dimension o
│ │ │ │ +0003b7f0: 6620 7468 6520 7061 7274 6961 6c20 6964  f the partial id
│ │ │ │ +0003b800: 6561 6c20 6f66 206d 696e 6f72 7320 666f  eal of minors fo
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│ │ │ │ +0003b830: 7468 6174 0a74 6872 6565 206d 696e 6f72  that.three minor
│ │ │ │ +0003b840: 7320 7765 7265 2063 6f6d 7075 7465 6420  s were computed 
│ │ │ │ +0003b850: 696e 2074 6865 2061 626f 7665 2065 7861  in the above exa
│ │ │ │ +0003b860: 6d70 6c65 2062 6566 6f72 6520 7765 2061  mple before we a
│ │ │ │ +0003b870: 7474 656d 7074 2074 6f20 636f 6d70 7574  ttempt to comput
│ │ │ │ +0003b880: 650a 636f 6469 6d65 6e73 696f 6e2e 0a0a  e.codimension...
│ │ │ │ +0003b890: 436f 6469 6d43 6865 636b 4675 6e63 7469  CodimCheckFuncti
│ │ │ │ +0003b8a0: 6f6e 2e20 5468 6520 6f70 7469 6f6e 2043  on. The option C
│ │ │ │ +0003b8b0: 6f64 696d 4368 6563 6b46 756e 6374 696f  odimCheckFunctio
│ │ │ │ +0003b8c0: 6e20 636f 6e74 726f 6c73 2068 6f77 2066  n controls how f
│ │ │ │ +0003b8d0: 7265 7175 656e 746c 7920 7468 650a 6469  requently the.di
│ │ │ │ +0003b8e0: 6d65 6e73 696f 6e20 6f66 2074 6865 2070  mension of the p
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│ │ │ │ +0003b900: 6d69 6e6f 7273 2069 7320 636f 6d70 7574  minors is comput
│ │ │ │ +0003b910: 6564 2e20 2046 6f72 2069 6e73 7461 6e63  ed.  For instanc
│ │ │ │ +0003b920: 652c 2073 6574 7469 6e67 0a43 6f64 696d  e, setting.Codim
│ │ │ │ +0003b930: 4368 6563 6b46 756e 6374 696f 6e20 3d3e  CheckFunction =>
│ │ │ │ +0003b940: 2074 202d 3e20 742f 3520 7769 6c6c 2073   t -> t/5 will s
│ │ │ │ +0003b950: 6179 2069 7420 7368 6f75 6c64 2063 6f6d  ay it should com
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│ │ │ │ +0003b970: 6674 6572 2065 7665 7279 0a35 206d 696e  fter every.5 min
│ │ │ │ +0003b980: 6f72 7320 6172 6520 6578 616d 696e 6564  ors are examined
│ │ │ │ +0003b990: 2e20 2049 6e20 6765 6e65 7261 6c2c 2061  .  In general, a
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│ │ │ │ +0003b9b0: 6f66 2074 6865 2043 6f64 696d 4368 6563  of the CodimChec
│ │ │ │ +0003b9c0: 6b46 756e 6374 696f 6e0a 696e 6372 6561  kFunction.increa
│ │ │ │ +0003b9d0: 7365 7320 6279 2061 6e20 696e 7465 6765  ses by an intege
│ │ │ │ +0003b9e0: 7220 7765 2063 6f6d 7075 7465 2074 6865  r we compute the
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│ │ │ │ +0003ba00: 696e 2e20 2054 6865 2064 6566 6175 6c74  in.  The default
│ │ │ │ +0003ba10: 2066 756e 6374 696f 6e0a 6861 7320 7468   function.has th
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│ │ │ │ +0003ba30: 636f 6d70 7574 6174 696f 6e73 2067 726f  computations gro
│ │ │ │ +0003ba40: 7720 6578 706f 6e65 6e74 6961 6c6c 792e  w exponentially.
│ │ │ │ +0003ba50: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │ +0003ba60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ba70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ba80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ba90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003baa0: 2b0a 7c69 3130 203a 2074 696d 6520 7265  +.|i10 : time re
│ │ │ │ +0003bab0: 6775 6c61 7249 6e43 6f64 696d 656e 7369  gularInCodimensi
│ │ │ │ +0003bac0: 6f6e 2831 2c20 532f 4a2c 204d 6178 4d69  on(1, S/J, MaxMi
│ │ │ │ +0003bad0: 6e6f 7273 3d3e 3235 2c20 436f 6469 6d43  nors=>25, CodimC
│ │ │ │ +0003bae0: 6865 636b 4675 6e63 7469 6f6e 203d 3e20  heckFunction => 
│ │ │ │ +0003baf0: 7c0a 7c20 2d2d 2075 7365 6420 302e 3735  |.| -- used 0.75
│ │ │ │ +0003bb00: 3439 3331 7320 2863 7075 293b 2030 2e35  4931s (cpu); 0.5
│ │ │ │ +0003bb10: 3430 3232 3273 2028 7468 7265 6164 293b  40222s (thread);
│ │ │ │ +0003bb20: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │ +0003bb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003bb40: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003bb50: 6d65 6e73 696f 6e3a 2072 696e 6720 6469  mension: ring di
│ │ │ │ +0003bb60: 6d65 6e73 696f 6e20 3d34 2c20 7468 6572  mension =4, ther
│ │ │ │ +0003bb70: 6520 6172 6520 3134 3635 3132 3820 706f  e are 1465128 po
│ │ │ │ +0003bb80: 7373 6962 6c65 2035 2062 7920 3520 6d69  ssible 5 by 5 mi
│ │ │ │ +0003bb90: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003bba0: 6d65 6e73 696f 6e3a 2041 626f 7574 2074  mension: About t
│ │ │ │ +0003bbb0: 6f20 656e 7465 7220 6c6f 6f70 2020 2020  o enter loop    
│ │ │ │ +0003bbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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
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│ │ │ │ -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       
│ │ │ │ -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       
│ │ │ │ +0003bc30: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
│ │ │ │ +0003bc40: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67  eMinor: Choosing
│ │ │ │ +0003bc50: 2047 5265 764c 6578 536d 616c 6c65 7374   GRevLexSmallest
│ │ │ │ +0003bc60: 5465 726d 2020 2020 2020 2020 2020 2020  Term            
│ │ │ │ +0003bc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003bc80: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003bc90: 6d65 6e73 696f 6e3a 2020 4c6f 6f70 2073  mension:  Loop s
│ │ │ │ +0003bca0: 7465 702c 2061 626f 7574 2074 6f20 636f  tep, about to co
│ │ │ │ +0003bcb0: 6d70 7574 6520 6469 6d65 6e73 696f 6e2e  mpute dimension.
│ │ │ │ +0003bcc0: 2020 5375 626d 6174 7269 6365 7320 636f    Submatrices co
│ │ │ │ +0003bcd0: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003bce0: 6d65 6e73 696f 6e3a 2020 6973 436f 6469  mension:  isCodi
│ │ │ │ +0003bcf0: 6d41 744c 6561 7374 2066 6169 6c65 642c  mAtLeast failed,
│ │ │ │ +0003bd00: 2063 6f6d 7075 7469 6e67 2063 6f64 696d   computing codim
│ │ │ │ +0003bd10: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ +0003bd20: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003bd30: 6d65 6e73 696f 6e3a 2020 7061 7274 6961  mension:  partia
│ │ │ │ +0003bd40: 6c20 7369 6e67 756c 6172 206c 6f63 7573  l singular locus
│ │ │ │ +0003bd50: 2064 696d 656e 7369 6f6e 2063 6f6d 7075   dimension compu
│ │ │ │ +0003bd60: 7465 642c 203d 2034 2020 2020 2020 2020  ted, = 4        
│ │ │ │ +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: 2047 5265 764c 6578 536d 616c 6c65 7374   GRevLexSmallest
│ │ │ │  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   
│ │ │ │ -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           
│ │ │ │ +0003be10: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
│ │ │ │ +0003be20: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67  eMinor: Choosing
│ │ │ │ +0003be30: 2052 616e 646f 6d20 2020 2020 2020 2020   Random         
│ │ │ │ +0003be40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003be50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003be60: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003be70: 6d65 6e73 696f 6e3a 2020 4c6f 6f70 2073  mension:  Loop s
│ │ │ │ +0003be80: 7465 702c 2061 626f 7574 2074 6f20 636f  tep, about to co
│ │ │ │ +0003be90: 6d70 7574 6520 6469 6d65 6e73 696f 6e2e  mpute dimension.
│ │ │ │ +0003bea0: 2020 5375 626d 6174 7269 6365 7320 636f    Submatrices co
│ │ │ │ +0003beb0: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003bec0: 6d65 6e73 696f 6e3a 2020 6973 436f 6469  mension:  isCodi
│ │ │ │ +0003bed0: 6d41 744c 6561 7374 2066 6169 6c65 642c  mAtLeast failed,
│ │ │ │ +0003bee0: 2063 6f6d 7075 7469 6e67 2063 6f64 696d   computing codim
│ │ │ │ +0003bef0: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ +0003bf00: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003bf10: 6d65 6e73 696f 6e3a 2020 7061 7274 6961  mension:  partia
│ │ │ │ +0003bf20: 6c20 7369 6e67 756c 6172 206c 6f63 7573  l singular locus
│ │ │ │ +0003bf30: 2064 696d 656e 7369 6f6e 2063 6f6d 7075   dimension compu
│ │ │ │ +0003bf40: 7465 642c 203d 2034 2020 2020 2020 2020  ted, = 4        
│ │ │ │ +0003bf50: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
│ │ │ │ +0003bf60: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67  eMinor: Choosing
│ │ │ │ +0003bf70: 2047 5265 764c 6578 536d 616c 6c65 7374   GRevLexSmallest
│ │ │ │ +0003bf80: 5465 726d 2020 2020 2020 2020 2020 2020  Term            
│ │ │ │  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: 204c 6578 536d 616c 6c65 7374 2020 2020   LexSmallest    
│ │ │ │ +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: 2052 616e 646f 6d4e 6f6e 5a65 726f 2020   RandomNonZero  
│ │ │ │ +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       
│ │ │ │ -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   
│ │ │ │ +0003c090: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
│ │ │ │ +0003c0a0: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67  eMinor: Choosing
│ │ │ │ +0003c0b0: 2047 5265 764c 6578 536d 616c 6c65 7374   GRevLexSmallest
│ │ │ │ +0003c0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c0e0: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003c0f0: 6d65 6e73 696f 6e3a 2020 4c6f 6f70 2073  mension:  Loop s
│ │ │ │ +0003c100: 7465 702c 2061 626f 7574 2074 6f20 636f  tep, about to co
│ │ │ │ +0003c110: 6d70 7574 6520 6469 6d65 6e73 696f 6e2e  mpute dimension.
│ │ │ │ +0003c120: 2020 5375 626d 6174 7269 6365 7320 636f    Submatrices co
│ │ │ │ +0003c130: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003c140: 6d65 6e73 696f 6e3a 2020 6973 436f 6469  mension:  isCodi
│ │ │ │ +0003c150: 6d41 744c 6561 7374 2066 6169 6c65 642c  mAtLeast failed,
│ │ │ │ +0003c160: 2063 6f6d 7075 7469 6e67 2063 6f64 696d   computing codim
│ │ │ │ +0003c170: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ +0003c180: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003c190: 6d65 6e73 696f 6e3a 2020 7061 7274 6961  mension:  partia
│ │ │ │ +0003c1a0: 6c20 7369 6e67 756c 6172 206c 6f63 7573  l singular locus
│ │ │ │ +0003c1b0: 2064 696d 656e 7369 6f6e 2063 6f6d 7075   dimension compu
│ │ │ │ +0003c1c0: 7465 642c 203d 2034 2020 2020 2020 2020  ted, = 4        
│ │ │ │ +0003c1d0: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
│ │ │ │ +0003c1e0: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67  eMinor: Choosing
│ │ │ │ +0003c1f0: 204c 6578 536d 616c 6c65 7374 5465 726d   LexSmallestTerm
│ │ │ │ +0003c200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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: 2047 5265 764c 6578 536d 616c 6c65 7374   GRevLexSmallest
│ │ │ │ +0003c250: 5465 726d 2020 2020 2020 2020 2020 2020  Term            
│ │ │ │  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: 204c 6578 536d 616c 6c65 7374 2020 2020   LexSmallest    
│ │ │ │ +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: 2047 5265 764c 6578 536d 616c 6c65 7374   GRevLexSmallest
│ │ │ │ +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           
│ │ │ │ -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       
│ │ │ │ +0003c310: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
│ │ │ │ +0003c320: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67  eMinor: Choosing
│ │ │ │ +0003c330: 2052 616e 646f 6d4e 6f6e 5a65 726f 2020   RandomNonZero  
│ │ │ │ +0003c340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c360: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003c370: 6d65 6e73 696f 6e3a 2020 4c6f 6f70 2073  mension:  Loop s
│ │ │ │ +0003c380: 7465 702c 2061 626f 7574 2074 6f20 636f  tep, about to co
│ │ │ │ +0003c390: 6d70 7574 6520 6469 6d65 6e73 696f 6e2e  mpute dimension.
│ │ │ │ +0003c3a0: 2020 5375 626d 6174 7269 6365 7320 636f    Submatrices co
│ │ │ │ +0003c3b0: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003c3c0: 6d65 6e73 696f 6e3a 2020 6973 436f 6469  mension:  isCodi
│ │ │ │ +0003c3d0: 6d41 744c 6561 7374 2066 6169 6c65 642c  mAtLeast failed,
│ │ │ │ +0003c3e0: 2063 6f6d 7075 7469 6e67 2063 6f64 696d   computing codim
│ │ │ │ +0003c3f0: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ +0003c400: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003c410: 6d65 6e73 696f 6e3a 2020 7061 7274 6961  mension:  partia
│ │ │ │ +0003c420: 6c20 7369 6e67 756c 6172 206c 6f63 7573  l singular locus
│ │ │ │ +0003c430: 2064 696d 656e 7369 6f6e 2063 6f6d 7075   dimension compu
│ │ │ │ +0003c440: 7465 642c 203d 2033 2020 2020 2020 2020  ted, = 3        
│ │ │ │ +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 2020 2020   LexSmallest    
│ │ │ │ +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: 204c 6578 536d 616c 6c65 7374 5465 726d   LexSmallestTerm
│ │ │ │ +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: 2052 616e 646f 6d4e 6f6e 5a65 726f 2020   RandomNonZero  
│ │ │ │ +0003c520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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           
│ │ │ │ -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         
│ │ │ │ +0003c590: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
│ │ │ │ +0003c5a0: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67  eMinor: Choosing
│ │ │ │ +0003c5b0: 2047 5265 764c 6578 536d 616c 6c65 7374   GRevLexSmallest
│ │ │ │ +0003c5c0: 5465 726d 2020 2020 2020 2020 2020 2020  Term            
│ │ │ │ +0003c5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c5e0: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003c5f0: 6d65 6e73 696f 6e3a 2020 4c6f 6f70 2073  mension:  Loop s
│ │ │ │ +0003c600: 7465 702c 2061 626f 7574 2074 6f20 636f  tep, about to co
│ │ │ │ +0003c610: 6d70 7574 6520 6469 6d65 6e73 696f 6e2e  mpute dimension.
│ │ │ │ +0003c620: 2020 5375 626d 6174 7269 6365 7320 636f    Submatrices co
│ │ │ │ +0003c630: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003c640: 6d65 6e73 696f 6e3a 2020 6973 436f 6469  mension:  isCodi
│ │ │ │ +0003c650: 6d41 744c 6561 7374 2066 6169 6c65 642c  mAtLeast failed,
│ │ │ │ +0003c660: 2063 6f6d 7075 7469 6e67 2063 6f64 696d   computing codim
│ │ │ │ +0003c670: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ +0003c680: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003c690: 6d65 6e73 696f 6e3a 2020 7061 7274 6961  mension:  partia
│ │ │ │ +0003c6a0: 6c20 7369 6e67 756c 6172 206c 6f63 7573  l singular locus
│ │ │ │ +0003c6b0: 2064 696d 656e 7369 6f6e 2063 6f6d 7075   dimension compu
│ │ │ │ +0003c6c0: 7465 642c 203d 2033 2020 2020 2020 2020  ted, = 3        
│ │ │ │ +0003c6d0: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
│ │ │ │ +0003c6e0: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67  eMinor: Choosing
│ │ │ │ +0003c6f0: 204c 6578 536d 616c 6c65 7374 2020 2020   LexSmallest    
│ │ │ │ +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: 2052 616e 646f 6d4e 6f6e 5a65 726f 2020   RandomNonZero  
│ │ │ │ +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: 204c 6578 536d 616c 6c65 7374 5465 726d   LexSmallestTerm
│ │ │ │ +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: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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|.|----
│ │ │ │ -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)   
│ │ │ │ -0003c9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c9f0: 2020 2020 2020 2020 207c 0a7c 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                  
│ │ │ │ +0003c810: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73  |.|internalChoos
│ │ │ │ +0003c820: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67  eMinor: Choosing
│ │ │ │ +0003c830: 2047 5265 764c 6578 536d 616c 6c65 7374   GRevLexSmallest
│ │ │ │ +0003c840: 5465 726d 2020 2020 2020 2020 2020 2020  Term            
│ │ │ │ +0003c850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c860: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003c870: 6d65 6e73 696f 6e3a 2020 4c6f 6f70 2073  mension:  Loop s
│ │ │ │ +0003c880: 7465 702c 2061 626f 7574 2074 6f20 636f  tep, about to co
│ │ │ │ +0003c890: 6d70 7574 6520 6469 6d65 6e73 696f 6e2e  mpute dimension.
│ │ │ │ +0003c8a0: 2020 5375 626d 6174 7269 6365 7320 636f    Submatrices co
│ │ │ │ +0003c8b0: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003c8c0: 6d65 6e73 696f 6e3a 2020 6973 436f 6469  mension:  isCodi
│ │ │ │ +0003c8d0: 6d41 744c 6561 7374 2066 6169 6c65 642c  mAtLeast failed,
│ │ │ │ +0003c8e0: 2063 6f6d 7075 7469 6e67 2063 6f64 696d   computing codim
│ │ │ │ +0003c8f0: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ +0003c900: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003c910: 6d65 6e73 696f 6e3a 2020 7061 7274 6961  mension:  partia
│ │ │ │ +0003c920: 6c20 7369 6e67 756c 6172 206c 6f63 7573  l singular locus
│ │ │ │ +0003c930: 2064 696d 656e 7369 6f6e 2063 6f6d 7075   dimension compu
│ │ │ │ +0003c940: 7465 642c 203d 2033 2020 2020 2020 2020  ted, = 3        
│ │ │ │ +0003c950: 7c0a 7c72 6567 756c 6172 496e 436f 6469  |.|regularInCodi
│ │ │ │ +0003c960: 6d65 6e73 696f 6e3a 2020 4c6f 6f70 2063  mension:  Loop c
│ │ │ │ +0003c970: 6f6d 706c 6574 6564 2c20 7375 626d 6174  ompleted, submat
│ │ │ │ +0003c980: 7269 6365 7320 636f 6e73 6964 6572 6564  rices considered
│ │ │ │ +0003c990: 203d 2032 352c 2061 6e64 2063 6f6d 7075   = 25, and compu
│ │ │ │ +0003c9a0: 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.|-------------
│ │ │ │ +0003c9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003c9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003c9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003c9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003c9f0: 7c0a 7c74 2d3e 742f 352c 204d 696e 4d69  |.|t->t/5, MinMi
│ │ │ │ +0003ca00: 6e6f 7273 4675 6e63 7469 6f6e 203d 3e20  norsFunction => 
│ │ │ │ +0003ca10: 742d 3e32 2c20 5665 7262 6f73 653d 3e74  t->2, Verbose=>t
│ │ │ │ +0003ca20: 7275 6529 2020 2020 2020 2020 2020 2020  rue)            
│ │ │ │  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 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003ca50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ca60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +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           |.|    
│ │ │ │ -0003caa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ca90: 7c0a 7c6e 6f72 732c 2077 6520 7769 6c6c  |.|nors, we will
│ │ │ │ +0003caa0: 2063 6f6d 7075 7465 2075 7020 746f 2032   compute up to 2
│ │ │ │ +0003cab0: 3520 6f66 2074 6865 6d2e 2020 2020 2020  5 of them.      
│ │ │ │  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 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003cb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +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           |.|    
│ │ │ │ -0003cbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003cbd0: 7c0a 7c6e 7369 6465 7265 643a 2032 2c20  |.|nsidered: 2, 
│ │ │ │ +0003cbe0: 616e 6420 636f 6d70 7574 6564 203d 2032  and computed = 2
│ │ │ │  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 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003cd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +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           |.|    
│ │ │ │ -0003cdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003cdb0: 7c0a 7c6e 7369 6465 7265 643a 2035 2c20  |.|nsidered: 5, 
│ │ │ │ +0003cdc0: 616e 6420 636f 6d70 7574 6564 203d 2035  and computed = 5
│ │ │ │  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 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003cff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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           |.|    
│ │ │ │ -0003d040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d030: 7c0a 7c6e 7369 6465 7265 643a 2031 302c  |.|nsidered: 10,
│ │ │ │ +0003d040: 2061 6e64 2063 6f6d 7075 7465 6420 3d20   and computed = 
│ │ │ │ +0003d050: 3130 2020 2020 2020 2020 2020 2020 2020  10              
│ │ │ │  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 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003d270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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           |.|    
│ │ │ │ -0003d2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d2b0: 7c0a 7c6e 7369 6465 7265 643a 2031 352c  |.|nsidered: 15,
│ │ │ │ +0003d2c0: 2061 6e64 2063 6f6d 7075 7465 6420 3d20   and computed = 
│ │ │ │ +0003d2d0: 3135 2020 2020 2020 2020 2020 2020 2020  15              
│ │ │ │  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 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003d4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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           |.|    
│ │ │ │ -0003d540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d530: 7c0a 7c6e 7369 6465 7265 643a 2032 302c  |.|nsidered: 20,
│ │ │ │ +0003d540: 2061 6e64 2063 6f6d 7075 7465 6420 3d20   and computed = 
│ │ │ │ +0003d550: 3138 2020 2020 2020 2020 2020 2020 2020  18              
│ │ │ │  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 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +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 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d7b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0003d7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d7b0: 7c0a 7c6e 7369 6465 7265 643a 2032 352c  |.|nsidered: 25,
│ │ │ │ +0003d7c0: 2061 6e64 2063 6f6d 7075 7465 6420 3d20   and computed = 
│ │ │ │ +0003d7d0: 3233 2020 2020 2020 2020 2020 2020 2020  23              
│ │ │ │  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           |.+----
│ │ │ │ -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
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│ │ │ │ -0003da30: 7469 616c 2069 6465 616c 2064 6566 696e  tial ideal defin
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│ │ │ │ -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
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│ │ │ │ -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
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│ │ │ │ -0003db20: 6973 436f 6469 6d41 744c 6561 7374 2061  isCodimAtLeast a
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│ │ │ │ -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=>
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│ │ │ │ -0003dc10: 2075 7365 6420 302e 3438 3230 3737 7320   used 0.482077s 
│ │ │ │ -0003dc20: 2863 7075 293b 2030 2e33 3134 3130 3773  (cpu); 0.314107s
│ │ │ │ -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
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│ │ │ │ -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          
│ │ │ │ -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      
│ │ │ │ +0003d850: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003d860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +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: 7c0a 7c74 6564 203d 2032 332e 2020 7369  |.|ted = 23.  si
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│ │ │ │ +0003d8d0: 6f20 6265 203d 2033 2020 2020 2020 2020  o be = 3        
│ │ │ │ +0003d8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d8f0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0003d900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003d910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003d920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003d930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003d940: 2b0a 0a69 7343 6f64 696d 4174 4c65 6173  +..isCodimAtLeas
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│ │ │ │ +0003d9a0: 7420 6973 436f 6469 6d41 744c 6561 7374  t isCodimAtLeast
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│ │ │ │ +0003da20: 6c6f 6375 730a 6469 6d65 6e73 696f 6e20  locus.dimension 
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│ │ │ │ +0003daa0: 486f 7720 6973 436f 6469 6d41 744c 6561  How isCodimAtLea
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│ │ │ │ +0003dac0: 2062 650a 636f 6e74 726f 6c6c 6564 2076   be.controlled v
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│ │ │ │ +0003dae0: 5061 6972 7346 756e 6374 696f 6e20 616e  PairsFunction an
│ │ │ │ +0003daf0: 6420 5061 6972 4c69 6d69 742c 2077 6869  d PairLimit, whi
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│ │ │ │ +0003db20: 436f 6469 6d41 744c 6561 7374 3a20 6973  CodimAtLeast: is
│ │ │ │ +0003db30: 436f 6469 6d41 744c 6561 7374 2c2e 2020  CodimAtLeast,.  
│ │ │ │ +0003db40: 596f 7520 6361 6e20 666f 7263 6520 7468  You can force th
│ │ │ │ +0003db50: 6520 6675 6e63 7469 6f6e 2074 6f0a 6f6e  e function to.on
│ │ │ │ +0003db60: 6c79 2075 7365 2069 7343 6f64 696d 4174  ly use isCodimAt
│ │ │ │ +0003db70: 4c65 6173 7420 616e 6420 6e6f 7420 6361  Least and not ca
│ │ │ │ +0003db80: 6c6c 2064 696d 656e 7369 6f6e 2062 7920  ll dimension by 
│ │ │ │ +0003db90: 7365 7474 696e 6720 5573 654f 6e6c 7946  setting UseOnlyF
│ │ │ │ +0003dba0: 6173 7443 6f64 696d 203d 3e0a 7472 7565  astCodim =>.true
│ │ │ │ +0003dbb0: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...+------------
│ │ │ │ +0003dbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003dbd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003dbe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003dbf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003dc00: 2d2b 0a7c 6931 3120 3a20 7469 6d65 2072  -+.|i11 : time r
│ │ │ │ +0003dc10: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ +0003dc20: 696f 6e28 312c 2053 2f4a 2c20 4d61 784d  ion(1, S/J, MaxM
│ │ │ │ +0003dc30: 696e 6f72 733d 3e32 352c 2055 7365 4f6e  inors=>25, UseOn
│ │ │ │ +0003dc40: 6c79 4661 7374 436f 6469 6d20 3d3e 2074  lyFastCodim => t
│ │ │ │ +0003dc50: 727c 0a7c 202d 2d20 7573 6564 2030 2e34  r|.| -- used 0.4
│ │ │ │ +0003dc60: 3933 3333 3673 2028 6370 7529 3b20 302e  93336s (cpu); 0.
│ │ │ │ +0003dc70: 3332 3735 3336 7320 2874 6872 6561 6429  327536s (thread)
│ │ │ │ +0003dc80: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ +0003dc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003dca0: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64   |.|regularInCod
│ │ │ │ +0003dcb0: 696d 656e 7369 6f6e 3a20 7269 6e67 2064  imension: ring d
│ │ │ │ +0003dcc0: 696d 656e 7369 6f6e 203d 342c 2074 6865  imension =4, the
│ │ │ │ +0003dcd0: 7265 2061 7265 2031 3436 3531 3238 2070  re are 1465128 p
│ │ │ │ +0003dce0: 6f73 7369 626c 6520 3520 6279 2035 206d  ossible 5 by 5 m
│ │ │ │ +0003dcf0: 697c 0a7c 7265 6775 6c61 7249 6e43 6f64  i|.|regularInCod
│ │ │ │ +0003dd00: 696d 656e 7369 6f6e 3a20 4162 6f75 7420  imension: About 
│ │ │ │ +0003dd10: 746f 2065 6e74 6572 206c 6f6f 7020 2020  to enter loop   
│ │ │ │ +0003dd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 4752 6576 4c65 7853 6d61 6c6c 6573  g GRevLexSmalles
│ │ │ │ +0003dd70: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │  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 4c65 7853 6d61 6c6c 6573 7420 2020  g LexSmallest   
│ │ │ │ +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 5261 6e64 6f6d 4e6f 6e5a 6572 6f20  g RandomNonZero 
│ │ │ │ +0003de60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 4752 6576 4c65 7853 6d61 6c6c 6573  g GRevLexSmalles
│ │ │ │ +0003deb0: 7454 6572 6d20 2020 2020 2020 2020 2020  tTerm           
│ │ │ │  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 4c65 7853 6d61 6c6c 6573 7420 2020  g LexSmallest   
│ │ │ │ +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          
│ │ │ │ -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  
│ │ │ │ +0003df20: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ +0003df30: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ +0003df40: 6720 5261 6e64 6f6d 2020 2020 2020 2020  g Random        
│ │ │ │ +0003df50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003df60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003df70: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64   |.|regularInCod
│ │ │ │ +0003df80: 696d 656e 7369 6f6e 3a20 204c 6f6f 7020  imension:  Loop 
│ │ │ │ +0003df90: 7374 6570 2c20 6162 6f75 7420 746f 2063  step, about to c
│ │ │ │ +0003dfa0: 6f6d 7075 7465 2064 696d 656e 7369 6f6e  ompute dimension
│ │ │ │ +0003dfb0: 2e20 2053 7562 6d61 7472 6963 6573 2063  .  Submatrices c
│ │ │ │ +0003dfc0: 6f7c 0a7c 7265 6775 6c61 7249 6e43 6f64  o|.|regularInCod
│ │ │ │ +0003dfd0: 696d 656e 7369 6f6e 3a20 2070 6172 7469  imension:  parti
│ │ │ │ +0003dfe0: 616c 2073 696e 6775 6c61 7220 6c6f 6375  al singular locu
│ │ │ │ +0003dff0: 7320 6469 6d65 6e73 696f 6e20 636f 6d70  s dimension comp
│ │ │ │ +0003e000: 7574 6564 2c20 3d20 3420 2020 2020 2020  uted, = 4       
│ │ │ │ +0003e010: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ +0003e020: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ +0003e030: 6720 4c65 7853 6d61 6c6c 6573 7420 2020  g LexSmallest   
│ │ │ │ +0003e040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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: 7454 6572 6d20 2020 2020 2020 2020 2020  tTerm           
│ │ │ │  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      
│ │ │ │ -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          
│ │ │ │ +0003e100: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ +0003e110: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ +0003e120: 6720 4752 6576 4c65 7853 6d61 6c6c 6573  g GRevLexSmalles
│ │ │ │ +0003e130: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ +0003e140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e150: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64   |.|regularInCod
│ │ │ │ +0003e160: 696d 656e 7369 6f6e 3a20 204c 6f6f 7020  imension:  Loop 
│ │ │ │ +0003e170: 7374 6570 2c20 6162 6f75 7420 746f 2063  step, about to c
│ │ │ │ +0003e180: 6f6d 7075 7465 2064 696d 656e 7369 6f6e  ompute dimension
│ │ │ │ +0003e190: 2e20 2053 7562 6d61 7472 6963 6573 2063  .  Submatrices c
│ │ │ │ +0003e1a0: 6f7c 0a7c 7265 6775 6c61 7249 6e43 6f64  o|.|regularInCod
│ │ │ │ +0003e1b0: 696d 656e 7369 6f6e 3a20 2070 6172 7469  imension:  parti
│ │ │ │ +0003e1c0: 616c 2073 696e 6775 6c61 7220 6c6f 6375  al singular locu
│ │ │ │ +0003e1d0: 7320 6469 6d65 6e73 696f 6e20 636f 6d70  s dimension comp
│ │ │ │ +0003e1e0: 7574 6564 2c20 3d20 3420 2020 2020 2020  uted, = 4       
│ │ │ │ +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 7454 6572  g LexSmallestTer
│ │ │ │ +0003e220: 6d20 2020 2020 2020 2020 2020 2020 2020  m               
│ │ │ │  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 4c65 7853 6d61 6c6c 6573 7420 2020  g LexSmallest   
│ │ │ │ +0003e270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 4752 6576 4c65 7853 6d61 6c6c 6573  g GRevLexSmalles
│ │ │ │ +0003e2c0: 7454 6572 6d20 2020 2020 2020 2020 2020  tTerm           
│ │ │ │  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|.|   
│ │ │ │ -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         
│ │ │ │ +0003e2e0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f   |.|internalChoo
│ │ │ │ +0003e2f0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e  seMinor: Choosin
│ │ │ │ +0003e300: 6720 4c65 7853 6d61 6c6c 6573 7454 6572  g LexSmallestTer
│ │ │ │ +0003e310: 6d20 2020 2020 2020 2020 2020 2020 2020  m               
│ │ │ │ +0003e320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e330: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64   |.|regularInCod
│ │ │ │ +0003e340: 696d 656e 7369 6f6e 3a20 204c 6f6f 7020  imension:  Loop 
│ │ │ │ +0003e350: 7374 6570 2c20 6162 6f75 7420 746f 2063  step, about to c
│ │ │ │ +0003e360: 6f6d 7075 7465 2064 696d 656e 7369 6f6e  ompute dimension
│ │ │ │ +0003e370: 2e20 2053 7562 6d61 7472 6963 6573 2063  .  Submatrices c
│ │ │ │ +0003e380: 6f7c 0a7c 7265 6775 6c61 7249 6e43 6f64  o|.|regularInCod
│ │ │ │ +0003e390: 696d 656e 7369 6f6e 3a20 2073 696e 6775  imension:  singu
│ │ │ │ +0003e3a0: 6c61 724c 6f63 7573 2064 696d 656e 7369  larLocus dimensi
│ │ │ │ +0003e3b0: 6f6e 2076 6572 6966 6965 6420 6279 2069  on verified by i
│ │ │ │ +0003e3c0: 7343 6f64 696d 4174 4c65 6173 7420 2020  sCodimAtLeast   
│ │ │ │ +0003e3d0: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64   |.|regularInCod
│ │ │ │ +0003e3e0: 696d 656e 7369 6f6e 3a20 2070 6172 7469  imension:  parti
│ │ │ │ +0003e3f0: 616c 2073 696e 6775 6c61 7220 6c6f 6375  al singular locu
│ │ │ │ +0003e400: 7320 6469 6d65 6e73 696f 6e20 636f 6d70  s dimension comp
│ │ │ │ +0003e410: 7574 6564 2c20 3d20 3220 2020 2020 2020  uted, = 2       
│ │ │ │ +0003e420: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64   |.|regularInCod
│ │ │ │ +0003e430: 696d 656e 7369 6f6e 3a20 204c 6f6f 7020  imension:  Loop 
│ │ │ │ +0003e440: 636f 6d70 6c65 7465 642c 2073 7562 6d61  completed, subma
│ │ │ │ +0003e450: 7472 6963 6573 2063 6f6e 7369 6465 7265  trices considere
│ │ │ │ +0003e460: 6420 3d20 3135 2c20 616e 6420 636f 6d70  d = 15, and comp
│ │ │ │ +0003e470: 757c 0a7c 2020 2020 2020 2020 2020 2020  u|.|            
│ │ │ │ +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            |.|---
│ │ │ │ -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) 
│ │ │ │ -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            |.|   
│ │ │ │ -0003e570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e4c0: 207c 0a7c 6f31 3120 3d20 7472 7565 2020   |.|o11 = true  
│ │ │ │ +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 2020                  
│ │ │ │ +0003e500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e510: 207c 0a7c 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: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003e550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003e560: 2d7c 0a7c 7565 2c20 5665 7262 6f73 653d  -|.|ue, Verbose=
│ │ │ │ +0003e570: 3e74 7275 6529 2020 2020 2020 2020 2020  >true)          
│ │ │ │  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 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0003e5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +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            |.|   
│ │ │ │ -0003e610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e600: 207c 0a7c 6e6f 7273 2c20 7765 2077 696c   |.|nors, we wil
│ │ │ │ +0003e610: 6c20 636f 6d70 7574 6520 7570 2074 6f20  l compute up to 
│ │ │ │ +0003e620: 3235 206f 6620 7468 656d 2e20 2020 2020  25 of them.     
│ │ │ │  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 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0003e890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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            |.|   
│ │ │ │ -0003e8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e8d0: 207c 0a7c 6e73 6964 6572 6564 3a20 372c   |.|nsidered: 7,
│ │ │ │ +0003e8e0: 2061 6e64 2063 6f6d 7075 7465 6420 3d20   and computed = 
│ │ │ │ +0003e8f0: 3720 2020 2020 2020 2020 2020 2020 2020  7               
│ │ │ │  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 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0003ea70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ea80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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            |.|   
│ │ │ │ -0003eac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ead0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003eab0: 207c 0a7c 6e73 6964 6572 6564 3a20 3131   |.|nsidered: 11
│ │ │ │ +0003eac0: 2c20 616e 6420 636f 6d70 7574 6564 203d  , and computed =
│ │ │ │ +0003ead0: 2031 3020 2020 2020 2020 2020 2020 2020   10             
│ │ │ │  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 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0003ec50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ec60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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            |.|   
│ │ │ │ -0003eca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ecb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ec90: 207c 0a7c 6e73 6964 6572 6564 3a20 3135   |.|nsidered: 15
│ │ │ │ +0003eca0: 2c20 616e 6420 636f 6d70 7574 6564 203d  , and computed =
│ │ │ │ +0003ecb0: 2031 3320 2020 2020 2020 2020 2020 2020   13             
│ │ │ │  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            |.+---
│ │ │ │ -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.
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│ │ │ │ -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  ****************
│ │ │ │ +0003ed30: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0003ed40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ed50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ed60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ed70: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0003edd0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
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│ │ │ │ +0003ee00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ee10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  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
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│ │ │ │ -0003f760: 7720 706f 6c79 6e6f 6d69 616c 2072 696e  w polynomial rin
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│ │ │ │ -0003f7b0: 6573 2069 7320 7261 6e64 6f6d 697a 6564  es is randomized
│ │ │ │ -0003f7c0: 2e0a 0a2b 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             |.+--
│ │ │ │ +0003f5a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0003f5b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0003f5c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0003f5d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0003f5e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a  ***********..  *
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│ │ │ │ +0003f600: 2052 3120 3d20 7265 6f72 6465 7250 6f6c   R1 = reorderPol
│ │ │ │ +0003f610: 796e 6f6d 6961 6c52 696e 6728 6f72 6465  ynomialRing(orde
│ │ │ │ +0003f620: 7254 7970 652c 2052 290a 2020 2a20 496e  rType, R).  * In
│ │ │ │ +0003f630: 7075 7473 3a0a 2020 2020 2020 2a20 522c  puts:.      * R,
│ │ │ │ +0003f640: 2061 202a 6e6f 7465 2072 696e 673a 2028   a *note ring: (
│ │ │ │ +0003f650: 4d61 6361 756c 6179 3244 6f63 2952 696e  Macaulay2Doc)Rin
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│ │ │ │ +0003f690: 2073 796d 626f 6c3a 2028 4d61 6361 756c   symbol: (Macaul
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│ │ │ │ +0003f6e0: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
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│ │ │ │ +0003f700: 7269 6e67 3a20 284d 6163 6175 6c61 7932  ring: (Macaulay2
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│ │ │ │ +0003f770: 5468 6973 2066 756e 6374 696f 6e20 7461  This function ta
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│ │ │ │  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 : 
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│ │ │ │ -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  
│ │ │ │ -0003f8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -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
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│ │ │ │ -0003f970: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003f990: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003f9b0: 7c0a 7c6f 3320 3a20 506f 6c79 6e6f 6d69  |.|o3 : Polynomi
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│ │ │ │ -0003f9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003fa00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420  ----------+.|i4 
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│ │ │ │ -0003fa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fa60: 2020 2020 7c0a 7c6f 3420 3d20 7472 7565      |.|o4 = true
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│ │ │ │ -0003fac0: 7c69 3520 3a20 7920 3e20 7a20 2020 2020  |i5 : y > z     
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│ │ │ │ -0003fae0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0003faf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fb10: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20          |.|o5 = 
│ │ │ │ -0003fb20: 6661 6c73 6520 2020 2020 2020 2020 2020  false           
│ │ │ │ -0003fb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0003fb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003fb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0003fb90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0003fba0: 0a7c 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
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│ │ │ │ -000400e0: 2046 6173 744d 696e 6f72 7353 7472 6174   FastMinorsStrat
│ │ │ │ -000400f0: 6567 7954 7574 6f72 6961 6c3a 0a46 6173  egyTutorial:.Fas
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│ │ │ │ -000401a0: 6e20 7468 6973 2063 6173 652c 2061 2073  n this case, a s
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│ │ │ │ -00041550: 5261 6e64 6f6d 3a20 6669 6e64 2072 616e  Random: find ran
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│ │ │ │ -000415c0: 202a 2050 6f69 6e74 733a 2066 696e 6420   * Points: find 
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│ │ │ │ -00041760: 2b2d 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
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│ │ │ │ -000417a0: 7944 6566 6175 6c74 2020 2020 2020 2020  yDefault        
│ │ │ │ -000417b0: 2020 2020 2020 2020 2020 7c0a 7c20 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
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│ │ │ │ -00041810: 2020 2020 207d 7c0a 7c20 2020 2020 2020       }|.|       
│ │ │ │ -00041820: 2020 2020 2020 2020 2020 4752 6576 4c65            GRevLe
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│ │ │ │ -00041840: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00041850: 2020 2020 2020 2020 4752 6576 4c65 7853          GRevLexS
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│ │ │ │ -00041870: 3620 7c0a 7c20 2020 2020 2020 2020 2020  6 |.|           
│ │ │ │ -00041880: 2020 2020 2020 4c65 784c 6172 6765 7374        LexLargest
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│ │ │ │ -000418a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -000418b0: 2020 2020 4c65 7853 6d61 6c6c 6573 7420      LexSmallest 
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│ │ │ │ -000418d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000418e0: 2020 4c65 7853 6d61 6c6c 6573 7454 6572    LexSmallestTer
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│ │ │ │ -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
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│ │ │ │ -00041980: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -00041990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000419b0: 2d2d 2d2d 2b0a 4561 6368 2073 7563 6820  ----+.Each such 
│ │ │ │ -000419c0: 6b65 7920 7368 6f75 6c64 2070 6f69 6e74  key should point
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│ │ │ │ -00041a80: 7261 6e64 6f6d 2073 7562 6d61 7472 6963  random submatric
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│ │ │ │ -00041aa0: 7661 6c75 6573 206f 6620 7468 6f73 6520  values of those 
│ │ │ │ -00041ab0: 6b65 7973 2e20 2046 6f72 2065 7861 6d70  keys.  For examp
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│ │ │ │ -00041f90: 7472 6174 6567 7952 616e 646f 6d3a 2063  trategyRandom: c
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│ │ │ │ -00041fc0: 2020 2a20 5374 7261 7465 6779 506f 696e    * StrategyPoin
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│ │ │ │ -00042050: 6d4e 6f6e 5a65 726f 2073 7562 6d61 7472  mNonZero submatr
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│ │ │ │ -000420b0: 6420 5374 7261 7465 6779 4375 7272 656e  d StrategyCurren
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│ │ │ │ -000420d0: 6564 2e20 2049 740a 6265 6769 6e73 2061  ed.  It.begins a
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│ │ │ │ -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
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│ │ │ │ -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
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│ │ │ │ -000421d0: 652e 2020 466f 7220 6578 616d 706c 653a  e.  For example:
│ │ │ │ -000421e0: 0a2b 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
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│ │ │ │ -00042250: 2831 2c20 542c 2053 7472 6174 6567 793d  (1, T, Strategy=
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│ │ │ │ -00042270: 297c 0a7c 202d 2d20 312e 3139 3036 3473  )|.| -- 1.19064s
│ │ │ │ -00042280: 2065 6c61 7073 6564 2020 2020 2020 2020   elapsed        
│ │ │ │ -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            |.|   
│ │ │ │ -000422c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040d30: 2020 7c0a 7c6f 3220 3d20 7472 7565 2020    |.|o2 = true  
│ │ │ │ +00040d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040d80: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00040d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00040da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00040db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00040dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00040dd0: 2d2d 2b0a 496e 2074 6869 7320 7061 7274  --+.In this part
│ │ │ │ +00040de0: 6963 756c 6172 2065 7861 6d70 6c65 2c20  icular example, 
│ │ │ │ +00040df0: 6f6e 206f 6e65 206d 6163 6869 6e65 2c20  on one machine, 
│ │ │ │ +00040e00: 7765 206c 6973 7420 6176 6572 6167 6520  we list average 
│ │ │ │ +00040e10: 7469 6d65 2074 6f20 636f 6d70 6c65 7469  time to completi
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│ │ │ │ +00040e30: 6520 6162 6f76 6520 7374 7261 7465 6769  e above strategi
│ │ │ │ +00040e40: 6573 2061 6674 6572 2031 3030 2072 756e  es after 100 run
│ │ │ │ +00040e50: 732e 0a20 202a 2053 7472 6174 6567 7944  s..  * StrategyD
│ │ │ │ +00040e60: 6566 6175 6c74 3a20 312e 3635 2073 6563  efault: 1.65 sec
│ │ │ │ +00040e70: 6f6e 6473 0a20 202a 2053 7472 6174 6567  onds.  * Strateg
│ │ │ │ +00040e80: 7952 616e 646f 6d3a 2038 2e33 3220 7365  yRandom: 8.32 se
│ │ │ │ +00040e90: 636f 6e64 730a 2020 2a20 5374 7261 7465  conds.  * Strate
│ │ │ │ +00040ea0: 6779 4465 6661 756c 744e 6f6e 5261 6e64  gyDefaultNonRand
│ │ │ │ +00040eb0: 6f6d 3a20 302e 3939 2073 6563 6f6e 6473  om: 0.99 seconds
│ │ │ │ +00040ec0: 0a20 202a 2053 7472 6174 6567 7950 6f69  .  * StrategyPoi
│ │ │ │ +00040ed0: 6e74 733a 2033 2e32 3720 7365 636f 6e64  nts: 3.27 second
│ │ │ │ +00040ee0: 730a 2020 2a20 5374 7261 7465 6779 4465  s.  * StrategyDe
│ │ │ │ +00040ef0: 6661 756c 7457 6974 6850 6f69 6e74 733a  faultWithPoints:
│ │ │ │ +00040f00: 2033 2e33 370a 526f 7567 686c 7920 7370   3.37.Roughly sp
│ │ │ │ +00040f10: 6561 6b69 6e67 2c20 6865 7572 6973 7469  eaking, heuristi
│ │ │ │ +00040f20: 6373 2074 656e 6420 746f 2070 726f 7669  cs tend to provi
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│ │ │ │ +00040fb0: 616c 736f 2c20 636f 6d70 7574 696e 6720  also, computing 
│ │ │ │ +00040fc0: 7261 6e64 6f6d 206f 7220 7261 7469 6f6e  random or ration
│ │ │ │ +00040fd0: 616c 2070 6f69 6e74 7320 646f 6573 2068  al points does h
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│ │ │ │ +00040ff0: 7320 7479 7069 6361 6c6c 7920 6665 7765  s typically fewe
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│ │ │ │ +00041020: 2069 6620 636f 6d70 7574 696e 670a 6d69   if computing.mi
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│ │ │ │ +00041040: 6174 6567 7950 6f69 6e74 7320 6973 2061  ategyPoints is a
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│ │ │ │ +00041060: 486f 7765 7665 722c 2073 6f6d 6574 696d  However, sometim
│ │ │ │ +00041070: 6573 2074 6861 740a 6e6f 6e2d 7472 6976  es that.non-triv
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│ │ │ │ +00041090: 6174 696f 6e20 7769 6c6c 2062 6563 6f6d  ation will becom
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│ │ │ │ +000410b0: 6162 6f76 6520 6578 616d 706c 652c 2074  above example, t
│ │ │ │ +000410c0: 6865 0a6d 6564 6961 6e20 7469 6d65 2066  he.median time f
│ │ │ │ +000410d0: 6f72 2053 7472 6174 6567 7950 6f69 6e74  or StrategyPoint
│ │ │ │ +000410e0: 7320 616e 6420 5374 7261 7465 6779 4465  s and StrategyDe
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│ │ │ │ +00041100: 7761 7320 636c 6f73 6520 746f 2031 2e35  was close to 1.5
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│ │ │ │ +00041120: 636f 7570 6c65 2072 756e 7320 696e 2065  couple runs in e
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│ │ │ │ +000411a0: 6173 2077 6520 6e6f 7720 6578 706c 6169  as we now explai
│ │ │ │ +000411b0: 6e2e 2020 496e 0a70 6172 7469 6375 6c61  n.  In.particula
│ │ │ │ +000411c0: 722c 2074 6865 2075 7365 7220 6361 6e20  r, the user can 
│ │ │ │ +000411d0: 6576 656e 2063 7573 746f 6d69 7a65 2074  even customize t
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│ │ │ │ +000411f0: 6564 2e20 2053 6565 2062 656c 6f77 2066  ed.  See below f
│ │ │ │ +00041200: 6f72 2068 6f77 0a74 6f20 6561 7369 6c79  or how.to easily
│ │ │ │ +00041210: 2075 7365 206f 6e6c 7920 6120 7369 6e67   use only a sing
│ │ │ │ +00041220: 6c65 2068 6575 7269 7374 6963 2e20 546f  le heuristic. To
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│ │ │ │ +00041250: 2061 0a48 6173 6854 6162 6c65 2077 6869   a.HashTable whi
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│ │ │ │ +00041270: 2066 6f6c 6c6f 7769 6e67 206b 6579 732e   following keys.
│ │ │ │ +00041280: 0a20 202a 2047 5265 764c 6578 4c61 7267  .  * GRevLexLarg
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│ │ │ │ +000412a0: 2073 7562 6d61 7472 6963 6573 2077 6865   submatrices whe
│ │ │ │ +000412b0: 7265 2065 6163 6820 726f 7720 616e 6420  re each row and 
│ │ │ │ +000412c0: 636f 6c75 6d6e 2068 6173 2061 0a20 2020  column has a.   
│ │ │ │ +000412d0: 206c 6172 6765 2065 6e74 7279 2077 6974   large entry wit
│ │ │ │ +000412e0: 6820 7265 7370 6563 7420 746f 2061 2072  h respect to a r
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│ │ │ │ +00041300: 6572 2e0a 2020 2a20 4752 6576 4c65 7853  er..  * GRevLexS
│ │ │ │ +00041310: 6d61 6c6c 6573 743a 2074 7279 2074 6f20  mallest: try to 
│ │ │ │ +00041320: 6669 6e64 2073 7562 6d61 7472 6963 6573  find submatrices
│ │ │ │ +00041330: 2077 6865 7265 2065 6163 6820 726f 7720   where each row 
│ │ │ │ +00041340: 616e 6420 636f 6c75 6d6e 2068 6173 2061  and column has a
│ │ │ │ +00041350: 0a20 2020 2073 6d61 6c6c 2065 6e74 7279  .    small entry
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│ │ │ │ +00041390: 4c65 7853 6d61 6c6c 6573 7454 6572 6d3a  LexSmallestTerm:
│ │ │ │ +000413a0: 2066 696e 6420 7375 626d 6174 7269 6365   find submatrice
│ │ │ │ +000413b0: 7320 7768 6572 6520 6561 6368 2072 6f77  s where each row
│ │ │ │ +000413c0: 2061 6e64 2063 6f6c 756d 6e20 6861 7320   and column has 
│ │ │ │ +000413d0: 616e 0a20 2020 2065 6e74 7279 2077 6974  an.    entry wit
│ │ │ │ +000413e0: 6820 6120 736d 616c 6c20 7465 726d 2077  h a small term w
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│ │ │ │ +00041400: 2072 616e 646f 6d20 4752 6576 4c65 786f   random GRevLexo
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│ │ │ │ +00041420: 6765 7374 3a20 7472 7920 746f 2066 696e  gest: try to fin
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│ │ │ │ +00041440: 6572 6520 6561 6368 2072 6f77 2061 6e64  ere each row and
│ │ │ │ +00041450: 2063 6f6c 756d 6e20 6861 7320 6120 6c61   column has a la
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│ │ │ │ +00041480: 7261 6e64 6f6d 204c 6578 6f72 6465 722e  random Lexorder.
│ │ │ │ +00041490: 0a20 202a 204c 6578 536d 616c 6c65 7374  .  * LexSmallest
│ │ │ │ +000414a0: 3a20 7472 7920 746f 2066 696e 6420 7375  : try to find su
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│ │ │ │ +000414c0: 6561 6368 2072 6f77 2061 6e64 2063 6f6c  each row and col
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│ │ │ │ +00041570: 7769 7468 2072 6573 7065 6374 2074 6f20  with respect to 
│ │ │ │ +00041580: 6120 7261 6e64 6f6d 204c 6578 6f72 6465  a random Lexorde
│ │ │ │ +00041590: 722e 0a20 202a 2052 616e 646f 6d3a 2066  r..  * Random: f
│ │ │ │ +000415a0: 696e 6420 7261 6e64 6f6d 2073 7562 6d61  ind random subma
│ │ │ │ +000415b0: 7472 6963 6573 200a 2020 2a20 5261 6e64  trices .  * Rand
│ │ │ │ +000415c0: 6f6d 4e6f 6e7a 6572 6f3a 2066 696e 6420  omNonzero: find 
│ │ │ │ +000415d0: 7261 6e64 6f6d 2073 7562 6d61 7472 6963  random submatric
│ │ │ │ +000415e0: 6573 2074 6861 7420 6861 7665 206e 6f6e  es that have non
│ │ │ │ +000415f0: 7a65 726f 2072 6f77 7320 616e 6420 636f  zero rows and co
│ │ │ │ +00041600: 6c75 6d6e 730a 2020 2a20 506f 696e 7473  lumns.  * Points
│ │ │ │ +00041610: 3a20 6669 6e64 2073 7562 6d61 7472 6963  : find submatric
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│ │ │ │ +00041630: 7369 6e67 756c 6172 2061 7420 7468 6520  singular at the 
│ │ │ │ +00041640: 6769 7665 6e20 6964 6561 6c20 6279 0a20  given ideal by. 
│ │ │ │ +00041650: 2020 2066 696e 6469 6e67 2061 2070 6f69     finding a poi
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│ │ │ │ +00041670: 6561 6c20 7661 6e69 7368 6573 2c20 616e  eal vanishes, an
│ │ │ │ +00041680: 6420 6576 616c 7561 7469 6e67 2074 6865  d evaluating the
│ │ │ │ +00041690: 206d 6174 7269 7820 6174 0a20 2020 2074   matrix at.    t
│ │ │ │ +000416a0: 6861 7420 706f 696e 7420 2876 6961 2074  hat point (via t
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│ │ │ │ +000416d0: 5261 6e64 6f6d 506f 696e 7473 2954 6f70  RandomPoints)Top
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│ │ │ │ +00041720: 7365 6c65 6374 2072 616e 646f 6d0a 2020  select random.  
│ │ │ │ +00041730: 2020 7375 626d 6174 7269 6365 732e 2020    submatrices.  
│ │ │ │ +00041740: 546f 2061 6363 6573 7320 6f70 7469 6f6e  To access option
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│ │ │ │ +00041790: 2c20 6f70 7469 6f6e 2e0a 466f 7220 6578  , option..For ex
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│ │ │ │ +000417b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000417c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000417d0: 2d2d 2d2b 0a7c 6933 203a 2070 6565 6b20  ---+.|i3 : peek 
│ │ │ │ +000417e0: 5374 7261 7465 6779 4465 6661 756c 7420  StrategyDefault 
│ │ │ │ +000417f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041800: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00041810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041820: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00041830: 0a7c 6f33 203d 204f 7074 696f 6e54 6162  .|o3 = OptionTab
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│ │ │ │ +00041860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041870: 2047 5265 764c 6578 536d 616c 6c65 7374   GRevLexSmallest
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│ │ │ │ +00041890: 2020 2020 2020 2020 2020 2020 2020 2047                 G
│ │ │ │ +000418a0: 5265 764c 6578 536d 616c 6c65 7374 5465  RevLexSmallestTe
│ │ │ │ +000418b0: 726d 203d 3e20 3136 207c 0a7c 2020 2020  rm => 16 |.|    
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│ │ │ │ -00042760: 746f 7269 616c 7f32 3134 3338 340a 4e6f  torial.214384.No
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│ │ │ │ +00042650: 7343 6f64 696d 4174 4c65 6173 747f 3936  sCodimAtLeast.96
│ │ │ │ +00042660: 3238 350a 4e6f 6465 3a20 6973 4469 6d41  285.Node: isDimA
│ │ │ │ +00042670: 744d 6f73 747f 3130 3435 3236 0a4e 6f64  tMost.104526.Nod
│ │ │ │ +00042680: 653a 2069 7352 616e 6b41 744c 6561 7374  e: isRankAtLeast
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│ │ │ │ +000426a0: 5261 6e6b 4174 4c65 6173 745f 6c70 5f70  RankAtLeast_lp_p
│ │ │ │ +000426b0: 645f 7064 5f70 645f 636d 5468 7265 6164  d_pd_pd_cmThread
│ │ │ │ +000426c0: 733d 3e5f 7064 5f70 645f 7064 5f72 707f  s=>_pd_pd_pd_rp.
│ │ │ │ +000426d0: 3131 3035 3632 0a4e 6f64 653a 204d 6178  110562.Node: Max
│ │ │ │ +000426e0: 4d69 6e6f 7273 7f31 3132 3130 370a 4e6f  Minors.112107.No
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│ │ │ │ +00042700: 7f31 3133 3638 390a 4e6f 6465 3a20 4d6f  .113689.Node: Mo
│ │ │ │ +00042710: 6475 6c75 737f 3131 3436 3833 0a4e 6f64  dulus.114683.Nod
│ │ │ │ +00042720: 653a 2050 6f69 6e74 4f70 7469 6f6e 737f  e: PointOptions.
│ │ │ │ +00042730: 3131 3537 3035 0a4e 6f64 653a 2070 726f  115705.Node: pro
│ │ │ │ +00042740: 6a44 696d 7f31 3139 3737 390a 4e6f 6465  jDim.119779.Node
│ │ │ │ +00042750: 3a20 7265 6375 7273 6976 654d 696e 6f72  : recursiveMinor
│ │ │ │ +00042760: 737f 3132 3532 3132 0a4e 6f64 653a 2072  s.125212.Node: r
│ │ │ │ +00042770: 6567 756c 6172 496e 436f 6469 6d65 6e73  egularInCodimens
│ │ │ │ +00042780: 696f 6e7f 3132 3834 3539 0a4e 6f64 653a  ion.128459.Node:
│ │ │ │ +00042790: 2052 6567 756c 6172 496e 436f 6469 6d65   RegularInCodime
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│ │ │ │ +000427e0: 7261 7465 6779 4465 6661 756c 747f 3236  rategyDefault.26
│ │ │ │ +000427f0: 3137 3337 0a1f 0a45 6e64 2054 6167 2054  1737...End Tag T
│ │ │ │ +00042800: 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 3238 3236 3135   used 0.00282615
│ │ │ │ -00004000: 7320 2863 7075 293b 2030 2e30 3032 3832  s (cpu); 0.00282
│ │ │ │ -00004010: 3235 3673 2028 7468 7265 6164 293b 2030  256s (thread); 0
│ │ │ │ +00003ff0: 2075 7365 6420 302e 3030 3238 3835 3632   used 0.00288562
│ │ │ │ +00004000: 7320 2863 7075 293b 2030 2e30 3032 3838  s (cpu); 0.00288
│ │ │ │ +00004010: 3231 3873 2028 7468 7265 6164 293b 2030  218s (thread); 0
│ │ │ │  00004020: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │  00004030: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00004040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004080: 7c0a 7c6f 3135 203d 2069 6465 616c 2028  |.|o15 = ideal (
│ │ │ │ @@ -1055,17 +1055,17 @@
│ │ │ │  000041e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000041f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20  -------+.|i16 : 
│ │ │ │  00004200: 7469 6d65 204a 3d66 6974 7469 6e67 4964  time J=fittingId
│ │ │ │  00004210: 6561 6c28 322d 312c 636f 6b65 7220 4b33  eal(2-1,coker K3
│ │ │ │  00004220: 293b 2020 2020 2020 2020 2020 2020 2020  );              
│ │ │ │  00004230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004240: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e    |.| -- used 0.
│ │ │ │ -00004250: 3030 3633 3134 3338 7320 2863 7075 293b  00631438s (cpu);
│ │ │ │ -00004260: 2030 2e30 3036 3331 3333 3173 2028 7468   0.00631331s (th
│ │ │ │ -00004270: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │ +00004250: 3030 3635 3234 3634 7320 2863 7075 293b  00652464s (cpu);
│ │ │ │ +00004260: 2030 2e30 3036 3532 3936 7320 2874 6872   0.0065296s (thr
│ │ │ │ +00004270: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 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 6634 3433  |.|o17 = 0x7f443
│ │ │ │ -0000db70: 6330 3661 3466 3020 2020 2020 2020 2020  c06a4f0         
│ │ │ │ +0000db60: 7c0a 7c6f 3137 203d 2030 7837 6639 6664  |.|o17 = 0x7f9fd
│ │ │ │ +0000db70: 3430 3661 3466 3020 2020 2020 2020 2020  406a4f0         
│ │ │ │  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 3766 3266 3665 3435  ary{0x7f7f2f6e45
│ │ │ │ +0000e3c0: 6172 797b 3078 3766 3434 3236 3837 3335  ary{0x7f44268735
│ │ │ │  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...+---
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│ │ │ │  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 6637     |.|o1 = 0x7f7
│ │ │ │ -00015180: 6631 6636 3965 3730 3020 2020 2020 2020  f1f69e700       
│ │ │ │ +00015170: 2020 207c 0a7c 6f31 203d 2030 7837 6634     |.|o1 = 0x7f4
│ │ │ │ +00015180: 3431 3531 3638 3666 3020 2020 2020 2020  4151686f0       
│ │ │ │  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: 6637 6631 6636 3965 3730 3020 2020 2020  f7f1f69e700     
│ │ │ │ +00015250: 6634 3431 3531 3638 3666 3020 2020 2020  f44151686f0     
│ │ │ │  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 3766    |.|o2 = 0x7f7f
│ │ │ │ -000157f0: 3238 3532 3638 6330 2020 2020 2020 2020  285268c0        
│ │ │ │ +000157e0: 2020 7c0a 7c6f 3220 3d20 3078 3766 3434    |.|o2 = 0x7f44
│ │ │ │ +000157f0: 3132 3962 6362 3230 2020 2020 2020 2020  129bcb20        
│ │ │ │  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 6637     |.|o1 = 0x7f7
│ │ │ │ -00016150: 6632 3835 3461 6137 3020 2020 2020 2020  f2854aa70       
│ │ │ │ +00016140: 2020 207c 0a7c 6f31 203d 2030 7837 6634     |.|o1 = 0x7f4
│ │ │ │ +00016150: 3431 3261 3033 6236 3020 2020 2020 2020  412a03b60       
│ │ │ │  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 3766 3262 3039 3332 3430  = 0x7f7f2b093240
│ │ │ │ +00016760: 3d20 3078 3766 3434 3231 3363 3134 3930  = 0x7f44213c1490
│ │ │ │  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 3766    |.|o2 = 0x7f7f
│ │ │ │ -000168e0: 3238 3532 3630 6130 2020 2020 2020 2020  285260a0        
│ │ │ │ +000168d0: 2020 7c0a 7c6f 3220 3d20 3078 3766 3434    |.|o2 = 0x7f44
│ │ │ │ +000168e0: 3132 3962 6332 3330 2020 2020 2020 2020  129bc230        
│ │ │ │  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 6637 6632 3835  .|o3 = 0x7f7f285
│ │ │ │ -00016aa0: 3461 6663 3020 2020 2020 2020 2020 2020  4afc0           
│ │ │ │ +00016a90: 0a7c 6f33 203d 2030 7837 6634 3431 3239  .|o3 = 0x7f44129
│ │ │ │ +00016aa0: 6263 3132 3020 2020 2020 2020 2020 2020  bc120           
│ │ │ │  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 3766 3238  .|o3 = {0x7f7f28
│ │ │ │ -00017540: 3534 6130 3330 2c20 3078 3766 3766 3238  54a030, 0x7f7f28
│ │ │ │ -00017550: 3534 6130 3230 2c20 3078 3766 3766 3238  54a020, 0x7f7f28
│ │ │ │ -00017560: 3534 6130 3130 7d20 2020 2020 2020 2020  54a010}         
│ │ │ │ +00017530: 0a7c 6f33 203d 207b 3078 3766 3434 3132  .|o3 = {0x7f4412
│ │ │ │ +00017540: 6130 3330 6130 2c20 3078 3766 3434 3132  a030a0, 0x7f4412
│ │ │ │ +00017550: 6130 3330 3930 2c20 3078 3766 3434 3132  a03090, 0x7f4412
│ │ │ │ +00017560: 6130 3330 3830 7d20 2020 2020 2020 2020  a03080}         
│ │ │ │  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 3766 3238 3536 3765 3230   {0x7f7f28567e20
│ │ │ │ -00019260: 2c20 3078 3766 3766 3238 3536 3765 3130  , 0x7f7f28567e10
│ │ │ │ -00019270: 2c20 3078 3766 3766 3238 3536 3765 3030  , 0x7f7f28567e00
│ │ │ │ +00019250: 207b 3078 3766 3434 3132 6134 6265 6530   {0x7f4412a4bee0
│ │ │ │ +00019260: 2c20 3078 3766 3434 3132 6134 6265 6430  , 0x7f4412a4bed0
│ │ │ │ +00019270: 2c20 3078 3766 3434 3132 6134 6265 6330  , 0x7f4412a4bec0
│ │ │ │  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 3235 3938  "bar" => 6.92598
│ │ │ │ +0001eeb0: 2262 6172 2220 3d3e 2036 2e39 3133 3436  "bar" => 6.91346
│ │ │ │  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 6637 6632 3835 3236 3566 307d 7c0a  x7f7f285265f0}|.
│ │ │ │ +0001fe60: 7837 6634 3431 3239 6263 3536 307d 7c0a  x7f44129bc560}|.
│ │ │ │  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: 3766 3238 3532 3635 6630 7c0a 7c20 2020  7f285265f0|.|   
│ │ │ │ +0001ff10: 3434 3132 3962 6335 3630 7c0a 7c20 2020  44129bc560|.|   
│ │ │ │  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 3766 3238  |.|o5 = 0x7f7f28
│ │ │ │ -00022b80: 3534 6164 3230 2020 2020 2020 2020 2020  54ad20          
│ │ │ │ +00022b70: 7c0a 7c6f 3520 3d20 3078 3766 3434 3132  |.|o5 = 0x7f4412
│ │ │ │ +00022b80: 6130 3362 6230 2020 2020 2020 2020 2020  a03bb0          
│ │ │ │  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 3766    |.|o6 = 0x7f7f
│ │ │ │ -00022c50: 3238 3534 6164 3230 2020 2020 2020 2020  2854ad20        
│ │ │ │ +00022c40: 2020 7c0a 7c6f 3620 3d20 3078 3766 3434    |.|o6 = 0x7f44
│ │ │ │ +00022c50: 3132 6130 3362 6230 2020 2020 2020 2020  12a03bb0        
│ │ │ │  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 3766 3134 3037 6639 3130  t 0x7f7f1407f910
│ │ │ │ +00024dc0: 7420 3078 3766 3433 6663 3037 6632 3530  t 0x7f43fc07f250
│ │ │ │  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 3766 3134  mory at 0x7f7f14
│ │ │ │ +00024e00: 6d6f 7279 2061 7420 3078 3766 3433 6663  mory at 0x7f43fc
│ │ │ │  00024e10: 3037 6639 3330 2020 2020 2020 2020 2020  07f930          
│ │ │ │  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 3766 3134 3037 6639 3530 2020 2020  7f7f1407f950    
│ │ │ │ +00024e50: 3766 3433 6663 3037 6638 6630 2020 2020  7f43fc07f8f0    
│ │ │ │  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 3766 3134 3037 6639   at 0x7f7f1407f9
│ │ │ │ -00024ea0: 3930 2020 2020 2020 2020 2020 2020 2020  90              
│ │ │ │ +00024e90: 2061 7420 3078 3766 3433 6663 3037 6632   at 0x7f43fc07f2
│ │ │ │ +00024ea0: 3330 2020 2020 2020 2020 2020 2020 2020  30              
│ │ │ │  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 3766  memory at 0x7f7f
│ │ │ │ -00024ee0: 3134 3037 6632 3530 2020 2020 2020 2020  1407f250        
│ │ │ │ +00024ed0: 6d65 6d6f 7279 2061 7420 3078 3766 3433  memory at 0x7f43
│ │ │ │ +00024ee0: 6663 3037 6639 3930 2020 2020 2020 2020  fc07f990        
│ │ │ │  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 3766 3134 3037 6632 3330 2020  0x7f7f1407f230  
│ │ │ │ +00024f20: 3078 3766 3433 6663 3037 6639 3730 2020  0x7f43fc07f970  
│ │ │ │  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 3766 3134 3037  ry at 0x7f7f1407
│ │ │ │ +00024f60: 7279 2061 7420 3078 3766 3433 6663 3037  ry at 0x7f43fc07
│ │ │ │  00024f70: 6639 6230 2020 2020 2020 2020 2020 2020  f9b0            
│ │ │ │  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: 3766 3134 3037 6639 3730 2020 2020 2020  7f1407f970      
│ │ │ │ +00024fb0: 3433 6663 3037 6639 3130 2020 2020 2020  43fc07f910      
│ │ │ │  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 3766 3134 3037 6638 6630  t 0x7f7f1407f8f0
│ │ │ │ +00024ff0: 7420 3078 3766 3433 6663 3037 6639 3530  t 0x7f43fc07f950
│ │ │ │  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: 3766 3238 3532 3634 6630 7d7c 0a2b 2d2d  7f285264f0}|.+--
│ │ │ │ +00025510: 3434 3132 3962 6335 6130 7d7c 0a2b 2d2d  44129bc5a0}|.+--
│ │ │ │  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 3766 3238 3532 3634 6630 207c  0x7f7f285264f0 |
│ │ │ │ +000255d0: 3078 3766 3434 3132 3962 6335 6130 207c  0x7f44129bc5a0 |
│ │ │ │  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 3766 3238 3532 3634 6635 7c0a  0x7f7f285264f5|.
│ │ │ │ +000256a0: 3078 3766 3434 3132 3962 6335 6135 7c0a  0x7f44129bc5a5|.
│ │ │ │  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 3766 3238 3532 3634  5 = 0x7f7f285264
│ │ │ │ -00025730: 6564 7c0a 7c20 2020 2020 2020 2020 2020  ed|.|           
│ │ │ │ +00025720: 3520 3d20 3078 3766 3434 3132 3962 6335  5 = 0x7f44129bc5
│ │ │ │ +00025730: 3964 7c0a 7c20 2020 2020 2020 2020 2020  9d|.|           
│ │ │ │  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 3363 3139 3537 3562 3430 7c0a  0x563c19575b40|.
│ │ │ │ +00026240: 3078 3535 6134 3362 6333 3762 3430 7c0a  0x55a43bc37b40|.
│ │ │ │  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 3766 3238 3532  |o2 = 0x7f7f2852
│ │ │ │ -00026360: 3661 3230 7c0a 7c20 2020 2020 2020 2020  6a20|.|         
│ │ │ │ +00026350: 7c6f 3220 3d20 3078 3766 3434 3132 3962  |o2 = 0x7f44129b
│ │ │ │ +00026360: 6339 3930 7c0a 7c20 2020 2020 2020 2020  c990|.|         
│ │ │ │  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 3766    |.|o2 = 0x7f7f
│ │ │ │ -00026910: 3238 3532 3663 6130 2020 2020 2020 2020  28526ca0        
│ │ │ │ +00026900: 2020 7c0a 7c6f 3220 3d20 3078 3766 3434    |.|o2 = 0x7f44
│ │ │ │ +00026910: 3132 3962 6337 3330 2020 2020 2020 2020  129bc730        
│ │ │ │  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: 3933 352d 302f 3020 2020 2020 2020 2020  935-0/0         
│ │ │ │ +000034c0: 7c6f 3220 3d20 2f74 6d70 2f4d 322d 3230  |o2 = /tmp/M2-20
│ │ │ │ +000034d0: 3935 352d 302f 3020 2020 2020 2020 2020  955-0/0         
│ │ │ │  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 3933 352d 302f 3020 2020 2020  2-15935-0/0     
│ │ │ │ +00003560: 322d 3230 3935 352d 302f 3020 2020 2020  2-20955-0/0     
│ │ │ │  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 3933 352d  = /tmp/M2-15935-
│ │ │ │ +00003680: 3d20 2f74 6d70 2f4d 322d 3230 3935 352d  = /tmp/M2-20955-
│ │ │ │  00003690: 302f 3020 2020 2020 2020 2020 2020 207c  0/0            |
│ │ │ │  000036a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000036b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000036c0: 2020 2020 7c0a 7c6f 3520 3a20 4669 6c65      |.|o5 : File
│ │ │ │  000036d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000036e0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  000036f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ ├── ./usr/share/info/FrobeniusThresholds.info.gz
│ │ │ ├── FrobeniusThresholds.info
│ │ │ │ @@ -2692,16 +2692,16 @@
│ │ │ │  0000a830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a850: 2d2d 2d2d 2d2b 0a7c 6932 3720 3a20 7469  -----+.|i27 : ti
│ │ │ │  0000a860: 6d65 206e 756d 6572 6963 2066 7074 2866  me numeric fpt(f
│ │ │ │  0000a870: 2c20 4465 7074 684f 6653 6561 7263 6820  , DepthOfSearch 
│ │ │ │  0000a880: 3d3e 2033 2c20 4669 6e61 6c41 7474 656d  => 3, FinalAttem
│ │ │ │  0000a890: 7074 203d 3e20 7472 7565 297c 0a7c 202d  pt => true)|.| -
│ │ │ │ -0000a8a0: 2d20 7573 6564 2032 2e32 3135 3532 7320  - used 2.21552s 
│ │ │ │ -0000a8b0: 2863 7075 293b 2031 2e32 3633 3132 7320  (cpu); 1.26312s 
│ │ │ │ +0000a8a0: 2d20 7573 6564 2032 2e34 3130 3039 7320  - used 2.41009s 
│ │ │ │ +0000a8b0: 2863 7075 293b 2031 2e34 3230 3634 7320  (cpu); 1.42064s 
│ │ │ │  0000a8c0: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │  0000a8d0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  0000a8e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0000a8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a920: 2020 2020 2020 207c 0a7c 6f32 3720 3d20         |.|o27 = 
│ │ │ │ @@ -2723,16 +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 3430 3332 7320 2863 7075 293b   1.34032s (cpu);
│ │ │ │ -0000aaa0: 2030 2e38 3139 3233 3873 2028 7468 7265   0.819238s (thre
│ │ │ │ +0000aa90: 2031 2e33 3830 3639 7320 2863 7075 293b   1.38069s (cpu);
│ │ │ │ +0000aaa0: 2030 2e38 3532 3238 3573 2028 7468 7265   0.852285s (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 2e30 3937 3739 7320  - used 1.09779s 
│ │ │ │ -0000ad10: 2863 7075 293b 2030 2e36 3937 3039 3373  (cpu); 0.697093s
│ │ │ │ +0000ad00: 2d20 7573 6564 2031 2e31 3234 3733 7320  - used 1.12473s 
│ │ │ │ +0000ad10: 2863 7075 293b 2030 2e36 3836 3036 3273  (cpu); 0.686062s
│ │ │ │  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 3339  | -- used 0.0039
│ │ │ │ -0000e5d0: 3939 3537 7320 2863 7075 293b 2030 2e30  9957s (cpu); 0.0
│ │ │ │ -0000e5e0: 3034 3134 3439 3973 2028 7468 7265 6164  0414499s (thread
│ │ │ │ +0000e5c0: 7c20 2d2d 2075 7365 6420 302e 3030 3430  | -- used 0.0040
│ │ │ │ +0000e5d0: 3435 3035 7320 2863 7075 293b 2030 2e30  4505s (cpu); 0.0
│ │ │ │ +0000e5e0: 3035 3439 3736 3873 2028 7468 7265 6164  0549768s (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                  
│ │ │ │ @@ -3690,17 +3690,17 @@
│ │ │ │  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 3531 3230 3337 7320 2863 7075 293b  0.512037s (cpu);
│ │ │ │ -0000e710: 2030 2e33 3439 3839 3173 2028 7468 7265   0.349891s (thre
│ │ │ │ -0000e720: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │ +0000e700: 302e 3530 3533 3237 7320 2863 7075 293b  0.505327s (cpu);
│ │ │ │ +0000e710: 2030 2e33 3031 3734 7320 2874 6872 6561   0.30174s (threa
│ │ │ │ +0000e720: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 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                  
│ │ │ │  0000e790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -3805,18 +3805,18 @@
│ │ │ │  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 3330  |.| -- used 0.30
│ │ │ │ -0000ee40: 3236 3739 7320 2863 7075 293b 2030 2e32  2679s (cpu); 0.2
│ │ │ │ -0000ee50: 3030 3137 3373 2028 7468 7265 6164 293b  00173s (thread);
│ │ │ │ -0000ee60: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │ +0000ee30: 7c0a 7c20 2d2d 2075 7365 6420 302e 3335  |.| -- used 0.35
│ │ │ │ +0000ee40: 3937 3936 7320 2863 7075 293b 2030 2e32  9796s (cpu); 0.2
│ │ │ │ +0000ee50: 3233 3135 7320 2874 6872 6561 6429 3b20  2315s (thread); 
│ │ │ │ +0000ee60: 3073 2028 6763 2920 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                  
│ │ │ │  0000eed0: 7c0a 7c6f 3139 203d 2034 3939 2020 2020  |.|o19 = 499    
│ │ │ │ @@ -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 3536  |.| -- used 1.56
│ │ │ │ -0000f070: 3234 3873 2028 6370 7529 3b20 312e 3230  248s (cpu); 1.20
│ │ │ │ -0000f080: 3538 3373 2028 7468 7265 6164 293b 2030  583s (thread); 0
│ │ │ │ -0000f090: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │ +0000f060: 7c0a 7c20 2d2d 2075 7365 6420 312e 3431  |.| -- used 1.41
│ │ │ │ +0000f070: 3736 7320 2863 7075 293b 2031 2e32 3030  76s (cpu); 1.200
│ │ │ │ +0000f080: 3139 7320 2874 6872 6561 6429 3b20 3073  19s (thread); 0s
│ │ │ │ +0000f090: 2028 6763 2920 2020 2020 2020 2020 2020   (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 3239  .| -- used 1.029
│ │ │ │ -0000fb60: 3634 7320 2863 7075 293b 2030 2e36 3333  64s (cpu); 0.633
│ │ │ │ -0000fb70: 3138 3873 2028 7468 7265 6164 293b 2030  188s (thread); 0
│ │ │ │ +0000fb50: 0a7c 202d 2d20 7573 6564 2031 2e32 3736  .| -- used 1.276
│ │ │ │ +0000fb60: 3239 7320 2863 7075 293b 2030 2e37 3333  29s (cpu); 0.733
│ │ │ │ +0000fb70: 3737 3473 2028 7468 7265 6164 293b 2030  774s (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 3332  .| -- used 1.532
│ │ │ │ -0000fcf0: 3939 7320 2863 7075 293b 2030 2e38 3939  99s (cpu); 0.899
│ │ │ │ -0000fd00: 3133 3373 2028 7468 7265 6164 293b 2030  133s (thread); 0
│ │ │ │ -0000fd10: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │ +0000fce0: 0a7c 202d 2d20 7573 6564 2031 2e38 3437  .| -- used 1.847
│ │ │ │ +0000fcf0: 3238 7320 2863 7075 293b 2031 2e30 3735  28s (cpu); 1.075
│ │ │ │ +0000fd00: 3439 7320 2874 6872 6561 6429 3b20 3073  49s (thread); 0s
│ │ │ │ +0000fd10: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │  0000fd20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000fd30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000fd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fd70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  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 2032 2e31 3534  .| -- used 2.154
│ │ │ │ -0000ffc0: 3236 7320 2863 7075 293b 2031 2e37 3938  26s (cpu); 1.798
│ │ │ │ -0000ffd0: 3231 7320 2874 6872 6561 6429 3b20 3073  21s (thread); 0s
│ │ │ │ +0000ffb0: 0a7c 202d 2d20 7573 6564 2031 2e37 3431  .| -- used 1.741
│ │ │ │ +0000ffc0: 3631 7320 2863 7075 293b 2031 2e34 3335  61s (cpu); 1.435
│ │ │ │ +0000ffd0: 3036 7320 2874 6872 6561 6429 3b20 3073  06s (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                 |
│ │ │ │ @@ -4110,18 +4110,18 @@
│ │ │ │  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 3932  .| -- used 0.692
│ │ │ │ -00010150: 3037 3573 2028 6370 7529 3b20 302e 3537  075s (cpu); 0.57
│ │ │ │ -00010160: 3136 3973 2028 7468 7265 6164 293b 2030  169s (thread); 0
│ │ │ │ -00010170: 7320 2867 6329 2020 2020 2020 2020 2020  s (gc)          
│ │ │ │ +00010140: 0a7c 202d 2d20 7573 6564 2030 2e35 3533  .| -- used 0.553
│ │ │ │ +00010150: 3436 3973 2028 6370 7529 3b20 302e 3439  469s (cpu); 0.49
│ │ │ │ +00010160: 3234 3832 7320 2874 6872 6561 6429 3b20  2482s (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                 |
│ │ │ │  000101e0: 0a7c 6f33 3120 3d20 3937 2020 2020 2020  .|o31 = 97
│ │ ├── ./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 2e35 3935 3737 3673 2028  used 0.595776s (
│ │ │ │ -00044a20: 6370 7529 3b20 302e 3335 3935 3435 7320  cpu); 0.359545s 
│ │ │ │ -00044a30: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ -00044a40: 2920 2020 2020 2020 207c 0a7c 2020 2020  )        |.|    
│ │ │ │ +00044a10: 7573 6564 2032 2e30 3133 3638 7320 2863  used 2.01368s (c
│ │ │ │ +00044a20: 7075 293b 2030 2e35 3336 3939 3473 2028  pu); 0.536994s (
│ │ │ │ +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 3031 3037 7320 2863  used 1.80107s (c
│ │ │ │ -00044be0: 7075 293b 2031 2e30 3336 3537 7320 2874  pu); 1.03657s (t
│ │ │ │ +00044bd0: 7573 6564 2033 2e32 3232 3035 7320 2863  used 3.22205s (c
│ │ │ │ +00044be0: 7075 293b 2031 2e30 3430 3634 7320 2874  pu); 1.04064s (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 6e65 7744 6967 7261 7068  {map, newDigraph
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│ │ │ │ +00046a40: 7b6d 6170 2c20 6469 6772 6170 682c 206e  {map, digraph, n
│ │ │ │ +00046a50: 6577 4469 6772 6170 687d 2020 2020 2020  ewDigraph}      
│ │ │ │  00046a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  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 @@
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│ │ │ │ @@ -8782,28 +8782,28 @@
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│ │ │ │ @@ -8830,79 +8830,79 @@
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│ │ │ │  00022bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +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 @@
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│ │ │ │  000230d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000230e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  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
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│ │ │ │ -000246e0: 7468 6573 6520 7265 7072 6573 656e 742e  these represent.
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│ │ │ │ -00024730: 6f74 6865 7220 7368 6f75 6c64 2062 6520  other should be 
│ │ │ │ -00024740: 3620 706f 696e 7473 2077 6974 6820 3320  6 points with 3 
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│ │ │ │ -00024780: 736b 6577 206c 696e 652e 0a0a 5365 6520  skew line...See 
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│ │ │ │ -000247a0: 202a 202a 6e6f 7465 2072 616e 646f 6d50   * *note randomP
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│ │ │ │ -000247c0: 7269 6574 793a 0a20 2020 2072 616e 646f  riety:.    rando
│ │ │ │ -000247d0: 6d50 6f69 6e74 4f6e 5261 7469 6f6e 616c  mPointOnRational
│ │ │ │ -000247e0: 5661 7269 6574 795f 6c70 4964 6561 6c5f  Variety_lpIdeal_
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│ │ │ │ -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
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│ │ │ │ -00024890: 6561 6c29 220a 0a46 6f72 2074 6865 2070  eal)"..For the p
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│ │ │ │ -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 726f 6475 6369 626c  uild/reproducibl
│ │ │ │ -000249b0: 652d 7061 7468 2f6d 6163 6175 6c61 7932  e-path/macaulay2
│ │ │ │ -000249c0: 2d31 2e32 352e 3131 2b64 732f 4d32 2f4d  -1.25.11+ds/M2/M
│ │ │ │ -000249d0: 6163 6175 6c61 7932 2f70 6163 6b61 6765  acaulay2/package
│ │ │ │ -000249e0: 732f 0a47 726f 6562 6e65 7253 7472 6174  s/.GroebnerStrat
│ │ │ │ -000249f0: 612e 6d32 3a31 3031 323a 302e 0a1f 0a46  a.m2:1012:0....F
│ │ │ │ -00024a00: 696c 653a 2047 726f 6562 6e65 7253 7472  ile: GroebnerStr
│ │ │ │ -00024a10: 6174 612e 696e 666f 2c20 4e6f 6465 3a20  ata.info, Node: 
│ │ │ │ -00024a20: 7261 6e64 6f6d 506f 696e 744f 6e52 6174  randomPointOnRat
│ │ │ │ -00024a30: 696f 6e61 6c56 6172 6965 7479 5f6c 7049  ionalVariety_lpI
│ │ │ │ -00024a40: 6465 616c 5f72 702c 204e 6578 743a 2072  deal_rp, Next: r
│ │ │ │ -00024a50: 616e 646f 6d50 6f69 6e74 734f 6e52 6174  andomPointsOnRat
│ │ │ │ -00024a60: 696f 6e61 6c56 6172 6965 7479 5f6c 7049  ionalVariety_lpI
│ │ │ │ -00024a70: 6465 616c 5f63 6d5a 5a5f 7270 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    |.|     ------
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│ │ │ │ +00027df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027e00: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +00027e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027e20: 2020 7c0a 7c20 2020 2020 7420 2074 2020    |.|     t  t  
│ │ │ │ +00027e30: 2d20 7420 7420 202d 2074 2074 2020 202b  - t t  - t t   +
│ │ │ │ +00027e40: 2074 2074 2020 202d 2074 2020 7420 2020   t t   - t  t   
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│ │ │ │ +00027e70: 2020 7c0a 7c20 2020 2020 2031 3720 3720    |.|      17 7 
│ │ │ │ +00027e80: 2020 2031 2039 2020 2020 3820 3136 2020     1 9    8 16  
│ │ │ │ +00027e90: 2020 3120 3232 2020 2020 3131 2031 3320    1 22    11 13 
│ │ │ │ +00027ea0: 2020 2037 2032 3220 3133 2020 2020 3720     7 22 13    7 
│ │ │ │ +00027eb0: 3135 2020 2020 3820 3133 2031 3520 2020  15    8 13 15   
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│ │ │ │ +00027f30: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +00027f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00028050: 2d2d 7c0a 7c20 2020 2020 2020 2032 2020  --|.|        2  
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│ │ │ │ +00028190: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020  --|.|           
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│ │ │ │ +00028270: 2020 2038 2037 2031 3520 2020 2020 2020     8 7 15       
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│ │ │ │ +00028310: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00028380: 3135 2020 2020 3120 3134 2032 3120 2020  15    1 14 21   
│ │ │ │ +00028390: 2037 2031 3420 3133 2032 3120 2020 2020   7 14 13 21     
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│ │ │ │ +000283b0: 2020 2031 3820 3720 2020 2034 2039 2020     18 7    4 9  
│ │ │ │ +000283c0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d    |.|     ------
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│ │ │ │ +00028460: 2020 7c0a 7c20 2020 2020 2031 3120 3136    |.|      11 16
│ │ │ │ +00028470: 2020 2020 3420 3232 2020 2020 3720 3136      4 22    7 16
│ │ │ │ +00028480: 2032 3220 2020 2031 3020 3720 3135 2020   22    10 7 15  
│ │ │ │ +00028490: 2020 3820 3136 2031 3520 2020 2031 2031    8 16 15    1 1
│ │ │ │ +000284a0: 3620 3231 2020 2020 2020 2020 2020 2020  6 21            
│ │ │ │  000284b0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d    |.|     ------
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│ │ │ │  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 2d2d 2d2d  ----------------
│ │ │ │ -00028500: 2d2d 7c0a 7c20 2020 2020 7420 2074 2020  --|.|     t  t  
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│ │ │ │ -00028530: 2074 2074 2020 7420 2020 2d20 7420 7420   t t  t   - t t 
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│ │ │ │ -00028560: 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
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│ │ │ │ +00028580: 3131 2031 3020 2020 2031 3120 3137 2020  11 10    11 17  
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│ │ │ │  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
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│ │ │ │ -00028610: 2020 7420 202d 2074 2020 7420 202d 2074    t  - t  t  - t
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│ │ │ │ -00028670: 3131 2031 3020 2020 2031 3120 3137 2020  11 10    11 17  
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│ │ │ │ +000285f0: 2d2d 7c0a 7c20 2020 2020 7420 2074 2074  --|.|     t  t t
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│ │ │ │ +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
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│ │ │ │ -00028740: 3136 2020 2020 3234 2037 2031 3320 2020  16    24 7 13   
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│ │ │ │ +00028760: 2020 2020 3234 2037 2031 3620 2020 2031      24 7 16    1
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│ │ │ │ +00028780: 3520 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d  5 |.|     ------
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│ │ │ │  000287c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00028840: 2020 2020 3132 2031 3020 2020 2036 2039      12 10    6 9
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│ │ │ │ -00028870: 3520 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d  5 |.|     ------
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│ │ │ │ -000288f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028910: 2020 7c0a 7c20 2020 2020 2020 2031 3820    |.|        18 
│ │ │ │ -00028920: 3120 3231 2020 2020 3138 2037 2031 3320  1 21    18 7 13 
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│ │ │ │ -00028940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028950: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -000289a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000289b0: 2020 7c0a 7c6f 3820 3a20 4964 6561 6c20    |.|o8 : Ideal 
│ │ │ │ -000289c0: 6f66 206b 6b5b 7420 2c20 7420 2c20 7420  of kk[t , t , t 
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│ │ │ │ -000289e0: 7420 202c 2074 2020 2c20 7420 2c20 7420  t  , t  , t , t 
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│ │ │ │ -00028a30: 2031 3820 2020 3234 2020 2031 2020 2033   18   24   1   3
│ │ │ │ -00028a40: 2020 2038 2020 2031 3020 2020 3137 2020     8   10   17  
│ │ │ │ -00028a50: 2032 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d   2|.|-----------
│ │ │ │ +000287d0: 2d2d 7c0a 7c20 2020 2020 2d20 7420 2074  --|.|     - t  t
│ │ │ │ +000287e0: 2074 2020 202d 2074 2020 7420 7420 2074   t   - t  t t  t
│ │ │ │ +000287f0: 2020 2920 2020 2020 2020 2020 2020 2020    )             
│ │ │ │ +00028800: 2020 2020 2020 2020 2020 2020 2020 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 (
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│ │ │ │ +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 
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│ │ │ │ +000299a0: 0a63 6f72 7265 7370 6f6e 6469 6e67 2069  .corresponding i
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│ │ │ │ +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                  
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│ │ │ │ +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 
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│ │ │ │ +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
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│ │ │ │ -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 |.|      --
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│ │ │ │ -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                  
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│ │ │ │ +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          
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│ │ │ │ -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                  
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│ │ │ │ +0002a0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +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                  
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│ │ │ │  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                  
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│ │ │ │ +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
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│ │ │ │ +0002a510: 7320 7468 6520 6964 6561 6c20 6f66 2032  s the ideal of 2
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│ │ │ │ +0002a550: 4361 7665 6174 0a3d 3d3d 3d3d 3d0a 0a54  Caveat.======..T
│ │ │ │ +0002a560: 6869 7320 726f 7574 696e 6520 6578 7065  his routine expe
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│ │ │ │ +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
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│ │ │ │ +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
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│ │ │ │ +0002a7a0: 756d 656e 7420 6973 2069 6e0a 2f62 7569  ument is in./bui
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│ │ │ │ +0002a810: 3a20 4772 6f65 626e 6572 5374 7261 7461  : GroebnerStrata
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│ │ │ │ +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  ---------------|
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│ │ │ │ +0002c080: 6465 636f 6d70 6f73 6520 4a3b 2020 2020  decompose J;    
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│ │ │ │ -0002c120: 3d3d 2032 2020 2020 2020 2020 2020 2020  == 2            
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│ │ │ │ -0002c260: 696d 2020 2020 2020 2020 2020 2020 2020  im              
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│ │ │ │ +0002c250: 0a7c 6f31 3120 3d20 7472 7565 2020 2020  .|o11 = true    
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│ │ │ │ +0002c390: 0a7c 6f31 3220 3d20 7b31 312c 2038 7d20  .|o12 = {11, 8} 
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│ │ │ │ -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
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│ │ │ │ -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 , 
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│ │ │ │ -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              |.+-
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│ │ │ │ +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                  
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│ │ │ │ +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        
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│ │ │ │ -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
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│ │ │ │ -0002ffa0: 6465 616c 2067 6976 6573 2074 6865 2073  deal gives the s
│ │ │ │ -0002ffb0: 7461 6e64 6172 640a 6d6f 6e6f 6d69 616c  tandard.monomial
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│ │ │ │ -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*
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│ │ │ │ -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
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│ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030e80: 2d2d 2d2d 2d2b 0a7c 6934 203a 2074 6169  -----+.|i4 : tai
│ │ │ │ -00030e90: 6c4d 6f6e 6f6d 6961 6c73 284d 2c20 416c  lMonomials(M, Al
│ │ │ │ -00030ea0: 6c53 7461 6e64 6172 6420 3d3e 2074 7275  lStandard => tru
│ │ │ │ -00030eb0: 6529 2020 2020 2020 2020 2020 2020 2020  e)              
│ │ │ │ +00030c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030c50: 2020 2020 207c 0a7c 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 2020 2020 2020 2020 2020                  
│ │ │ │ +00030ca0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00030cb0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00030cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030cd0: 2020 3220 2020 2020 2020 2020 2020 2020    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  ----------------
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│ │ │ │ -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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00031080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00031090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000310a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000310b0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ +000310c0: 2020 2032 2020 2020 2020 2032 2020 2020     2       2    
│ │ │ │ +000310d0: 2032 2020 2033 2020 2020 2020 2020 2020   2   3          
│ │ │ │ +000310e0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +000310f0: 2020 2020 2032 2020 2020 2032 2020 2020       2     2    
│ │ │ │ +00031100: 2032 2020 207c 0a7c 2020 2020 2063 2a64   2   |.|     c*d
│ │ │ │ +00031110: 2c20 6420 7d2c 207b 612a 6320 2c20 622a  , d }, {a*c , b*
│ │ │ │ +00031120: 6320 2c20 6320 2c20 612a 622a 642c 2061  c , c , a*b*d, a
│ │ │ │ +00031130: 2a63 2a64 2c20 622a 632a 642c 2063 2064  *c*d, b*c*d, c d
│ │ │ │ +00031140: 2c20 612a 6420 2c20 622a 6420 2c20 632a  , a*d , b*d , c*
│ │ │ │ +00031150: 6420 2c20 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  ----------------
│ │ │ │ +00031310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00031320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00031330: 2d2d 2d2d 2d2b 0a7c 6935 203a 2074 6169  -----+.|i5 : tai
│ │ │ │ +00031340: 6c4d 6f6e 6f6d 6961 6c73 284d 2c20 625e  lMonomials(M, b^
│ │ │ │ +00031350: 3229 2020 2020 2020 2020 2020 2020 2020  2)              
│ │ │ │  00031360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031380: 2020 2020 207c 0a7c 6f35 203d 207b 612a       |.|o5 = {a*
│ │ │ │ -00031390: 632c 2062 2a63 2c20 6320 2c20 612a 642c  c, b*c, c , a*d,
│ │ │ │ -000313a0: 2062 2a64 2c20 632a 642c 2064 207d 2020   b*d, c*d, d }  
│ │ │ │ +00031380: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00031390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000313a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000313b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000313c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000313d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000313e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000313f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000313e0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +000313f0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │  00031400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031420: 2020 2020 207c 0a7c 6f35 203a 204c 6973       |.|o5 : Lis
│ │ │ │ -00031430: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ -00031440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031420: 2020 2020 207c 0a7c 6f35 203d 207b 612a       |.|o5 = {a*
│ │ │ │ +00031430: 632c 2062 2a63 2c20 6320 2c20 612a 642c  c, b*c, c , a*d,
│ │ │ │ +00031440: 2062 2a64 2c20 632a 642c 2064 207d 2020   b*d, c*d, d }  
│ │ │ │  00031450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031470: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00031480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000314a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000314b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000314c0: 2d2d 2d2d 2d2b 0a7c 6936 203a 2074 6169  -----+.|i6 : tai
│ │ │ │ -000314d0: 6c4d 6f6e 6f6d 6961 6c73 284d 2c20 625e  lMonomials(M, b^
│ │ │ │ -000314e0: 322c 2041 6c6c 5374 616e 6461 7264 3d3e  2, AllStandard=>
│ │ │ │ -000314f0: 7472 7565 2920 2020 2020 2020 2020 2020  true)           
│ │ │ │ +00031470: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00031480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000314a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000314b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000314c0: 2020 2020 207c 0a7c 6f35 203a 204c 6973       |.|o5 : Lis
│ │ │ │ +000314d0: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ +000314e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000314f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031510: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00031520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031560: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00031570: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -00031580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031590: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00031510: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00031520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00031530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00031540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00031550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00031560: 2d2d 2d2d 2d2b 0a7c 6936 203a 2074 6169  -----+.|i6 : tai
│ │ │ │ +00031570: 6c4d 6f6e 6f6d 6961 6c73 284d 2c20 625e  lMonomials(M, b^
│ │ │ │ +00031580: 322c 2041 6c6c 5374 616e 6461 7264 3d3e  2, AllStandard=>
│ │ │ │ +00031590: 7472 7565 2920 2020 2020 2020 2020 2020  true)           
│ │ │ │  000315a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000315b0: 2020 2020 207c 0a7c 6f36 203d 207b 612a       |.|o6 = {a*
│ │ │ │ -000315c0: 622c 2061 2a63 2c20 622a 632c 2063 202c  b, a*c, b*c, c ,
│ │ │ │ -000315d0: 2061 2a64 2c20 622a 642c 2063 2a64 2c20   a*d, b*d, c*d, 
│ │ │ │ -000315e0: 6420 7d20 2020 2020 2020 2020 2020 2020  d }             
│ │ │ │ +000315b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +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 2020 2020 2020 2020 2020                  
│ │ │ │  00031600: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00031610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031610: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  00031620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031630: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  00031640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031650: 2020 2020 207c 0a7c 6f36 203a 204c 6973       |.|o6 : Lis
│ │ │ │ -00031660: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ -00031670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031650: 2020 2020 207c 0a7c 6f36 203d 207b 612a       |.|o6 = {a*
│ │ │ │ +00031660: 622c 2061 2a63 2c20 622a 632c 2063 202c  b, a*c, b*c, c ,
│ │ │ │ +00031670: 2061 2a64 2c20 622a 642c 2063 2a64 2c20   a*d, b*d, c*d, 
│ │ │ │ +00031680: 6420 7d20 2020 2020 2020 2020 2020 2020  d }             
│ │ │ │  00031690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000316a0: 2020 2020 207c 0a2b 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 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000316f0: 2d2d 2d2d 2d2b 0a0a 5365 6520 616c 736f  -----+..See also
│ │ │ │ -00031700: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  .========..  * *
│ │ │ │ -00031710: 6e6f 7465 2073 7461 6e64 6172 644d 6f6e  note standardMon
│ │ │ │ -00031720: 6f6d 6961 6c73 3a20 7374 616e 6461 7264  omials: standard
│ │ │ │ -00031730: 4d6f 6e6f 6d69 616c 732c 202d 2d20 636f  Monomials, -- co
│ │ │ │ -00031740: 6d70 7574 6573 2073 7461 6e64 6172 6420  mputes standard 
│ │ │ │ -00031750: 6d6f 6e6f 6d69 616c 730a 2020 2a20 2a6e  monomials.  * *n
│ │ │ │ -00031760: 6f74 6520 736d 616c 6c65 724d 6f6e 6f6d  ote smallerMonom
│ │ │ │ -00031770: 6961 6c73 3a20 736d 616c 6c65 724d 6f6e  ials: smallerMon
│ │ │ │ -00031780: 6f6d 6961 6c73 2c20 2d2d 2072 6574 7572  omials, -- retur
│ │ │ │ -00031790: 6e73 2074 6865 2073 7461 6e64 6172 6420  ns the standard 
│ │ │ │ -000317a0: 6d6f 6e6f 6d69 616c 730a 2020 2020 736d  monomials.    sm
│ │ │ │ -000317b0: 616c 6c65 7220 6275 7420 6f66 2074 6865  aller but of the
│ │ │ │ -000317c0: 2073 616d 6520 6465 6772 6565 2061 7320   same degree as 
│ │ │ │ -000317d0: 6769 7665 6e20 6d6f 6e6f 6d69 616c 2873  given monomial(s
│ │ │ │ -000317e0: 290a 0a57 6179 7320 746f 2075 7365 2074  )..Ways to use t
│ │ │ │ -000317f0: 6169 6c4d 6f6e 6f6d 6961 6c73 3a0a 3d3d  ailMonomials:.==
│ │ │ │ -00031800: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00031810: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2274  ========..  * "t
│ │ │ │ -00031820: 6169 6c4d 6f6e 6f6d 6961 6c73 2849 6465  ailMonomials(Ide
│ │ │ │ -00031830: 616c 2922 0a20 202a 2022 7461 696c 4d6f  al)".  * "tailMo
│ │ │ │ -00031840: 6e6f 6d69 616c 7328 4964 6561 6c2c 5269  nomials(Ideal,Ri
│ │ │ │ -00031850: 6e67 456c 656d 656e 7429 220a 0a46 6f72  ngElement)"..For
│ │ │ │ -00031860: 2074 6865 2070 726f 6772 616d 6d65 720a   the programmer.
│ │ │ │ -00031870: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00031880: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a  ==..The object *
│ │ │ │ -00031890: 6e6f 7465 2074 6169 6c4d 6f6e 6f6d 6961  note tailMonomia
│ │ │ │ -000318a0: 6c73 3a20 7461 696c 4d6f 6e6f 6d69 616c  ls: tailMonomial
│ │ │ │ -000318b0: 732c 2069 7320 6120 2a6e 6f74 6520 6d65  s, is a *note me
│ │ │ │ -000318c0: 7468 6f64 2066 756e 6374 696f 6e20 7769  thod function wi
│ │ │ │ -000318d0: 7468 0a6f 7074 696f 6e73 3a20 284d 6163  th.options: (Mac
│ │ │ │ -000318e0: 6175 6c61 7932 446f 6329 4d65 7468 6f64  aulay2Doc)Method
│ │ │ │ -000318f0: 4675 6e63 7469 6f6e 5769 7468 4f70 7469  FunctionWithOpti
│ │ │ │ -00031900: 6f6e 732c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d  ons,...---------
│ │ │ │ -00031910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031950: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572  ------..The sour
│ │ │ │ -00031960: 6365 206f 6620 7468 6973 2064 6f63 756d  ce of this docum
│ │ │ │ -00031970: 656e 7420 6973 2069 6e0a 2f62 7569 6c64  ent is in./build
│ │ │ │ -00031980: 2f72 6570 726f 6475 6369 626c 652d 7061  /reproducible-pa
│ │ │ │ -00031990: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32  th/macaulay2-1.2
│ │ │ │ -000319a0: 352e 3131 2b64 732f 4d32 2f4d 6163 6175  5.11+ds/M2/Macau
│ │ │ │ -000319b0: 6c61 7932 2f70 6163 6b61 6765 732f 0a47  lay2/packages/.G
│ │ │ │ -000319c0: 726f 6562 6e65 7253 7472 6174 612e 6d32  roebnerStrata.m2
│ │ │ │ -000319d0: 3a35 3635 3a30 2e0a 1f0a 5461 6720 5461  :565:0....Tag Ta
│ │ │ │ -000319e0: 626c 653a 0a4e 6f64 653a 2054 6f70 7f32  ble:.Node: Top.2
│ │ │ │ -000319f0: 3537 0a4e 6f64 653a 2041 6c6c 5374 616e  57.Node: AllStan
│ │ │ │ -00031a00: 6461 7264 7f35 3037 3839 0a4e 6f64 653a  dard.50789.Node:
│ │ │ │ -00031a10: 2066 696e 6457 6569 6768 7443 6f6e 7374   findWeightConst
│ │ │ │ -00031a20: 7261 696e 7473 7f35 3237 3430 0a4e 6f64  raints.52740.Nod
│ │ │ │ -00031a30: 653a 2066 696e 6457 6569 6768 7456 6563  e: findWeightVec
│ │ │ │ -00031a40: 746f 727f 3631 3535 300a 4e6f 6465 3a20  tor.61550.Node: 
│ │ │ │ -00031a50: 6772 6f65 626e 6572 4661 6d69 6c79 7f36  groebnerFamily.6
│ │ │ │ -00031a60: 3632 3630 0a4e 6f64 653a 2067 726f 6562  6260.Node: groeb
│ │ │ │ -00031a70: 6e65 7253 7472 6174 756d 7f31 3130 3331  nerStratum.11031
│ │ │ │ -00031a80: 370a 4e6f 6465 3a20 6c69 6e65 6172 5061  7.Node: linearPa
│ │ │ │ -00031a90: 7274 7f31 3139 3538 330a 4e6f 6465 3a20  rt.119583.Node: 
│ │ │ │ -00031aa0: 4d69 6e69 6d61 6c69 7a65 7f31 3231 3638  Minimalize.12168
│ │ │ │ -00031ab0: 340a 4e6f 6465 3a20 6e6f 6e6d 696e 696d  4.Node: nonminim
│ │ │ │ -00031ac0: 616c 4d61 7073 7f31 3233 3039 320a 4e6f  alMaps.123092.No
│ │ │ │ -00031ad0: 6465 3a20 7261 6e64 6f6d 506f 696e 744f  de: randomPointO
│ │ │ │ -00031ae0: 6e52 6174 696f 6e61 6c56 6172 6965 7479  nRationalVariety
│ │ │ │ -00031af0: 5f6c 7049 6465 616c 5f72 707f 3135 3030  _lpIdeal_rp.1500
│ │ │ │ -00031b00: 3133 0a4e 6f64 653a 2072 616e 646f 6d50  13.Node: randomP
│ │ │ │ -00031b10: 6f69 6e74 734f 6e52 6174 696f 6e61 6c56  ointsOnRationalV
│ │ │ │ -00031b20: 6172 6965 7479 5f6c 7049 6465 616c 5f63  ariety_lpIdeal_c
│ │ │ │ -00031b30: 6d5a 5a5f 7270 7f31 3733 3933 300a 4e6f  mZZ_rp.173930.No
│ │ │ │ -00031b40: 6465 3a20 736d 616c 6c65 724d 6f6e 6f6d  de: smallerMonom
│ │ │ │ -00031b50: 6961 6c73 7f31 3839 3638 330a 4e6f 6465  ials.189683.Node
│ │ │ │ -00031b60: 3a20 7374 616e 6461 7264 4d6f 6e6f 6d69  : standardMonomi
│ │ │ │ -00031b70: 616c 737f 3139 3334 3236 0a4e 6f64 653a  als.193426.Node:
│ │ │ │ -00031b80: 2074 6169 6c4d 6f6e 6f6d 6961 6c73 7f31   tailMonomials.1
│ │ │ │ -00031b90: 3937 3635 340a 1f0a 456e 6420 5461 6720  97654...End Tag 
│ │ │ │ -00031ba0: 5461 626c 650a                           Table.
│ │ │ │ +000316a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000316b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000316c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000316d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000316e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000316f0: 2020 2020 207c 0a7c 6f36 203a 204c 6973       |.|o6 : Lis
│ │ │ │ +00031700: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ +00031710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031740: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00031750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00031760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00031770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00031780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00031790: 2d2d 2d2d 2d2b 0a0a 5365 6520 616c 736f  -----+..See also
│ │ │ │ +000317a0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  .========..  * *
│ │ │ │ +000317b0: 6e6f 7465 2073 7461 6e64 6172 644d 6f6e  note standardMon
│ │ │ │ +000317c0: 6f6d 6961 6c73 3a20 7374 616e 6461 7264  omials: standard
│ │ │ │ +000317d0: 4d6f 6e6f 6d69 616c 732c 202d 2d20 636f  Monomials, -- co
│ │ │ │ +000317e0: 6d70 7574 6573 2073 7461 6e64 6172 6420  mputes standard 
│ │ │ │ +000317f0: 6d6f 6e6f 6d69 616c 730a 2020 2a20 2a6e  monomials.  * *n
│ │ │ │ +00031800: 6f74 6520 736d 616c 6c65 724d 6f6e 6f6d  ote smallerMonom
│ │ │ │ +00031810: 6961 6c73 3a20 736d 616c 6c65 724d 6f6e  ials: smallerMon
│ │ │ │ +00031820: 6f6d 6961 6c73 2c20 2d2d 2072 6574 7572  omials, -- retur
│ │ │ │ +00031830: 6e73 2074 6865 2073 7461 6e64 6172 6420  ns the standard 
│ │ │ │ +00031840: 6d6f 6e6f 6d69 616c 730a 2020 2020 736d  monomials.    sm
│ │ │ │ +00031850: 616c 6c65 7220 6275 7420 6f66 2074 6865  aller but of the
│ │ │ │ +00031860: 2073 616d 6520 6465 6772 6565 2061 7320   same degree as 
│ │ │ │ +00031870: 6769 7665 6e20 6d6f 6e6f 6d69 616c 2873  given monomial(s
│ │ │ │ +00031880: 290a 0a57 6179 7320 746f 2075 7365 2074  )..Ways to use t
│ │ │ │ +00031890: 6169 6c4d 6f6e 6f6d 6961 6c73 3a0a 3d3d  ailMonomials:.==
│ │ │ │ +000318a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000318b0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2274  ========..  * "t
│ │ │ │ +000318c0: 6169 6c4d 6f6e 6f6d 6961 6c73 2849 6465  ailMonomials(Ide
│ │ │ │ +000318d0: 616c 2922 0a20 202a 2022 7461 696c 4d6f  al)".  * "tailMo
│ │ │ │ +000318e0: 6e6f 6d69 616c 7328 4964 6561 6c2c 5269  nomials(Ideal,Ri
│ │ │ │ +000318f0: 6e67 456c 656d 656e 7429 220a 0a46 6f72  ngElement)"..For
│ │ │ │ +00031900: 2074 6865 2070 726f 6772 616d 6d65 720a   the 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  ----------------
│ │ │ │ +000319e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000319f0: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572  ------..The sour
│ │ │ │ +00031a00: 6365 206f 6620 7468 6973 2064 6f63 756d  ce of this docum
│ │ │ │ +00031a10: 656e 7420 6973 2069 6e0a 2f62 7569 6c64  ent is in./build
│ │ │ │ +00031a20: 2f72 6570 726f 6475 6369 626c 652d 7061  /reproducible-pa
│ │ │ │ +00031a30: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32  th/macaulay2-1.2
│ │ │ │ +00031a40: 352e 3131 2b64 732f 4d32 2f4d 6163 6175  5.11+ds/M2/Macau
│ │ │ │ +00031a50: 6c61 7932 2f70 6163 6b61 6765 732f 0a47  lay2/packages/.G
│ │ │ │ +00031a60: 726f 6562 6e65 7253 7472 6174 612e 6d32  roebnerStrata.m2
│ │ │ │ +00031a70: 3a35 3635 3a30 2e0a 1f0a 5461 6720 5461  :565:0....Tag Ta
│ │ │ │ +00031a80: 626c 653a 0a4e 6f64 653a 2054 6f70 7f32  ble:.Node: Top.2
│ │ │ │ +00031a90: 3537 0a4e 6f64 653a 2041 6c6c 5374 616e  57.Node: AllStan
│ │ │ │ +00031aa0: 6461 7264 7f35 3037 3839 0a4e 6f64 653a  dard.50789.Node:
│ │ │ │ +00031ab0: 2066 696e 6457 6569 6768 7443 6f6e 7374   findWeightConst
│ │ │ │ +00031ac0: 7261 696e 7473 7f35 3237 3430 0a4e 6f64  raints.52740.Nod
│ │ │ │ +00031ad0: 653a 2066 696e 6457 6569 6768 7456 6563  e: findWeightVec
│ │ │ │ +00031ae0: 746f 727f 3631 3535 300a 4e6f 6465 3a20  tor.61550.Node: 
│ │ │ │ +00031af0: 6772 6f65 626e 6572 4661 6d69 6c79 7f36  groebnerFamily.6
│ │ │ │ +00031b00: 3632 3630 0a4e 6f64 653a 2067 726f 6562  6260.Node: groeb
│ │ │ │ +00031b10: 6e65 7253 7472 6174 756d 7f31 3130 3331  nerStratum.11031
│ │ │ │ +00031b20: 370a 4e6f 6465 3a20 6c69 6e65 6172 5061  7.Node: linearPa
│ │ │ │ +00031b30: 7274 7f31 3139 3538 330a 4e6f 6465 3a20  rt.119583.Node: 
│ │ │ │ +00031b40: 4d69 6e69 6d61 6c69 7a65 7f31 3231 3638  Minimalize.12168
│ │ │ │ +00031b50: 340a 4e6f 6465 3a20 6e6f 6e6d 696e 696d  4.Node: nonminim
│ │ │ │ +00031b60: 616c 4d61 7073 7f31 3233 3039 320a 4e6f  alMaps.123092.No
│ │ │ │ +00031b70: 6465 3a20 7261 6e64 6f6d 506f 696e 744f  de: randomPointO
│ │ │ │ +00031b80: 6e52 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 2e30        |.| -- 3.0
│ │ │ │ -00000d60: 3031 3931 7320 656c 6170 7365 6420 2020  0191s elapsed   
│ │ │ │ +00000d50: 2020 2020 2020 7c0a 7c20 2d2d 2031 2e39        |.| -- 1.9
│ │ │ │ +00000d60: 3836 3434 7320 656c 6170 7365 6420 2020  8644s 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 2e30 3839 3136 7320 656c  | -- 2.08916s el
│ │ │ │ +00000ff0: 7c20 2d2d 2031 2e36 3336 3032 7320 656c  | -- 1.63602s 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/Hadamard.info.gz
│ │ │ ├── Hadamard.info
│ │ │ │ @@ -580,1432 +580,1442 @@
│ │ │ │  00002430: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00002440: 2020 2020 2020 2020 2031 2020 2020 2020           1      
│ │ │ │  00002450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002480: 2020 2020 7c0a 7c6f 3220 3d20 7b50 6f69      |.|o2 = {Poi
│ │ │ │  00002490: 6e74 7b31 2c20 302c 202d 7d2c 2050 6f69  nt{1, 0, -}, Poi
│ │ │ │ -000024a0: 6e74 7b31 2c20 382c 2036 347d 2c20 506f  nt{1, 8, 64}, Po
│ │ │ │ -000024b0: 696e 747b 312c 2034 2c20 387d 2c20 506f  int{1, 4, 8}, Po
│ │ │ │ -000024c0: 696e 747b 312c 2030 2c20 3136 7d2c 2020  int{1, 0, 16},  
│ │ │ │ +000024a0: 6e74 7b31 2c20 302c 2032 7d2c 2050 6f69  nt{1, 0, 2}, Poi
│ │ │ │ +000024b0: 6e74 7b31 2c20 322c 2031 7d2c 2050 6f69  nt{1, 2, 1}, Poi
│ │ │ │ +000024c0: 6e74 7b31 2c20 302c 2034 7d2c 2020 2020  nt{1, 0, 4},    
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│ │ │ │ -000024e0: 2020 2020 2020 2020 2034 2020 2020 2020           4      
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│ │ │ │  00002500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002520: 2020 2020 7c0a 7c20 2020 2020 2d2d 2d2d      |.|     ----
│ │ │ │  00002530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002570: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020  ----|.|         
│ │ │ │ -00002580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002590: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ +00002580: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ +00002590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000025a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000025b0: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ +000025b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000025c0: 2020 2020 7c0a 7c20 2020 2020 506f 696e      |.|     Poin
│ │ │ │ -000025d0: 747b 312c 2030 2c20 317d 2c20 506f 696e  t{1, 0, 1}, Poin
│ │ │ │ -000025e0: 747b 312c 2031 2c20 2d7d 2c20 506f 696e  t{1, 1, -}, Poin
│ │ │ │ -000025f0: 747b 312c 2030 2c20 327d 2c20 506f 696e  t{1, 0, 2}, Poin
│ │ │ │ -00002600: 747b 312c 2030 2c20 2d7d 2c20 506f 696e  t{1, 0, -}, Poin
│ │ │ │ -00002610: 747b 312c 7c0a 7c20 2020 2020 2020 2020  t{1,|.|         
│ │ │ │ -00002620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002630: 2020 2020 2020 2020 3820 2020 2020 2020          8       
│ │ │ │ +000025d0: 747b 312c 2030 2c20 2d7d 2c20 506f 696e  t{1, 0, -}, Poin
│ │ │ │ +000025e0: 747b 312c 2038 2c20 3634 7d2c 2050 6f69  t{1, 8, 64}, Poi
│ │ │ │ +000025f0: 6e74 7b31 2c20 342c 2038 7d2c 2050 6f69  nt{1, 4, 8}, Poi
│ │ │ │ +00002600: 6e74 7b31 2c20 302c 2031 367d 2c20 2020  nt{1, 0, 16},   
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│ │ │ │ +00002620: 2020 2020 2020 2020 3420 2020 2020 2020          4       
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│ │ │ │ -00002650: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +00002650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002660: 2020 2020 7c0a 7c20 2020 2020 2d2d 2d2d      |.|     ----
│ │ │ │  00002670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000026a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000026b0: 2d2d 2d2d 7c0a 7c20 2020 2020 322c 2031  ----|.|     2, 1
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│ │ │ │ +000026b0: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020  ----|.|         
│ │ │ │ +000026c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000026d0: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │  000026e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000026f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002700: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00002710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00002700: 2020 2020 7c0a 7c20 2020 2020 506f 696e      |.|     Poin
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│ │ │ │ +00002720: 747b 312c 2031 2c20 2d7d 7d20 2020 2020  t{1, 1, -}}     
│ │ │ │  00002730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002750: 2020 2020 7c0a 7c6f 3220 3a20 4c69 7374      |.|o2 : List
│ │ │ │ +00002750: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00002760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00002770: 2020 2020 2020 2020 3820 2020 2020 2020          8       
│ │ │ │  00002780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000027a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -000027b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000027c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000027d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000027e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000027f0: 2d2d 2d2d 2b0a 0a57 6179 7320 746f 2075  ----+..Ways to u
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│ │ │ │ -00002810: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00002820: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ -00002830: 6f74 6520 6861 6461 6d61 7264 506f 7765  ote hadamardPowe
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│ │ │ │ -00002850: 6d61 7264 506f 7765 725f 6c70 4c69 7374  mardPower_lpList
│ │ │ │ -00002860: 5f63 6d5a 5a5f 7270 2c20 2d2d 2063 6f6d  _cmZZ_rp, -- com
│ │ │ │ -00002870: 7075 7465 7320 7468 650a 2020 2020 2472  putes the.    $r
│ │ │ │ -00002880: 242d 7468 2048 6164 6d61 7264 2070 6f77  $-th Hadmard pow
│ │ │ │ -00002890: 6572 7320 6f66 2061 2073 6574 2070 6f69  ers of a set poi
│ │ │ │ -000028a0: 6e74 730a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  nts.------------
│ │ │ │ -000028b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000028c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000028d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000028e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000028f0: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520  ---..The source 
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│ │ │ │ -00002940: 312b 6473 2f4d 322f 4d61 6361 756c 6179  1+ds/M2/Macaulay
│ │ │ │ -00002950: 322f 7061 636b 6167 6573 2f48 6164 616d  2/packages/Hadam
│ │ │ │ -00002960: 6172 642e 6d32 0a3a 3339 383a 302e 0a1f  ard.m2.:398:0...
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│ │ │ │ -000029b0: 6374 5f6c 7049 6465 616c 5f63 6d49 6465  ct_lpIdeal_cmIde
│ │ │ │ -000029c0: 616c 5f72 702c 2050 7265 763a 2068 6164  al_rp, Prev: had
│ │ │ │ -000029d0: 616d 6172 6450 6f77 6572 5f6c 704c 6973  amardPower_lpLis
│ │ │ │ -000029e0: 745f 636d 5a5a 5f72 702c 2055 703a 2054  t_cmZZ_rp, Up: T
│ │ │ │ -000029f0: 6f70 0a0a 6861 6461 6d61 7264 5072 6f64  op..hadamardProd
│ │ │ │ -00002a00: 7563 7420 2d2d 2063 6f6d 7075 7465 7320  uct -- computes 
│ │ │ │ -00002a10: 7468 6520 4861 6461 6d61 7264 2070 726f  the Hadamard pro
│ │ │ │ -00002a20: 6475 6374 206f 6620 7661 7269 6574 6965  duct of varietie
│ │ │ │ -00002a30: 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  s.**************
│ │ │ │ -00002a40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00002a50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00002a60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ -00002a70: 0a57 6179 7320 746f 2075 7365 2068 6164  .Ways to use had
│ │ │ │ -00002a80: 616d 6172 6450 726f 6475 6374 3a0a 3d3d  amardProduct:.==
│ │ │ │ -00002a90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00002aa0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ -00002ab0: 2a6e 6f74 6520 6861 6461 6d61 7264 5072  *note hadamardPr
│ │ │ │ -00002ac0: 6f64 7563 7428 4964 6561 6c2c 4964 6561  oduct(Ideal,Idea
│ │ │ │ -00002ad0: 6c29 3a20 6861 6461 6d61 7264 5072 6f64  l): hadamardProd
│ │ │ │ -00002ae0: 7563 745f 6c70 4964 6561 6c5f 636d 4964  uct_lpIdeal_cmId
│ │ │ │ -00002af0: 6561 6c5f 7270 2c20 2d2d 0a20 2020 2048  eal_rp, --.    H
│ │ │ │ -00002b00: 6164 616d 6172 6420 7072 6f64 7563 7420  adamard product 
│ │ │ │ -00002b10: 6f66 2074 776f 2068 6f6d 6f67 656e 656f  of two homogeneo
│ │ │ │ -00002b20: 7573 2069 6465 616c 730a 2020 2a20 2a6e  us ideals.  * *n
│ │ │ │ -00002b30: 6f74 6520 6861 6461 6d61 7264 5072 6f64  ote hadamardProd
│ │ │ │ -00002b40: 7563 7428 4c69 7374 293a 2068 6164 616d  uct(List): hadam
│ │ │ │ -00002b50: 6172 6450 726f 6475 6374 5f6c 704c 6973  ardProduct_lpLis
│ │ │ │ -00002b60: 745f 7270 2c20 2d2d 2048 6164 616d 6172  t_rp, -- Hadamar
│ │ │ │ -00002b70: 6420 7072 6f64 7563 740a 2020 2020 6f66  d product.    of
│ │ │ │ -00002b80: 2061 206c 6973 7420 6f66 2068 6f6d 6f67   a list of homog
│ │ │ │ -00002b90: 656e 656f 7573 2069 6465 616c 732c 206f  eneous ideals, o
│ │ │ │ -00002ba0: 7220 706f 696e 7473 0a20 202a 202a 6e6f  r points.  * *no
│ │ │ │ -00002bb0: 7465 2068 6164 616d 6172 6450 726f 6475  te hadamardProdu
│ │ │ │ -00002bc0: 6374 284c 6973 742c 4c69 7374 293a 2068  ct(List,List): h
│ │ │ │ -00002bd0: 6164 616d 6172 6450 726f 6475 6374 5f6c  adamardProduct_l
│ │ │ │ -00002be0: 704c 6973 745f 636d 4c69 7374 5f72 702c  pList_cmList_rp,
│ │ │ │ -00002bf0: 202d 2d0a 2020 2020 4861 6461 6d61 7264   --.    Hadamard
│ │ │ │ -00002c00: 2070 726f 6475 6374 206f 6620 7477 6f20   product of two 
│ │ │ │ -00002c10: 7365 7473 206f 6620 706f 696e 7473 0a0a  sets of points..
│ │ │ │ -00002c20: 466f 7220 7468 6520 7072 6f67 7261 6d6d  For the programm
│ │ │ │ -00002c30: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  er.=============
│ │ │ │ -00002c40: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563  =====..The objec
│ │ │ │ -00002c50: 7420 2a6e 6f74 6520 6861 6461 6d61 7264  t *note hadamard
│ │ │ │ -00002c60: 5072 6f64 7563 743a 2068 6164 616d 6172  Product: hadamar
│ │ │ │ -00002c70: 6450 726f 6475 6374 2c20 6973 2061 202a  dProduct, is a *
│ │ │ │ -00002c80: 6e6f 7465 206d 6574 686f 6420 6675 6e63  note method func
│ │ │ │ -00002c90: 7469 6f6e 3a0a 284d 6163 6175 6c61 7932  tion:.(Macaulay2
│ │ │ │ -00002ca0: 446f 6329 4d65 7468 6f64 4675 6e63 7469  Doc)MethodFuncti
│ │ │ │ -00002cb0: 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d  on,...----------
│ │ │ │ -00002cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -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 2d2d  ----------------
│ │ │ │ -00002d00: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263  -----..The sourc
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│ │ │ │ -00002d50: 2e31 312b 6473 2f4d 322f 4d61 6361 756c  .11+ds/M2/Macaul
│ │ │ │ -00002d60: 6179 322f 7061 636b 6167 6573 2f48 6164  ay2/packages/Had
│ │ │ │ -00002d70: 616d 6172 642e 6d32 0a3a 3235 313a 302e  amard.m2.:251:0.
│ │ │ │ -00002d80: 0a1f 0a46 696c 653a 2048 6164 616d 6172  ...File: Hadamar
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│ │ │ │ -00002da0: 6461 6d61 7264 5072 6f64 7563 745f 6c70  damardProduct_lp
│ │ │ │ -00002db0: 4964 6561 6c5f 636d 4964 6561 6c5f 7270  Ideal_cmIdeal_rp
│ │ │ │ -00002dc0: 2c20 4e65 7874 3a20 6861 6461 6d61 7264  , Next: hadamard
│ │ │ │ -00002dd0: 5072 6f64 7563 745f 6c70 4c69 7374 5f72  Product_lpList_r
│ │ │ │ -00002de0: 702c 2050 7265 763a 2068 6164 616d 6172  p, Prev: hadamar
│ │ │ │ -00002df0: 6450 726f 6475 6374 2c20 5570 3a20 546f  dProduct, Up: To
│ │ │ │ -00002e00: 700a 0a68 6164 616d 6172 6450 726f 6475  p..hadamardProdu
│ │ │ │ -00002e10: 6374 2849 6465 616c 2c49 6465 616c 2920  ct(Ideal,Ideal) 
│ │ │ │ -00002e20: 2d2d 2048 6164 616d 6172 6420 7072 6f64  -- Hadamard prod
│ │ │ │ -00002e30: 7563 7420 6f66 2074 776f 2068 6f6d 6f67  uct of two homog
│ │ │ │ -00002e40: 656e 656f 7573 2069 6465 616c 730a 2a2a  eneous ideals.**
│ │ │ │ -00002e50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00002e60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00002e70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00002e80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00002e90: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 4675  ********..  * Fu
│ │ │ │ -00002ea0: 6e63 7469 6f6e 3a20 2a6e 6f74 6520 6861  nction: *note ha
│ │ │ │ -00002eb0: 6461 6d61 7264 5072 6f64 7563 743a 2068  damardProduct: h
│ │ │ │ -00002ec0: 6164 616d 6172 6450 726f 6475 6374 2c0a  adamardProduct,.
│ │ │ │ -00002ed0: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ -00002ee0: 2020 2020 6861 6461 6d61 7264 5072 6f64      hadamardProd
│ │ │ │ -00002ef0: 7563 7428 492c 4a29 0a20 202a 2049 6e70  uct(I,J).  * Inp
│ │ │ │ -00002f00: 7574 733a 0a20 2020 2020 202a 2049 2c20  uts:.      * I, 
│ │ │ │ -00002f10: 616e 202a 6e6f 7465 2069 6465 616c 3a20  an *note ideal: 
│ │ │ │ -00002f20: 284d 6163 6175 6c61 7932 446f 6329 4964  (Macaulay2Doc)Id
│ │ │ │ -00002f30: 6561 6c2c 2c20 2868 6f6d 6f67 656e 656f  eal,, (homogeneo
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│ │ │ │ -00002f50: 6e20 2a6e 6f74 6520 6964 6561 6c3a 2028  n *note ideal: (
│ │ │ │ -00002f60: 4d61 6361 756c 6179 3244 6f63 2949 6465  Macaulay2Doc)Ide
│ │ │ │ -00002f70: 616c 2c2c 2028 686f 6d6f 6765 6e65 6f75  al,, (homogeneou
│ │ │ │ -00002f80: 7329 0a20 202a 204f 7574 7075 7473 3a0a  s).  * Outputs:.
│ │ │ │ -00002f90: 2020 2020 2020 2a20 616e 202a 6e6f 7465        * an *note
│ │ │ │ -00002fa0: 2069 6465 616c 3a20 284d 6163 6175 6c61   ideal: (Macaula
│ │ │ │ -00002fb0: 7932 446f 6329 4964 6561 6c2c 2c20 0a0a  y2Doc)Ideal,, ..
│ │ │ │ -00002fc0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ -00002fd0: 3d3d 3d3d 3d3d 3d0a 0a47 6976 656e 2074  =======..Given t
│ │ │ │ -00002fe0: 776f 2070 726f 6a65 6374 6976 6520 7375  wo projective su
│ │ │ │ -00002ff0: 6276 6172 6965 7469 6573 2024 5824 2061  bvarieties $X$ a
│ │ │ │ -00003000: 6e64 2024 5924 2c20 7468 6569 7220 4861  nd $Y$, their Ha
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│ │ │ │ -00003020: 730a 6465 6669 6e65 6420 6173 2074 6865  s.defined as the
│ │ │ │ -00003030: 205a 6172 6973 6b69 2063 6c6f 7375 7265   Zariski closure
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│ │ │ │ -00003050: 7765 6c6c 2d64 6566 696e 6564 2920 656e  well-defined) en
│ │ │ │ -00003060: 7472 7977 6973 6520 7072 6f64 7563 7473  trywise products
│ │ │ │ -00003070: 0a6f 6620 7061 6972 7320 6f66 2070 6f69  .of pairs of poi
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│ │ │ │ -00003090: 7369 616e 2070 726f 6475 6374 2024 5820  sian product $X 
│ │ │ │ -000030a0: 5c74 696d 6573 2059 242e 2054 6869 7320  \times Y$. This 
│ │ │ │ -000030b0: 6361 6e20 616c 736f 2062 650a 7265 6761  can also be.rega
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│ │ │ │ -000031b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000031d0: 515b 782c 792c 7a2c 745d 3b20 2020 2020  Q[x,y,z,t];     
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│ │ │ │ -00003200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00003230: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000027a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
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│ │ │ │ +000027c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000027d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000027e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00002810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00002820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00002830: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00002860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00002870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00002880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00002890: 2d2d 2d2d 2b0a 0a57 6179 7320 746f 2075  ----+..Ways to u
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│ │ │ │ +000028b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000028c0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
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│ │ │ │ +00002920: 242d 7468 2048 6164 6d61 7264 2070 6f77  $-th Hadmard pow
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│ │ │ │ +00002960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00002970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00002980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00002990: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520  ---..The source 
│ │ │ │ +000029a0: 6f66 2074 6869 7320 646f 6375 6d65 6e74  of this document
│ │ │ │ +000029b0: 2069 7320 696e 0a2f 6275 696c 642f 7265   is in./build/re
│ │ │ │ +000029c0: 7072 6f64 7563 6962 6c65 2d70 6174 682f  producible-path/
│ │ │ │ +000029d0: 6d61 6361 756c 6179 322d 312e 3235 2e31  macaulay2-1.25.1
│ │ │ │ +000029e0: 312b 6473 2f4d 322f 4d61 6361 756c 6179  1+ds/M2/Macaulay
│ │ │ │ +000029f0: 322f 7061 636b 6167 6573 2f48 6164 616d  2/packages/Hadam
│ │ │ │ +00002a00: 6172 642e 6d32 0a3a 3339 383a 302e 0a1f  ard.m2.:398:0...
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│ │ │ │ +00002a30: 6d61 7264 5072 6f64 7563 742c 204e 6578  mardProduct, Nex
│ │ │ │ +00002a40: 743a 2068 6164 616d 6172 6450 726f 6475  t: hadamardProdu
│ │ │ │ +00002a50: 6374 5f6c 7049 6465 616c 5f63 6d49 6465  ct_lpIdeal_cmIde
│ │ │ │ +00002a60: 616c 5f72 702c 2050 7265 763a 2068 6164  al_rp, Prev: had
│ │ │ │ +00002a70: 616d 6172 6450 6f77 6572 5f6c 704c 6973  amardPower_lpLis
│ │ │ │ +00002a80: 745f 636d 5a5a 5f72 702c 2055 703a 2054  t_cmZZ_rp, Up: T
│ │ │ │ +00002a90: 6f70 0a0a 6861 6461 6d61 7264 5072 6f64  op..hadamardProd
│ │ │ │ +00002aa0: 7563 7420 2d2d 2063 6f6d 7075 7465 7320  uct -- computes 
│ │ │ │ +00002ab0: 7468 6520 4861 6461 6d61 7264 2070 726f  the Hadamard pro
│ │ │ │ +00002ac0: 6475 6374 206f 6620 7661 7269 6574 6965  duct of varietie
│ │ │ │ +00002ad0: 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  s.**************
│ │ │ │ +00002ae0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00002af0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00002b00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ +00002b10: 0a57 6179 7320 746f 2075 7365 2068 6164  .Ways to use had
│ │ │ │ +00002b20: 616d 6172 6450 726f 6475 6374 3a0a 3d3d  amardProduct:.==
│ │ │ │ +00002b30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00002b40: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ +00002b50: 2a6e 6f74 6520 6861 6461 6d61 7264 5072  *note hadamardPr
│ │ │ │ +00002b60: 6f64 7563 7428 4964 6561 6c2c 4964 6561  oduct(Ideal,Idea
│ │ │ │ +00002b70: 6c29 3a20 6861 6461 6d61 7264 5072 6f64  l): hadamardProd
│ │ │ │ +00002b80: 7563 745f 6c70 4964 6561 6c5f 636d 4964  uct_lpIdeal_cmId
│ │ │ │ +00002b90: 6561 6c5f 7270 2c20 2d2d 0a20 2020 2048  eal_rp, --.    H
│ │ │ │ +00002ba0: 6164 616d 6172 6420 7072 6f64 7563 7420  adamard product 
│ │ │ │ +00002bb0: 6f66 2074 776f 2068 6f6d 6f67 656e 656f  of two homogeneo
│ │ │ │ +00002bc0: 7573 2069 6465 616c 730a 2020 2a20 2a6e  us ideals.  * *n
│ │ │ │ +00002bd0: 6f74 6520 6861 6461 6d61 7264 5072 6f64  ote hadamardProd
│ │ │ │ +00002be0: 7563 7428 4c69 7374 293a 2068 6164 616d  uct(List): hadam
│ │ │ │ +00002bf0: 6172 6450 726f 6475 6374 5f6c 704c 6973  ardProduct_lpLis
│ │ │ │ +00002c00: 745f 7270 2c20 2d2d 2048 6164 616d 6172  t_rp, -- Hadamar
│ │ │ │ +00002c10: 6420 7072 6f64 7563 740a 2020 2020 6f66  d product.    of
│ │ │ │ +00002c20: 2061 206c 6973 7420 6f66 2068 6f6d 6f67   a list of homog
│ │ │ │ +00002c30: 656e 656f 7573 2069 6465 616c 732c 206f  eneous ideals, o
│ │ │ │ +00002c40: 7220 706f 696e 7473 0a20 202a 202a 6e6f  r points.  * *no
│ │ │ │ +00002c50: 7465 2068 6164 616d 6172 6450 726f 6475  te hadamardProdu
│ │ │ │ +00002c60: 6374 284c 6973 742c 4c69 7374 293a 2068  ct(List,List): h
│ │ │ │ +00002c70: 6164 616d 6172 6450 726f 6475 6374 5f6c  adamardProduct_l
│ │ │ │ +00002c80: 704c 6973 745f 636d 4c69 7374 5f72 702c  pList_cmList_rp,
│ │ │ │ +00002c90: 202d 2d0a 2020 2020 4861 6461 6d61 7264   --.    Hadamard
│ │ │ │ +00002ca0: 2070 726f 6475 6374 206f 6620 7477 6f20   product of two 
│ │ │ │ +00002cb0: 7365 7473 206f 6620 706f 696e 7473 0a0a  sets of points..
│ │ │ │ +00002cc0: 466f 7220 7468 6520 7072 6f67 7261 6d6d  For the programm
│ │ │ │ +00002cd0: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  er.=============
│ │ │ │ +00002ce0: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563  =====..The objec
│ │ │ │ +00002cf0: 7420 2a6e 6f74 6520 6861 6461 6d61 7264  t *note hadamard
│ │ │ │ +00002d00: 5072 6f64 7563 743a 2068 6164 616d 6172  Product: hadamar
│ │ │ │ +00002d10: 6450 726f 6475 6374 2c20 6973 2061 202a  dProduct, is a *
│ │ │ │ +00002d20: 6e6f 7465 206d 6574 686f 6420 6675 6e63  note method func
│ │ │ │ +00002d30: 7469 6f6e 3a0a 284d 6163 6175 6c61 7932  tion:.(Macaulay2
│ │ │ │ +00002d40: 446f 6329 4d65 7468 6f64 4675 6e63 7469  Doc)MethodFuncti
│ │ │ │ +00002d50: 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d  on,...----------
│ │ │ │ +00002d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00002d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00002d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00002d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00002da0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263  -----..The sourc
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│ │ │ │ +00002de0: 682f 6d61 6361 756c 6179 322d 312e 3235  h/macaulay2-1.25
│ │ │ │ +00002df0: 2e31 312b 6473 2f4d 322f 4d61 6361 756c  .11+ds/M2/Macaul
│ │ │ │ +00002e00: 6179 322f 7061 636b 6167 6573 2f48 6164  ay2/packages/Had
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│ │ │ │ +00002e40: 6461 6d61 7264 5072 6f64 7563 745f 6c70  damardProduct_lp
│ │ │ │ +00002e50: 4964 6561 6c5f 636d 4964 6561 6c5f 7270  Ideal_cmIdeal_rp
│ │ │ │ +00002e60: 2c20 4e65 7874 3a20 6861 6461 6d61 7264  , Next: hadamard
│ │ │ │ +00002e70: 5072 6f64 7563 745f 6c70 4c69 7374 5f72  Product_lpList_r
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│ │ │ │ +00002e90: 6450 726f 6475 6374 2c20 5570 3a20 546f  dProduct, Up: To
│ │ │ │ +00002ea0: 700a 0a68 6164 616d 6172 6450 726f 6475  p..hadamardProdu
│ │ │ │ +00002eb0: 6374 2849 6465 616c 2c49 6465 616c 2920  ct(Ideal,Ideal) 
│ │ │ │ +00002ec0: 2d2d 2048 6164 616d 6172 6420 7072 6f64  -- Hadamard prod
│ │ │ │ +00002ed0: 7563 7420 6f66 2074 776f 2068 6f6d 6f67  uct of two homog
│ │ │ │ +00002ee0: 656e 656f 7573 2069 6465 616c 730a 2a2a  eneous ideals.**
│ │ │ │ +00002ef0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00002f00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00002f10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00002f20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00002f30: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 4675  ********..  * Fu
│ │ │ │ +00002f40: 6e63 7469 6f6e 3a20 2a6e 6f74 6520 6861  nction: *note ha
│ │ │ │ +00002f50: 6461 6d61 7264 5072 6f64 7563 743a 2068  damardProduct: h
│ │ │ │ +00002f60: 6164 616d 6172 6450 726f 6475 6374 2c0a  adamardProduct,.
│ │ │ │ +00002f70: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ +00002f80: 2020 2020 6861 6461 6d61 7264 5072 6f64      hadamardProd
│ │ │ │ +00002f90: 7563 7428 492c 4a29 0a20 202a 2049 6e70  uct(I,J).  * Inp
│ │ │ │ +00002fa0: 7574 733a 0a20 2020 2020 202a 2049 2c20  uts:.      * I, 
│ │ │ │ +00002fb0: 616e 202a 6e6f 7465 2069 6465 616c 3a20  an *note ideal: 
│ │ │ │ +00002fc0: 284d 6163 6175 6c61 7932 446f 6329 4964  (Macaulay2Doc)Id
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│ │ │ │ +00002ff0: 6e20 2a6e 6f74 6520 6964 6561 6c3a 2028  n *note ideal: (
│ │ │ │ +00003000: 4d61 6361 756c 6179 3244 6f63 2949 6465  Macaulay2Doc)Ide
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│ │ │ │ +00003060: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
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│ │ │ │ +00003110: 0a6f 6620 7061 6972 7320 6f66 2070 6f69  .of pairs of poi
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│ │ │ │ +00003130: 7369 616e 2070 726f 6475 6374 2024 5820  sian product $X 
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│ │ │ │ +00003180: 726f 6475 6374 206f 6620 2458 205c 7469  roduct of $X \ti
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│ │ │ │ +000031c0: 2063 6f6f 7264 696e 6174 6573 2e20 5468   coordinates. Th
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│ │ │ │  00003250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00003270: 515b 782c 792c 7a2c 745d 3b20 2020 2020  Q[x,y,z,t];     
│ │ │ │ +00003280: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  00003290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000032b0: 2d2d 2d2b 0a7c 6934 203a 2069 6465 616c  ---+.|i4 : ideal
│ │ │ │ -000032c0: 4f66 5072 6f6a 6563 7469 7665 506f 696e  OfProjectivePoin
│ │ │ │ -000032d0: 7473 287b 702a 717d 2c53 297c 0a7c 2020  ts({p*q},S)|.|  
│ │ │ │ -000032e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000032f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003300: 2020 207c 0a7c 6f34 203d 2069 6465 616c     |.|o4 = ideal
│ │ │ │ -00003310: 2028 327a 202d 2074 2c20 3279 202d 2074   (2z - t, 2y - t
│ │ │ │ -00003320: 2c20 3278 202b 2074 2920 207c 0a7c 2020  , 2x + t)  |.|  
│ │ │ │ -00003330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003350: 2020 207c 0a7c 6f34 203a 2049 6465 616c     |.|o4 : Ideal
│ │ │ │ -00003360: 206f 6620 5320 2020 2020 2020 2020 2020   of S           
│ │ │ │ -00003370: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -00003380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000033a0: 2d2d 2d2b 0a0a 5468 6973 2063 616e 2062  ---+..This can b
│ │ │ │ -000033b0: 6520 636f 6d70 7574 6564 2061 6c73 6f20  e computed also 
│ │ │ │ -000033c0: 6672 6f6d 2074 6865 6972 2064 6566 696e  from their defin
│ │ │ │ -000033d0: 696e 6720 6964 6561 6c73 2061 7320 6578  ing ideals as ex
│ │ │ │ -000033e0: 706c 6169 6e65 642e 0a0a 2b2d 2d2d 2d2d  plained...+-----
│ │ │ │ -000033f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00003410: 7c69 3520 3a20 4950 203d 2069 6465 616c  |i5 : IP = ideal
│ │ │ │ -00003420: 2878 2d79 2c78 2d7a 2c32 2a78 2d74 2920  (x-y,x-z,2*x-t) 
│ │ │ │ -00003430: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00003440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003450: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ -00003460: 3d20 6964 6561 6c20 2878 202d 2079 2c20  = ideal (x - y, 
│ │ │ │ -00003470: 7820 2d20 7a2c 2032 7820 2d20 7429 2020  x - z, 2x - t)  
│ │ │ │ -00003480: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00003490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000034a0: 2020 2020 2020 7c0a 7c6f 3520 3a20 4964        |.|o5 : Id
│ │ │ │ -000034b0: 6561 6c20 6f66 2053 2020 2020 2020 2020  eal of S        
│ │ │ │ -000034c0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -000034d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000034e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000034f0: 2d2d 2b0a 7c69 3620 3a20 4951 203d 2069  --+.|i6 : IQ = i
│ │ │ │ -00003500: 6465 616c 2878 2b79 2c78 2b7a 2c78 2b74  deal(x+y,x+z,x+t
│ │ │ │ -00003510: 2920 2020 2020 2020 7c0a 7c20 2020 2020  )       |.|     
│ │ │ │ -00003520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003530: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00003540: 7c6f 3620 3d20 6964 6561 6c20 2878 202b  |o6 = ideal (x +
│ │ │ │ -00003550: 2079 2c20 7820 2b20 7a2c 2078 202b 2074   y, x + z, x + t
│ │ │ │ -00003560: 2920 2020 7c0a 7c20 2020 2020 2020 2020  )   |.|         
│ │ │ │ -00003570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003580: 2020 2020 2020 2020 2020 7c0a 7c6f 3620            |.|o6 
│ │ │ │ -00003590: 3a20 4964 6561 6c20 6f66 2053 2020 2020  : Ideal of S    
│ │ │ │ -000035a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000035b0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -000035c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000035d0: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 6861  ------+.|i7 : ha
│ │ │ │ -000035e0: 6461 6d61 7264 5072 6f64 7563 7428 4950  damardProduct(IP
│ │ │ │ -000035f0: 2c49 5129 2020 2020 2020 2020 7c0a 7c20  ,IQ)        |.| 
│ │ │ │ -00003600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000032b0: 2d2d 2d2b 0a7c 6932 203a 2070 203d 2070  ---+.|i2 : p = p
│ │ │ │ +000032c0: 6f69 6e74 207b 312c 312c 312c 327d 3b20  oint {1,1,1,2}; 
│ │ │ │ +000032d0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000032e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000032f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003300: 2d2d 2d2b 0a7c 6933 203a 2071 203d 2070  ---+.|i3 : q = p
│ │ │ │ +00003310: 6f69 6e74 207b 312c 2d31 2c2d 312c 2d31  oint {1,-1,-1,-1
│ │ │ │ +00003320: 7d3b 2020 2020 2020 2020 207c 0a2b 2d2d  };         |.+--
│ │ │ │ +00003330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003350: 2d2d 2d2b 0a7c 6934 203a 2069 6465 616c  ---+.|i4 : ideal
│ │ │ │ +00003360: 4f66 5072 6f6a 6563 7469 7665 506f 696e  OfProjectivePoin
│ │ │ │ +00003370: 7473 287b 702a 717d 2c53 297c 0a7c 2020  ts({p*q},S)|.|  
│ │ │ │ +00003380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000033a0: 2020 207c 0a7c 6f34 203d 2069 6465 616c     |.|o4 = ideal
│ │ │ │ +000033b0: 2028 327a 202d 2074 2c20 3279 202d 2074   (2z - t, 2y - t
│ │ │ │ +000033c0: 2c20 3278 202b 2074 2920 207c 0a7c 2020  , 2x + t)  |.|  
│ │ │ │ +000033d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000033e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000033f0: 2020 207c 0a7c 6f34 203a 2049 6465 616c     |.|o4 : Ideal
│ │ │ │ +00003400: 206f 6620 5320 2020 2020 2020 2020 2020   of S           
│ │ │ │ +00003410: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00003420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003440: 2d2d 2d2b 0a0a 5468 6973 2063 616e 2062  ---+..This can b
│ │ │ │ +00003450: 6520 636f 6d70 7574 6564 2061 6c73 6f20  e computed also 
│ │ │ │ +00003460: 6672 6f6d 2074 6865 6972 2064 6566 696e  from their defin
│ │ │ │ +00003470: 696e 6720 6964 6561 6c73 2061 7320 6578  ing ideals as ex
│ │ │ │ +00003480: 706c 6169 6e65 642e 0a0a 2b2d 2d2d 2d2d  plained...+-----
│ │ │ │ +00003490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000034a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000034b0: 7c69 3520 3a20 4950 203d 2069 6465 616c  |i5 : IP = ideal
│ │ │ │ +000034c0: 2878 2d79 2c78 2d7a 2c32 2a78 2d74 2920  (x-y,x-z,2*x-t) 
│ │ │ │ +000034d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000034e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000034f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ +00003500: 3d20 6964 6561 6c20 2878 202d 2079 2c20  = ideal (x - y, 
│ │ │ │ +00003510: 7820 2d20 7a2c 2032 7820 2d20 7429 2020  x - z, 2x - t)  
│ │ │ │ +00003520: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00003530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003540: 2020 2020 2020 7c0a 7c6f 3520 3a20 4964        |.|o5 : Id
│ │ │ │ +00003550: 6561 6c20 6f66 2053 2020 2020 2020 2020  eal of S        
│ │ │ │ +00003560: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00003570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003590: 2d2d 2b0a 7c69 3620 3a20 4951 203d 2069  --+.|i6 : IQ = i
│ │ │ │ +000035a0: 6465 616c 2878 2b79 2c78 2b7a 2c78 2b74  deal(x+y,x+z,x+t
│ │ │ │ +000035b0: 2920 2020 2020 2020 7c0a 7c20 2020 2020  )       |.|     
│ │ │ │ +000035c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000035d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000035e0: 7c6f 3620 3d20 6964 6561 6c20 2878 202b  |o6 = ideal (x +
│ │ │ │ +000035f0: 2079 2c20 7820 2b20 7a2c 2078 202b 2074   y, x + z, x + t
│ │ │ │ +00003600: 2920 2020 7c0a 7c20 2020 2020 2020 2020  )   |.|         
│ │ │ │  00003610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003620: 2020 7c0a 7c6f 3720 3d20 6964 6561 6c20    |.|o7 = ideal 
│ │ │ │ -00003630: 2832 7a20 2d20 742c 2032 7920 2d20 742c  (2z - t, 2y - t,
│ │ │ │ -00003640: 2032 7820 2b20 7429 7c0a 7c20 2020 2020   2x + t)|.|     
│ │ │ │ -00003650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003660: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00003670: 7c6f 3720 3a20 4964 6561 6c20 6f66 2053  |o7 : Ideal of S
│ │ │ │ -00003680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003690: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -000036a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000036b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 6520  ----------+..We 
│ │ │ │ -000036c0: 6361 6e20 616c 736f 2063 6f6e 7369 6465  can also conside
│ │ │ │ -000036d0: 7220 4861 6461 6d61 7264 2070 726f 6475  r Hadamard produ
│ │ │ │ -000036e0: 6374 206f 6620 6869 6768 6572 2064 696d  ct of higher dim
│ │ │ │ -000036f0: 656e 7369 6f6e 616c 2076 6172 6965 7469  ensional varieti
│ │ │ │ -00003700: 6573 2e20 466f 720a 6578 616d 706c 652c  es. For.example,
│ │ │ │ -00003710: 2074 6865 2048 6164 616d 6172 6420 7072   the Hadamard pr
│ │ │ │ -00003720: 6f64 7563 7420 6f66 2074 776f 206c 696e  oduct of two lin
│ │ │ │ -00003730: 6573 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  es...+----------
│ │ │ │ +00003620: 2020 2020 2020 2020 2020 7c0a 7c6f 3620            |.|o6 
│ │ │ │ +00003630: 3a20 4964 6561 6c20 6f66 2053 2020 2020  : Ideal of S    
│ │ │ │ +00003640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003650: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00003660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003670: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 6861  ------+.|i7 : ha
│ │ │ │ +00003680: 6461 6d61 7264 5072 6f64 7563 7428 4950  damardProduct(IP
│ │ │ │ +00003690: 2c49 5129 2020 2020 2020 2020 7c0a 7c20  ,IQ)        |.| 
│ │ │ │ +000036a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000036b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000036c0: 2020 7c0a 7c6f 3720 3d20 6964 6561 6c20    |.|o7 = ideal 
│ │ │ │ +000036d0: 2832 7a20 2d20 742c 2032 7920 2d20 742c  (2z - t, 2y - t,
│ │ │ │ +000036e0: 2032 7820 2b20 7429 7c0a 7c20 2020 2020   2x + t)|.|     
│ │ │ │ +000036f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003700: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00003710: 7c6f 3720 3a20 4964 6561 6c20 6f66 2053  |o7 : Ideal of S
│ │ │ │ +00003720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003730: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  00003740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003780: 2d2d 2d2b 0a7c 6938 203a 2049 203d 2069  ---+.|i8 : I = i
│ │ │ │ -00003790: 6465 616c 2872 616e 646f 6d28 312c 5329  deal(random(1,S)
│ │ │ │ -000037a0: 2c72 616e 646f 6d28 312c 5329 293b 2020  ,random(1,S));  
│ │ │ │ -000037b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000037c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000037d0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -000037e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000037f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003820: 2020 207c 0a7c 6f38 203a 2049 6465 616c     |.|o8 : Ideal
│ │ │ │ -00003830: 206f 6620 5320 2020 2020 2020 2020 2020   of S           
│ │ │ │ -00003840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003750: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 6520  ----------+..We 
│ │ │ │ +00003760: 6361 6e20 616c 736f 2063 6f6e 7369 6465  can also conside
│ │ │ │ +00003770: 7220 4861 6461 6d61 7264 2070 726f 6475  r Hadamard produ
│ │ │ │ +00003780: 6374 206f 6620 6869 6768 6572 2064 696d  ct of higher dim
│ │ │ │ +00003790: 656e 7369 6f6e 616c 2076 6172 6965 7469  ensional varieti
│ │ │ │ +000037a0: 6573 2e20 466f 720a 6578 616d 706c 652c  es. For.example,
│ │ │ │ +000037b0: 2074 6865 2048 6164 616d 6172 6420 7072   the Hadamard pr
│ │ │ │ +000037c0: 6f64 7563 7420 6f66 2074 776f 206c 696e  oduct of two lin
│ │ │ │ +000037d0: 6573 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  es...+----------
│ │ │ │ +000037e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000037f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003820: 2d2d 2d2b 0a7c 6938 203a 2049 203d 2069  ---+.|i8 : I = i
│ │ │ │ +00003830: 6465 616c 2872 616e 646f 6d28 312c 5329  deal(random(1,S)
│ │ │ │ +00003840: 2c72 616e 646f 6d28 312c 5329 293b 2020  ,random(1,S));  
│ │ │ │  00003850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003870: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -00003880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000038a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000038b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000038c0: 2d2d 2d2b 0a7c 6939 203a 204a 203d 2069  ---+.|i9 : J = i
│ │ │ │ -000038d0: 6465 616c 2872 616e 646f 6d28 312c 5329  deal(random(1,S)
│ │ │ │ -000038e0: 2c72 616e 646f 6d28 312c 5329 293b 2020  ,random(1,S));  
│ │ │ │ +00003870: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00003880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000038a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000038b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000038c0: 2020 207c 0a7c 6f38 203a 2049 6465 616c     |.|o8 : Ideal
│ │ │ │ +000038d0: 206f 6620 5320 2020 2020 2020 2020 2020   of S           
│ │ │ │ +000038e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000038f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003910: 2020 207c 0a7c 2020 2020 2020 2020 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 2020                  
│ │ │ │ -00003950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00003960: 2020 207c 0a7c 6f39 203a 2049 6465 616c     |.|o9 : Ideal
│ │ │ │ -00003970: 206f 6620 5320 2020 2020 2020 2020 2020   of S           
│ │ │ │ -00003980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003910: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00003920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003960: 2d2d 2d2b 0a7c 6939 203a 204a 203d 2069  ---+.|i9 : J = i
│ │ │ │ +00003970: 6465 616c 2872 616e 646f 6d28 312c 5329  deal(random(1,S)
│ │ │ │ +00003980: 2c72 616e 646f 6d28 312c 5329 293b 2020  ,random(1,S));  
│ │ │ │  00003990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000039a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000039b0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -000039c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000039d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000039e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000039f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00003a00: 2d2d 2d2b 0a7c 6931 3020 3a20 6861 6461  ---+.|i10 : hada
│ │ │ │ -00003a10: 6d61 7264 5072 6f64 7563 7428 492c 4a29  mardProduct(I,J)
│ │ │ │ +000039b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000039c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000039d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000039e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000039f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00003a00: 2020 207c 0a7c 6f39 203a 2049 6465 616c     |.|o9 : Ideal
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│ │ │ │ +00003ee0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f  =======..  * *no
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│ │ │ │ +00003f00: 6374 2849 6465 616c 2c49 6465 616c 293a  ct(Ideal,Ideal):
│ │ │ │ +00003f10: 2068 6164 616d 6172 6450 726f 6475 6374   hadamardProduct
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│ │ │ │ +00003f30: 5f72 702c 202d 2d0a 2020 2020 4861 6461  _rp, --.    Hada
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│ │ │ │ +00003f60: 6964 6561 6c73 0a2d 2d2d 2d2d 2d2d 2d2d  ideals.---------
│ │ │ │ +00003f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00003fb0: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572  ------..The sour
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│ │ │ │ +00004000: 352e 3131 2b64 732f 4d32 2f4d 6163 6175  5.11+ds/M2/Macau
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│ │ │ │ +00004020: 6461 6d61 7264 2e6d 320a 3a32 3933 3a30  damard.m2.:293:0
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│ │ │ │ +00004040: 7264 2e69 6e66 6f2c 204e 6f64 653a 2068  rd.info, Node: h
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│ │ │ │ +00004060: 704c 6973 745f 7270 2c20 4e65 7874 3a20  pList_rp, Next: 
│ │ │ │ +00004070: 6861 6461 6d61 7264 5072 6f64 7563 745f  hadamardProduct_
│ │ │ │ +00004080: 6c70 4c69 7374 5f63 6d4c 6973 745f 7270  lpList_cmList_rp
│ │ │ │ +00004090: 2c20 5072 6576 3a20 6861 6461 6d61 7264  , Prev: hadamard
│ │ │ │ +000040a0: 5072 6f64 7563 745f 6c70 4964 6561 6c5f  Product_lpIdeal_
│ │ │ │ +000040b0: 636d 4964 6561 6c5f 7270 2c20 5570 3a20  cmIdeal_rp, Up: 
│ │ │ │ +000040c0: 546f 700a 0a68 6164 616d 6172 6450 726f  Top..hadamardPro
│ │ │ │ +000040d0: 6475 6374 284c 6973 7429 202d 2d20 4861  duct(List) -- Ha
│ │ │ │ +000040e0: 6461 6d61 7264 2070 726f 6475 6374 206f  damard product o
│ │ │ │ +000040f0: 6620 6120 6c69 7374 206f 6620 686f 6d6f  f a list of homo
│ │ │ │ +00004100: 6765 6e65 6f75 7320 6964 6561 6c73 2c20  geneous ideals, 
│ │ │ │ +00004110: 6f72 2070 6f69 6e74 730a 2a2a 2a2a 2a2a  or points.******
│ │ │ │ +00004120: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00004130: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00004140: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00004150: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00004160: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ +00004170: 2020 2a20 4675 6e63 7469 6f6e 3a20 2a6e    * Function: *n
│ │ │ │ +00004180: 6f74 6520 6861 6461 6d61 7264 5072 6f64  ote hadamardProd
│ │ │ │ +00004190: 7563 743a 2068 6164 616d 6172 6450 726f  uct: hadamardPro
│ │ │ │ +000041a0: 6475 6374 2c0a 2020 2a20 5573 6167 653a  duct,.  * Usage:
│ │ │ │ +000041b0: 200a 2020 2020 2020 2020 6861 6461 6d61   .        hadama
│ │ │ │ +000041c0: 7264 5072 6f64 7563 7428 4c29 0a20 202a  rdProduct(L).  *
│ │ │ │ +000041d0: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ +000041e0: 204c 2c20 6120 2a6e 6f74 6520 6c69 7374   L, a *note list
│ │ │ │ +000041f0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00004200: 4c69 7374 2c2c 206f 6620 2a6e 6f74 6520  List,, of *note 
│ │ │ │ +00004210: 2868 6f6d 6f67 656e 656f 7573 2920 6964  (homogeneous) id
│ │ │ │ +00004220: 6561 6c73 3a0a 2020 2020 2020 2020 284d  eals:.        (M
│ │ │ │ +00004230: 6163 6175 6c61 7932 446f 6329 4964 6561  acaulay2Doc)Idea
│ │ │ │ +00004240: 6c2c 206f 7220 2a6e 6f74 6520 2870 726f  l, or *note (pro
│ │ │ │ +00004250: 6a65 6374 6976 6529 2070 6f69 6e74 733a  jective) points:
│ │ │ │ +00004260: 2050 6f69 6e74 2c0a 2020 2a20 4f75 7470   Point,.  * Outp
│ │ │ │ +00004270: 7574 733a 0a20 2020 2020 202a 2061 6e20  uts:.      * an 
│ │ │ │ +00004280: 2a6e 6f74 6520 6964 6561 6c3a 2028 4d61  *note ideal: (Ma
│ │ │ │ +00004290: 6361 756c 6179 3244 6f63 2949 6465 616c  caulay2Doc)Ideal
│ │ │ │ +000042a0: 2c2c 200a 2020 2020 2020 2a20 616e 2069  ,, .      * an i
│ │ │ │ +000042b0: 6e73 7461 6e63 6520 6f66 2074 6865 2074  nstance of the t
│ │ │ │ +000042c0: 7970 6520 2a6e 6f74 6520 506f 696e 743a  ype *note Point:
│ │ │ │ +000042d0: 2050 6f69 6e74 2c2c 200a 0a44 6573 6372   Point,, ..Descr
│ │ │ │ +000042e0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ +000042f0: 3d3d 0a0a 5468 6520 4861 6461 6d61 7264  ==..The Hadamard
│ │ │ │ +00004300: 2070 726f 6475 6374 206f 6620 6120 6c69   product of a li
│ │ │ │ +00004310: 7374 206f 6620 6964 6561 6c73 206f 7220  st of ideals or 
│ │ │ │ +00004320: 706f 696e 7473 2063 6f6e 7374 7275 6374  points construct
│ │ │ │ +00004330: 6564 2062 7920 7573 696e 670a 6974 6572  ed by using.iter
│ │ │ │ +00004340: 6174 6976 656c 7920 7468 6520 6269 6e61  atively the bina
│ │ │ │ +00004350: 7279 2066 756e 6374 696f 6e20 2a6e 6f74  ry function *not
│ │ │ │ +00004360: 6520 6861 6461 6d61 7264 5072 6f64 7563  e hadamardProduc
│ │ │ │ +00004370: 7428 4964 6561 6c2c 4964 6561 6c29 3a0a  t(Ideal,Ideal):.
│ │ │ │ +00004380: 6861 6461 6d61 7264 5072 6f64 7563 745f  hadamardProduct_
│ │ │ │ +00004390: 6c70 4964 6561 6c5f 636d 4964 6561 6c5f  lpIdeal_cmIdeal_
│ │ │ │ +000043a0: 7270 2c2c 206f 7220 202a 6e6f 7465 2050  rp,, or  *note P
│ │ │ │ +000043b0: 6f69 6e74 202a 2050 6f69 6e74 3a20 506f  oint * Point: Po
│ │ │ │ +000043c0: 696e 7420 5f73 7420 506f 696e 742c 2e0a  int _st Point,..
│ │ │ │  000043d0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  000043e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000043f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00004420: 0a7c 6932 203a 2049 203d 2069 6465 616c  .|i2 : I = ideal
│ │ │ │ -00004430: 2872 616e 646f 6d28 312c 5329 2c72 616e  (random(1,S),ran
│ │ │ │ -00004440: 646f 6d28 312c 5329 293b 2020 2020 2020  dom(1,S));      
│ │ │ │ +00004420: 0a7c 6931 203a 2053 203d 2051 515b 782c  .|i1 : S = QQ[x,
│ │ │ │ +00004430: 792c 7a2c 745d 3b20 2020 2020 2020 2020  y,z,t];         
│ │ │ │ +00004440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004460: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004470: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00004480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000044a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000044b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000044c0: 0a7c 6f32 203a 2049 6465 616c 206f 6620  .|o2 : Ideal of 
│ │ │ │ -000044d0: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ -000044e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004470: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00004480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00004490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000044a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000044b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +000044c0: 0a7c 6932 203a 2049 203d 2069 6465 616c  .|i2 : I = ideal
│ │ │ │ +000044d0: 2872 616e 646f 6d28 312c 5329 2c72 616e  (random(1,S),ran
│ │ │ │ +000044e0: 646f 6d28 312c 5329 293b 2020 2020 2020  dom(1,S));      
│ │ │ │  000044f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004500: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004510: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00004520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00004560: 0a7c 6933 203a 204a 203d 2069 6465 616c  .|i3 : J = ideal
│ │ │ │ -00004570: 2872 616e 646f 6d28 312c 5329 2c72 616e  (random(1,S),ran
│ │ │ │ -00004580: 646f 6d28 312c 5329 293b 2020 2020 2020  dom(1,S));      
│ │ │ │ +00004510: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +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 207c                 |
│ │ │ │ +00004560: 0a7c 6f32 203a 2049 6465 616c 206f 6620  .|o2 : Ideal of 
│ │ │ │ +00004570: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ +00004580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000045a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000045b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000045c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000045d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000045e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000045f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004600: 0a7c 6f33 203a 2049 6465 616c 206f 6620  .|o3 : Ideal of 
│ │ │ │ -00004610: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ -00004620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000045b0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000045c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000045d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000045e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000045f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00004600: 0a7c 6933 203a 204a 203d 2069 6465 616c  .|i3 : J = ideal
│ │ │ │ +00004610: 2872 616e 646f 6d28 312c 5329 2c72 616e  (random(1,S),ran
│ │ │ │ +00004620: 646f 6d28 312c 5329 293b 2020 2020 2020  dom(1,S));      
│ │ │ │  00004630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004640: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004650: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00004660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000046a0: 0a7c 6934 203a 204c 203d 207b 492c 4a7d  .|i4 : L = {I,J}
│ │ │ │ -000046b0: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ +00004650: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00004660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004690: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000046a0: 0a7c 6f33 203a 2049 6465 616c 206f 6620  .|o3 : Ideal of 
│ │ │ │ +000046b0: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │  000046c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000046d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000046e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  000046f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00004700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00004740: 0a7c 6935 203a 2068 6164 616d 6172 6450  .|i5 : hadamardP
│ │ │ │ -00004750: 726f 6475 6374 284c 2920 2020 2020 2020  roduct(L)       
│ │ │ │ +00004740: 0a7c 6934 203a 204c 203d 207b 492c 4a7d  .|i4 : L = {I,J}
│ │ │ │ +00004750: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │  00004760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004780: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004790: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000047a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000047b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000047c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000047d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000047e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000047f0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +00004790: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000047a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000047b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000047c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000047d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +000047e0: 0a7c 6935 203a 2068 6164 616d 6172 6450  .|i5 : hadamardP
│ │ │ │ +000047f0: 726f 6475 6374 284c 2920 2020 2020 2020  roduct(L)       
│ │ │ │  00004800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004810: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +00004810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004820: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004830: 0a7c 6f35 203d 2069 6465 616c 2831 3634  .|o5 = ideal(164
│ │ │ │ -00004840: 3036 3636 3431 3230 3030 7820 202d 2031  0666412000x  - 1
│ │ │ │ -00004850: 3237 3938 3431 3838 3135 3078 2a79 202d  27984188150x*y -
│ │ │ │ -00004860: 2034 3534 3532 3633 3435 3079 2020 2b20   4545263450y  + 
│ │ │ │ -00004870: 3338 3630 3239 3838 3432 3430 782a 7a7c  386029884240x*z|
│ │ │ │ -00004880: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d  .|     ---------
│ │ │ │ -00004890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000048a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000048b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000048c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000048d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000048e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000048f0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -00004900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004910: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004920: 0a7c 2020 2020 202b 2031 3039 3832 3131  .|     + 1098211
│ │ │ │ -00004930: 3433 3235 792a 7a20 2d20 3133 3636 3035  4325y*z - 136605
│ │ │ │ -00004940: 3236 3838 7a20 202b 2036 3934 3832 3939  2688z  + 6948299
│ │ │ │ -00004950: 3235 3536 3235 782a 7420 2d20 3637 3636  255625x*t - 6766
│ │ │ │ -00004960: 3331 3536 3536 3735 792a 7420 2b20 207c  31565675y*t +  |
│ │ │ │ -00004970: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d  .|     ---------
│ │ │ │ -00004980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000049a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000049b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000049c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000049d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000049e0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -000049f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004a00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004a10: 0a7c 2020 2020 2036 3734 3231 3133 3132  .|     674211312
│ │ │ │ -00004a20: 3435 7a2a 7420 2b20 3135 3134 3932 3536  45z*t + 15149256
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│ │ │ │ +00004970: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00004980: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00004a00: 3331 3536 3536 3735 792a 7420 2b20 207c  31565675y*t +  |
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│ │ │ │ +00004a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │  00004a60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
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│ │ │ │ -00004a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004a80: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  00004a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004aa0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004ab0: 0a7c 6f35 203a 2049 6465 616c 206f 6620  .|o5 : Ideal of 
│ │ │ │ -00004ac0: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ -00004ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004ab0: 0a7c 2020 2020 2036 3734 3231 3133 3132  .|     674211312
│ │ │ │ +00004ac0: 3435 7a2a 7420 2b20 3135 3134 3932 3536  45z*t + 15149256
│ │ │ │ +00004ad0: 3936 3337 3574 2029 2020 2020 2020 2020  96375t )        
│ │ │ │  00004ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00004b00: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00004b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00004b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00004b50: 0a7c 6936 203a 2050 203d 2070 6f69 6e74  .|i6 : P = point
│ │ │ │ -00004b60: 5c7b 7b31 2c32 2c33 7d2c 7b2d 312c 312c  \{{1,2,3},{-1,1,
│ │ │ │ -00004b70: 317d 2c7b 312c 312f 322c 2d31 2f33 7d7d  1},{1,1/2,-1/3}}
│ │ │ │ +00004b00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00004b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004b40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ +00004b60: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ +00004b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004b90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004ba0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00004bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004be0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004bf0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00004c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004c20: 2020 3120 2020 2031 2020 2020 2020 2020    1    1        
│ │ │ │ +00004ba0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00004bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00004bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00004bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00004be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00004bf0: 0a7c 6936 203a 2050 203d 2070 6f69 6e74  .|i6 : P = point
│ │ │ │ +00004c00: 5c7b 7b31 2c32 2c33 7d2c 7b2d 312c 312c  \{{1,2,3},{-1,1,
│ │ │ │ +00004c10: 317d 2c7b 312c 312f 322c 2d31 2f33 7d7d  1},{1,1/2,-1/3}}
│ │ │ │ +00004c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004c30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004c40: 0a7c 6f36 203d 207b 506f 696e 747b 312c  .|o6 = {Point{1,
│ │ │ │ -00004c50: 2032 2c20 337d 2c20 506f 696e 747b 2d31   2, 3}, Point{-1
│ │ │ │ -00004c60: 2c20 312c 2031 7d2c 2050 6f69 6e74 7b31  , 1, 1}, Point{1
│ │ │ │ -00004c70: 2c20 2d2c 202d 202d 7d7d 2020 2020 2020  , -, - -}}      
│ │ │ │ +00004c40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00004c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004c80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00004c90: 0a7c 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 3220 2020 2033 2020 2020 2020 2020    2    3        
│ │ │ │ +00004cc0: 2020 3120 2020 2031 2020 2020 2020 2020    1    1        
│ │ │ │  00004cd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004ce0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -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                  
│ │ │ │ +00004ce0: 0a7c 6f36 203d 207b 506f 696e 747b 312c  .|o6 = {Point{1,
│ │ │ │ +00004cf0: 2032 2c20 337d 2c20 506f 696e 747b 2d31   2, 3}, Point{-1
│ │ │ │ +00004d00: 2c20 312c 2031 7d2c 2050 6f69 6e74 7b31  , 1, 1}, Point{1
│ │ │ │ +00004d10: 2c20 2d2c 202d 202d 7d7d 2020 2020 2020  , -, - -}}      
│ │ │ │  00004d20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004d30: 0a7c 6f36 203a 204c 6973 7420 2020 2020  .|o6 : List     
│ │ │ │ +00004d30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00004d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004d60: 2020 3220 2020 2033 2020 2020 2020 2020    2    3        
│ │ │ │  00004d70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004d80: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00004d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00004dd0: 0a7c 6937 203a 2068 6164 616d 6172 6450  .|i7 : hadamardP
│ │ │ │ -00004de0: 726f 6475 6374 2850 2920 2020 2020 2020  roduct(P)       
│ │ │ │ +00004d80: 0a7c 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 2020 2020 207c                 |
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│ │ │ │ +00004de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004e10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004e20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00004e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00004e20: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00004e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00004e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00004e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00004e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00004e70: 0a7c 6937 203a 2068 6164 616d 6172 6450  .|i7 : hadamardP
│ │ │ │ +00004e80: 726f 6475 6374 2850 2920 2020 2020 2020  roduct(P)       
│ │ │ │  00004e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004eb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00004ec0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00004ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004f00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │  00004f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004f50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00004f60: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00004f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00004fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
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│ │ │ │ -00005080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000050a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -000051c0: 726f 6475 6374 284c 6973 742c 4c69 7374  roduct(List,List
│ │ │ │ -000051d0: 2920 2d2d 2048 6164 616d 6172 6420 7072  ) -- Hadamard pr
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│ │ │ │ -00005200: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00005210: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00005220: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00005230: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -00005350: 6f74 6520 6c69 7374 3a20 284d 6163 6175  ote list: (Macau
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│ │ │ │ -00005380: 6f69 6e74 2c0a 0a44 6573 6372 6970 7469  oint,..Descripti
│ │ │ │ -00005390: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │ -000053a0: 4769 7665 6e20 7477 6f20 7365 7473 206f  Given two sets o
│ │ │ │ -000053b0: 6620 706f 696e 7473 2024 4c24 2061 6e64  f points $L$ and
│ │ │ │ -000053c0: 2024 4d24 2072 6574 7572 6e73 2074 6865   $M$ returns the
│ │ │ │ -000053d0: 206c 6973 7420 6f66 2028 7765 6c6c 2d64   list of (well-d
│ │ │ │ -000053e0: 6566 696e 6564 290a 656e 7472 7977 6973  efined).entrywis
│ │ │ │ -000053f0: 6520 6d75 6c74 6970 6c69 6361 7469 6f6e  e multiplication
│ │ │ │ -00005400: 206f 6620 7061 6972 7320 6f66 2070 6f69   of pairs of poi
│ │ │ │ -00005410: 6e74 7320 696e 2074 6865 2063 6172 7465  nts in the carte
│ │ │ │ -00005420: 7369 616e 2070 726f 6475 6374 2024 4c5c  sian product $L\
│ │ │ │ -00005430: 7469 6d65 730a 4d24 2e0a 0a2b 2d2d 2d2d  times.M$...+----
│ │ │ │ -00005440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005460: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ -00005470: 4c20 3d20 7b70 6f69 6e74 7b30 2c31 7d2c  L = {point{0,1},
│ │ │ │ -00005480: 2070 6f69 6e74 7b31 2c32 7d7d 3b20 2020   point{1,2}};   
│ │ │ │ -00005490: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -000054a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000054b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000054c0: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 4d20  ------+.|i2 : M 
│ │ │ │ -000054d0: 3d20 7b70 6f69 6e74 7b31 2c30 7d2c 2070  = {point{1,0}, p
│ │ │ │ -000054e0: 6f69 6e74 7b32 2c32 7d7d 3b20 2020 2020  oint{2,2}};     
│ │ │ │ -000054f0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00005500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005520: 2d2d 2d2d 2b0a 7c69 3320 3a20 6861 6461  ----+.|i3 : hada
│ │ │ │ -00005530: 6d61 7264 5072 6f64 7563 7428 4c2c 4d29  mardProduct(L,M)
│ │ │ │ -00005540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005550: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00005560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005580: 2020 7c0a 7c6f 3320 3d20 7b50 6f69 6e74    |.|o3 = {Point
│ │ │ │ -00005590: 7b31 2c20 307d 2c20 506f 696e 747b 302c  {1, 0}, Point{0,
│ │ │ │ -000055a0: 2032 7d2c 2050 6f69 6e74 7b32 2c20 347d   2}, Point{2, 4}
│ │ │ │ -000055b0: 7d7c 0a7c 2020 2020 2020 2020 2020 2020  }|.|            
│ │ │ │ -000055c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000055d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000055e0: 7c0a 7c6f 3320 3a20 4c69 7374 2020 2020  |.|o3 : List    
│ │ │ │ -000055f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005600: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00005610: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00005620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00005640: 0a57 6179 7320 746f 2075 7365 2074 6869  .Ways to use thi
│ │ │ │ -00005650: 7320 6d65 7468 6f64 3a0a 3d3d 3d3d 3d3d  s method:.======
│ │ │ │ -00005660: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00005670: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6861  ==..  * *note ha
│ │ │ │ -00005680: 6461 6d61 7264 5072 6f64 7563 7428 4c69  damardProduct(Li
│ │ │ │ -00005690: 7374 2c4c 6973 7429 3a20 6861 6461 6d61  st,List): hadama
│ │ │ │ -000056a0: 7264 5072 6f64 7563 745f 6c70 4c69 7374  rdProduct_lpList
│ │ │ │ -000056b0: 5f63 6d4c 6973 745f 7270 2c20 2d2d 0a20  _cmList_rp, --. 
│ │ │ │ -000056c0: 2020 2048 6164 616d 6172 6420 7072 6f64     Hadamard prod
│ │ │ │ -000056d0: 7563 7420 6f66 2074 776f 2073 6574 7320  uct of two sets 
│ │ │ │ -000056e0: 6f66 2070 6f69 6e74 730a 2d2d 2d2d 2d2d  of points.------
│ │ │ │ -000056f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005730: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073  ---------..The s
│ │ │ │ -00005740: 6f75 7263 6520 6f66 2074 6869 7320 646f  ource of this do
│ │ │ │ -00005750: 6375 6d65 6e74 2069 7320 696e 0a2f 6275  cument is in./bu
│ │ │ │ -00005760: 696c 642f 7265 7072 6f64 7563 6962 6c65  ild/reproducible
│ │ │ │ -00005770: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ -00005780: 312e 3235 2e31 312b 6473 2f4d 322f 4d61  1.25.11+ds/M2/Ma
│ │ │ │ -00005790: 6361 756c 6179 322f 7061 636b 6167 6573  caulay2/packages
│ │ │ │ -000057a0: 2f48 6164 616d 6172 642e 6d32 0a3a 3334  /Hadamard.m2.:34
│ │ │ │ -000057b0: 353a 302e 0a1f 0a46 696c 653a 2048 6164  5:0....File: Had
│ │ │ │ -000057c0: 616d 6172 642e 696e 666f 2c20 4e6f 6465  amard.info, Node
│ │ │ │ -000057d0: 3a20 6964 6561 6c4f 6650 726f 6a65 6374  : idealOfProject
│ │ │ │ -000057e0: 6976 6550 6f69 6e74 732c 204e 6578 743a  ivePoints, Next:
│ │ │ │ -000057f0: 2050 6f69 6e74 2c20 5072 6576 3a20 6861   Point, Prev: ha
│ │ │ │ -00005800: 6461 6d61 7264 5072 6f64 7563 745f 6c70  damardProduct_lp
│ │ │ │ -00005810: 4c69 7374 5f63 6d4c 6973 745f 7270 2c20  List_cmList_rp, 
│ │ │ │ -00005820: 5570 3a20 546f 700a 0a69 6465 616c 4f66  Up: Top..idealOf
│ │ │ │ -00005830: 5072 6f6a 6563 7469 7665 506f 696e 7473  ProjectivePoints
│ │ │ │ -00005840: 202d 2d20 636f 6d70 7574 6573 2074 6865   -- computes the
│ │ │ │ -00005850: 2069 6465 616c 206f 6620 7365 7420 6f66   ideal of set of
│ │ │ │ -00005860: 2070 6f69 6e74 730a 2a2a 2a2a 2a2a 2a2a   points.********
│ │ │ │ -00005870: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00005880: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00005890: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000058a0: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167  ******..  * Usag
│ │ │ │ -000058b0: 653a 200a 2020 2020 2020 2020 6964 6561  e: .        idea
│ │ │ │ -000058c0: 6c4f 6650 726f 6a65 6374 6976 6550 6f69  lOfProjectivePoi
│ │ │ │ -000058d0: 6e74 7328 4c2c 5329 0a20 202a 2049 6e70  nts(L,S).  * Inp
│ │ │ │ -000058e0: 7574 733a 0a20 2020 2020 202a 204c 2c20  uts:.      * L, 
│ │ │ │ -000058f0: 6120 2a6e 6f74 6520 6c69 7374 3a20 284d  a *note list: (M
│ │ │ │ -00005900: 6163 6175 6c61 7932 446f 6329 4c69 7374  acaulay2Doc)List
│ │ │ │ -00005910: 2c2c 206f 6620 2a6e 6f74 6520 2870 726f  ,, of *note (pro
│ │ │ │ -00005920: 6a65 6374 6976 6529 2070 6f69 6e74 733a  jective) points:
│ │ │ │ -00005930: 0a20 2020 2020 2020 2050 6f69 6e74 2c0a  .        Point,.
│ │ │ │ -00005940: 2020 2020 2020 2a20 532c 2061 202a 6e6f        * S, a *no
│ │ │ │ -00005950: 7465 2072 696e 673a 2028 4d61 6361 756c  te ring: (Macaul
│ │ │ │ -00005960: 6179 3244 6f63 2952 696e 672c 2c20 0a20  ay2Doc)Ring,, . 
│ │ │ │ -00005970: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ -00005980: 2020 2a20 492c 2061 6e20 2a6e 6f74 6520    * I, an *note 
│ │ │ │ -00005990: 6964 6561 6c3a 2028 4d61 6361 756c 6179  ideal: (Macaulay
│ │ │ │ -000059a0: 3244 6f63 2949 6465 616c 2c2c 200a 0a44  2Doc)Ideal,, ..D
│ │ │ │ -000059b0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ -000059c0: 3d3d 3d3d 3d3d 0a0a 4769 7665 6e20 6120  ======..Given a 
│ │ │ │ -000059d0: 7365 7420 706f 696e 7473 2024 5824 2c20  set points $X$, 
│ │ │ │ -000059e0: 6974 2072 6574 7572 6e73 2074 6865 2064  it returns the d
│ │ │ │ -000059f0: 6566 696e 696e 6720 6964 6561 6c20 6f66  efining ideal of
│ │ │ │ -00005a00: 2024 4928 5829 240a 0a2b 2d2d 2d2d 2d2d   $I(X)$..+------
│ │ │ │ -00005a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005a50: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2053  -------+.|i1 : S
│ │ │ │ -00005a60: 203d 2051 515b 782c 792c 7a5d 2020 2020   = QQ[x,y,z]    
│ │ │ │ -00005a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005aa0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00005ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005af0: 2020 2020 2020 207c 0a7c 6f31 203d 2053         |.|o1 = S
│ │ │ │ -00005b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004f60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00004f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004fa0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00004fb0: 0a7c 6f37 203a 2050 6f69 6e74 2020 2020  .|o7 : Point    
│ │ │ │ +00004fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004ff0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00005000: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00005010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00005050: 0a0a 5761 7973 2074 6f20 7573 6520 7468  ..Ways to use th
│ │ │ │ +00005060: 6973 206d 6574 686f 643a 0a3d 3d3d 3d3d  is method:.=====
│ │ │ │ +00005070: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00005080: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2068  ===..  * *note h
│ │ │ │ +00005090: 6164 616d 6172 6450 726f 6475 6374 284c  adamardProduct(L
│ │ │ │ +000050a0: 6973 7429 3a20 6861 6461 6d61 7264 5072  ist): hadamardPr
│ │ │ │ +000050b0: 6f64 7563 745f 6c70 4c69 7374 5f72 702c  oduct_lpList_rp,
│ │ │ │ +000050c0: 202d 2d20 4861 6461 6d61 7264 2070 726f   -- Hadamard pro
│ │ │ │ +000050d0: 6475 6374 0a20 2020 206f 6620 6120 6c69  duct.    of a li
│ │ │ │ +000050e0: 7374 206f 6620 686f 6d6f 6765 6e65 6f75  st of homogeneou
│ │ │ │ +000050f0: 7320 6964 6561 6c73 2c20 6f72 2070 6f69  s ideals, or poi
│ │ │ │ +00005100: 6e74 730a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  nts.------------
│ │ │ │ +00005110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005150: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520  ---..The source 
│ │ │ │ +00005160: 6f66 2074 6869 7320 646f 6375 6d65 6e74  of this document
│ │ │ │ +00005170: 2069 7320 696e 0a2f 6275 696c 642f 7265   is in./build/re
│ │ │ │ +00005180: 7072 6f64 7563 6962 6c65 2d70 6174 682f  producible-path/
│ │ │ │ +00005190: 6d61 6361 756c 6179 322d 312e 3235 2e31  macaulay2-1.25.1
│ │ │ │ +000051a0: 312b 6473 2f4d 322f 4d61 6361 756c 6179  1+ds/M2/Macaulay
│ │ │ │ +000051b0: 322f 7061 636b 6167 6573 2f48 6164 616d  2/packages/Hadam
│ │ │ │ +000051c0: 6172 642e 6d32 0a3a 3332 303a 302e 0a1f  ard.m2.:320:0...
│ │ │ │ +000051d0: 0a46 696c 653a 2048 6164 616d 6172 642e  .File: Hadamard.
│ │ │ │ +000051e0: 696e 666f 2c20 4e6f 6465 3a20 6861 6461  info, Node: hada
│ │ │ │ +000051f0: 6d61 7264 5072 6f64 7563 745f 6c70 4c69  mardProduct_lpLi
│ │ │ │ +00005200: 7374 5f63 6d4c 6973 745f 7270 2c20 4e65  st_cmList_rp, Ne
│ │ │ │ +00005210: 7874 3a20 6964 6561 6c4f 6650 726f 6a65  xt: idealOfProje
│ │ │ │ +00005220: 6374 6976 6550 6f69 6e74 732c 2050 7265  ctivePoints, Pre
│ │ │ │ +00005230: 763a 2068 6164 616d 6172 6450 726f 6475  v: hadamardProdu
│ │ │ │ +00005240: 6374 5f6c 704c 6973 745f 7270 2c20 5570  ct_lpList_rp, Up
│ │ │ │ +00005250: 3a20 546f 700a 0a68 6164 616d 6172 6450  : Top..hadamardP
│ │ │ │ +00005260: 726f 6475 6374 284c 6973 742c 4c69 7374  roduct(List,List
│ │ │ │ +00005270: 2920 2d2d 2048 6164 616d 6172 6420 7072  ) -- Hadamard pr
│ │ │ │ +00005280: 6f64 7563 7420 6f66 2074 776f 2073 6574  oduct of two set
│ │ │ │ +00005290: 7320 6f66 2070 6f69 6e74 730a 2a2a 2a2a  s of points.****
│ │ │ │ +000052a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000052b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000052c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000052d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000052e0: 0a0a 2020 2a20 4675 6e63 7469 6f6e 3a20  ..  * Function: 
│ │ │ │ +000052f0: 2a6e 6f74 6520 6861 6461 6d61 7264 5072  *note hadamardPr
│ │ │ │ +00005300: 6f64 7563 743a 2068 6164 616d 6172 6450  oduct: hadamardP
│ │ │ │ +00005310: 726f 6475 6374 2c0a 2020 2a20 5573 6167  roduct,.  * Usag
│ │ │ │ +00005320: 653a 200a 2020 2020 2020 2020 6861 6461  e: .        hada
│ │ │ │ +00005330: 6d61 7264 5072 6f64 7563 7428 4c2c 4d29  mardProduct(L,M)
│ │ │ │ +00005340: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ +00005350: 2020 202a 204c 2c20 6120 2a6e 6f74 6520     * L, a *note 
│ │ │ │ +00005360: 6c69 7374 3a20 284d 6163 6175 6c61 7932  list: (Macaulay2
│ │ │ │ +00005370: 446f 6329 4c69 7374 2c2c 206f 6620 2a6e  Doc)List,, of *n
│ │ │ │ +00005380: 6f74 6520 506f 696e 743a 2050 6f69 6e74  ote Point: Point
│ │ │ │ +00005390: 2c0a 2020 2020 2020 2a20 4d2c 2061 202a  ,.      * M, a *
│ │ │ │ +000053a0: 6e6f 7465 206c 6973 743a 2028 4d61 6361  note list: (Maca
│ │ │ │ +000053b0: 756c 6179 3244 6f63 294c 6973 742c 2c20  ulay2Doc)List,, 
│ │ │ │ +000053c0: 6f66 202a 6e6f 7465 2050 6f69 6e74 3a20  of *note Point: 
│ │ │ │ +000053d0: 506f 696e 742c 0a20 202a 204f 7574 7075  Point,.  * Outpu
│ │ │ │ +000053e0: 7473 3a0a 2020 2020 2020 2a20 6120 2a6e  ts:.      * a *n
│ │ │ │ +000053f0: 6f74 6520 6c69 7374 3a20 284d 6163 6175  ote list: (Macau
│ │ │ │ +00005400: 6c61 7932 446f 6329 4c69 7374 2c2c 206f  lay2Doc)List,, o
│ │ │ │ +00005410: 6620 2a6e 6f74 6520 506f 696e 743a 2050  f *note Point: P
│ │ │ │ +00005420: 6f69 6e74 2c0a 0a44 6573 6372 6970 7469  oint,..Descripti
│ │ │ │ +00005430: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │ +00005440: 4769 7665 6e20 7477 6f20 7365 7473 206f  Given two sets o
│ │ │ │ +00005450: 6620 706f 696e 7473 2024 4c24 2061 6e64  f points $L$ and
│ │ │ │ +00005460: 2024 4d24 2072 6574 7572 6e73 2074 6865   $M$ returns the
│ │ │ │ +00005470: 206c 6973 7420 6f66 2028 7765 6c6c 2d64   list of (well-d
│ │ │ │ +00005480: 6566 696e 6564 290a 656e 7472 7977 6973  efined).entrywis
│ │ │ │ +00005490: 6520 6d75 6c74 6970 6c69 6361 7469 6f6e  e multiplication
│ │ │ │ +000054a0: 206f 6620 7061 6972 7320 6f66 2070 6f69   of pairs of poi
│ │ │ │ +000054b0: 6e74 7320 696e 2074 6865 2063 6172 7465  nts in the carte
│ │ │ │ +000054c0: 7369 616e 2070 726f 6475 6374 2024 4c5c  sian product $L\
│ │ │ │ +000054d0: 7469 6d65 730a 4d24 2e0a 0a2b 2d2d 2d2d  times.M$...+----
│ │ │ │ +000054e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00005570: 3d20 7b70 6f69 6e74 7b31 2c30 7d2c 2070  = {point{1,0}, p
│ │ │ │ +00005580: 6f69 6e74 7b32 2c32 7d7d 3b20 2020 2020  oint{2,2}};     
│ │ │ │ +00005590: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000055a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000055b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000055c0: 2d2d 2d2d 2b0a 7c69 3320 3a20 6861 6461  ----+.|i3 : hada
│ │ │ │ +000055d0: 6d61 7264 5072 6f64 7563 7428 4c2c 4d29  mardProduct(L,M)
│ │ │ │ +000055e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000055f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00005600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005620: 2020 7c0a 7c6f 3320 3d20 7b50 6f69 6e74    |.|o3 = {Point
│ │ │ │ +00005630: 7b32 2c20 347d 2c20 506f 696e 747b 312c  {2, 4}, Point{1,
│ │ │ │ +00005640: 2030 7d2c 2050 6f69 6e74 7b30 2c20 327d   0}, Point{0, 2}
│ │ │ │ +00005650: 7d7c 0a7c 2020 2020 2020 2020 2020 2020  }|.|            
│ │ │ │ +00005660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005680: 7c0a 7c6f 3320 3a20 4c69 7374 2020 2020  |.|o3 : List    
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│ │ │ │ +000056a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000056b0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000056c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +000056e0: 0a57 6179 7320 746f 2075 7365 2074 6869  .Ways to use thi
│ │ │ │ +000056f0: 7320 6d65 7468 6f64 3a0a 3d3d 3d3d 3d3d  s method:.======
│ │ │ │ +00005700: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ +00005730: 7374 2c4c 6973 7429 3a20 6861 6461 6d61  st,List): hadama
│ │ │ │ +00005740: 7264 5072 6f64 7563 745f 6c70 4c69 7374  rdProduct_lpList
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│ │ │ │ +00005760: 2020 2048 6164 616d 6172 6420 7072 6f64     Hadamard prod
│ │ │ │ +00005770: 7563 7420 6f66 2074 776f 2073 6574 7320  uct of two sets 
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│ │ │ │ +000057a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000057b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000057c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000057d0: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073  ---------..The s
│ │ │ │ +000057e0: 6f75 7263 6520 6f66 2074 6869 7320 646f  ource of this do
│ │ │ │ +000057f0: 6375 6d65 6e74 2069 7320 696e 0a2f 6275  cument is in./bu
│ │ │ │ +00005800: 696c 642f 7265 7072 6f64 7563 6962 6c65  ild/reproducible
│ │ │ │ +00005810: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ +00005820: 312e 3235 2e31 312b 6473 2f4d 322f 4d61  1.25.11+ds/M2/Ma
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│ │ │ │ +00005880: 6976 6550 6f69 6e74 732c 204e 6578 743a  ivePoints, Next:
│ │ │ │ +00005890: 2050 6f69 6e74 2c20 5072 6576 3a20 6861   Point, Prev: ha
│ │ │ │ +000058a0: 6461 6d61 7264 5072 6f64 7563 745f 6c70  damardProduct_lp
│ │ │ │ +000058b0: 4c69 7374 5f63 6d4c 6973 745f 7270 2c20  List_cmList_rp, 
│ │ │ │ +000058c0: 5570 3a20 546f 700a 0a69 6465 616c 4f66  Up: Top..idealOf
│ │ │ │ +000058d0: 5072 6f6a 6563 7469 7665 506f 696e 7473  ProjectivePoints
│ │ │ │ +000058e0: 202d 2d20 636f 6d70 7574 6573 2074 6865   -- computes the
│ │ │ │ +000058f0: 2069 6465 616c 206f 6620 7365 7420 6f66   ideal of set of
│ │ │ │ +00005900: 2070 6f69 6e74 730a 2a2a 2a2a 2a2a 2a2a   points.********
│ │ │ │ +00005910: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00005920: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00005930: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00005940: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167  ******..  * Usag
│ │ │ │ +00005950: 653a 200a 2020 2020 2020 2020 6964 6561  e: .        idea
│ │ │ │ +00005960: 6c4f 6650 726f 6a65 6374 6976 6550 6f69  lOfProjectivePoi
│ │ │ │ +00005970: 6e74 7328 4c2c 5329 0a20 202a 2049 6e70  nts(L,S).  * Inp
│ │ │ │ +00005980: 7574 733a 0a20 2020 2020 202a 204c 2c20  uts:.      * L, 
│ │ │ │ +00005990: 6120 2a6e 6f74 6520 6c69 7374 3a20 284d  a *note list: (M
│ │ │ │ +000059a0: 6163 6175 6c61 7932 446f 6329 4c69 7374  acaulay2Doc)List
│ │ │ │ +000059b0: 2c2c 206f 6620 2a6e 6f74 6520 2870 726f  ,, of *note (pro
│ │ │ │ +000059c0: 6a65 6374 6976 6529 2070 6f69 6e74 733a  jective) points:
│ │ │ │ +000059d0: 0a20 2020 2020 2020 2050 6f69 6e74 2c0a  .        Point,.
│ │ │ │ +000059e0: 2020 2020 2020 2a20 532c 2061 202a 6e6f        * S, a *no
│ │ │ │ +000059f0: 7465 2072 696e 673a 2028 4d61 6361 756c  te ring: (Macaul
│ │ │ │ +00005a00: 6179 3244 6f63 2952 696e 672c 2c20 0a20  ay2Doc)Ring,, . 
│ │ │ │ +00005a10: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ +00005a20: 2020 2a20 492c 2061 6e20 2a6e 6f74 6520    * I, an *note 
│ │ │ │ +00005a30: 6964 6561 6c3a 2028 4d61 6361 756c 6179  ideal: (Macaulay
│ │ │ │ +00005a40: 3244 6f63 2949 6465 616c 2c2c 200a 0a44  2Doc)Ideal,, ..D
│ │ │ │ +00005a50: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ +00005a60: 3d3d 3d3d 3d3d 0a0a 4769 7665 6e20 6120  ======..Given a 
│ │ │ │ +00005a70: 7365 7420 706f 696e 7473 2024 5824 2c20  set points $X$, 
│ │ │ │ +00005a80: 6974 2072 6574 7572 6e73 2074 6865 2064  it returns the d
│ │ │ │ +00005a90: 6566 696e 696e 6720 6964 6561 6c20 6f66  efining ideal of
│ │ │ │ +00005aa0: 2024 4928 5829 240a 0a2b 2d2d 2d2d 2d2d   $I(X)$..+------
│ │ │ │ +00005ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005af0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2053  -------+.|i1 : S
│ │ │ │ +00005b00: 203d 2051 515b 782c 792c 7a5d 2020 2020   = QQ[x,y,z]    
│ │ │ │  00005b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005b40: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00005b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005b90: 2020 2020 2020 207c 0a7c 6f31 203a 2050         |.|o1 : P
│ │ │ │ -00005ba0: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020  olynomialRing   
│ │ │ │ +00005b90: 2020 2020 2020 207c 0a7c 6f31 203d 2053         |.|o1 = S
│ │ │ │ +00005ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005be0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -00005bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005c30: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2058  -------+.|i2 : X
│ │ │ │ -00005c40: 203d 207b 706f 696e 747b 312c 312c 307d   = {point{1,1,0}
│ │ │ │ -00005c50: 2c70 6f69 6e74 7b30 2c31 2c31 7d2c 706f  ,point{0,1,1},po
│ │ │ │ -00005c60: 696e 747b 312c 322c 2d31 7d7d 2020 2020  int{1,2,-1}}    
│ │ │ │ +00005be0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00005bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00005c30: 2020 2020 2020 207c 0a7c 6f31 203a 2050         |.|o1 : P
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│ │ │ │ +00005c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005c80: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00005c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005cd0: 2020 2020 2020 207c 0a7c 6f32 203d 207b         |.|o2 = {
│ │ │ │ -00005ce0: 506f 696e 747b 312c 2031 2c20 307d 2c20  Point{1, 1, 0}, 
│ │ │ │ -00005cf0: 506f 696e 747b 302c 2031 2c20 317d 2c20  Point{0, 1, 1}, 
│ │ │ │ -00005d00: 506f 696e 747b 312c 2032 2c20 2d31 7d7d  Point{1, 2, -1}}
│ │ │ │ +00005c80: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00005c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005cd0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2058  -------+.|i2 : X
│ │ │ │ +00005ce0: 203d 207b 706f 696e 747b 312c 312c 307d   = {point{1,1,0}
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│ │ │ │  00005d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005d20: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00005d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00005d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005d70: 2020 2020 2020 207c 0a7c 6f32 203a 204c         |.|o2 : L
│ │ │ │ -00005d80: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ -00005d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005d70: 2020 2020 2020 207c 0a7c 6f32 203d 207b         |.|o2 = {
│ │ │ │ +00005d80: 506f 696e 747b 312c 2031 2c20 307d 2c20  Point{1, 1, 0}, 
│ │ │ │ +00005d90: 506f 696e 747b 302c 2031 2c20 317d 2c20  Point{0, 1, 1}, 
│ │ │ │ +00005da0: 506f 696e 747b 312c 2032 2c20 2d31 7d7d  Point{1, 2, -1}}
│ │ │ │  00005db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005dc0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -00005dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005e10: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2049  -------+.|i3 : I
│ │ │ │ -00005e20: 203d 2069 6465 616c 4f66 5072 6f6a 6563   = idealOfProjec
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│ │ │ │ +00005dc0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
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│ │ │ │ -00005e60: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00005e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00005e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005eb0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00005ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005ed0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -00005ee0: 3220 2020 2020 2020 2020 2020 3220 2020  2           2   
│ │ │ │ -00005ef0: 2032 2020 2020 2032 2020 2020 2020 2020   2     2        
│ │ │ │ -00005f00: 2020 2020 3220 207c 0a7c 6f33 203d 2069      2  |.|o3 = i
│ │ │ │ -00005f10: 6465 616c 2028 3378 2a7a 202d 2079 2a7a  deal (3x*z - y*z
│ │ │ │ -00005f20: 202b 207a 202c 2033 782a 7920 2d20 3379   + z , 3x*y - 3y
│ │ │ │ -00005f30: 2020 2d20 792a 7a20 2b20 347a 202c 2033    - y*z + 4z , 3
│ │ │ │ -00005f40: 7820 202d 2033 7920 202d 2032 792a 7a20  x  - 3y  - 2y*z 
│ │ │ │ -00005f50: 2b20 357a 202c 207c 0a7c 2020 2020 202d  + 5z , |.|     -
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│ │ │ │ -00005f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00005fa0: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2020  -------|.|      
│ │ │ │ -00005fb0: 3220 2020 2020 2020 3220 2020 2020 3320  2       2     3 
│ │ │ │ -00005fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00005ff0: 2020 2020 2020 207c 0a7c 2020 2020 2079         |.|     y
│ │ │ │ -00006000: 207a 202b 2079 2a7a 2020 2d20 327a 2029   z + y*z  - 2z )
│ │ │ │ -00006010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006040: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00006050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005e60: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00005e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00005eb0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2049  -------+.|i3 : I
│ │ │ │ +00005ec0: 203d 2069 6465 616c 4f66 5072 6f6a 6563   = idealOfProjec
│ │ │ │ +00005ed0: 7469 7665 506f 696e 7473 2858 2c53 2920  tivePoints(X,S) 
│ │ │ │ +00005ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005f00: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00005f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005f50: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00005f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00005f70: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00005f80: 3220 2020 2020 2020 2020 2020 3220 2020  2           2   
│ │ │ │ +00005f90: 2032 2020 2020 2032 2020 2020 2020 2020   2     2        
│ │ │ │ +00005fa0: 2020 2020 3220 207c 0a7c 6f33 203d 2069      2  |.|o3 = i
│ │ │ │ +00005fb0: 6465 616c 2028 3378 2a7a 202d 2079 2a7a  deal (3x*z - y*z
│ │ │ │ +00005fc0: 202b 207a 202c 2033 782a 7920 2d20 3379   + z , 3x*y - 3y
│ │ │ │ +00005fd0: 2020 2d20 792a 7a20 2b20 347a 202c 2033    - y*z + 4z , 3
│ │ │ │ +00005fe0: 7820 202d 2033 7920 202d 2032 792a 7a20  x  - 3y  - 2y*z 
│ │ │ │ +00005ff0: 2b20 357a 202c 207c 0a7c 2020 2020 202d  + 5z , |.|     -
│ │ │ │ +00006000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006040: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2020  -------|.|      
│ │ │ │ +00006050: 3220 2020 2020 2020 3220 2020 2020 3320  2       2     3 
│ │ │ │  00006060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006090: 2020 2020 2020 207c 0a7c 6f33 203a 2049         |.|o3 : I
│ │ │ │ -000060a0: 6465 616c 206f 6620 5320 2020 2020 2020  deal of S       
│ │ │ │ +00006090: 2020 2020 2020 207c 0a7c 2020 2020 2079         |.|     y
│ │ │ │ +000060a0: 207a 202b 2079 2a7a 2020 2d20 327a 2029   z + y*z  - 2z )
│ │ │ │  000060b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000060c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000060d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000060e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -000060f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006130: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2049  -------+.|i4 : I
│ │ │ │ -00006140: 3220 3d20 6861 6461 6d61 7264 506f 7765  2 = hadamardPowe
│ │ │ │ -00006150: 7228 492c 3229 2020 2020 2020 2020 2020  r(I,2)          
│ │ │ │ +000060e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000060f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006130: 2020 2020 2020 207c 0a7c 6f33 203a 2049         |.|o3 : I
│ │ │ │ +00006140: 6465 616c 206f 6620 5320 2020 2020 2020  deal of S       
│ │ │ │ +00006150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006180: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00006190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000061a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000061b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000061c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000061d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000061e0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -000061f0: 2032 2020 2020 2020 3220 2020 2020 3320   2      2     3 
│ │ │ │ -00006200: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -00006210: 2032 2020 2020 2020 2032 2020 2020 3320   2       2    3 
│ │ │ │ -00006220: 2020 2020 3220 207c 0a7c 6f34 203d 2069      2  |.|o4 = i
│ │ │ │ -00006230: 6465 616c 2028 7920 7a20 2d20 3138 782a  deal (y z - 18x*
│ │ │ │ -00006240: 7a20 202b 2079 2a7a 2020 2d20 327a 202c  z  + y*z  - 2z ,
│ │ │ │ -00006250: 2078 2a79 2a7a 202d 2034 782a 7a20 2c20   x*y*z - 4x*z , 
│ │ │ │ -00006260: 7820 7a20 2d20 782a 7a20 2c20 3278 2020  x z - x*z , 2x  
│ │ │ │ -00006270: 2d20 3378 2079 207c 0a7c 2020 2020 202d  - 3x y |.|     -
│ │ │ │ -00006280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000062a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000062b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000062c0: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2020  -------|.|      
│ │ │ │ -000062d0: 2020 2020 3220 2020 2020 2020 3220 2020      2       2   
│ │ │ │ -000062e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000062f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006310: 2020 2020 2020 207c 0a7c 2020 2020 202b         |.|     +
│ │ │ │ -00006320: 2078 2a79 2020 2d20 3678 2a7a 2029 2020   x*y  - 6x*z )  
│ │ │ │ -00006330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006360: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00006370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006180: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00006190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000061a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000061b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000061c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000061d0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2049  -------+.|i4 : I
│ │ │ │ +000061e0: 3220 3d20 6861 6461 6d61 7264 506f 7765  2 = hadamardPowe
│ │ │ │ +000061f0: 7228 492c 3229 2020 2020 2020 2020 2020  r(I,2)          
│ │ │ │ +00006200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006220: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00006230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006270: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00006280: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +00006290: 2032 2020 2020 2020 3220 2020 2020 3320   2      2     3 
│ │ │ │ +000062a0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +000062b0: 2032 2020 2020 2020 2032 2020 2020 3320   2       2    3 
│ │ │ │ +000062c0: 2020 2020 3220 207c 0a7c 6f34 203d 2069      2  |.|o4 = i
│ │ │ │ +000062d0: 6465 616c 2028 7920 7a20 2d20 3138 782a  deal (y z - 18x*
│ │ │ │ +000062e0: 7a20 202b 2079 2a7a 2020 2d20 327a 202c  z  + y*z  - 2z ,
│ │ │ │ +000062f0: 2078 2a79 2a7a 202d 2034 782a 7a20 2c20   x*y*z - 4x*z , 
│ │ │ │ +00006300: 7820 7a20 2d20 782a 7a20 2c20 3278 2020  x z - x*z , 2x  
│ │ │ │ +00006310: 2d20 3378 2079 207c 0a7c 2020 2020 202d  - 3x y |.|     -
│ │ │ │ +00006320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006360: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2020  -------|.|      
│ │ │ │ +00006370: 2020 2020 3220 2020 2020 2020 3220 2020      2       2   
│ │ │ │  00006380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000063a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000063b0: 2020 2020 2020 207c 0a7c 6f34 203a 2049         |.|o4 : I
│ │ │ │ -000063c0: 6465 616c 206f 6620 5320 2020 2020 2020  deal of S       
│ │ │ │ +000063b0: 2020 2020 2020 207c 0a7c 2020 2020 202b         |.|     +
│ │ │ │ +000063c0: 2078 2a79 2020 2d20 3678 2a7a 2029 2020   x*y  - 6x*z )  
│ │ │ │  000063d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000063e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000063f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006400: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -00006410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006450: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2058  -------+.|i5 : X
│ │ │ │ -00006460: 3220 3d20 6861 6461 6d61 7264 506f 7765  2 = hadamardPowe
│ │ │ │ -00006470: 7228 582c 3229 2020 2020 2020 2020 2020  r(X,2)          
│ │ │ │ +00006400: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00006410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006450: 2020 2020 2020 207c 0a7c 6f34 203a 2049         |.|o4 : I
│ │ │ │ +00006460: 6465 616c 206f 6620 5320 2020 2020 2020  deal of S       
│ │ │ │ +00006470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000064a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000064b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000064c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000064d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000064e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000064f0: 2020 2020 2020 207c 0a7c 6f35 203d 207b         |.|o5 = {
│ │ │ │ -00006500: 506f 696e 747b 312c 2034 2c20 317d 2c20  Point{1, 4, 1}, 
│ │ │ │ -00006510: 506f 696e 747b 302c 2032 2c20 2d31 7d2c  Point{0, 2, -1},
│ │ │ │ -00006520: 2050 6f69 6e74 7b30 2c20 312c 2030 7d2c   Point{0, 1, 0},
│ │ │ │ -00006530: 2050 6f69 6e74 7b30 2c20 312c 2031 7d2c   Point{0, 1, 1},
│ │ │ │ -00006540: 2020 2020 2020 207c 0a7c 2020 2020 202d         |.|     -
│ │ │ │ -00006550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006590: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2050  -------|.|     P
│ │ │ │ -000065a0: 6f69 6e74 7b31 2c20 312c 2030 7d2c 2050  oint{1, 1, 0}, P
│ │ │ │ -000065b0: 6f69 6e74 7b31 2c20 322c 2030 7d7d 2020  oint{1, 2, 0}}  
│ │ │ │ -000065c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000065d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000065e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000065f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006630: 2020 2020 2020 207c 0a7c 6f35 203a 204c         |.|o5 : L
│ │ │ │ -00006640: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ -00006650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000064a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000064b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000064c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000064d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000064e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000064f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2058  -------+.|i5 : X
│ │ │ │ +00006500: 3220 3d20 6861 6461 6d61 7264 506f 7765  2 = hadamardPowe
│ │ │ │ +00006510: 7228 582c 3229 2020 2020 2020 2020 2020  r(X,2)          
│ │ │ │ +00006520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006540: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00006550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006590: 2020 2020 2020 207c 0a7c 6f35 203d 207b         |.|o5 = {
│ │ │ │ +000065a0: 506f 696e 747b 312c 2034 2c20 317d 2c20  Point{1, 4, 1}, 
│ │ │ │ +000065b0: 506f 696e 747b 302c 2032 2c20 2d31 7d2c  Point{0, 2, -1},
│ │ │ │ +000065c0: 2050 6f69 6e74 7b30 2c20 312c 2030 7d2c   Point{0, 1, 0},
│ │ │ │ +000065d0: 2050 6f69 6e74 7b30 2c20 312c 2031 7d2c   Point{0, 1, 1},
│ │ │ │ +000065e0: 2020 2020 2020 207c 0a7c 2020 2020 202d         |.|     -
│ │ │ │ +000065f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006630: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2050  -------|.|     P
│ │ │ │ +00006640: 6f69 6e74 7b31 2c20 312c 2030 7d2c 2050  oint{1, 1, 0}, P
│ │ │ │ +00006650: 6f69 6e74 7b31 2c20 322c 2030 7d7d 2020  oint{1, 2, 0}}  
│ │ │ │  00006660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006680: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -00006690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000066a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000066b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000066c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000066d0: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2049  -------+.|i6 : I
│ │ │ │ -000066e0: 3220 3d3d 2069 6465 616c 4f66 5072 6f6a  2 == idealOfProj
│ │ │ │ -000066f0: 6563 7469 7665 506f 696e 7473 2858 322c  ectivePoints(X2,
│ │ │ │ -00006700: 5329 2020 2020 2020 2020 2020 2020 2020  S)              
│ │ │ │ +00006680: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00006690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000066a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000066b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000066c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000066d0: 2020 2020 2020 207c 0a7c 6f35 203a 204c         |.|o5 : L
│ │ │ │ +000066e0: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +000066f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006720: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00006730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00006770: 2020 2020 2020 207c 0a7c 6f36 203d 2074         |.|o6 = t
│ │ │ │ -00006780: 7275 6520 2020 2020 2020 2020 2020 2020  rue             
│ │ │ │ -00006790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000067a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006720: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00006730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006770: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2049  -------+.|i6 : I
│ │ │ │ +00006780: 3220 3d3d 2069 6465 616c 4f66 5072 6f6a  2 == idealOfProj
│ │ │ │ +00006790: 6563 7469 7665 506f 696e 7473 2858 322c  ectivePoints(X2,
│ │ │ │ +000067a0: 5329 2020 2020 2020 2020 2020 2020 2020  S)              
│ │ │ │  000067b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000067c0: 2020 2020 2020 207c 0a2b 2d2d 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006810: 2d2d 2d2d 2d2d 2d2b 0a0a 5761 7973 2074  -------+..Ways t
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│ │ │ │ -00006840: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00006850: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00006860: 3d3d 3d0a 0a20 202a 2022 6964 6561 6c4f  ===..  * "idealO
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│ │ │ │ -000068b0: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ -000068c0: 202a 6e6f 7465 2069 6465 616c 4f66 5072   *note idealOfPr
│ │ │ │ -000068d0: 6f6a 6563 7469 7665 506f 696e 7473 3a20  ojectivePoints: 
│ │ │ │ -000068e0: 6964 6561 6c4f 6650 726f 6a65 6374 6976  idealOfProjectiv
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│ │ │ │ -00006950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00006a90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00006aa0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00006ab0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ -00006ac0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ -00006ad0: 3d3d 3d3d 3d3d 3d0a 0a41 2070 6f69 6e74  =======..A point
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│ │ │ │ -00006ba0: 2050 6f69 6e74 3a0a 3d3d 3d3d 3d3d 3d3d   Point:.========
│ │ │ │ -00006bb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00006bc0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00006bd0: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 506f  ==..  * *note Po
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│ │ │ │ -00006c30: 2020 2a20 2a6e 6f74 6520 506f 696e 7420    * *note Point 
│ │ │ │ -00006c40: 3d3d 2050 6f69 6e74 3a20 506f 696e 7420  == Point: Point 
│ │ │ │ -00006c50: 3d3d 2050 6f69 6e74 2c20 2d2d 2063 6865  == Point, -- che
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│ │ │ │ -00006c70: 776f 2070 726f 6a65 6374 6976 650a 2020  wo projective.  
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│ │ │ │ -00006ca0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -00006cb0: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ -00006cc0: 6520 506f 696e 743a 2050 6f69 6e74 2c20  e Point: Point, 
│ │ │ │ -00006cd0: 6973 2061 202a 6e6f 7465 2074 7970 653a  is a *note type:
│ │ │ │ -00006ce0: 2028 4d61 6361 756c 6179 3244 6f63 2954   (Macaulay2Doc)T
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│ │ │ │ -00006d00: 746f 7220 636c 6173 7365 7320 2a6e 6f74  tor classes *not
│ │ │ │ -00006d10: 6520 4261 7369 634c 6973 743a 2028 4d61  e BasicList: (Ma
│ │ │ │ -00006d20: 6361 756c 6179 3244 6f63 2942 6173 6963  caulay2Doc)Basic
│ │ │ │ -00006d30: 4c69 7374 2c20 3c20 2a6e 6f74 6520 5468  List, < *note Th
│ │ │ │ -00006d40: 696e 673a 0a28 4d61 6361 756c 6179 3244  ing:.(Macaulay2D
│ │ │ │ -00006d50: 6f63 2954 6869 6e67 2c2e 0a0a 2d2d 2d2d  oc)Thing,...----
│ │ │ │ -00006d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00006da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
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│ │ │ │ -00006e00: 4d61 6361 756c 6179 322f 7061 636b 6167  Macaulay2/packag
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│ │ │ │ -00006e60: 2c20 5072 6576 3a20 506f 696e 742c 2055  , Prev: Point, U
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│ │ │ │ -00006e80: 2063 6f6e 7374 7275 6374 7320 6120 7072   constructs a pr
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│ │ │ │ -00006eb0: 2061 7272 6179 2920 6f66 2063 6f6f 7264   array) of coord
│ │ │ │ -00006ec0: 696e 6174 6573 2e0a 2a2a 2a2a 2a2a 2a2a  inates..********
│ │ │ │ -00006ed0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00006ee0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00006ef0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00006f00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00006f10: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361  *******..  * Usa
│ │ │ │ -00006f20: 6765 3a20 0a20 2020 2020 2020 2070 6f69  ge: .        poi
│ │ │ │ -00006f30: 6e74 284c 290a 2020 2a20 496e 7075 7473  nt(L).  * Inputs
│ │ │ │ -00006f40: 3a0a 2020 2020 2020 2a20 4c2c 2061 202a  :.      * L, a *
│ │ │ │ -00006f50: 6e6f 7465 206c 6973 743a 2028 4d61 6361  note list: (Maca
│ │ │ │ -00006f60: 756c 6179 3244 6f63 294c 6973 742c 2c20  ulay2Doc)List,, 
│ │ │ │ -00006f70: 6f72 202a 6e6f 7465 2061 7272 6179 3a0a  or *note array:.
│ │ │ │ -00006f80: 2020 2020 2020 2020 284d 6163 6175 6c61          (Macaula
│ │ │ │ -00006f90: 7932 446f 6329 4172 7261 792c 206f 7220  y2Doc)Array, or 
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│ │ │ │ -00006fc0: 6329 5669 7369 626c 654c 6973 742c 0a20  c)VisibleList,. 
│ │ │ │ -00006fd0: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ -00006fe0: 2020 2a20 616e 2069 6e73 7461 6e63 6520    * an instance 
│ │ │ │ -00006ff0: 6f66 2074 6865 2074 7970 6520 2a6e 6f74  of the type *not
│ │ │ │ -00007000: 6520 506f 696e 743a 2050 6f69 6e74 2c2c  e Point: Point,,
│ │ │ │ -00007010: 200a 0a44 6573 6372 6970 7469 6f6e 0a3d   ..Description.=
│ │ │ │ -00007020: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2b2d 2d2d  ==========..+---
│ │ │ │ -00007030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007040: 2b0a 7c69 3120 3a20 706f 696e 7420 7b31  +.|i1 : point {1
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│ │ │ │ -00007060: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00007070: 3120 3d20 506f 696e 747b 312c 2032 2c20  1 = Point{1, 2, 
│ │ │ │ -00007080: 337d 7c0a 7c20 2020 2020 2020 2020 2020  3}|.|           
│ │ │ │ -00007090: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20          |.|o1 : 
│ │ │ │ -000070a0: 506f 696e 7420 2020 2020 2020 2020 7c0a  Point         |.
│ │ │ │ -000070b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -000070c0: 2d2d 2d2d 2b0a 7c69 3220 3a20 706f 696e  ----+.|i2 : poin
│ │ │ │ -000070d0: 7420 5b31 2c34 2c36 5d20 7c0a 7c20 2020  t [1,4,6] |.|   
│ │ │ │ -000070e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000070f0: 7c0a 7c6f 3220 3d20 506f 696e 747b 312c  |.|o2 = Point{1,
│ │ │ │ -00007100: 2034 2c20 367d 7c0a 7c20 2020 2020 2020   4, 6}|.|       
│ │ │ │ -00007110: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00007120: 3220 3a20 506f 696e 7420 2020 2020 2020  2 : Point       
│ │ │ │ -00007130: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -00007140: 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 6179 7320  --------+..Ways 
│ │ │ │ -00007150: 746f 2075 7365 2070 6f69 6e74 3a0a 3d3d  to use point:.==
│ │ │ │ -00007160: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00007170: 0a0a 2020 2a20 2270 6f69 6e74 2856 6973  ..  * "point(Vis
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│ │ │ │ -000071a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000071b0: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
│ │ │ │ -000071c0: 6f74 6520 706f 696e 743a 2070 6f69 6e74  ote point: point
│ │ │ │ -000071d0: 2c20 6973 2061 202a 6e6f 7465 206d 6574  , is a *note met
│ │ │ │ -000071e0: 686f 6420 6675 6e63 7469 6f6e 3a0a 284d  hod function:.(M
│ │ │ │ -000071f0: 6163 6175 6c61 7932 446f 6329 4d65 7468  acaulay2Doc)Meth
│ │ │ │ -00007200: 6f64 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d  odFunction,...--
│ │ │ │ -00007210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
│ │ │ │ -00007260: 6865 2073 6f75 7263 6520 6f66 2074 6869  he source of thi
│ │ │ │ -00007270: 7320 646f 6375 6d65 6e74 2069 7320 696e  s document is in
│ │ │ │ -00007280: 0a2f 6275 696c 642f 7265 7072 6f64 7563  ./build/reproduc
│ │ │ │ -00007290: 6962 6c65 2d70 6174 682f 6d61 6361 756c  ible-path/macaul
│ │ │ │ -000072a0: 6179 322d 312e 3235 2e31 312b 6473 2f4d  ay2-1.25.11+ds/M
│ │ │ │ -000072b0: 322f 4d61 6361 756c 6179 322f 7061 636b  2/Macaulay2/pack
│ │ │ │ -000072c0: 6167 6573 2f48 6164 616d 6172 642e 6d32  ages/Hadamard.m2
│ │ │ │ -000072d0: 0a3a 3234 343a 302e 0a1f 0a46 696c 653a  .:244:0....File:
│ │ │ │ -000072e0: 2048 6164 616d 6172 642e 696e 666f 2c20   Hadamard.info, 
│ │ │ │ -000072f0: 4e6f 6465 3a20 506f 696e 7420 5f73 7420  Node: Point _st 
│ │ │ │ -00007300: 506f 696e 742c 204e 6578 743a 2050 6f69  Point, Next: Poi
│ │ │ │ -00007310: 6e74 203d 3d20 506f 696e 742c 2050 7265  nt == Point, Pre
│ │ │ │ -00007320: 763a 2070 6f69 6e74 2c20 5570 3a20 546f  v: point, Up: To
│ │ │ │ -00007330: 700a 0a50 6f69 6e74 202a 2050 6f69 6e74  p..Point * Point
│ │ │ │ -00007340: 202d 2d20 656e 7472 7977 6973 6520 7072   -- entrywise pr
│ │ │ │ -00007350: 6f64 7563 7420 6f66 2074 776f 2070 726f  oduct of two pro
│ │ │ │ -00007360: 6a65 6374 6976 6520 706f 696e 7473 0a2a  jective points.*
│ │ │ │ -00007370: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00007380: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00007390: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000073a0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ -000073b0: 4f70 6572 6174 6f72 3a20 2a6e 6f74 6520  Operator: *note 
│ │ │ │ -000073c0: 2a3a 2028 4d61 6361 756c 6179 3244 6f63  *: (Macaulay2Doc
│ │ │ │ -000073d0: 295f 7374 2c0a 2020 2a20 5573 6167 653a  )_st,.  * Usage:
│ │ │ │ -000073e0: 200a 2020 2020 2020 2020 7020 2a20 710a   .        p * q.
│ │ │ │ -000073f0: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ -00007400: 2020 2a20 702c 2061 6e20 696e 7374 616e    * p, an instan
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│ │ │ │ -00007420: 6e6f 7465 2050 6f69 6e74 3a20 506f 696e  note Point: Poin
│ │ │ │ -00007430: 742c 2c20 0a20 2020 2020 202a 2071 2c20  t,, .      * q, 
│ │ │ │ -00007440: 616e 2069 6e73 7461 6e63 6520 6f66 2074  an instance of t
│ │ │ │ -00007450: 6865 2074 7970 6520 2a6e 6f74 6520 506f  he type *note Po
│ │ │ │ -00007460: 696e 743a 2050 6f69 6e74 2c2c 200a 2020  int: Point,, .  
│ │ │ │ -00007470: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ -00007480: 202a 2061 6e20 696e 7374 616e 6365 206f   * an instance o
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│ │ │ │ -00007540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00007560: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00007570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00007580: 2020 2020 2020 7c0a 7c6f 3320 3d20 506f        |.|o3 = Po
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│ │ │ │ -000075a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000075b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -000075c0: 3320 3a20 506f 696e 7420 2020 2020 2020  3 : Point       
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│ │ │ │ +000067c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000067d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000067e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000067f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006810: 2020 2020 2020 207c 0a7c 6f36 203d 2074         |.|o6 = t
│ │ │ │ +00006820: 7275 6520 2020 2020 2020 2020 2020 2020  rue             
│ │ │ │ +00006830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00006860: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00006870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00006890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000068a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000068b0: 2d2d 2d2d 2d2d 2d2b 0a0a 5761 7973 2074  -------+..Ways t
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│ │ │ │ +000068f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ +000069f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00006b30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00006b40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00006b50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ +00006b60: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
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│ │ │ │ +00006c40: 2050 6f69 6e74 3a0a 3d3d 3d3d 3d3d 3d3d   Point:.========
│ │ │ │ +00006c50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00006c60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ +00006cd0: 2020 2a20 2a6e 6f74 6520 506f 696e 7420    * *note Point 
│ │ │ │ +00006ce0: 3d3d 2050 6f69 6e74 3a20 506f 696e 7420  == Point: Point 
│ │ │ │ +00006cf0: 3d3d 2050 6f69 6e74 2c20 2d2d 2063 6865  == Point, -- che
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│ │ │ │ +00006d40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
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│ │ │ │ +00006d60: 6520 506f 696e 743a 2050 6f69 6e74 2c20  e Point: Point, 
│ │ │ │ +00006d70: 6973 2061 202a 6e6f 7465 2074 7970 653a  is a *note type:
│ │ │ │ +00006d80: 2028 4d61 6361 756c 6179 3244 6f63 2954   (Macaulay2Doc)T
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│ │ │ │ +00006da0: 746f 7220 636c 6173 7365 7320 2a6e 6f74  tor classes *not
│ │ │ │ +00006db0: 6520 4261 7369 634c 6973 743a 2028 4d61  e BasicList: (Ma
│ │ │ │ +00006dc0: 6361 756c 6179 3244 6f63 2942 6173 6963  caulay2Doc)Basic
│ │ │ │ +00006dd0: 4c69 7374 2c20 3c20 2a6e 6f74 6520 5468  List, < *note Th
│ │ │ │ +00006de0: 696e 673a 0a28 4d61 6361 756c 6179 3244  ing:.(Macaulay2D
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│ │ │ │ +00006e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00006e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00006e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
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│ │ │ │ +00006f50: 2061 7272 6179 2920 6f66 2063 6f6f 7264   array) of coord
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│ │ │ │ +00006f70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00006f80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00006f90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00006fa0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00006fb0: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361  *******..  * Usa
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│ │ │ │ +00006fe0: 3a0a 2020 2020 2020 2a20 4c2c 2061 202a  :.      * L, a *
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│ │ │ │ +00007010: 6f72 202a 6e6f 7465 2061 7272 6179 3a0a  or *note array:.
│ │ │ │ +00007020: 2020 2020 2020 2020 284d 6163 6175 6c61          (Macaula
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│ │ │ │ +000070d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00007100: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
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│ │ │ │ +00007130: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20          |.|o1 : 
│ │ │ │ +00007140: 506f 696e 7420 2020 2020 2020 2020 7c0a  Point         |.
│ │ │ │ +00007150: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
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│ │ │ │ +00007180: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +000071b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
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│ │ │ │ +000072f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
│ │ │ │ +00007300: 6865 2073 6f75 7263 6520 6f66 2074 6869  he source of thi
│ │ │ │ +00007310: 7320 646f 6375 6d65 6e74 2069 7320 696e  s document is in
│ │ │ │ +00007320: 0a2f 6275 696c 642f 7265 7072 6f64 7563  ./build/reproduc
│ │ │ │ +00007330: 6962 6c65 2d70 6174 682f 6d61 6361 756c  ible-path/macaul
│ │ │ │ +00007340: 6179 322d 312e 3235 2e31 312b 6473 2f4d  ay2-1.25.11+ds/M
│ │ │ │ +00007350: 322f 4d61 6361 756c 6179 322f 7061 636b  2/Macaulay2/pack
│ │ │ │ +00007360: 6167 6573 2f48 6164 616d 6172 642e 6d32  ages/Hadamard.m2
│ │ │ │ +00007370: 0a3a 3234 343a 302e 0a1f 0a46 696c 653a  .:244:0....File:
│ │ │ │ +00007380: 2048 6164 616d 6172 642e 696e 666f 2c20   Hadamard.info, 
│ │ │ │ +00007390: 4e6f 6465 3a20 506f 696e 7420 5f73 7420  Node: Point _st 
│ │ │ │ +000073a0: 506f 696e 742c 204e 6578 743a 2050 6f69  Point, Next: Poi
│ │ │ │ +000073b0: 6e74 203d 3d20 506f 696e 742c 2050 7265  nt == Point, Pre
│ │ │ │ +000073c0: 763a 2070 6f69 6e74 2c20 5570 3a20 546f  v: point, Up: To
│ │ │ │ +000073d0: 700a 0a50 6f69 6e74 202a 2050 6f69 6e74  p..Point * Point
│ │ │ │ +000073e0: 202d 2d20 656e 7472 7977 6973 6520 7072   -- entrywise pr
│ │ │ │ +000073f0: 6f64 7563 7420 6f66 2074 776f 2070 726f  oduct of two pro
│ │ │ │ +00007400: 6a65 6374 6976 6520 706f 696e 7473 0a2a  jective points.*
│ │ │ │ +00007410: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00007420: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00007430: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00007440: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ +00007450: 4f70 6572 6174 6f72 3a20 2a6e 6f74 6520  Operator: *note 
│ │ │ │ +00007460: 2a3a 2028 4d61 6361 756c 6179 3244 6f63  *: (Macaulay2Doc
│ │ │ │ +00007470: 295f 7374 2c0a 2020 2a20 5573 6167 653a  )_st,.  * Usage:
│ │ │ │ +00007480: 200a 2020 2020 2020 2020 7020 2a20 710a   .        p * q.
│ │ │ │ +00007490: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ +000074a0: 2020 2a20 702c 2061 6e20 696e 7374 616e    * p, an instan
│ │ │ │ +000074b0: 6365 206f 6620 7468 6520 7479 7065 202a  ce of the type *
│ │ │ │ +000074c0: 6e6f 7465 2050 6f69 6e74 3a20 506f 696e  note Point: Poin
│ │ │ │ +000074d0: 742c 2c20 0a20 2020 2020 202a 2071 2c20  t,, .      * q, 
│ │ │ │ +000074e0: 616e 2069 6e73 7461 6e63 6520 6f66 2074  an instance of t
│ │ │ │ +000074f0: 6865 2074 7970 6520 2a6e 6f74 6520 506f  he type *note Po
│ │ │ │ +00007500: 696e 743a 2050 6f69 6e74 2c2c 200a 2020  int: Point,, .  
│ │ │ │ +00007510: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ +00007520: 202a 2061 6e20 696e 7374 616e 6365 206f   * an instance o
│ │ │ │ +00007530: 6620 7468 6520 7479 7065 202a 6e6f 7465  f the type *note
│ │ │ │ +00007540: 2050 6f69 6e74 3a20 506f 696e 742c 2c20   Point: Point,, 
│ │ │ │ +00007550: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ +00007560: 3d3d 3d3d 3d3d 3d3d 3d0a 0a2b 2d2d 2d2d  =========..+----
│ │ │ │ +00007570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00007580: 2d2d 2d2d 2b0a 7c69 3120 3a20 7020 3d20  ----+.|i1 : p = 
│ │ │ │ +00007590: 706f 696e 7420 7b31 2c32 2c33 7d3b 207c  point {1,2,3}; |
│ │ │ │ +000075a0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000075b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220  ----------+.|i2 
│ │ │ │ +000075c0: 3a20 7120 3d20 706f 696e 7420 7b2d 312c  : q = point {-1,
│ │ │ │ +000075d0: 322c 357d 3b7c 0a2b 2d2d 2d2d 2d2d 2d2d  2,5};|.+--------
│ │ │ │  000075e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000075f0: 2d2d 2b0a 0a4e 6f74 6520 7468 6174 2074  --+..Note that t
│ │ │ │ -00007600: 6869 7320 6f70 6572 6174 696f 6e20 6973  his operation is
│ │ │ │ -00007610: 206e 6f74 2061 6c77 6179 7320 7765 6c6c   not always well
│ │ │ │ -00007620: 2d64 6566 696e 6564 2069 6e20 7072 6f6a  -defined in proj
│ │ │ │ -00007630: 6563 7469 7665 2073 7061 6365 2c20 652e  ective space, e.
│ │ │ │ -00007640: 672e 2c0a 7468 6520 4861 6461 6d61 7264  g.,.the Hadamard
│ │ │ │ -00007650: 2070 726f 6475 6374 206f 6620 7468 6520   product of the 
│ │ │ │ -00007660: 706f 696e 7473 2024 5b31 3a30 5d24 2061  points $[1:0]$ a
│ │ │ │ -00007670: 6e64 2024 5b30 3a31 5d24 2069 7320 6e6f  nd $[0:1]$ is no
│ │ │ │ -00007680: 7420 7765 6c6c 2d64 6566 696e 6564 0a0a  t well-defined..
│ │ │ │ -00007690: 5761 7973 2074 6f20 7573 6520 7468 6973  Ways to use this
│ │ │ │ -000076a0: 206d 6574 686f 643a 0a3d 3d3d 3d3d 3d3d   method:.=======
│ │ │ │ -000076b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000076c0: 3d0a 0a20 202a 202a 6e6f 7465 2050 6f69  =..  * *note Poi
│ │ │ │ -000076d0: 6e74 202a 2050 6f69 6e74 3a20 506f 696e  nt * Point: Poin
│ │ │ │ -000076e0: 7420 5f73 7420 506f 696e 742c 202d 2d20  t _st Point, -- 
│ │ │ │ -000076f0: 656e 7472 7977 6973 6520 7072 6f64 7563  entrywise produc
│ │ │ │ -00007700: 7420 6f66 2074 776f 0a20 2020 2070 726f  t of two.    pro
│ │ │ │ -00007710: 6a65 6374 6976 6520 706f 696e 7473 0a2d  jective points.-
│ │ │ │ -00007720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ -00007770: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
│ │ │ │ -00007780: 6973 2064 6f63 756d 656e 7420 6973 2069  is document is i
│ │ │ │ -00007790: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475  n./build/reprodu
│ │ │ │ -000077a0: 6369 626c 652d 7061 7468 2f6d 6163 6175  cible-path/macau
│ │ │ │ -000077b0: 6c61 7932 2d31 2e32 352e 3131 2b64 732f  lay2-1.25.11+ds/
│ │ │ │ -000077c0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163  M2/Macaulay2/pac
│ │ │ │ -000077d0: 6b61 6765 732f 4861 6461 6d61 7264 2e6d  kages/Hadamard.m
│ │ │ │ -000077e0: 320a 3a32 3035 3a30 2e0a 1f0a 4669 6c65  2.:205:0....File
│ │ │ │ -000077f0: 3a20 4861 6461 6d61 7264 2e69 6e66 6f2c  : Hadamard.info,
│ │ │ │ -00007800: 204e 6f64 653a 2050 6f69 6e74 203d 3d20   Node: Point == 
│ │ │ │ -00007810: 506f 696e 742c 2050 7265 763a 2050 6f69  Point, Prev: Poi
│ │ │ │ -00007820: 6e74 205f 7374 2050 6f69 6e74 2c20 5570  nt _st Point, Up
│ │ │ │ -00007830: 3a20 546f 700a 0a50 6f69 6e74 203d 3d20  : Top..Point == 
│ │ │ │ -00007840: 506f 696e 7420 2d2d 2063 6865 636b 2065  Point -- check e
│ │ │ │ -00007850: 7175 616c 6974 7920 6f66 2074 776f 2070  quality of two p
│ │ │ │ -00007860: 726f 6a65 6374 6976 6520 706f 696e 7473  rojective points
│ │ │ │ -00007870: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │ -00007880: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00007890: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000078a0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ -000078b0: 4f70 6572 6174 6f72 3a20 2a6e 6f74 6520  Operator: *note 
│ │ │ │ -000078c0: 3d3d 3a20 284d 6163 6175 6c61 7932 446f  ==: (Macaulay2Do
│ │ │ │ -000078d0: 6329 3d3d 2c0a 2020 2a20 5573 6167 653a  c)==,.  * Usage:
│ │ │ │ -000078e0: 200a 2020 2020 2020 2020 7020 3d3d 2071   .        p == q
│ │ │ │ -000078f0: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ -00007900: 2020 202a 2070 2c20 616e 2069 6e73 7461     * p, an insta
│ │ │ │ -00007910: 6e63 6520 6f66 2074 6865 2074 7970 6520  nce of the type 
│ │ │ │ -00007920: 2a6e 6f74 6520 506f 696e 743a 2050 6f69  *note Point: Poi
│ │ │ │ -00007930: 6e74 2c2c 200a 2020 2020 2020 2a20 712c  nt,, .      * q,
│ │ │ │ -00007940: 2061 6e20 696e 7374 616e 6365 206f 6620   an instance of 
│ │ │ │ -00007950: 7468 6520 7479 7065 202a 6e6f 7465 2050  the type *note P
│ │ │ │ -00007960: 6f69 6e74 3a20 506f 696e 742c 2c20 0a20  oint: Point,, . 
│ │ │ │ -00007970: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ -00007980: 2020 2a20 6120 2a6e 6f74 6520 426f 6f6c    * a *note Bool
│ │ │ │ -00007990: 6561 6e20 7661 6c75 653a 2028 4d61 6361  ean value: (Maca
│ │ │ │ -000079a0: 756c 6179 3244 6f63 2942 6f6f 6c65 616e  ulay2Doc)Boolean
│ │ │ │ -000079b0: 2c2c 200a 0a44 6573 6372 6970 7469 6f6e  ,, ..Description
│ │ │ │ -000079c0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2b2d  .===========..+-
│ │ │ │ -000079d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000079e0: 2d2d 2d2d 2b0a 7c69 3120 3a20 7020 3d20  ----+.|i1 : p = 
│ │ │ │ -000079f0: 706f 696e 7420 7b31 2c31 7d3b 7c0a 2b2d  point {1,1};|.+-
│ │ │ │ -00007a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007a10: 2d2d 2d2d 2b0a 7c69 3220 3a20 7120 3d20  ----+.|i2 : q = 
│ │ │ │ -00007a20: 706f 696e 7420 7b32 2c32 7d3b 7c0a 2b2d  point {2,2};|.+-
│ │ │ │ -00007a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007a40: 2d2d 2d2d 2b0a 7c69 3320 3a20 7020 3d3d  ----+.|i3 : p ==
│ │ │ │ -00007a50: 2071 2020 2020 2020 2020 2020 7c0a 7c20   q          |.| 
│ │ │ │ -00007a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00007a70: 2020 2020 7c0a 7c6f 3320 3d20 7472 7565      |.|o3 = true
│ │ │ │ -00007a80: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -00007a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007aa0: 2d2d 2d2d 2b0a 0a57 6179 7320 746f 2075  ----+..Ways to u
│ │ │ │ -00007ab0: 7365 2074 6869 7320 6d65 7468 6f64 3a0a  se this method:.
│ │ │ │ -00007ac0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00007ad0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ -00007ae0: 6f74 6520 506f 696e 7420 3d3d 2050 6f69  ote Point == Poi
│ │ │ │ -00007af0: 6e74 3a20 506f 696e 7420 3d3d 2050 6f69  nt: Point == Poi
│ │ │ │ -00007b00: 6e74 2c20 2d2d 2063 6865 636b 2065 7175  nt, -- check equ
│ │ │ │ -00007b10: 616c 6974 7920 6f66 2074 776f 2070 726f  ality of two pro
│ │ │ │ -00007b20: 6a65 6374 6976 650a 2020 2020 706f 696e  jective.    poin
│ │ │ │ -00007b30: 7473 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ts.-------------
│ │ │ │ -00007b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00007b80: 2d2d 0a0a 5468 6520 736f 7572 6365 206f  --..The source o
│ │ │ │ -00007b90: 6620 7468 6973 2064 6f63 756d 656e 7420  f this document 
│ │ │ │ -00007ba0: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570  is in./build/rep
│ │ │ │ -00007bb0: 726f 6475 6369 626c 652d 7061 7468 2f6d  roducible-path/m
│ │ │ │ -00007bc0: 6163 6175 6c61 7932 2d31 2e32 352e 3131  acaulay2-1.25.11
│ │ │ │ -00007bd0: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932  +ds/M2/Macaulay2
│ │ │ │ -00007be0: 2f70 6163 6b61 6765 732f 4861 6461 6d61  /packages/Hadama
│ │ │ │ -00007bf0: 7264 2e6d 320a 3a32 3234 3a30 2e0a 1f0a  rd.m2.:224:0....
│ │ │ │ -00007c00: 5461 6720 5461 626c 653a 0a4e 6f64 653a  Tag Table:.Node:
│ │ │ │ -00007c10: 2054 6f70 7f32 3438 0a4e 6f64 653a 2068   Top.248.Node: h
│ │ │ │ -00007c20: 6164 616d 6172 6450 6f77 6572 7f33 3735  adamardPower.375
│ │ │ │ -00007c30: 350a 4e6f 6465 3a20 6861 6461 6d61 7264  5.Node: hadamard
│ │ │ │ -00007c40: 506f 7765 725f 6c70 4964 6561 6c5f 636d  Power_lpIdeal_cm
│ │ │ │ -00007c50: 5a5a 5f72 707f 3436 3233 0a4e 6f64 653a  ZZ_rp.4623.Node:
│ │ │ │ -00007c60: 2068 6164 616d 6172 6450 6f77 6572 5f6c   hadamardPower_l
│ │ │ │ -00007c70: 704c 6973 745f 636d 5a5a 5f72 707f 3736  pList_cmZZ_rp.76
│ │ │ │ -00007c80: 3234 0a4e 6f64 653a 2068 6164 616d 6172  24.Node: hadamar
│ │ │ │ -00007c90: 6450 726f 6475 6374 7f31 3036 3037 0a4e  dProduct.10607.N
│ │ │ │ -00007ca0: 6f64 653a 2068 6164 616d 6172 6450 726f  ode: hadamardPro
│ │ │ │ -00007cb0: 6475 6374 5f6c 7049 6465 616c 5f63 6d49  duct_lpIdeal_cmI
│ │ │ │ -00007cc0: 6465 616c 5f72 707f 3131 3634 390a 4e6f  deal_rp.11649.No
│ │ │ │ -00007cd0: 6465 3a20 6861 6461 6d61 7264 5072 6f64  de: hadamardProd
│ │ │ │ -00007ce0: 7563 745f 6c70 4c69 7374 5f72 707f 3136  uct_lpList_rp.16
│ │ │ │ -00007cf0: 3237 340a 4e6f 6465 3a20 6861 6461 6d61  274.Node: hadama
│ │ │ │ -00007d00: 7264 5072 6f64 7563 745f 6c70 4c69 7374  rdProduct_lpList
│ │ │ │ -00007d10: 5f63 6d4c 6973 745f 7270 7f32 3037 3833  _cmList_rp.20783
│ │ │ │ -00007d20: 0a4e 6f64 653a 2069 6465 616c 4f66 5072  .Node: idealOfPr
│ │ │ │ -00007d30: 6f6a 6563 7469 7665 506f 696e 7473 7f32  ojectivePoints.2
│ │ │ │ -00007d40: 3234 3533 0a4e 6f64 653a 2050 6f69 6e74  2453.Node: Point
│ │ │ │ -00007d50: 7f32 3731 3336 0a4e 6f64 653a 2070 6f69  .27136.Node: poi
│ │ │ │ -00007d60: 6e74 7f32 3831 3939 0a4e 6f64 653a 2050  nt.28199.Node: P
│ │ │ │ -00007d70: 6f69 6e74 205f 7374 2050 6f69 6e74 7f32  oint _st Point.2
│ │ │ │ -00007d80: 3934 3031 0a4e 6f64 653a 2050 6f69 6e74  9401.Node: Point
│ │ │ │ -00007d90: 203d 3d20 506f 696e 747f 3330 3639 380a   == Point.30698.
│ │ │ │ -00007da0: 1f0a 456e 6420 5461 6720 5461 626c 650a  ..End Tag Table.
│ │ │ │ +000075f0: 2b0a 7c69 3320 3a20 7020 2a20 7120 2020  +.|i3 : p * q   
│ │ │ │ +00007600: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00007610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007620: 2020 2020 2020 7c0a 7c6f 3320 3d20 506f        |.|o3 = Po
│ │ │ │ +00007630: 696e 747b 2d31 2c20 342c 2031 357d 2020  int{-1, 4, 15}  
│ │ │ │ +00007640: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00007650: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00007660: 3320 3a20 506f 696e 7420 2020 2020 2020  3 : Point       
│ │ │ │ +00007670: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00007680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00007690: 2d2d 2b0a 0a4e 6f74 6520 7468 6174 2074  --+..Note that t
│ │ │ │ +000076a0: 6869 7320 6f70 6572 6174 696f 6e20 6973  his operation is
│ │ │ │ +000076b0: 206e 6f74 2061 6c77 6179 7320 7765 6c6c   not always well
│ │ │ │ +000076c0: 2d64 6566 696e 6564 2069 6e20 7072 6f6a  -defined in proj
│ │ │ │ +000076d0: 6563 7469 7665 2073 7061 6365 2c20 652e  ective space, e.
│ │ │ │ +000076e0: 672e 2c0a 7468 6520 4861 6461 6d61 7264  g.,.the Hadamard
│ │ │ │ +000076f0: 2070 726f 6475 6374 206f 6620 7468 6520   product of the 
│ │ │ │ +00007700: 706f 696e 7473 2024 5b31 3a30 5d24 2061  points $[1:0]$ a
│ │ │ │ +00007710: 6e64 2024 5b30 3a31 5d24 2069 7320 6e6f  nd $[0:1]$ is no
│ │ │ │ +00007720: 7420 7765 6c6c 2d64 6566 696e 6564 0a0a  t well-defined..
│ │ │ │ +00007730: 5761 7973 2074 6f20 7573 6520 7468 6973  Ways to use this
│ │ │ │ +00007740: 206d 6574 686f 643a 0a3d 3d3d 3d3d 3d3d   method:.=======
│ │ │ │ +00007750: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00007760: 3d0a 0a20 202a 202a 6e6f 7465 2050 6f69  =..  * *note Poi
│ │ │ │ +00007770: 6e74 202a 2050 6f69 6e74 3a20 506f 696e  nt * Point: Poin
│ │ │ │ +00007780: 7420 5f73 7420 506f 696e 742c 202d 2d20  t _st Point, -- 
│ │ │ │ +00007790: 656e 7472 7977 6973 6520 7072 6f64 7563  entrywise produc
│ │ │ │ +000077a0: 7420 6f66 2074 776f 0a20 2020 2070 726f  t of two.    pro
│ │ │ │ +000077b0: 6a65 6374 6976 6520 706f 696e 7473 0a2d  jective points.-
│ │ │ │ +000077c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000077d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00007800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ +00007810: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
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│ │ │ │ +00007890: 3a20 4861 6461 6d61 7264 2e69 6e66 6f2c  : Hadamard.info,
│ │ │ │ +000078a0: 204e 6f64 653a 2050 6f69 6e74 203d 3d20   Node: Point == 
│ │ │ │ +000078b0: 506f 696e 742c 2050 7265 763a 2050 6f69  Point, Prev: Poi
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│ │ │ │ +000078d0: 3a20 546f 700a 0a50 6f69 6e74 203d 3d20  : Top..Point == 
│ │ │ │ +000078e0: 506f 696e 7420 2d2d 2063 6865 636b 2065  Point -- check e
│ │ │ │ +000078f0: 7175 616c 6974 7920 6f66 2074 776f 2070  quality of two p
│ │ │ │ +00007900: 726f 6a65 6374 6976 6520 706f 696e 7473  rojective points
│ │ │ │ +00007910: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │ +00007920: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00007930: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00007940: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ +00007950: 4f70 6572 6174 6f72 3a20 2a6e 6f74 6520  Operator: *note 
│ │ │ │ +00007960: 3d3d 3a20 284d 6163 6175 6c61 7932 446f  ==: (Macaulay2Do
│ │ │ │ +00007970: 6329 3d3d 2c0a 2020 2a20 5573 6167 653a  c)==,.  * Usage:
│ │ │ │ +00007980: 200a 2020 2020 2020 2020 7020 3d3d 2071   .        p == q
│ │ │ │ +00007990: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ +000079a0: 2020 202a 2070 2c20 616e 2069 6e73 7461     * p, an insta
│ │ │ │ +000079b0: 6e63 6520 6f66 2074 6865 2074 7970 6520  nce of the type 
│ │ │ │ +000079c0: 2a6e 6f74 6520 506f 696e 743a 2050 6f69  *note Point: Poi
│ │ │ │ +000079d0: 6e74 2c2c 200a 2020 2020 2020 2a20 712c  nt,, .      * q,
│ │ │ │ +000079e0: 2061 6e20 696e 7374 616e 6365 206f 6620   an instance of 
│ │ │ │ +000079f0: 7468 6520 7479 7065 202a 6e6f 7465 2050  the type *note P
│ │ │ │ +00007a00: 6f69 6e74 3a20 506f 696e 742c 2c20 0a20  oint: Point,, . 
│ │ │ │ +00007a10: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ +00007a20: 2020 2a20 6120 2a6e 6f74 6520 426f 6f6c    * a *note Bool
│ │ │ │ +00007a30: 6561 6e20 7661 6c75 653a 2028 4d61 6361  ean value: (Maca
│ │ │ │ +00007a40: 756c 6179 3244 6f63 2942 6f6f 6c65 616e  ulay2Doc)Boolean
│ │ │ │ +00007a50: 2c2c 200a 0a44 6573 6372 6970 7469 6f6e  ,, ..Description
│ │ │ │ +00007a60: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2b2d  .===========..+-
│ │ │ │ +00007a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00007a80: 2d2d 2d2d 2b0a 7c69 3120 3a20 7020 3d20  ----+.|i1 : p = 
│ │ │ │ +00007a90: 706f 696e 7420 7b31 2c31 7d3b 7c0a 2b2d  point {1,1};|.+-
│ │ │ │ +00007aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00007ab0: 2d2d 2d2d 2b0a 7c69 3220 3a20 7120 3d20  ----+.|i2 : q = 
│ │ │ │ +00007ac0: 706f 696e 7420 7b32 2c32 7d3b 7c0a 2b2d  point {2,2};|.+-
│ │ │ │ +00007ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00007ae0: 2d2d 2d2d 2b0a 7c69 3320 3a20 7020 3d3d  ----+.|i3 : p ==
│ │ │ │ +00007af0: 2071 2020 2020 2020 2020 2020 7c0a 7c20   q          |.| 
│ │ │ │ +00007b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00007b10: 2020 2020 7c0a 7c6f 3320 3d20 7472 7565      |.|o3 = true
│ │ │ │ +00007b20: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00007b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00007b40: 2d2d 2d2d 2b0a 0a57 6179 7320 746f 2075  ----+..Ways to u
│ │ │ │ +00007b50: 7365 2074 6869 7320 6d65 7468 6f64 3a0a  se this method:.
│ │ │ │ +00007b60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00007b70: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ +00007b80: 6f74 6520 506f 696e 7420 3d3d 2050 6f69  ote Point == Poi
│ │ │ │ +00007b90: 6e74 3a20 506f 696e 7420 3d3d 2050 6f69  nt: Point == Poi
│ │ │ │ +00007ba0: 6e74 2c20 2d2d 2063 6865 636b 2065 7175  nt, -- check equ
│ │ │ │ +00007bb0: 616c 6974 7920 6f66 2074 776f 2070 726f  ality of two pro
│ │ │ │ +00007bc0: 6a65 6374 6976 650a 2020 2020 706f 696e  jective.    poin
│ │ │ │ +00007bd0: 7473 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ts.-------------
│ │ │ │ +00007be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00007bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00007c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00007c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00007c20: 2d2d 0a0a 5468 6520 736f 7572 6365 206f  --..The source o
│ │ │ │ +00007c30: 6620 7468 6973 2064 6f63 756d 656e 7420  f this document 
│ │ │ │ +00007c40: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570  is in./build/rep
│ │ │ │ +00007c50: 726f 6475 6369 626c 652d 7061 7468 2f6d  roducible-path/m
│ │ │ │ +00007c60: 6163 6175 6c61 7932 2d31 2e32 352e 3131  acaulay2-1.25.11
│ │ │ │ +00007c70: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932  +ds/M2/Macaulay2
│ │ │ │ +00007c80: 2f70 6163 6b61 6765 732f 4861 6461 6d61  /packages/Hadama
│ │ │ │ +00007c90: 7264 2e6d 320a 3a32 3234 3a30 2e0a 1f0a  rd.m2.:224:0....
│ │ │ │ +00007ca0: 5461 6720 5461 626c 653a 0a4e 6f64 653a  Tag Table:.Node:
│ │ │ │ +00007cb0: 2054 6f70 7f32 3438 0a4e 6f64 653a 2068   Top.248.Node: h
│ │ │ │ +00007cc0: 6164 616d 6172 6450 6f77 6572 7f33 3735  adamardPower.375
│ │ │ │ +00007cd0: 350a 4e6f 6465 3a20 6861 6461 6d61 7264  5.Node: hadamard
│ │ │ │ +00007ce0: 506f 7765 725f 6c70 4964 6561 6c5f 636d  Power_lpIdeal_cm
│ │ │ │ +00007cf0: 5a5a 5f72 707f 3436 3233 0a4e 6f64 653a  ZZ_rp.4623.Node:
│ │ │ │ +00007d00: 2068 6164 616d 6172 6450 6f77 6572 5f6c   hadamardPower_l
│ │ │ │ +00007d10: 704c 6973 745f 636d 5a5a 5f72 707f 3736  pList_cmZZ_rp.76
│ │ │ │ +00007d20: 3234 0a4e 6f64 653a 2068 6164 616d 6172  24.Node: hadamar
│ │ │ │ +00007d30: 6450 726f 6475 6374 7f31 3037 3637 0a4e  dProduct.10767.N
│ │ │ │ +00007d40: 6f64 653a 2068 6164 616d 6172 6450 726f  ode: hadamardPro
│ │ │ │ +00007d50: 6475 6374 5f6c 7049 6465 616c 5f63 6d49  duct_lpIdeal_cmI
│ │ │ │ +00007d60: 6465 616c 5f72 707f 3131 3830 390a 4e6f  deal_rp.11809.No
│ │ │ │ +00007d70: 6465 3a20 6861 6461 6d61 7264 5072 6f64  de: hadamardProd
│ │ │ │ +00007d80: 7563 745f 6c70 4c69 7374 5f72 707f 3136  uct_lpList_rp.16
│ │ │ │ +00007d90: 3433 340a 4e6f 6465 3a20 6861 6461 6d61  434.Node: hadama
│ │ │ │ +00007da0: 7264 5072 6f64 7563 745f 6c70 4c69 7374  rdProduct_lpList
│ │ │ │ +00007db0: 5f63 6d4c 6973 745f 7270 7f32 3039 3433  _cmList_rp.20943
│ │ │ │ +00007dc0: 0a4e 6f64 653a 2069 6465 616c 4f66 5072  .Node: idealOfPr
│ │ │ │ +00007dd0: 6f6a 6563 7469 7665 506f 696e 7473 7f32  ojectivePoints.2
│ │ │ │ +00007de0: 3236 3133 0a4e 6f64 653a 2050 6f69 6e74  2613.Node: Point
│ │ │ │ +00007df0: 7f32 3732 3936 0a4e 6f64 653a 2070 6f69  .27296.Node: poi
│ │ │ │ +00007e00: 6e74 7f32 3833 3539 0a4e 6f64 653a 2050  nt.28359.Node: P
│ │ │ │ +00007e10: 6f69 6e74 205f 7374 2050 6f69 6e74 7f32  oint _st Point.2
│ │ │ │ +00007e20: 3935 3631 0a4e 6f64 653a 2050 6f69 6e74  9561.Node: Point
│ │ │ │ +00007e30: 203d 3d20 506f 696e 747f 3330 3835 380a   == Point.30858.
│ │ │ │ +00007e40: 1f0a 456e 6420 5461 6720 5461 626c 650a  ..End Tag Table.
│ │ ├── ./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 3434   |.| -- .0000044
│ │ │ │ -0000faf0: 3739 7320 656c 6170 7365 6420 2020 2020  79s elapsed     
│ │ │ │ +0000fae0: 207c 0a7c 202d 2d20 2e30 3030 3030 3931   |.| -- .0000091
│ │ │ │ +0000faf0: 3532 7320 656c 6170 7365 6420 2020 2020  52s 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 3432   |.| -- .0000042
│ │ │ │ -0000fb40: 3338 7320 656c 6170 7365 6420 2020 2020  38s elapsed     
│ │ │ │ +0000fb30: 207c 0a7c 202d 2d20 2e30 3030 3030 3739   |.| -- .0000079
│ │ │ │ +0000fb40: 3538 7320 656c 6170 7365 6420 2020 2020  58s 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 3339   |.| -- .0000039
│ │ │ │ -0000fb90: 3737 7320 656c 6170 7365 6420 2020 2020  77s elapsed     
│ │ │ │ +0000fb80: 207c 0a7c 202d 2d20 2e30 3030 3030 3834   |.| -- .0000084
│ │ │ │ +0000fb90: 3173 2065 6c61 7073 6564 2020 2020 2020  1s 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 3138   |.| -- .0000018
│ │ │ │ -0000fbe0: 3833 7320 656c 6170 7365 6420 2020 2020  83s elapsed     
│ │ │ │ +0000fbd0: 207c 0a7c 202d 2d20 2e30 3030 3030 3834   |.| -- .0000084
│ │ │ │ +0000fbe0: 3032 7320 656c 6170 7365 6420 2020 2020  02s 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 3133   |.| -- .0000013
│ │ │ │ -0000fc30: 3932 7320 656c 6170 7365 6420 2020 2020  92s elapsed     
│ │ │ │ +0000fc20: 207c 0a7c 202d 2d20 2e30 3030 3030 3933   |.| -- .0000093
│ │ │ │ +0000fc30: 3832 7320 656c 6170 7365 6420 2020 2020  82s 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 3038 3873 2065  -- .000004088s e
│ │ │ │ +00015bc0: 2d2d 202e 3030 3030 3036 3039 3173 2065  -- .000006091s 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
│ │ │ │ @@ -5678,16 +5678,16 @@
│ │ │ │  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 3035 3335 7320 656c  -- .00000535s el
│ │ │ │ -00016350: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │ +00016340: 2d2d 202e 3030 3030 3036 3333 3973 2065  -- .000006339s 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
│ │ │ │  000163c0: 2063 6f65 6666 6963 6965 6e74 2072 696e   coefficient rin
│ │ ├── ./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    |.|           
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│ │ │ │  000085a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000085b0: 2020 7c0a 7c6f 3135 203d 207b 287b 5420    |.|o15 = {({T 
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│ │ │ │ -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  - 
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│ │ │ │ +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   
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│ │ │ │  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 ,
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│ │ │ │ +00009700: 2020 2031 3120 2020 2020 2032 2020 2036     11      2   6
│ │ │ │ +00009710: 2020 2020 3220 3620 2020 2033 2038 2020      2 6    3 8  
│ │ │ │ +00009720: 2020 3420 3920 2020 2020 2031 3420 2020    4 9      14   
│ │ │ │  00009730: 2020 7c0a 7c20 2020 2020 202d 2d2d 2d2d    |.|      -----
│ │ │ │  00009740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  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  
│ │ │ │ +00009790: 292c 2028 7b54 202c 2054 2020 7d2c 2054  ), ({T , T  }, T
│ │ │ │ +000097a0: 2054 2020 2d20 5420 5420 2020 2d20 7a2a   T  - T T   - z*
│ │ │ │ +000097b0: 5420 2020 2b20 7a2a 5420 2029 2c20 287b  T   + z*T  ), ({
│ │ │ │ +000097c0: 5420 2c20 5420 7d2c 2054 2054 2020 2b20  T , T }, T T  + 
│ │ │ │ +000097d0: 2020 7c0a 7c20 2020 2020 2020 2020 3137    |.|         17
│ │ │ │ +000097e0: 2020 2020 2020 3520 2020 3130 2020 2020        5   10    
│ │ │ │ +000097f0: 3420 3920 2020 2035 2031 3020 2020 2020  4 9    5 10     
│ │ │ │ +00009800: 2031 3720 2020 2020 2031 3920 2020 2020   17      19     
│ │ │ │ +00009810: 2033 2020 2038 2020 2020 3220 3620 2020   3   8    2 6   
│ │ │ │  00009820: 2020 7c0a 7c20 2020 2020 202d 2d2d 2d2d    |.|      -----
│ │ │ │  00009830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00009870: 2d2d 7c0a 7c20 2020 2020 207a 2a54 2020  --|.|      z*T  
│ │ │ │ -00009880: 292c 2028 7b54 202c 2054 2020 7d2c 2054  ), ({T , T  }, T
│ │ │ │ -00009890: 2054 2020 2d20 5420 5420 2020 2d20 7a2a   T  - T T   - z*
│ │ │ │ -000098a0: 5420 2020 2b20 7a2a 5420 2029 2c20 287b  T   + z*T  ), ({
│ │ │ │ -000098b0: 5420 2c20 5420 7d2c 2054 2054 2020 2b20  T , T }, T T  + 
│ │ │ │ -000098c0: 2020 7c0a 7c20 2020 2020 2020 2020 3137    |.|         17
│ │ │ │ -000098d0: 2020 2020 2020 3520 2020 3130 2020 2020        5   10    
│ │ │ │ -000098e0: 3420 3920 2020 2035 2031 3020 2020 2020  4 9    5 10     
│ │ │ │ -000098f0: 2031 3720 2020 2020 2031 3920 2020 2020   17      19     
│ │ │ │ -00009900: 2033 2020 2038 2020 2020 3220 3620 2020   3   8    2 6   
│ │ │ │ +00009870: 2d2d 7c0a 7c20 2020 2020 2054 2054 2020  --|.|      T T  
│ │ │ │ +00009880: 2b20 5420 5420 202b 2079 2a54 2020 202d  + T T  + y*T   -
│ │ │ │ +00009890: 207a 2a54 2020 292c 2028 7b54 202c 2054   z*T  ), ({T , T
│ │ │ │ +000098a0: 207d 2c20 5420 5420 202b 2079 2a54 2020   }, T T  + y*T  
│ │ │ │ +000098b0: 202d 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
│ │ │ │ @@ -4491,16 +4491,16 @@
│ │ │ │  000118a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000118b0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20  --------+.|i4 : 
│ │ │ │  000118c0: 7469 6d65 2052 2720 3d20 696e 7465 6772  time R' = integr
│ │ │ │  000118d0: 616c 436c 6f73 7572 6520 5220 2020 2020  alClosure R     
│ │ │ │  000118e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000118f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011900: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ -00011910: 7365 6420 302e 3639 3838 3373 2028 6370  sed 0.69883s (cp
│ │ │ │ -00011920: 7529 3b20 302e 3433 3032 3433 7320 2874  u); 0.430243s (t
│ │ │ │ +00011910: 7365 6420 302e 3737 3336 3234 7320 2863  sed 0.773624s (c
│ │ │ │ +00011920: 7075 293b 2030 2e34 3039 3037 7320 2874  pu); 0.40907s (t
│ │ │ │  00011930: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │  00011940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011950: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00011960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4981,16 +4981,16 @@
│ │ │ │  00013740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013750: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a  --------+.|i10 :
│ │ │ │  00013760: 2074 696d 6520 5227 203d 2069 6e74 6567   time R' = integ
│ │ │ │  00013770: 7261 6c43 6c6f 7375 7265 2852 2c20 5374  ralClosure(R, St
│ │ │ │  00013780: 7261 7465 6779 203d 3e20 5261 6469 6361  rategy => Radica
│ │ │ │  00013790: 6c29 2020 2020 2020 2020 2020 2020 2020  l)              
│ │ │ │  000137a0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ -000137b0: 7365 6420 302e 3738 3834 3734 7320 2863  sed 0.788474s (c
│ │ │ │ -000137c0: 7075 293b 2030 2e34 3138 3436 3373 2028  pu); 0.418463s (
│ │ │ │ +000137b0: 7365 6420 302e 3833 3730 3932 7320 2863  sed 0.837092s (c
│ │ │ │ +000137c0: 7075 293b 2030 2e34 3331 3137 3173 2028  pu); 0.431171s (
│ │ │ │  000137d0: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │  000137e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000137f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00013800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -5471,16 +5471,16 @@
│ │ │ │  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 3837 3437 3939 7320 2863  sed 0.874799s (c
│ │ │ │ -00015660: 7075 293b 2030 2e34 3937 3536 3373 2028  pu); 0.497563s (
│ │ │ │ +00015650: 7365 6420 302e 3834 3131 3137 7320 2863  sed 0.841117s (c
│ │ │ │ +00015660: 7075 293b 2030 2e34 3133 3136 3973 2028  pu); 0.413169s (
│ │ │ │  00015670: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │  00015680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015690: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  000156a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000156b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000156c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000156d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -5916,16 +5916,16 @@
│ │ │ │  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 3934 3835 3636 7320 2863  sed 0.948566s (c
│ │ │ │ -00017230: 7075 293b 2030 2e35 3236 3932 3973 2028  pu); 0.526929s (
│ │ │ │ +00017220: 7365 6420 302e 3933 3732 3138 7320 2863  sed 0.937218s (c
│ │ │ │ +00017230: 7075 293b 2030 2e34 3639 3737 3373 2028  pu); 0.469773s (
│ │ │ │  00017240: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 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                  
│ │ │ │ @@ -6361,16 +6361,16 @@
│ │ │ │  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 3734 3835 3473 2028 6370  sed 1.74854s (cp
│ │ │ │ -00018e00: 7529 3b20 302e 3837 3332 3632 7320 2874  u); 0.873262s (t
│ │ │ │ +00018df0: 7365 6420 312e 3830 3235 3573 2028 6370  sed 1.80255s (cp
│ │ │ │ +00018e00: 7529 3b20 302e 3831 3132 3933 7320 2874  u); 0.811293s (t
│ │ │ │  00018e10: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 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                  
│ │ │ │ @@ -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 3534 3934 3639 7320 2863  sed 0.549469s (c
│ │ │ │ -0001a9d0: 7075 293b 2030 2e34 3430 3539 3673 2028  pu); 0.440596s (
│ │ │ │ +0001a9c0: 7365 6420 302e 3538 3639 3739 7320 2863  sed 0.586979s (c
│ │ │ │ +0001a9d0: 7075 293b 2030 2e34 3130 3239 3773 2028  pu); 0.410297s (
│ │ │ │  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: 3032 3373 2028 6370 7529 3b20 302e 3034  023s (cpu); 0.04
│ │ │ │ -0001c3f0: 3430 3230 3173 2028 7468 7265 6164 293b  40201s (thread);
│ │ │ │ -0001c400: 2030 7320 2867 6329 2020 7c0a 7c20 2020   0s (gc)  |.|   
│ │ │ │ +0001c3d0: 0a7c 202d 2d20 7573 6564 2030 2e30 3532  .| -- used 0.052
│ │ │ │ +0001c3e0: 3833 3731 7320 2863 7075 293b 2030 2e30  8371s (cpu); 0.0
│ │ │ │ +0001c3f0: 3532 3833 3531 7320 2874 6872 6561 6429  528351s (thread)
│ │ │ │ +0001c400: 3b20 3073 2028 6763 2920 7c0a 7c20 2020  ; 0s (gc) |.|   
│ │ │ │  0001c410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c440: 2020 2020 207c 0a7c 6f33 3620 3d20 5227       |.|o36 = R'
│ │ │ │  0001c450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  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 3433 3635 3231 7320 2863 7075 293b  .0436521s (cpu);
│ │ │ │ -0001cfc0: 2030 2e30 3433 3635 3237 7320 2874 6872   0.0436527s (thr
│ │ │ │ +0001cfb0: 2e30 3531 3533 3738 7320 2863 7075 293b  .0515378s (cpu);
│ │ │ │ +0001cfc0: 2030 2e30 3531 3533 3737 7320 2874 6872   0.0515377s (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                  
│ │ │ │ @@ -7555,17 +7555,17 @@
│ │ │ │  0001d820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d840: 2b0a 7c69 3436 203a 2074 696d 6520 5227  +.|i46 : time R'
│ │ │ │  0001d850: 203d 2069 6e74 6567 7261 6c43 6c6f 7375   = integralClosu
│ │ │ │  0001d860: 7265 2852 2c20 5374 7261 7465 6779 203d  re(R, Strategy =
│ │ │ │  0001d870: 3e20 416c 6c43 6f64 696d 656e 7369 6f6e  > AllCodimension
│ │ │ │  0001d880: 7329 7c0a 7c20 2d2d 2075 7365 6420 302e  s)|.| -- used 0.
│ │ │ │ -0001d890: 3036 3231 3831 3473 2028 6370 7529 3b20  0621814s (cpu); 
│ │ │ │ -0001d8a0: 302e 3036 3231 3832 3873 2028 7468 7265  0.0621828s (thre
│ │ │ │ -0001d8b0: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │ +0001d890: 3037 3436 3831 3173 2028 6370 7529 3b20  0746811s (cpu); 
│ │ │ │ +0001d8a0: 302e 3037 3436 3736 7320 2874 6872 6561  0.074676s (threa
│ │ │ │ +0001d8b0: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 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  '               
│ │ │ │  0001d920: 2020 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 3432 3638 3131 7320 2863 7075 293b  .0426811s (cpu);
│ │ │ │ -0001e190: 2030 2e30 3432 3637 3638 7320 2874 6872   0.0426768s (thr
│ │ │ │ +0001e180: 2e30 3537 3833 3736 7320 2863 7075 293b  .0578376s (cpu);
│ │ │ │ +0001e190: 2030 2e30 3537 3833 3434 7320 2874 6872   0.0578344s (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 3536 3535   -- used 0.05655
│ │ │ │ -0001ea80: 3831 7320 2863 7075 293b 2030 2e30 3536  81s (cpu); 0.056
│ │ │ │ -0001ea90: 3535 3931 7320 2874 6872 6561 6429 3b20  5591s (thread); 
│ │ │ │ +0001ea70: 202d 2d20 7573 6564 2030 2e30 3730 3732   -- used 0.07072
│ │ │ │ +0001ea80: 3936 7320 2863 7075 293b 2030 2e30 3730  96s (cpu); 0.070
│ │ │ │ +0001ea90: 3732 3931 7320 2874 6872 6561 6429 3b20  7291s (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 3036 3031 3831 3773 2028 6370 7529  0.0601817s (cpu)
│ │ │ │ -00020ad0: 3b20 302e 3036 3031 3830 3873 2028 7468  ; 0.0601808s (th
│ │ │ │ -00020ae0: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │ +00020ac0: 302e 3139 3237 3032 7320 2863 7075 293b  0.192702s (cpu);
│ │ │ │ +00020ad0: 2030 2e31 3031 3739 3773 2028 7468 7265   0.101797s (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' 
│ │ │ │ @@ -8853,16 +8853,16 @@
│ │ │ │  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 3339 3936 3837 7320 2863 7075 293b  0.399687s (cpu);
│ │ │ │ -000229c0: 2030 2e33 3437 3737 3373 2028 7468 7265   0.347773s (thre
│ │ │ │ +000229b0: 302e 3436 3039 3135 7320 2863 7075 293b  0.460915s (cpu);
│ │ │ │ +000229c0: 2030 2e33 3834 3830 3573 2028 7468 7265   0.384805s (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                  
│ │ │ │ @@ -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 3531  |.| -- used 0.51
│ │ │ │ -00023200: 3133 3937 7320 2863 7075 293b 2030 2e33  1397s (cpu); 0.3
│ │ │ │ -00023210: 3733 3437 3973 2028 7468 7265 6164 293b  73479s (thread);
│ │ │ │ +000231f0: 7c0a 7c20 2d2d 2075 7365 6420 302e 3534  |.| -- used 0.54
│ │ │ │ +00023200: 3335 3531 7320 2863 7075 293b 2030 2e33  3551s (cpu); 0.3
│ │ │ │ +00023210: 3937 3230 3673 2028 7468 7265 6164 293b  97206s (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                  
│ │ │ │ @@ -9118,41 +9118,41 @@
│ │ │ │  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 3638 3638 3620 7365 6320   .000568686 sec 
│ │ │ │ +00023a40: 202e 3030 3036 3033 3238 3420 7365 6320   .000603284 sec 
│ │ │ │  00023a50: 236d 696e 6f72 7320 345d 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 3035 3639 3220 7365 6320 2023  e .205692 sec  #
│ │ │ │ +00023b30: 6520 2e32 3133 3734 3420 7365 6320 2023  e .213744 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 3331 3832 3320 7365 6320 2023  e .231823 sec  #
│ │ │ │ +00023b80: 6520 2e32 3539 3639 3420 7365 6320 2023  e .259694 sec  #
│ │ │ │  00023b90: 6672 6163 7469 6f6e 7320 365d 2020 2020  fractions 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 3434 3135  | -- used 0.4415
│ │ │ │ -00023bd0: 3435 7320 2863 7075 293b 2030 2e33 3131  45s (cpu); 0.311
│ │ │ │ -00023be0: 3837 3673 2028 7468 7265 6164 293b 2030  876s (thread); 0
│ │ │ │ +00023bc0: 7c20 2d2d 2075 7365 6420 302e 3437 3737  | -- used 0.4777
│ │ │ │ +00023bd0: 3839 7320 2863 7075 293b 2030 2e33 3233  89s (cpu); 0.323
│ │ │ │ +00023be0: 3231 3573 2028 7468 7265 6164 293b 2030  215s (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                |.
│ │ │ │ @@ -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 3331 3037 3620 7365  me .000531076 se
│ │ │ │ -00024580: 6320 236d 696e 6f72 7320 345d 2020 2020  c #minors 4]    
│ │ │ │ +00024570: 6d65 202e 3030 3035 3735 3234 2073 6563  me .00057524 sec
│ │ │ │ +00024580: 2023 6d69 6e6f 7273 2034 5d20 2020 2020   #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 3039 3030 3138 3720 7365 6320 2023   .0900187 sec  #
│ │ │ │ -00024670: 6672 6163 7469 6f6e 7320 365d 2020 2020  fractions 6]    
│ │ │ │ +00024660: 202e 3130 3534 3932 2073 6563 2020 2366   .105492 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 3631 3535 3120 7365 6320 2023 6672  .361551 sec  #fr
│ │ │ │ +000246b0: 2e34 3631 3136 3820 7365 6320 2023 6672  .461168 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 3535 3530 3873  - used 0.455508s
│ │ │ │ -00024700: 2028 6370 7529 3b20 302e 3333 3234 3836   (cpu); 0.332486
│ │ │ │ +000246f0: 2d20 7573 6564 2030 2e35 3730 3736 3573  - used 0.570765s
│ │ │ │ +00024700: 2028 6370 7529 3b20 302e 3339 3435 3337   (cpu); 0.394537
│ │ │ │  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                  
│ │ │ │ @@ -9474,42 +9474,42 @@
│ │ │ │  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 3134 3334  n time .00061434
│ │ │ │ -00025090: 3220 7365 6320 236d 696e 6f72 7320 315d  2 sec #minors 1]
│ │ │ │ +00025080: 6e20 7469 6d65 202e 3030 3037 3636 3639  n time .00076669
│ │ │ │ +00025090: 3520 7365 6320 236d 696e 6f72 7320 315d  5 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 3135 3334 3920     time .115349 
│ │ │ │ +00025170: 2020 2074 696d 6520 2e31 3336 3632 3620     time .136626 
│ │ │ │  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 3736 3539 3720     time .476597 
│ │ │ │ +000251c0: 2020 2074 696d 6520 2e35 3235 3434 3720     time .525447 
│ │ │ │  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 3539 3535 3736 7320 2863 7075 293b  0.595576s (cpu);
│ │ │ │ -00025220: 2030 2e34 3334 3438 7320 2874 6872 6561   0.43448s (threa
│ │ │ │ -00025230: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +00025210: 302e 3636 3635 3535 7320 2863 7075 293b  0.666555s (cpu);
│ │ │ │ +00025220: 2030 2e34 3836 3832 3773 2028 7468 7265   0.486827s (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                  
│ │ │ │  000252a0: 2020 2020 7c0a 7c6f 3934 203d 2052 2720      |.|o94 = R' 
│ │ │ │ @@ -10305,15 +10305,15 @@
│ │ │ │  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 3930 3031 3720  time .000590017 
│ │ │ │ +00028470: 7469 6d65 202e 3030 3036 3037 3333 3220  time .000607332 
│ │ │ │  00028480: 7365 6320 236d 696e 6f72 7320 335d 2020  sec #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  
│ │ │ │ @@ -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 3236 3136 3238 2073 6563  s) .00261628 sec
│ │ │ │ +000285c0: 7329 202e 3030 3334 3634 3638 2073 6563  s) .00346468 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 3039 3634 3134 3120  er1:  .00964141 
│ │ │ │ -00028610: 7365 636f 6e64 7320 2020 2020 2020 2020  seconds         
│ │ │ │ +00028600: 6572 313a 2020 2e30 3039 3738 3037 2073  er1:  .0097807 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 3835 3336 3920  er2:  .00985369 
│ │ │ │ -00028660: 7365 636f 6e64 7320 2020 2020 2020 2020  seconds         
│ │ │ │ +00028650: 6572 323a 2020 2e30 3131 3038 3439 2073  er2:  .0110849 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 3637 3634 3620  es:   .00867646 
│ │ │ │ -000286b0: 7365 636f 6e64 7320 2020 2020 2020 2020  seconds         
│ │ │ │ +000286a0: 6573 3a20 2020 2e30 3130 3631 3339 2073  es:   .0106139 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 3432    |.|  time .042
│ │ │ │ -000286f0: 3535 3937 2073 6563 2020 2366 7261 6374  5597 sec  #fract
│ │ │ │ +000286e0: 2020 7c0a 7c20 2074 696d 6520 2e30 3438    |.|  time .048
│ │ │ │ +000286f0: 3137 3839 2073 6563 2020 2366 7261 6374  1789 sec  #fract
│ │ │ │  00028700: 696f 6e73 2034 5d20 2020 2020 2020 2020  ions 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 3233 3933 3737 2073 6563  s) .00239377 sec
│ │ │ │ +000287a0: 7329 202e 3030 3237 3535 3131 2073 6563  s) .00275511 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 3131 3135 3431 2073  er1:  .0111541 s
│ │ │ │ -000287f0: 6563 6f6e 6473 2020 2020 2020 2020 2020  econds          
│ │ │ │ +000287e0: 6572 313a 2020 2e30 3134 3132 3620 7365  er1:  .014126 se
│ │ │ │ +000287f0: 636f 6e64 7320 2020 2020 2020 2020 2020  conds           
│ │ │ │  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 3039 3932 3832 3820  er2:  .00992828 
│ │ │ │ -00028840: 7365 636f 6e64 7320 2020 2020 2020 2020  seconds         
│ │ │ │ +00028830: 6572 323a 2020 2e30 3132 3532 3231 2073  er2:  .0125221 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 3139 3338 2073  es:   .0111938 s
│ │ │ │ +00028880: 6573 3a20 2020 2e30 3134 3631 3133 2073  es:   .0146113 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 3435    |.|  time .045
│ │ │ │ -000288d0: 3433 3337 2073 6563 2020 2366 7261 6374  4337 sec  #fract
│ │ │ │ +000288c0: 2020 7c0a 7c20 2074 696d 6520 2e30 3537    |.|  time .057
│ │ │ │ +000288d0: 3130 3239 2073 6563 2020 2366 7261 6374  1029 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 3233 3933 3435 2073 6563  s) .00239345 sec
│ │ │ │ +00028980: 7329 202e 3030 3238 3539 3738 2073 6563  s) .00285978 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 3531 3232 2073  er1:  .0115122 s
│ │ │ │ +000289c0: 6572 313a 2020 2e30 3134 3133 3239 2073  er1:  .0141329 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 3731 3836 3120  er2:  .00971861 
│ │ │ │ -00028a20: 7365 636f 6e64 7320 2020 2020 2020 2020  seconds         
│ │ │ │ +00028a10: 6572 323a 2020 2e30 3131 3734 3734 2073  er2:  .0117474 s
│ │ │ │ +00028a20: 6563 6f6e 6473 2020 2020 2020 2020 2020  econds          
│ │ │ │  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 3038 3930 3233 3220  es:   .00890232 
│ │ │ │ -00028a70: 7365 636f 6e64 7320 2020 2020 2020 2020  seconds         
│ │ │ │ +00028a60: 6573 3a20 2020 2e30 3131 3032 3534 2073  es:   .0110254 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: 3431 3539 2073 6563 2020 2366 7261 6374  4159 sec  #fract
│ │ │ │ +00028aa0: 2020 7c0a 7c20 2074 696d 6520 2e30 3532    |.|  time .052
│ │ │ │ +00028ab0: 3731 3732 2073 6563 2020 2366 7261 6374  7172 sec  #fract
│ │ │ │  00028ac0: 696f 6e73 2035 5d20 2020 2020 2020 2020  ions 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 3235 3238 3438 2073 6563  s) .00252848 sec
│ │ │ │ +00028b60: 7329 202e 3030 3239 3734 3131 2073 6563  s) .00297411 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 3138 3434 3220 7365  er1:  .118442 se
│ │ │ │ +00028ba0: 6572 313a 2020 2e31 3337 3531 3220 7365  er1:  .137512 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 3133 3332 3834 2073  er2:  .0133284 s
│ │ │ │ +00028bf0: 6572 323a 2020 2e30 3135 3037 3236 2073  er2:  .0150726 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 3639 3235 2073  es:   .0156925 s
│ │ │ │ +00028c40: 6573 3a20 2020 2e30 3138 3535 3733 2073  es:   .0185573 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 3632    |.|  time .162
│ │ │ │ -00028c90: 3036 3720 7365 6320 2023 6672 6163 7469  067 sec  #fracti
│ │ │ │ +00028c80: 2020 7c0a 7c20 2074 696d 6520 2e31 3838    |.|  time .188
│ │ │ │ +00028c90: 3339 3720 7365 6320 2023 6672 6163 7469  397 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 3238 3238 3237 2073 6563  s) .00282827 sec
│ │ │ │ +00028d40: 7329 202e 3030 3238 3538 3536 2073 6563  s) .00285856 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 3039 3234 3335 3120  er1:  .00924351 
│ │ │ │ -00028d90: 7365 636f 6e64 7320 2020 2020 2020 2020  seconds         
│ │ │ │ +00028d80: 6572 313a 2020 2e30 3131 3230 3433 2073  er1:  .0112043 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 3136 3233 3739 2073  er2:  .0162379 s
│ │ │ │ +00028dd0: 6572 323a 2020 2e30 3138 3236 3333 2073  er2:  .0182633 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 3032 3632 2073  es:   .0120262 s
│ │ │ │ +00028e20: 6573 3a20 2020 2e30 3139 3034 3736 2073  es:   .0190476 s
│ │ │ │  00028e30: 6563 6f6e 6473 2020 2020 2020 2020 2020  econds          
│ │ │ │  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 3533    |.|  time .053
│ │ │ │ -00028e70: 3632 3233 2073 6563 2020 2366 7261 6374  6223 sec  #fract
│ │ │ │ +00028e60: 2020 7c0a 7c20 2074 696d 6520 2e30 3637    |.|  time .067
│ │ │ │ +00028e70: 3132 3932 2073 6563 2020 2366 7261 6374  1292 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 3233 3231 3439 2073 6563  s) .00232149 sec
│ │ │ │ +00028f20: 7329 202e 3030 3238 3337 3139 2073 6563  s) .00283719 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 3037 3839 3533 3420  er1:  .00789534 
│ │ │ │ -00028f70: 7365 636f 6e64 7320 2020 2020 2020 2020  seconds         
│ │ │ │ +00028f60: 6572 313a 2020 2e30 3130 3336 3132 2073  er1:  .0103612 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 3136    |.|  time .016
│ │ │ │ -00028fb0: 3930 3839 2073 6563 2020 2366 7261 6374  9089 sec  #fract
│ │ │ │ +00028fa0: 2020 7c0a 7c20 2074 696d 6520 2e30 3231    |.|  time .021
│ │ │ │ +00028fb0: 3732 3735 2073 6563 2020 2366 7261 6374  7275 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: 3336 3832 3631 7320 2863 7075 293b 2030  368261s (cpu); 0
│ │ │ │ -00029010: 2e33 3031 3538 3573 2028 7468 7265 6164  .301585s (thread
│ │ │ │ +00029000: 3433 3937 3933 7320 2863 7075 293b 2030  439793s (cpu); 0
│ │ │ │ +00029010: 2e33 3630 3635 3873 2028 7468 7265 6164  .360658s (thread
│ │ │ │  00029020: 293b 2030 7320 2867 6329 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                  
│ │ │ │ @@ -10998,17 +10998,17 @@
│ │ │ │  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 2031 2e30 3035  .| -- used 1.005
│ │ │ │ -0002afd0: 3332 7320 2863 7075 293b 2030 2e37 3137  32s (cpu); 0.717
│ │ │ │ -0002afe0: 3039 3573 2028 7468 7265 6164 293b 2030  095s (thread); 0
│ │ │ │ +0002afc0: 0a7c 202d 2d20 7573 6564 2031 2e34 3131  .| -- used 1.411
│ │ │ │ +0002afd0: 3934 7320 2863 7075 293b 2030 2e38 3038  94s (cpu); 0.808
│ │ │ │ +0002afe0: 3534 3273 2028 7468 7265 6164 293b 2030  542s (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                 |
│ │ │ │ @@ -11053,18 +11053,18 @@
│ │ │ │  0002b2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0002b2e0: 0a7c 6935 203a 2074 696d 6520 696e 7465  .|i5 : time inte
│ │ │ │  0002b2f0: 6772 616c 436c 6f73 7572 6528 4a2c 2053  gralClosure(J, S
│ │ │ │  0002b300: 7472 6174 6567 793d 3e7b 5261 6469 6361  trategy=>{Radica
│ │ │ │  0002b310: 6c43 6f64 696d 317d 2920 2020 2020 2020  lCodim1})       
│ │ │ │  0002b320: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002b330: 0a7c 202d 2d20 7573 6564 2030 2e36 3337  .| -- used 0.637
│ │ │ │ -0002b340: 3832 3673 2028 6370 7529 3b20 302e 3438  826s (cpu); 0.48
│ │ │ │ -0002b350: 3437 7320 2874 6872 6561 6429 3b20 3073  47s (thread); 0s
│ │ │ │ -0002b360: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │ +0002b330: 0a7c 202d 2d20 7573 6564 2030 2e39 3532  .| -- used 0.952
│ │ │ │ +0002b340: 3735 3573 2028 6370 7529 3b20 302e 3535  755s (cpu); 0.55
│ │ │ │ +0002b350: 3639 3238 7320 2874 6872 6561 6429 3b20  6928s (thread); 
│ │ │ │ +0002b360: 3073 2028 6763 2920 2020 2020 2020 2020  0s (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 3237 3437    |.| -- .002747
│ │ │ │ -0000cec0: 3032 7320 656c 6170 7365 6420 2020 2020  02s elapsed     
│ │ │ │ +0000ceb0: 2020 7c0a 7c20 2d2d 202e 3030 3334 3338    |.| -- .003438
│ │ │ │ +0000cec0: 3173 2065 6c61 7073 6564 2020 2020 2020  1s 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 3237    |.| -- .000427
│ │ │ │ -0000d6e0: 3739 3873 2065 6c61 7073 6564 2020 2020  798s elapsed    
│ │ │ │ +0000d6d0: 2020 7c0a 7c20 2d2d 202e 3030 3035 3836    |.| -- .000586
│ │ │ │ +0000d6e0: 3534 3273 2065 6c61 7073 6564 2020 2020  542s 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  ----------------
│ │ │ │ @@ -7199,17 +7199,17 @@
│ │ │ │  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 2e37 3937  .| -- used 0.797
│ │ │ │ -0001c260: 3635 3973 2028 6370 7529 3b20 302e 3531  659s (cpu); 0.51
│ │ │ │ -0001c270: 3939 3439 7320 2874 6872 6561 6429 3b20  9949s (thread); 
│ │ │ │ +0001c250: 0a7c 202d 2d20 7573 6564 2030 2e39 3433  .| -- used 0.943
│ │ │ │ +0001c260: 3434 3273 2028 6370 7529 3b20 302e 3633  442s (cpu); 0.63
│ │ │ │ +0001c270: 3831 3139 7320 2874 6872 6561 6429 3b20  8119s (thread); 
│ │ │ │  0001c280: 3073 2028 6763 2920 2020 2020 2020 2020  0s (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                 |
│ │ │ │ @@ -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 3731  .| -- used 0.671
│ │ │ │ -0001c4e0: 3432 3273 2028 6370 7529 3b20 302e 3336  422s (cpu); 0.36
│ │ │ │ -0001c4f0: 3434 3634 7320 2874 6872 6561 6429 3b20  4464s (thread); 
│ │ │ │ -0001c500: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │ +0001c4d0: 0a7c 202d 2d20 7573 6564 2030 2e37 3834  .| -- used 0.784
│ │ │ │ +0001c4e0: 3332 3773 2028 6370 7529 3b20 302e 3431  327s (cpu); 0.41
│ │ │ │ +0001c4f0: 3932 3473 2028 7468 7265 6164 293b 2030  924s (thread); 0
│ │ │ │ +0001c500: 7320 2867 6329 2020 2020 2020 2020 2020  s (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  .|              
│ │ │ │ @@ -7558,17 +7558,17 @@
│ │ │ │  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 3231  .| -- used 0.021
│ │ │ │ -0001d8d0: 3931 3137 7320 2863 7075 293b 2030 2e30  9117s (cpu); 0.0
│ │ │ │ -0001d8e0: 3231 3931 3638 7320 2874 6872 6561 6429  219168s (thread)
│ │ │ │ +0001d8c0: 0a7c 202d 2d20 7573 6564 2030 2e30 3235  .| -- used 0.025
│ │ │ │ +0001d8d0: 3636 3538 7320 2863 7075 293b 2030 2e30  6658s (cpu); 0.0
│ │ │ │ +0001d8e0: 3235 3636 3639 7320 2874 6872 6561 6429  256669s (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                 |
│ │ │ │ @@ -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 2032 2e30 3033  .| -- used 2.003
│ │ │ │ -0001db50: 3433 7320 2863 7075 293b 2031 2e32 3637  43s (cpu); 1.267
│ │ │ │ -0001db60: 3935 7320 2874 6872 6561 6429 3b20 3073  95s (thread); 0s
│ │ │ │ +0001db40: 0a7c 202d 2d20 7573 6564 2032 2e36 3836  .| -- used 2.686
│ │ │ │ +0001db50: 3334 7320 2863 7075 293b 2031 2e35 3237  34s (cpu); 1.527
│ │ │ │ +0001db60: 3632 7320 2874 6872 6561 6429 3b20 3073  62s (thread); 0s
│ │ │ │  0001db70: 2028 6763 2920 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                 |
│ │ │ │ @@ -8867,15 +8867,15 @@
│ │ │ │  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 3739 3935     |.| -- .77995
│ │ │ │ +00022a90: 2020 207c 0a7c 202d 2d20 2e35 3939 3135     |.| -- .59915
│ │ │ │  00022aa0: 3673 2065 6c61 7073 6564 2020 2020 2020  6s 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                  
│ │ │ │ @@ -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 3234 3632     |.| -- .52462
│ │ │ │ -00023040: 3573 2065 6c61 7073 6564 2020 2020 2020  5s elapsed      
│ │ │ │ +00023030: 2020 207c 0a7c 202d 2d20 2e34 3830 3339     |.| -- .48039
│ │ │ │ +00023040: 3773 2065 6c61 7073 6564 2020 2020 2020  7s 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                  
│ │ │ │ @@ -9894,15 +9894,15 @@
│ │ │ │  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 3737 3139 3473 2065  .| -- .677194s e
│ │ │ │ +00026ac0: 0a7c 202d 2d20 2e35 3730 3435 3473 2065  .| -- .570454s e
│ │ │ │  00026ad0: 6c61 7073 6564 2020 2020 2020 2020 2020  lapsed          
│ │ │ │  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                  
│ │ │ │ @@ -9959,16 +9959,16 @@
│ │ │ │  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 2e31 3034 3936 3473 2065  .| -- .104964s e
│ │ │ │ -00026ee0: 6c61 7073 6564 2020 2020 2020 2020 2020  lapsed          
│ │ │ │ +00026ed0: 0a7c 202d 2d20 2e30 3732 3137 3931 7320  .| -- .0721791s 
│ │ │ │ +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                  
│ │ │ │  00026f50: 2020 2020 2020 2020 2020 2020 2020 2020
│ │ ├── ./usr/share/info/Isomorphism.info.gz
│ │ │ ├── Isomorphism.info
│ │ │ │ @@ -4544,15 +4544,15 @@
│ │ │ │  00011bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011c00: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3320 3a20  -------+.|i23 : 
│ │ │ │  00011c10: 656c 6170 7365 6454 696d 6520 6973 4973  elapsedTime isIs
│ │ │ │  00011c20: 6f6d 6f72 7068 6963 2854 312c 2054 3229  omorphic(T1, T2)
│ │ │ │  00011c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011c50: 2020 2020 2020 207c 0a7c 202d 2d20 312e         |.| -- 1.
│ │ │ │ -00011c60: 3430 3034 7320 656c 6170 7365 6420 2020  4004s elapsed   
│ │ │ │ +00011c60: 3630 3437 3373 2065 6c61 7073 6564 2020  60473s elapsed  
│ │ │ │  00011c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011ca0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00011cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4569,16 +4569,16 @@
│ │ │ │  00011d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011d90: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3420 3a20  -------+.|i24 : 
│ │ │ │  00011da0: 656c 6170 7365 6454 696d 6520 6973 6f6d  elapsedTime isom
│ │ │ │  00011db0: 6f72 7068 6973 6d28 5431 2c20 5432 2920  orphism(T1, T2) 
│ │ │ │  00011dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011de0: 2020 2020 2020 207c 0a7c 202d 2d20 2e30         |.| -- .0
│ │ │ │ -00011df0: 3030 3032 3038 3773 2065 6c61 7073 6564  0002087s elapsed
│ │ │ │ -00011e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00011df0: 3030 3033 3732 3831 7320 656c 6170 7365  00037281s elapse
│ │ │ │ +00011e00: 6420 2020 2020 2020 2020 2020 2020 2020  d               
│ │ │ │  00011e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011e30: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00011e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011e70: 2020 2020 2020 2020 2020 2020 2020 2020
│ │ ├── ./usr/share/info/JSON.info.gz
│ │ │ ├── JSON.info
│ │ │ │ @@ -476,27 +476,27 @@
│ │ │ │  00001db0: 2d2d 2d2d 2d2b 0a7c 6939 203a 206a 736f  -----+.|i9 : jso
│ │ │ │  00001dc0: 6e46 696c 6520 3d20 7465 6d70 6f72 6172  nFile = temporar
│ │ │ │  00001dd0: 7946 696c 654e 616d 6528 2920 7c20 222e  yFileName() | ".
│ │ │ │  00001de0: 6a73 6f6e 2220 7c0a 7c20 2020 2020 2020  json" |.|       
│ │ │ │  00001df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001e10: 2020 2020 2020 207c 0a7c 6f39 203d 202f         |.|o9 = /
│ │ │ │ -00001e20: 746d 702f 4d32 2d35 3034 3132 2d30 2f30  tmp/M2-50412-0/0
│ │ │ │ +00001e20: 746d 702f 4d32 2d37 3934 3032 2d30 2f30  tmp/M2-79402-0/0
│ │ │ │  00001e30: 2e6a 736f 6e20 2020 2020 2020 2020 2020  .json           
│ │ │ │  00001e40: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │  00001e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00001e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00001e70: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020  ---------+.|i10 
│ │ │ │  00001e80: 3a20 6a73 6f6e 4669 6c65 203c 3c20 225b  : jsonFile << "[
│ │ │ │  00001e90: 312c 2032 2c20 335d 2220 3c3c 2065 6e64  1, 2, 3]" << end
│ │ │ │  00001ea0: 6c20 3c3c 2063 6c6f 7365 7c0a 7c20 2020  l << close|.|   
│ │ │ │  00001eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001ed0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -00001ee0: 3020 3d20 2f74 6d70 2f4d 322d 3530 3431  0 = /tmp/M2-5041
│ │ │ │ +00001ee0: 3020 3d20 2f74 6d70 2f4d 322d 3739 3430  0 = /tmp/M2-7940
│ │ │ │  00001ef0: 322d 302f 302e 6a73 6f6e 2020 2020 2020  2-0/0.json      
│ │ │ │  00001f00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00001f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001f30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00001f40: 6f31 3020 3a20 4669 6c65 2020 2020 2020  o10 : File      
│ │ │ │  00001f50: 2020 2020 2020 2020 2020 2020 2020 2020
│ │ ├── ./usr/share/info/Jets.info.gz
│ │ │ ├── Jets.info
│ │ │ │ @@ -5260,15 +5260,15 @@
│ │ │ │  000148b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000148c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20  --------+.|i4 : 
│ │ │ │  000148d0: 656c 6170 7365 6454 696d 6520 6a65 7473  elapsedTime jets
│ │ │ │  000148e0: 5261 6469 6361 6c28 322c 4929 2020 2020  Radical(2,I)    
│ │ │ │  000148f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014910: 2020 2020 2020 2020 7c0a 7c20 2d2d 202e          |.| -- .
│ │ │ │ -00014920: 3030 3232 3632 3839 7320 656c 6170 7365  00226289s elapse
│ │ │ │ +00014920: 3030 3236 3833 3738 7320 656c 6170 7365  00268378s elapse
│ │ │ │  00014930: 6420 2020 2020 2020 2020 2020 2020 2020  d               
│ │ │ │  00014940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014960: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00014970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -5305,15 +5305,15 @@
│ │ │ │  00014b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014b90: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20  --------+.|i5 : 
│ │ │ │  00014ba0: 656c 6170 7365 6454 696d 6520 7261 6469  elapsedTime radi
│ │ │ │  00014bb0: 6361 6c20 4a32 4920 2020 2020 2020 2020  cal J2I         
│ │ │ │  00014bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014be0: 2020 2020 2020 2020 7c0a 7c20 2d2d 202e          |.| -- .
│ │ │ │ -00014bf0: 3239 3730 3039 7320 656c 6170 7365 6420  297009s elapsed 
│ │ │ │ +00014bf0: 3235 3532 3937 7320 656c 6170 7365 6420  255297s elapsed 
│ │ │ │  00014c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014c30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00014c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -8104,15 +8104,15 @@
│ │ │ │  0001fa70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fa80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fa90: 2d2d 2b0a 7c69 3820 3a20 656c 6170 7365  --+.|i8 : elapse
│ │ │ │  0001faa0: 6454 696d 6520 6a65 7473 2833 2c49 2920  dTime jets(3,I) 
│ │ │ │  0001fab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fae0: 207c 0a7c 202d 2d20 2e30 3234 3739 3933   |.| -- .0247993
│ │ │ │ +0001fae0: 207c 0a7c 202d 2d20 2e30 3131 3037 3434   |.| -- .0110744
│ │ │ │  0001faf0: 7320 656c 6170 7365 6420 2020 2020 2020  s elapsed       
│ │ │ │  0001fb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fb30: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0001fb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -8203,16 +8203,16 @@
│ │ │ │  000200a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000200b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  000200c0: 7c69 3131 203a 2065 6c61 7073 6564 5469  |i11 : elapsedTi
│ │ │ │  000200d0: 6d65 206a 6574 7328 332c 4929 2020 2020  me jets(3,I)    
│ │ │ │  000200e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000200f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020100: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00020110: 202d 2d20 2e30 3134 3631 3434 7320 656c   -- .0146144s el
│ │ │ │ -00020120: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │ +00020110: 202d 2d20 2e30 3033 3038 3632 3673 2065   -- .00308626s e
│ │ │ │ +00020120: 6c61 7073 6564 2020 2020 2020 2020 2020  lapsed          
│ │ │ │  00020130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020150: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00020160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -8242,16 +8242,16 @@
│ │ │ │  00020310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020330: 2d2d 2d2d 2d2d 2b0a 7c69 3132 203a 2065  ------+.|i12 : e
│ │ │ │  00020340: 6c61 7073 6564 5469 6d65 206a 6574 7328  lapsedTime jets(
│ │ │ │  00020350: 322c 4929 2020 2020 2020 2020 2020 2020  2,I)            
│ │ │ │  00020360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020380: 2020 2020 207c 0a7c 202d 2d20 2e30 3036       |.| -- .006
│ │ │ │ -00020390: 3334 3333 3573 2065 6c61 7073 6564 2020  34335s elapsed  
│ │ │ │ +00020380: 2020 2020 207c 0a7c 202d 2d20 2e30 3032       |.| -- .002
│ │ │ │ +00020390: 3732 3433 7320 656c 6170 7365 6420 2020  7243s elapsed   
│ │ │ │  000203a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000203b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000203c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000203d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  000203e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000203f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -8754,15 +8754,15 @@
│ │ │ │  00022310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  00022330: 0a7c 6932 3420 3a20 656c 6170 7365 6454  .|i24 : elapsedT
│ │ │ │  00022340: 696d 6520 6a65 7473 2833 2c66 2920 2020  ime jets(3,f)   
│ │ │ │  00022350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022370: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022380: 0a7c 202d 2d20 2e30 3131 3638 3632 7320  .| -- .0116862s 
│ │ │ │ +00022380: 0a7c 202d 2d20 2e30 3135 3037 3334 7320  .| -- .0150734s 
│ │ │ │  00022390: 656c 6170 7365 6420 2020 2020 2020 2020  elapsed         
│ │ │ │  000223a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000223b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000223c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  000223d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000223e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000223f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -8944,15 +8944,15 @@
│ │ │ │  00022ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  00022f10: 0a7c 6932 3720 3a20 656c 6170 7365 6454  .|i27 : elapsedT
│ │ │ │  00022f20: 696d 6520 6a65 7473 2832 2c66 2920 2020  ime jets(2,f)   
│ │ │ │  00022f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022f50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022f60: 0a7c 202d 2d20 2e30 3030 3631 3337 3935  .| -- .000613795
│ │ │ │ +00022f60: 0a7c 202d 2d20 2e30 3030 3739 3836 3534  .| -- .000798654
│ │ │ │  00022f70: 7320 656c 6170 7365 6420 2020 2020 2020  s elapsed       
│ │ │ │  00022f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022fa0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00022fb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00022fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022fd0: 2020 2020 2020 2020 2020 2020 2020 2020
│ │ ├── ./usr/share/info/K3Carpets.info.gz
│ │ │ ├── K3Carpets.info
│ │ │ │ @@ -1860,15 +1860,15 @@
│ │ │ │  00007430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00007440: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 204c  -------+.|i4 : L
│ │ │ │  00007450: 203d 2061 6e61 6c79 7a65 5374 7261 6e64   = analyzeStrand
│ │ │ │  00007460: 2846 2c61 293b 2023 4c20 2020 2020 2020  (F,a); #L       
│ │ │ │  00007470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007490: 2020 2020 2020 207c 0a7c 202d 2d20 2e30         |.| -- .0
│ │ │ │ -000074a0: 3236 3933 3936 7320 656c 6170 7365 6420  269396s elapsed 
│ │ │ │ +000074a0: 3238 3139 3938 7320 656c 6170 7365 6420  281998s elapsed 
│ │ │ │  000074b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000074c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000074d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000074e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000074f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -1995,35 +1995,35 @@
│ │ │ │  00007ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00007cb0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20  -------+.|i10 : 
│ │ │ │  00007cc0: 6361 7270 6574 4265 7474 6954 6162 6c65  carpetBettiTable
│ │ │ │  00007cd0: 2861 2c62 2c33 2920 2020 2020 2020 2020  (a,b,3)         
│ │ │ │  00007ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007d00: 2020 2020 2020 207c 0a7c 202d 2d20 2e30         |.| -- .0
│ │ │ │ -00007d10: 3032 3430 3432 7320 656c 6170 7365 6420  024042s elapsed 
│ │ │ │ +00007d10: 3032 3636 3136 3673 2065 6c61 7073 6564  0266166s elapsed
│ │ │ │  00007d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007d50: 2020 2020 2020 207c 0a7c 202d 2d20 2e30         |.| -- .0
│ │ │ │ -00007d60: 3036 3530 3737 3773 2065 6c61 7073 6564  0650777s elapsed
│ │ │ │ +00007d60: 3037 3530 3336 3473 2065 6c61 7073 6564  0750364s elapsed
│ │ │ │  00007d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007da0: 2020 2020 2020 207c 0a7c 202d 2d20 2e30         |.| -- .0
│ │ │ │ -00007db0: 3532 3730 3634 7320 656c 6170 7365 6420  527064s elapsed 
│ │ │ │ +00007db0: 3237 3130 3635 7320 656c 6170 7365 6420  271065s elapsed 
│ │ │ │  00007dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007df0: 2020 2020 2020 207c 0a7c 202d 2d20 2e30         |.| -- .0
│ │ │ │ -00007e00: 3733 3637 3531 7320 656c 6170 7365 6420  736751s elapsed 
│ │ │ │ +00007e00: 3432 3137 3237 7320 656c 6170 7365 6420  421727s elapsed 
│ │ │ │  00007e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007e40: 2020 2020 2020 207c 0a7c 202d 2d20 2e30         |.| -- .0
│ │ │ │ -00007e50: 3136 3232 3173 2065 6c61 7073 6564 2020  16221s elapsed  
│ │ │ │ +00007e50: 3033 3637 3930 3773 2065 6c61 7073 6564  0367907s elapsed
│ │ │ │  00007e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007e90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00007ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -3676,35 +3676,35 @@
│ │ │ │  0000e5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000e5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000e5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000e5e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2065  -------+.|i2 : e
│ │ │ │  0000e5f0: 6c61 7073 6564 5469 6d65 2054 3d63 6172  lapsedTime T=car
│ │ │ │  0000e600: 7065 7442 6574 7469 5461 626c 6528 612c  petBettiTable(a,
│ │ │ │  0000e610: 622c 3329 2020 2020 2020 2020 2020 2020  b,3)            
│ │ │ │ -0000e620: 2020 2020 7c0a 7c20 2d2d 202e 3030 3232      |.| -- .0022
│ │ │ │ -0000e630: 3336 3739 7320 656c 6170 7365 6420 2020  3679s elapsed   
│ │ │ │ +0000e620: 2020 2020 7c0a 7c20 2d2d 202e 3030 3238      |.| -- .0028
│ │ │ │ +0000e630: 3132 3932 7320 656c 6170 7365 6420 2020  1292s elapsed   
│ │ │ │  0000e640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000e660: 207c 0a7c 202d 2d20 2e30 3038 3034 3035   |.| -- .0080405
│ │ │ │ -0000e670: 3573 2065 6c61 7073 6564 2020 2020 2020  5s elapsed      
│ │ │ │ +0000e660: 207c 0a7c 202d 2d20 2e30 3037 3234 3435   |.| -- .0072445
│ │ │ │ +0000e670: 3173 2065 6c61 7073 6564 2020 2020 2020  1s elapsed      
│ │ │ │  0000e680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e690: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000e6a0: 7c20 2d2d 202e 3032 3236 3237 3373 2065  | -- .0226273s e
│ │ │ │ +0000e6a0: 7c20 2d2d 202e 3032 3633 3832 3273 2065  | -- .0263822s e
│ │ │ │  0000e6b0: 6c61 7073 6564 2020 2020 2020 2020 2020  lapsed          
│ │ │ │  0000e6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e6d0: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -0000e6e0: 2d20 2e30 3039 3633 3632 3173 2065 6c61  - .00963621s ela
│ │ │ │ -0000e6f0: 7073 6564 2020 2020 2020 2020 2020 2020  psed            
│ │ │ │ +0000e6e0: 2d20 2e30 3039 3833 3635 7320 656c 6170  - .0098365s elap
│ │ │ │ +0000e6f0: 7365 6420 2020 2020 2020 2020 2020 2020  sed             
│ │ │ │  0000e700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e710: 2020 2020 2020 2020 7c0a 7c20 2d2d 202e          |.| -- .
│ │ │ │ -0000e720: 3030 3334 3932 3331 7320 656c 6170 7365  00349231s elapse
│ │ │ │ +0000e720: 3030 3430 3131 3835 7320 656c 6170 7365  00401185s elapse
│ │ │ │  0000e730: 6420 2020 2020 2020 2020 2020 2020 2020  d               
│ │ │ │  0000e740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000e750: 2020 2020 207c 0a7c 202d 2d20 2e34 3734       |.| -- .474
│ │ │ │ -0000e760: 3430 3773 2065 6c61 7073 6564 2020 2020  407s elapsed    
│ │ │ │ +0000e750: 2020 2020 207c 0a7c 202d 2d20 2e34 3234       |.| -- .424
│ │ │ │ +0000e760: 3132 3473 2065 6c61 7073 6564 2020 2020  124s elapsed    
│ │ │ │  0000e770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e790: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0000e7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e7c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000e7d0: 0a7c 2020 2020 2020 2020 2020 2020 3020  .|            0 
│ │ │ │ @@ -3764,15 +3764,15 @@
│ │ │ │  0000eb30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000eb40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000eb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000eb60: 2d2d 2b0a 7c69 3420 3a20 656c 6170 7365  --+.|i4 : elapse
│ │ │ │  0000eb70: 6454 696d 6520 5427 3d6d 696e 696d 616c  dTime T'=minimal
│ │ │ │  0000eb80: 4265 7474 6920 4a20 2020 2020 2020 2020  Betti J         
│ │ │ │  0000eb90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000eba0: 0a7c 202d 2d20 2e32 3437 3730 3273 2065  .| -- .247702s e
│ │ │ │ +0000eba0: 0a7c 202d 2d20 2e32 3034 3331 3673 2065  .| -- .204316s e
│ │ │ │  0000ebb0: 6c61 7073 6564 2020 2020 2020 2020 2020  lapsed          
│ │ │ │  0000ebc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ebd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0000ebe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ebf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ec00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ec10: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ @@ -3848,42 +3848,42 @@
│ │ │ │  0000f070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f0a0: 2b0a 7c69 3620 3a20 656c 6170 7365 6454  +.|i6 : elapsedT
│ │ │ │  0000f0b0: 696d 6520 683d 6361 7270 6574 4265 7474  ime h=carpetBett
│ │ │ │  0000f0c0: 6954 6162 6c65 7328 362c 3629 3b20 2020  iTables(6,6);   
│ │ │ │  0000f0d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0000f0e0: 202d 2d20 2e30 3034 3530 3830 3373 2065   -- .00450803s e
│ │ │ │ +0000f0e0: 202d 2d20 2e30 3034 3836 3137 3473 2065   -- .00486174s e
│ │ │ │  0000f0f0: 6c61 7073 6564 2020 2020 2020 2020 2020  lapsed          
│ │ │ │  0000f100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f110: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ -0000f120: 202e 3031 3731 3435 7320 656c 6170 7365   .017145s elapse
│ │ │ │ -0000f130: 6420 2020 2020 2020 2020 2020 2020 2020  d               
│ │ │ │ +0000f120: 202e 3031 3839 3639 3273 2065 6c61 7073   .0189692s elaps
│ │ │ │ +0000f130: 6564 2020 2020 2020 2020 2020 2020 2020  ed              
│ │ │ │  0000f140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f150: 2020 2020 2020 207c 0a7c 202d 2d20 2e31         |.| -- .1
│ │ │ │ -0000f160: 3332 3237 3873 2065 6c61 7073 6564 2020  32278s elapsed  
│ │ │ │ +0000f160: 3036 3538 3273 2065 6c61 7073 6564 2020  06582s elapsed  
│ │ │ │  0000f170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f190: 2020 2020 7c0a 7c20 2d2d 2031 2e31 3431      |.| -- 1.141
│ │ │ │ -0000f1a0: 3339 7320 656c 6170 7365 6420 2020 2020  39s elapsed     
│ │ │ │ +0000f190: 2020 2020 7c0a 7c20 2d2d 202e 3934 3933      |.| -- .9493
│ │ │ │ +0000f1a0: 3473 2065 6c61 7073 6564 2020 2020 2020  4s elapsed      
│ │ │ │  0000f1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f1d0: 207c 0a7c 202d 2d20 2e35 3137 3134 3373   |.| -- .517143s
│ │ │ │ +0000f1d0: 207c 0a7c 202d 2d20 2e34 3132 3635 3373   |.| -- .412653s
│ │ │ │  0000f1e0: 2065 6c61 7073 6564 2020 2020 2020 2020   elapsed        
│ │ │ │  0000f1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f200: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0000f210: 7c20 2d2d 202e 3037 3130 3930 3273 2065  | -- .0710902s e
│ │ │ │ +0000f210: 7c20 2d2d 202e 3034 3037 3233 3973 2065  | -- .0407239s e
│ │ │ │  0000f220: 6c61 7073 6564 2020 2020 2020 2020 2020  lapsed          
│ │ │ │  0000f230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f240: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -0000f250: 2d20 2e30 3036 3532 3236 3773 2065 6c61  - .00652267s ela
│ │ │ │ +0000f250: 2d20 2e30 3037 3137 3634 3473 2065 6c61  - .00717644s ela
│ │ │ │  0000f260: 7073 6564 2020 2020 2020 2020 2020 2020  psed            
│ │ │ │  0000f270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000f280: 2020 2020 2020 2020 7c0a 7c20 2d2d 2036          |.| -- 6
│ │ │ │ -0000f290: 2e34 3736 3738 7320 656c 6170 7365 6420  .47678s elapsed 
│ │ │ │ +0000f280: 2020 2020 2020 2020 7c0a 7c20 2d2d 2035          |.| -- 5
│ │ │ │ +0000f290: 2e36 3433 3239 7320 656c 6170 7365 6420  .64329s elapsed 
│ │ │ │  0000f2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f2c0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  0000f2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000f300: 2d2d 2b0a 7c69 3720 3a20 6361 7270 6574  --+.|i7 : carpet
│ │ │ │ @@ -4107,32 +4107,32 @@
│ │ │ │  000100a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000100b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000100c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000100d0: 2d2d 2d2d 2d2b 0a7c 6932 203a 2068 3d63  -----+.|i2 : h=c
│ │ │ │  000100e0: 6172 7065 7442 6574 7469 5461 626c 6573  arpetBettiTables
│ │ │ │  000100f0: 2861 2c62 2920 2020 2020 2020 2020 2020  (a,b)           
│ │ │ │  00010100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010110: 2020 2020 207c 0a7c 202d 2d20 2e30 3130       |.| -- .010
│ │ │ │ -00010120: 3336 3835 7320 656c 6170 7365 6420 2020  3685s elapsed   
│ │ │ │ +00010110: 2020 2020 207c 0a7c 202d 2d20 2e30 3032       |.| -- .002
│ │ │ │ +00010120: 3838 3739 3773 2065 6c61 7073 6564 2020  88797s elapsed  
│ │ │ │  00010130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010150: 2020 2020 207c 0a7c 202d 2d20 2e30 3036       |.| -- .006
│ │ │ │ -00010160: 3632 3637 3773 2065 6c61 7073 6564 2020  62677s elapsed  
│ │ │ │ +00010150: 2020 2020 207c 0a7c 202d 2d20 2e30 3039       |.| -- .009
│ │ │ │ +00010160: 3934 3738 3373 2065 6c61 7073 6564 2020  94783s elapsed  
│ │ │ │  00010170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010190: 2020 2020 207c 0a7c 202d 2d20 2e30 3232       |.| -- .022
│ │ │ │ -000101a0: 3834 3335 7320 656c 6170 7365 6420 2020  8435s elapsed   
│ │ │ │ +00010190: 2020 2020 207c 0a7c 202d 2d20 2e30 3235       |.| -- .025
│ │ │ │ +000101a0: 3438 3336 7320 656c 6170 7365 6420 2020  4836s elapsed   
│ │ │ │  000101b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000101c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000101d0: 2020 2020 207c 0a7c 202d 2d20 2e30 3333       |.| -- .033
│ │ │ │ -000101e0: 3738 3532 7320 656c 6170 7365 6420 2020  7852s elapsed   
│ │ │ │ +000101d0: 2020 2020 207c 0a7c 202d 2d20 2e30 3130       |.| -- .010
│ │ │ │ +000101e0: 3036 3539 7320 656c 6170 7365 6420 2020  0659s elapsed   
│ │ │ │  000101f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010210: 2020 2020 207c 0a7c 202d 2d20 2e30 3033       |.| -- .003
│ │ │ │ -00010220: 3633 3137 3573 2065 6c61 7073 6564 2020  63175s elapsed  
│ │ │ │ +00010210: 2020 2020 207c 0a7c 202d 2d20 2e30 3034       |.| -- .004
│ │ │ │ +00010220: 3138 3139 3773 2065 6c61 7073 6564 2020  18197s elapsed  
│ │ │ │  00010230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010250: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00010260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010290: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ @@ -4287,16 +4287,16 @@
│ │ │ │  00010be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010c10: 2d2d 2d2d 2d2b 0a7c 6935 203a 2065 6c61  -----+.|i5 : ela
│ │ │ │  00010c20: 7073 6564 5469 6d65 2054 273d 6d69 6e69  psedTime T'=mini
│ │ │ │  00010c30: 6d61 6c42 6574 7469 204a 2020 2020 2020  malBetti J      
│ │ │ │  00010c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010c50: 2020 2020 207c 0a7c 202d 2d20 2e32 3438       |.| -- .248
│ │ │ │ -00010c60: 3733 3673 2065 6c61 7073 6564 2020 2020  736s elapsed    
│ │ │ │ +00010c50: 2020 2020 207c 0a7c 202d 2d20 2e32 3132       |.| -- .212
│ │ │ │ +00010c60: 3133 3173 2065 6c61 7073 6564 2020 2020  131s elapsed    
│ │ │ │  00010c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010c90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00010ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010cd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ @@ -4375,44 +4375,44 @@
│ │ │ │  00011160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011190: 2d2d 2d2d 2d2b 0a7c 6937 203a 2065 6c61  -----+.|i7 : ela
│ │ │ │  000111a0: 7073 6564 5469 6d65 2068 3d63 6172 7065  psedTime h=carpe
│ │ │ │  000111b0: 7442 6574 7469 5461 626c 6573 2836 2c36  tBettiTables(6,6
│ │ │ │  000111c0: 293b 2020 2020 2020 2020 2020 2020 2020  );              
│ │ │ │ -000111d0: 2020 2020 207c 0a7c 202d 2d20 2e30 3035       |.| -- .005
│ │ │ │ -000111e0: 3336 3131 3573 2065 6c61 7073 6564 2020  36115s elapsed  
│ │ │ │ +000111d0: 2020 2020 207c 0a7c 202d 2d20 2e30 3034       |.| -- .004
│ │ │ │ +000111e0: 3934 3237 3973 2065 6c61 7073 6564 2020  94279s elapsed  
│ │ │ │  000111f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011210: 2020 2020 207c 0a7c 202d 2d20 2e30 3337       |.| -- .037
│ │ │ │ -00011220: 3030 3035 7320 656c 6170 7365 6420 2020  0005s elapsed   
│ │ │ │ +00011210: 2020 2020 207c 0a7c 202d 2d20 2e30 3138       |.| -- .018
│ │ │ │ +00011220: 3535 3235 7320 656c 6170 7365 6420 2020  5525s elapsed   
│ │ │ │  00011230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011250: 2020 2020 207c 0a7c 202d 2d20 2e32 3034       |.| -- .204
│ │ │ │ -00011260: 3938 3373 2065 6c61 7073 6564 2020 2020  983s elapsed    
│ │ │ │ +00011250: 2020 2020 207c 0a7c 202d 2d20 2e31 3035       |.| -- .105
│ │ │ │ +00011260: 3736 3873 2065 6c61 7073 6564 2020 2020  768s elapsed    
│ │ │ │  00011270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011290: 2020 2020 207c 0a7c 202d 2d20 312e 3332       |.| -- 1.32
│ │ │ │ -000112a0: 3933 3373 2065 6c61 7073 6564 2020 2020  933s elapsed    
│ │ │ │ +00011290: 2020 2020 207c 0a7c 202d 2d20 2e39 3738       |.| -- .978
│ │ │ │ +000112a0: 3430 3573 2065 6c61 7073 6564 2020 2020  405s elapsed    
│ │ │ │  000112b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000112c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000112d0: 2020 2020 207c 0a7c 202d 2d20 2e34 3330       |.| -- .430
│ │ │ │ -000112e0: 3936 3973 2065 6c61 7073 6564 2020 2020  969s elapsed    
│ │ │ │ +000112d0: 2020 2020 207c 0a7c 202d 2d20 2e34 3431       |.| -- .441
│ │ │ │ +000112e0: 3935 3573 2065 6c61 7073 6564 2020 2020  955s elapsed    
│ │ │ │  000112f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011310: 2020 2020 207c 0a7c 202d 2d20 2e30 3531       |.| -- .051
│ │ │ │ -00011320: 3136 3534 7320 656c 6170 7365 6420 2020  1654s elapsed   
│ │ │ │ +00011310: 2020 2020 207c 0a7c 202d 2d20 2e30 3430       |.| -- .040
│ │ │ │ +00011320: 3836 3835 7320 656c 6170 7365 6420 2020  8685s elapsed   
│ │ │ │  00011330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011350: 2020 2020 207c 0a7c 202d 2d20 2e30 3036       |.| -- .006
│ │ │ │ -00011360: 3534 3939 3273 2065 6c61 7073 6564 2020  54992s elapsed  
│ │ │ │ +00011350: 2020 2020 207c 0a7c 202d 2d20 2e30 3037       |.| -- .007
│ │ │ │ +00011360: 3738 3130 3873 2065 6c61 7073 6564 2020  78108s elapsed  
│ │ │ │  00011370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00011390: 2020 2020 207c 0a7c 202d 2d20 362e 3834       |.| -- 6.84
│ │ │ │ -000113a0: 3134 3973 2065 6c61 7073 6564 2020 2020  149s elapsed    
│ │ │ │ +00011390: 2020 2020 207c 0a7c 202d 2d20 352e 3737       |.| -- 5.77
│ │ │ │ +000113a0: 3837 3273 2065 6c61 7073 6564 2020 2020  872s elapsed    
│ │ │ │  000113b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000113c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000113d0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  000113e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000113f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011410: 2d2d 2d2d 2d2b 0a7c 6938 203a 206b 6579  -----+.|i8 : key
│ │ │ │ @@ -4634,162 +4634,162 @@
│ │ │ │  00012190: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │  000121a0: 203a 2053 6571 7565 6e63 6520 2020 2020   : Sequence     
│ │ │ │  000121b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  000121c0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000121d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000121e0: 2d2b 0a7c 6932 203a 2064 3d63 6172 7065  -+.|i2 : d=carpe
│ │ │ │  000121f0: 7444 6574 2861 2c62 2920 2020 2020 2020  tDet(a,b)       
│ │ │ │ -00012200: 2020 2020 7c0a 7c20 2d2d 202e 3030 3639      |.| -- .0069
│ │ │ │ -00012210: 3436 3339 7320 656c 6170 7365 6420 2020  4639s elapsed   
│ │ │ │ +00012200: 2020 2020 7c0a 7c20 2d2d 202e 3030 3938      |.| -- .0098
│ │ │ │ +00012210: 3734 7320 656c 6170 7365 6420 2020 2020  74s elapsed     
│ │ │ │  00012220: 2020 2020 2020 207c 0a7c 202d 2d20 2e30         |.| -- .0
│ │ │ │ -00012230: 3132 3334 3831 7320 656c 6170 7365 6420  123481s elapsed 
│ │ │ │ +00012230: 3132 3635 3837 7320 656c 6170 7365 6420  126587s elapsed 
│ │ │ │  00012240: 2020 2020 2020 2020 2020 7c0a 7c28 6e75            |.|(nu
│ │ │ │  00012250: 6d62 6572 204f 6620 626c 6f63 6b73 2c20  mber Of blocks, 
│ │ │ │  00012260: 3236 2920 2020 2020 2020 2020 207c 0a7c  26)          |.|
│ │ │ │ -00012270: 202d 2d20 2e30 3030 3239 3731 3834 7320   -- .000297184s 
│ │ │ │ -00012280: 656c 6170 7365 6420 2020 2020 2020 2020  elapsed         
│ │ │ │ +00012270: 202d 2d20 2e30 3030 3237 3435 3573 2065   -- .00027455s e
│ │ │ │ +00012280: 6c61 7073 6564 2020 2020 2020 2020 2020  lapsed          
│ │ │ │  00012290: 7c0a 7c31 2020 2020 2020 2020 2020 2020  |.|1            
│ │ │ │  000122a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000122b0: 2020 207c 0a7c 202d 2d20 2e30 3030 3134     |.| -- .00014
│ │ │ │ -000122c0: 3334 3337 7320 656c 6170 7365 6420 2020  3437s elapsed   
│ │ │ │ +000122b0: 2020 207c 0a7c 202d 2d20 2e30 3030 3138     |.| -- .00018
│ │ │ │ +000122c0: 3536 3838 7320 656c 6170 7365 6420 2020  5688s elapsed   
│ │ │ │  000122d0: 2020 2020 2020 7c0a 7c31 2020 2020 2020        |.|1      
│ │ │ │  000122e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000122f0: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ -00012300: 2e30 3030 3133 3132 3935 7320 656c 6170  .000131295s elap
│ │ │ │ +00012300: 2e30 3030 3137 3833 3739 7320 656c 6170  .000178379s elap
│ │ │ │  00012310: 7365 6420 2020 2020 2020 2020 7c0a 7c31  sed         |.|1
│ │ │ │  00012320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012330: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00012340: 0a7c 202d 2d20 2e30 3030 3132 3835 3739  .| -- .000128579
│ │ │ │ +00012340: 0a7c 202d 2d20 2e30 3030 3137 3734 3831  .| -- .000177481
│ │ │ │  00012350: 7320 656c 6170 7365 6420 2020 2020 2020  s elapsed       
│ │ │ │  00012360: 2020 7c0a 7c31 2020 2020 2020 2020 2020    |.|1          
│ │ │ │  00012370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012380: 2020 2020 207c 0a7c 202d 2d20 2e30 3030       |.| -- .000
│ │ │ │ -00012390: 3134 3333 3237 7320 656c 6170 7365 6420  143327s elapsed 
│ │ │ │ +00012390: 3136 3932 3039 7320 656c 6170 7365 6420  169209s elapsed 
│ │ │ │  000123a0: 2020 2020 2020 2020 7c0a 7c32 2020 2020          |.|2    
│ │ │ │  000123b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000123c0: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -000123d0: 2d20 2e30 3030 3134 3236 3835 7320 656c  - .000142685s el
│ │ │ │ +000123d0: 2d20 2e30 3030 3136 3937 3134 7320 656c  - .000169714s el
│ │ │ │  000123e0: 6170 7365 6420 2020 2020 2020 2020 7c0a  apsed         |.
│ │ │ │  000123f0: 7c20 3220 2020 2020 2020 2020 2020 2020  | 2             
│ │ │ │  00012400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012410: 207c 0a7c 3220 2020 2020 2020 2020 2020   |.|2           
│ │ │ │  00012420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012430: 2020 2020 7c0a 7c20 2d2d 202e 3030 3031      |.| -- .0001
│ │ │ │ -00012440: 3534 3037 3873 2065 6c61 7073 6564 2020  54078s elapsed  
│ │ │ │ +00012440: 3837 3937 3873 2065 6c61 7073 6564 2020  87978s elapsed  
│ │ │ │  00012450: 2020 2020 2020 207c 0a7c 2032 2020 2020         |.| 2    
│ │ │ │  00012460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012470: 2020 2020 2020 2020 2020 7c0a 7c32 2020            |.|2  
│ │ │ │  00012480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012490: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000124a0: 202d 2d20 2e30 3030 3136 3030 3939 7320   -- .000160099s 
│ │ │ │ -000124b0: 656c 6170 7365 6420 2020 2020 2020 2020  elapsed         
│ │ │ │ +000124a0: 202d 2d20 2e30 3030 3233 3830 3273 2065   -- .00023802s e
│ │ │ │ +000124b0: 6c61 7073 6564 2020 2020 2020 2020 2020  lapsed          
│ │ │ │  000124c0: 7c0a 7c20 3220 2020 2020 2020 2020 2020  |.| 2           
│ │ │ │  000124d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000124e0: 2020 207c 0a7c 3220 3320 2020 2020 2020     |.|2 3       
│ │ │ │  000124f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012500: 2020 2020 2020 7c0a 7c20 2d2d 202e 3030        |.| -- .00
│ │ │ │ -00012510: 3031 3430 3436 3273 2065 6c61 7073 6564  0140462s elapsed
│ │ │ │ +00012510: 3031 3637 3739 3373 2065 6c61 7073 6564  0167793s elapsed
│ │ │ │  00012520: 2020 2020 2020 2020 207c 0a7c 2032 2020           |.| 2  
│ │ │ │  00012530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012540: 2020 2020 2020 2020 2020 2020 7c0a 7c32              |.|2
│ │ │ │  00012550: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │  00012560: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00012570: 0a7c 202d 2d20 2e30 3030 3133 3937 3173  .| -- .00013971s
│ │ │ │ -00012580: 2065 6c61 7073 6564 2020 2020 2020 2020   elapsed        
│ │ │ │ +00012570: 0a7c 202d 2d20 2e30 3030 3138 3637 3534  .| -- .000186754
│ │ │ │ +00012580: 7320 656c 6170 7365 6420 2020 2020 2020  s elapsed       
│ │ │ │  00012590: 2020 7c0a 7c20 3220 2020 2020 2020 2020    |.| 2         
│ │ │ │  000125a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000125b0: 2020 2020 207c 0a7c 3220 3320 2020 2020       |.|2 3     
│ │ │ │  000125c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000125d0: 2020 2020 2020 2020 7c0a 7c20 2d2d 202e          |.| -- .
│ │ │ │ -000125e0: 3030 3031 3338 3032 3873 2065 6c61 7073  000138028s elaps
│ │ │ │ +000125e0: 3030 3031 3935 3234 3773 2065 6c61 7073  000195247s elaps
│ │ │ │  000125f0: 6564 2020 2020 2020 2020 207c 0a7c 2032  ed         |.| 2
│ │ │ │  00012600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012610: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00012620: 7c32 2020 2020 2020 2020 2020 2020 2020  |2              
│ │ │ │  00012630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012640: 207c 0a7c 202d 2d20 2e30 3030 3133 3236   |.| -- .0001326
│ │ │ │ -00012650: 3738 7320 656c 6170 7365 6420 2020 2020  78s elapsed     
│ │ │ │ +00012640: 207c 0a7c 202d 2d20 2e30 3030 3137 3837   |.| -- .0001787
│ │ │ │ +00012650: 3333 7320 656c 6170 7365 6420 2020 2020  33s elapsed     
│ │ │ │  00012660: 2020 2020 7c0a 7c20 3220 2020 2020 2020      |.| 2       
│ │ │ │  00012670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012680: 2020 2020 2020 207c 0a7c 3220 2020 2020         |.|2     
│ │ │ │  00012690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000126a0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ -000126b0: 202e 3030 3031 3230 3538 3473 2065 6c61   .000120584s ela
│ │ │ │ +000126b0: 202e 3030 3031 3437 3235 3873 2065 6c61   .000147258s ela
│ │ │ │  000126c0: 7073 6564 2020 2020 2020 2020 207c 0a7c  psed         |.|
│ │ │ │  000126d0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  000126e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000126f0: 7c0a 7c20 2d2d 202e 3030 3031 3232 3931  |.| -- .00012291
│ │ │ │ +000126f0: 7c0a 7c20 2d2d 202e 3030 3031 3530 3537  |.| -- .00015057
│ │ │ │  00012700: 3973 2065 6c61 7073 6564 2020 2020 2020  9s elapsed      
│ │ │ │  00012710: 2020 207c 0a7c 3220 2020 2020 2020 2020     |.|2         
│ │ │ │  00012720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012730: 2020 2020 2020 7c0a 7c20 2d2d 202e 3030        |.| -- .00
│ │ │ │ -00012740: 3031 3334 3932 3273 2065 6c61 7073 6564  0134922s elapsed
│ │ │ │ +00012740: 3031 3737 3038 3373 2065 6c61 7073 6564  0177083s elapsed
│ │ │ │  00012750: 2020 2020 2020 2020 207c 0a7c 2032 2020           |.| 2  
│ │ │ │  00012760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012770: 2020 2020 2020 2020 2020 2020 7c0a 7c32              |.|2
│ │ │ │  00012780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012790: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000127a0: 0a7c 202d 2d20 2e30 3030 3132 3331 3773  .| -- .00012317s
│ │ │ │ -000127b0: 2065 6c61 7073 6564 2020 2020 2020 2020   elapsed        
│ │ │ │ +000127a0: 0a7c 202d 2d20 2e30 3030 3136 3133 3338  .| -- .000161338
│ │ │ │ +000127b0: 7320 656c 6170 7365 6420 2020 2020 2020  s elapsed       
│ │ │ │  000127c0: 2020 7c0a 7c20 3220 2020 2020 2020 2020    |.| 2         
│ │ │ │  000127d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000127e0: 2020 2020 207c 0a7c 3220 2020 2020 2020       |.|2       
│ │ │ │  000127f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012800: 2020 2020 2020 2020 7c0a 7c20 2d2d 202e          |.| -- .
│ │ │ │ -00012810: 3030 3031 3430 3831 3273 2065 6c61 7073  000140812s elaps
│ │ │ │ +00012810: 3030 3031 3735 3533 3673 2065 6c61 7073  000175536s elaps
│ │ │ │  00012820: 6564 2020 2020 2020 2020 207c 0a7c 2032  ed         |.| 2
│ │ │ │  00012830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012840: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00012850: 7c32 2033 2020 2020 2020 2020 2020 2020  |2 3            
│ │ │ │  00012860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012870: 207c 0a7c 202d 2d20 2e30 3030 3133 3338   |.| -- .0001338
│ │ │ │ -00012880: 3539 7320 656c 6170 7365 6420 2020 2020  59s elapsed     
│ │ │ │ +00012870: 207c 0a7c 202d 2d20 2e30 3030 3135 3737   |.| -- .0001577
│ │ │ │ +00012880: 3133 7320 656c 6170 7365 6420 2020 2020  13s elapsed     
│ │ │ │  00012890: 2020 2020 7c0a 7c20 3220 2020 2020 2020      |.| 2       
│ │ │ │  000128a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000128b0: 2020 2020 2020 207c 0a7c 3220 3320 2020         |.|2 3   
│ │ │ │  000128c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000128d0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ -000128e0: 202e 3030 3031 3432 3236 3573 2065 6c61   .000142265s ela
│ │ │ │ +000128e0: 202e 3030 3031 3636 3238 3473 2065 6c61   .000166284s ela
│ │ │ │  000128f0: 7073 6564 2020 2020 2020 2020 207c 0a7c  psed         |.|
│ │ │ │  00012900: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  00012910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012920: 7c0a 7c32 2033 2020 2020 2020 2020 2020  |.|2 3          
│ │ │ │  00012930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012940: 2020 207c 0a7c 202d 2d20 2e30 3030 3133     |.| -- .00013
│ │ │ │ -00012950: 3036 3534 7320 656c 6170 7365 6420 2020  0654s elapsed   
│ │ │ │ +00012940: 2020 207c 0a7c 202d 2d20 2e30 3030 3137     |.| -- .00017
│ │ │ │ +00012950: 3833 3533 7320 656c 6170 7365 6420 2020  8353s elapsed   
│ │ │ │  00012960: 2020 2020 2020 7c0a 7c20 3220 2020 2020        |.| 2     
│ │ │ │  00012970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012980: 2020 2020 2020 2020 207c 0a7c 3220 2020           |.|2   
│ │ │ │  00012990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000129a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000129b0: 2d2d 202e 3030 3031 3332 3131 3673 2065  -- .000132116s e
│ │ │ │ +000129b0: 2d2d 202e 3030 3031 3637 3438 3673 2065  -- .000167486s e
│ │ │ │  000129c0: 6c61 7073 6564 2020 2020 2020 2020 207c  lapsed         |
│ │ │ │  000129d0: 0a7c 2032 2020 2020 2020 2020 2020 2020  .| 2            
│ │ │ │  000129e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000129f0: 2020 7c0a 7c32 2020 2020 2020 2020 2020    |.|2          
│ │ │ │  00012a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012a10: 2020 2020 207c 0a7c 202d 2d20 2e30 3030       |.| -- .000
│ │ │ │ -00012a20: 3131 3733 3038 7320 656c 6170 7365 6420  117308s elapsed 
│ │ │ │ +00012a20: 3138 3430 3037 7320 656c 6170 7365 6420  184007s elapsed 
│ │ │ │  00012a30: 2020 2020 2020 2020 7c0a 7c32 2020 2020          |.|2    
│ │ │ │  00012a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012a50: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -00012a60: 2d20 2e30 3030 3131 3937 3334 7320 656c  - .000119734s el
│ │ │ │ -00012a70: 6170 7365 6420 2020 2020 2020 2020 7c0a  apsed         |.
│ │ │ │ +00012a60: 2d20 2e30 3030 3136 3239 7320 656c 6170  - .0001629s elap
│ │ │ │ +00012a70: 7365 6420 2020 2020 2020 2020 2020 7c0a  sed           |.
│ │ │ │  00012a80: 7c31 2020 2020 2020 2020 2020 2020 2020  |1              
│ │ │ │  00012a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00012aa0: 207c 0a7c 202d 2d20 2e30 3030 3132 3634   |.| -- .0001264
│ │ │ │ -00012ab0: 3035 7320 656c 6170 7365 6420 2020 2020  05s elapsed     
│ │ │ │ +00012aa0: 207c 0a7c 202d 2d20 2e30 3030 3134 3834   |.| -- .0001484
│ │ │ │ +00012ab0: 3932 7320 656c 6170 7365 6420 2020 2020  92s elapsed     
│ │ │ │  00012ac0: 2020 2020 7c0a 7c31 2020 2020 2020 2020      |.|1        
│ │ │ │  00012ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012ae0: 2020 2020 2020 207c 0a7c 202d 2d20 2e30         |.| -- .0
│ │ │ │ -00012af0: 3030 3132 3734 3338 7320 656c 6170 7365  00127438s elapse
│ │ │ │ +00012af0: 3030 3138 3435 3338 7320 656c 6170 7365  00184538s elapse
│ │ │ │  00012b00: 6420 2020 2020 2020 2020 7c0a 7c31 2020  d         |.|1  
│ │ │ │  00012b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012b20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00012b30: 202d 2d20 2e30 3030 3133 3335 3439 7320   -- .000133549s 
│ │ │ │ +00012b30: 202d 2d20 2e30 3030 3136 3233 3136 7320   -- .000162316s 
│ │ │ │  00012b40: 656c 6170 7365 6420 2020 2020 2020 2020  elapsed         
│ │ │ │  00012b50: 7c0a 7c31 2020 2020 2020 2020 2020 2020  |.|1            
│ │ │ │  00012b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012b70: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00012b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012b90: 2020 2020 2020 7c0a 7c6f 3220 3d20 3331        |.|o2 = 31
│ │ │ │  00012ba0: 3331 3033 3131 3538 3738 3420 2020 2020  31031158784     
│ │ │ │ @@ -4930,26 +4930,26 @@
│ │ │ │  00013410: 2020 2020 7c0a 7c6f 3120 3a20 5365 7175      |.|o1 : Sequ
│ │ │ │  00013420: 656e 6365 2020 2020 2020 2020 2020 2020  ence            
│ │ │ │  00013430: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  00013440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013450: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 636f  ------+.|i2 : co
│ │ │ │  00013460: 6d70 7574 6542 6f75 6e64 2033 2020 2020  mputeBound 3    
│ │ │ │  00013470: 2020 2020 2020 207c 0a7c 202d 2d20 2e31         |.| -- .1
│ │ │ │ -00013480: 3933 3434 3773 2065 6c61 7073 6564 2020  93447s elapsed  
│ │ │ │ +00013480: 3530 3833 3273 2065 6c61 7073 6564 2020  50832s elapsed  
│ │ │ │  00013490: 2020 2020 2020 2020 7c0a 7c20 2d2d 202e          |.| -- .
│ │ │ │ -000134a0: 3236 3533 3738 7320 656c 6170 7365 6420  265378s elapsed 
│ │ │ │ +000134a0: 3139 3236 3634 7320 656c 6170 7365 6420  192664s elapsed 
│ │ │ │  000134b0: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ -000134c0: 2e32 3131 3736 3173 2065 6c61 7073 6564  .211761s elapsed
│ │ │ │ +000134c0: 2e31 3635 3033 3373 2065 6c61 7073 6564  .165033s elapsed
│ │ │ │  000134d0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ -000134e0: 202e 3238 3431 3735 7320 656c 6170 7365   .284175s elapse
│ │ │ │ +000134e0: 202e 3138 3537 3533 7320 656c 6170 7365   .185753s elapse
│ │ │ │  000134f0: 6420 2020 2020 2020 2020 207c 0a7c 202d  d          |.| -
│ │ │ │ -00013500: 2d20 2e33 3232 3233 3873 2065 6c61 7073  - .322238s elaps
│ │ │ │ +00013500: 2d20 2e32 3232 3031 3473 2065 6c61 7073  - .222014s elaps
│ │ │ │  00013510: 6564 2020 2020 2020 2020 2020 7c0a 7c20  ed          |.| 
│ │ │ │ -00013520: 2d2d 202e 3431 3031 3635 7320 656c 6170  -- .410165s elap
│ │ │ │ -00013530: 7365 6420 2020 2020 2020 2020 207c 0a7c  sed          |.|
│ │ │ │ +00013520: 2d2d 202e 3234 3134 3473 2065 6c61 7073  -- .24144s elaps
│ │ │ │ +00013530: 6564 2020 2020 2020 2020 2020 207c 0a7c  ed           |.|
│ │ │ │  00013540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013550: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00013560: 7c6f 3220 3d20 3620 2020 2020 2020 2020  |o2 = 6         
│ │ │ │  00013570: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00013580: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00013590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000135a0: 2b0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d  +..See also.====
│ │ │ │ @@ -6949,33 +6949,33 @@
│ │ │ │  0001b240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b260: 2b0a 7c69 3320 3a20 683d 6465 6765 6e65  +.|i3 : h=degene
│ │ │ │  0001b270: 7261 7465 4b33 4265 7474 6954 6162 6c65  rateK3BettiTable
│ │ │ │  0001b280: 7328 612c 622c 6529 2020 2020 2020 2020  s(a,b,e)        
│ │ │ │  0001b290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b2a0: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ -0001b2b0: 2e30 3633 3438 3933 7320 656c 6170 7365  .0634893s elapse
│ │ │ │ +0001b2b0: 2e30 3232 3037 3034 7320 656c 6170 7365  .0220704s elapse
│ │ │ │  0001b2c0: 6420 2020 2020 2020 2020 2020 2020 2020  d               
│ │ │ │  0001b2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b2f0: 2020 7c0a 7c20 2d2d 202e 3030 3635 3734    |.| -- .006574
│ │ │ │ -0001b300: 3333 7320 656c 6170 7365 6420 2020 2020  33s elapsed     
│ │ │ │ +0001b2f0: 2020 7c0a 7c20 2d2d 202e 3030 3737 3435    |.| -- .007745
│ │ │ │ +0001b300: 3233 7320 656c 6170 7365 6420 2020 2020  23s elapsed     
│ │ │ │  0001b310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b330: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -0001b340: 2d20 2e30 3235 3435 3233 7320 656c 6170  - .0254523s elap
│ │ │ │ +0001b340: 2d20 2e30 3236 3835 3934 7320 656c 6170  - .0268594s elap
│ │ │ │  0001b350: 7365 6420 2020 2020 2020 2020 2020 2020  sed             
│ │ │ │  0001b360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b380: 2020 2020 7c0a 7c20 2d2d 202e 3030 3837      |.| -- .0087
│ │ │ │ -0001b390: 3933 3237 7320 656c 6170 7365 6420 2020  9327s elapsed   
│ │ │ │ +0001b380: 2020 2020 7c0a 7c20 2d2d 202e 3031 3034      |.| -- .0104
│ │ │ │ +0001b390: 3732 3873 2065 6c61 7073 6564 2020 2020  728s elapsed    
│ │ │ │  0001b3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b3c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001b3d0: 202d 2d20 2e30 3033 3336 3738 3873 2065   -- .00336788s e
│ │ │ │ +0001b3d0: 202d 2d20 2e30 3034 3032 3033 3373 2065   -- .00402033s e
│ │ │ │  0001b3e0: 6c61 7073 6564 2020 2020 2020 2020 2020  lapsed          
│ │ │ │  0001b3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b410: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001b420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -7131,15 +7131,15 @@
│ │ │ │  0001bda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001bdb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001bdc0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20  --------+.|i5 : 
│ │ │ │  0001bdd0: 656c 6170 7365 6454 696d 6520 543d 206d  elapsedTime T= m
│ │ │ │  0001bde0: 696e 696d 616c 4265 7474 6920 6465 6765  inimalBetti dege
│ │ │ │  0001bdf0: 6e65 7261 7465 4b33 2861 2c62 2c65 2c43  nerateK3(a,b,e,C
│ │ │ │  0001be00: 6861 7261 6374 6572 6973 7469 633d 3e35  haracteristic=>5
│ │ │ │ -0001be10: 297c 0a7c 202d 2d20 2e33 3039 3136 3673  )|.| -- .309166s
│ │ │ │ +0001be10: 297c 0a7c 202d 2d20 2e33 3138 3136 3573  )|.| -- .318165s
│ │ │ │  0001be20: 2065 6c61 7073 6564 2020 2020 2020 2020   elapsed        
│ │ │ │  0001be30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001be40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001be50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0001be60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001be70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001be80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -7277,33 +7277,33 @@
│ │ │ │  0001c6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  0001c6f0: 6938 203a 2068 3d64 6567 656e 6572 6174  i8 : h=degenerat
│ │ │ │  0001c700: 654b 3342 6574 7469 5461 626c 6573 2861  eK3BettiTables(a
│ │ │ │  0001c710: 2c62 2c65 2920 2020 2020 2020 2020 2020  ,b,e)           
│ │ │ │  0001c720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c730: 2020 7c0a 7c20 2d2d 202e 3030 3235 3035    |.| -- .002505
│ │ │ │ -0001c740: 3935 7320 656c 6170 7365 6420 2020 2020  95s elapsed     
│ │ │ │ +0001c730: 2020 7c0a 7c20 2d2d 202e 3030 3330 3230    |.| -- .003020
│ │ │ │ +0001c740: 3631 7320 656c 6170 7365 6420 2020 2020  61s elapsed     
│ │ │ │  0001c750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c770: 2020 2020 2020 207c 0a7c 202d 2d20 2e30         |.| -- .0
│ │ │ │ -0001c780: 3036 3539 3335 3873 2065 6c61 7073 6564  0659358s elapsed
│ │ │ │ +0001c780: 3036 3639 3537 3573 2065 6c61 7073 6564  0669575s elapsed
│ │ │ │  0001c790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c7b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001c7c0: 2d2d 202e 3032 3331 3832 3373 2065 6c61  -- .0231823s ela
│ │ │ │ +0001c7c0: 2d2d 202e 3032 3538 3935 3173 2065 6c61  -- .0258951s ela
│ │ │ │  0001c7d0: 7073 6564 2020 2020 2020 2020 2020 2020  psed            
│ │ │ │  0001c7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c800: 207c 0a7c 202d 2d20 2e30 3130 3230 3139   |.| -- .0102019
│ │ │ │ +0001c800: 207c 0a7c 202d 2d20 2e30 3139 3237 3433   |.| -- .0192743
│ │ │ │  0001c810: 7320 656c 6170 7365 6420 2020 2020 2020  s elapsed       
│ │ │ │  0001c820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c840: 2020 2020 2020 7c0a 7c20 2d2d 202e 3030        |.| -- .00
│ │ │ │ -0001c850: 3337 3735 3733 7320 656c 6170 7365 6420  377573s elapsed 
│ │ │ │ +0001c850: 3433 3837 3031 7320 656c 6170 7365 6420  438701s elapsed 
│ │ │ │  0001c860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c880: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0001c890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -9875,16 +9875,16 @@
│ │ │ │  00026920: 3d20 3420 2020 2020 2020 2020 2020 2020  = 4             
│ │ │ │  00026930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026940: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00026950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026960: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220  ----------+.|i2 
│ │ │ │  00026970: 3a20 2864 312c 6432 293d 7265 736f 6e61  : (d1,d2)=resona
│ │ │ │  00026980: 6e63 6544 6574 2861 2920 2020 2020 2020  nceDet(a)       
│ │ │ │ -00026990: 2020 7c0a 7c20 2d2d 202e 3031 3737 3437    |.| -- .017747
│ │ │ │ -000269a0: 3773 2065 6c61 7073 6564 2020 2020 2020  7s elapsed      
│ │ │ │ +00026990: 2020 7c0a 7c20 2d2d 202e 3031 3833 3373    |.| -- .01833s
│ │ │ │ +000269a0: 2065 6c61 7073 6564 2020 2020 2020 2020   elapsed        
│ │ │ │  000269b0: 2020 2020 2020 2020 2020 7c0a 7c28 6e75            |.|(nu
│ │ │ │  000269c0: 6d62 6572 206f 6620 626c 6f63 6b73 3d20  mber of blocks= 
│ │ │ │  000269d0: 2c20 3138 2920 2020 2020 2020 2020 2020  , 18)           
│ │ │ │  000269e0: 2020 7c0a 7c28 7369 7a65 206f 6620 7468    |.|(size of th
│ │ │ │  000269f0: 6520 6d61 7472 6963 6573 2c20 5461 6c6c  e matrices, Tall
│ │ │ │  00026a00: 797b 3120 3d3e 2034 7d29 7c0a 7c20 2020  y{1 => 4})|.|   
│ │ │ │  00026a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -9898,15 +9898,15 @@
│ │ │ │  00026a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026aa0: 2020 2020 2020 2020 2020 7c0a 7c74 6f74            |.|tot
│ │ │ │  00026ab0: 616c 3a20 3120 3120 2020 2020 2020 2020  al: 1 1         
│ │ │ │  00026ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026ad0: 2020 7c0a 7c20 2020 2037 3a20 3120 3120    |.|    7: 1 1 
│ │ │ │  00026ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026af0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ -00026b00: 202e 3030 3030 3531 3332 3573 2065 6c61   .000051325s ela
│ │ │ │ +00026b00: 202e 3030 3030 3337 3932 3973 2065 6c61   .000037929s ela
│ │ │ │  00026b10: 7073 6564 2020 2020 2020 2020 2020 2020  psed            
│ │ │ │  00026b20: 2020 7c0a 7c28 6520 2928 2d31 2920 2020    |.|(e )(-1)   
│ │ │ │  00026b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026b40: 2020 2020 2020 2020 2020 7c0a 7c20 2031            |.|  1
│ │ │ │  00026b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026b70: 2020 7c0a 7c20 2020 2020 2020 3020 3120    |.|       0 1 
│ │ │ │ @@ -9915,16 +9915,16 @@
│ │ │ │  00026ba0: 616c 3a20 3220 3220 2020 2020 2020 2020  al: 2 2         
│ │ │ │  00026bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026bc0: 2020 7c0a 7c20 2020 2037 3a20 3220 2e20    |.|    7: 2 . 
│ │ │ │  00026bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026be0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00026bf0: 2038 3a20 2e20 3220 2020 2020 2020 2020   8: . 2         
│ │ │ │  00026c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026c10: 2020 7c0a 7c20 2d2d 202e 3030 3030 3935    |.| -- .000095
│ │ │ │ -00026c20: 3633 3973 2065 6c61 7073 6564 2020 2020  639s elapsed    
│ │ │ │ +00026c10: 2020 7c0a 7c20 2d2d 202e 3030 3030 3833    |.| -- .000083
│ │ │ │ +00026c20: 3532 3473 2065 6c61 7073 6564 2020 2020  524s elapsed    
│ │ │ │  00026c30: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00026c40: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  00026c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026c60: 2020 7c0a 7c28 6520 2920 2865 2029 282d    |.|(e ) (e )(-
│ │ │ │  00026c70: 3129 2020 2020 2020 2020 2020 2020 2020  1)              
│ │ │ │  00026c80: 2020 2020 2020 2020 2020 7c0a 7c20 2031            |.|  1
│ │ │ │  00026c90: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ @@ -9938,15 +9938,15 @@
│ │ │ │  00026d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026d20: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00026d30: 2038 3a20 2e20 2e20 2020 2020 2020 2020   8: . .         
│ │ │ │  00026d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026d50: 2020 7c0a 7c20 2020 2039 3a20 2e20 3220    |.|    9: . 2 
│ │ │ │  00026d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026d70: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ -00026d80: 202e 3030 3030 3736 3734 3373 2065 6c61   .000076743s ela
│ │ │ │ +00026d80: 202e 3030 3030 3934 3938 3173 2065 6c61   .000094981s ela
│ │ │ │  00026d90: 7073 6564 2020 2020 2020 2020 2020 2020  psed            
│ │ │ │  00026da0: 2020 7c0a 7c20 2020 2032 2020 2020 3220    |.|    2    2 
│ │ │ │  00026db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026dc0: 2020 2020 2020 2020 2020 7c0a 7c28 6520            |.|(e 
│ │ │ │  00026dd0: 2920 2865 2029 2020 2020 2020 2020 2020  ) (e )          
│ │ │ │  00026de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026df0: 2020 7c0a 7c20 2031 2020 2020 3220 2020    |.|  1    2   
│ │ │ │ @@ -9963,15 +9963,15 @@
│ │ │ │  00026ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026eb0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00026ec0: 2039 3a20 2e20 3120 2020 2020 2020 2020   9: . 1         
│ │ │ │  00026ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026ee0: 2020 7c0a 7c20 2020 3130 3a20 2e20 3220    |.|   10: . 2 
│ │ │ │  00026ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026f00: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ -00026f10: 202e 3030 3030 3739 3936 3973 2065 6c61   .000079969s ela
│ │ │ │ +00026f10: 202e 3030 3031 3033 3930 3773 2065 6c61   .000103907s ela
│ │ │ │  00026f20: 7073 6564 2020 2020 2020 2020 2020 2020  psed            
│ │ │ │  00026f30: 2020 7c0a 7c20 2020 2032 2020 2020 3420    |.|    2    4 
│ │ │ │  00026f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026f50: 2020 2020 2020 2020 2020 7c0a 7c28 6520            |.|(e 
│ │ │ │  00026f60: 2920 2865 2029 2028 2d33 2920 2020 2020  ) (e ) (-3)     
│ │ │ │  00026f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026f80: 2020 7c0a 7c20 2031 2020 2020 3220 2020    |.|  1    2   
│ │ │ │ @@ -9990,16 +9990,16 @@
│ │ │ │  00027050: 2039 3a20 3220 3220 2020 2020 2020 2020   9: 2 2         
│ │ │ │  00027060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027070: 2020 7c0a 7c20 2020 3130 3a20 2e20 3120    |.|   10: . 1 
│ │ │ │  00027080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027090: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000270a0: 3131 3a20 2e20 3120 2020 2020 2020 2020  11: . 1         
│ │ │ │  000270b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000270c0: 2020 7c0a 7c20 2d2d 202e 3030 3030 3833    |.| -- .000083
│ │ │ │ -000270d0: 3632 3573 2065 6c61 7073 6564 2020 2020  625s elapsed    
│ │ │ │ +000270c0: 2020 7c0a 7c20 2d2d 202e 3030 3031 3036    |.| -- .000106
│ │ │ │ +000270d0: 3632 3273 2065 6c61 7073 6564 2020 2020  622s elapsed    
│ │ │ │  000270e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000270f0: 2032 2020 2020 3420 2020 2020 2020 2020   2    4         
│ │ │ │  00027100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027110: 2020 7c0a 7c28 6520 2920 2865 2029 2028    |.|(e ) (e ) (
│ │ │ │  00027120: 3329 2020 2020 2020 2020 2020 2020 2020  3)              
│ │ │ │  00027130: 2020 2020 2020 2020 2020 7c0a 7c20 2031            |.|  1
│ │ │ │  00027140: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ @@ -10015,16 +10015,16 @@
│ │ │ │  000271e0: 2039 3a20 3220 3120 2020 2020 2020 2020   9: 2 1         
│ │ │ │  000271f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027200: 2020 7c0a 7c20 2020 3130 3a20 3120 3220    |.|   10: 1 2 
│ │ │ │  00027210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027220: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00027230: 3131 3a20 2e20 3120 2020 2020 2020 2020  11: . 1         
│ │ │ │  00027240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027250: 2020 7c0a 7c20 2d2d 202e 3030 3030 3933    |.| -- .000093
│ │ │ │ -00027260: 3433 3473 2065 6c61 7073 6564 2020 2020  434s elapsed    
│ │ │ │ +00027250: 2020 7c0a 7c20 2d2d 202e 3030 3030 3839    |.| -- .000089
│ │ │ │ +00027260: 3837 3373 2065 6c61 7073 6564 2020 2020  873s elapsed    
│ │ │ │  00027270: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00027280: 2032 2020 2020 3320 2020 2020 2020 2020   2    3         
│ │ │ │  00027290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000272a0: 2020 7c0a 7c28 6520 2920 2865 2029 2028    |.|(e ) (e ) (
│ │ │ │  000272b0: 3329 2020 2020 2020 2020 2020 2020 2020  3)              
│ │ │ │  000272c0: 2020 2020 2020 2020 2020 7c0a 7c20 2031            |.|  1
│ │ │ │  000272d0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ @@ -10033,15 +10033,15 @@
│ │ │ │  00027300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027310: 2020 2020 2020 2020 2020 7c0a 7c74 6f74            |.|tot
│ │ │ │  00027320: 616c 3a20 3120 3120 2020 2020 2020 2020  al: 1 1         
│ │ │ │  00027330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027340: 2020 7c0a 7c20 2020 2039 3a20 3120 3120    |.|    9: 1 1 
│ │ │ │  00027350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027360: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ -00027370: 202e 3030 3030 3234 3332 3673 2065 6c61   .000024326s ela
│ │ │ │ +00027370: 202e 3030 3030 3237 3235 3673 2065 6c61   .000027256s ela
│ │ │ │  00027380: 7073 6564 2020 2020 2020 2020 2020 2020  psed            
│ │ │ │  00027390: 2020 7c0a 7c28 6520 2928 2d31 2920 2020    |.|(e )(-1)   
│ │ │ │  000273a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000273b0: 2020 2020 2020 2020 2020 7c0a 7c20 2031            |.|  1
│ │ │ │  000273c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000273d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000273e0: 2020 7c0a 7c20 2020 2020 2020 3020 3120    |.|       0 1 
│ │ │ │ @@ -10050,16 +10050,16 @@
│ │ │ │  00027410: 616c 3a20 3220 3220 2020 2020 2020 2020  al: 2 2         
│ │ │ │  00027420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027430: 2020 7c0a 7c20 2020 2039 3a20 3120 3120    |.|    9: 1 1 
│ │ │ │  00027440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027450: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00027460: 3130 3a20 3120 3120 2020 2020 2020 2020  10: 1 1         
│ │ │ │  00027470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027480: 2020 7c0a 7c20 2d2d 202e 3030 3030 3637    |.| -- .000067
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│ │ │ │  000274a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000274b0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  000274c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000274d0: 2020 7c0a 7c28 6520 2920 2020 2020 2020    |.|(e )       
│ │ │ │  000274e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000274f0: 2020 2020 2020 2020 2020 7c0a 7c20 2031            |.|  1
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│ │ │ │  00027630: 2020 2020 2020 2020 2020 7c0a 7c28 6520            |.|(e 
│ │ │ │  00027640: 2920 2865 2029 2028 2d31 2920 2020 2020  ) (e ) (-1)     
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│ │ │ │  00027770: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ -00027780: 202e 3030 3030 3935 3936 3973 2065 6c61   .000095969s ela
│ │ │ │ +00027780: 202e 3030 3031 3031 3436 3573 2065 6c61   .000101465s ela
│ │ │ │  00027790: 7073 6564 2020 2020 2020 2020 2020 2020  psed            
│ │ │ │  000277a0: 2020 7c0a 7c20 2020 2032 2020 2020 3320    |.|    2    3 
│ │ │ │  000277b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000277c0: 2020 2020 2020 2020 2020 7c0a 7c28 6520            |.|(e 
│ │ │ │  000277d0: 2920 2865 2029 2028 3329 2020 2020 2020  ) (e ) (3)      
│ │ │ │  000277e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000277f0: 2020 7c0a 7c20 2031 2020 2020 3220 2020    |.|  1    2   
│ │ │ │ @@ -10125,16 +10125,16 @@
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│ │ │ │  00027920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027930: 2020 7c0a 7c20 2d2d 202e 3030 3030 3930    |.| -- .000090
│ │ │ │ -00027940: 3438 3873 2065 6c61 7073 6564 2020 2020  488s elapsed    
│ │ │ │ +00027930: 2020 7c0a 7c20 2d2d 202e 3030 3031 3030    |.| -- .000100
│ │ │ │ +00027940: 3139 3373 2065 6c61 7073 6564 2020 2020  193s elapsed    
│ │ │ │  00027950: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00027960: 2032 2020 2020 3420 2020 2020 2020 2020   2    4         
│ │ │ │  00027970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027980: 2020 7c0a 7c28 6520 2920 2865 2029 2028    |.|(e ) (e ) (
│ │ │ │  00027990: 3329 2020 2020 2020 2020 2020 2020 2020  3)              
│ │ │ │  000279a0: 2020 2020 2020 2020 2020 7c0a 7c20 2031            |.|  1
│ │ │ │  000279b0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ @@ -10148,15 +10148,15 @@
│ │ │ │  00027a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027a40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
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│ │ │ │  00027a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027a70: 2020 7c0a 7c20 2020 3131 3a20 3120 3220    |.|   11: 1 2 
│ │ │ │  00027a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027a90: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ -00027aa0: 202e 3030 3030 3831 3439 3273 2065 6c61   .000081492s ela
│ │ │ │ +00027aa0: 202e 3030 3030 3833 3738 3273 2065 6c61   .000083782s ela
│ │ │ │  00027ab0: 7073 6564 2020 2020 2020 2020 2020 2020  psed            
│ │ │ │  00027ac0: 2020 7c0a 7c20 2020 2032 2020 2020 3220    |.|    2    2 
│ │ │ │  00027ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027ae0: 2020 2020 2020 2020 2020 7c0a 7c28 6520            |.|(e 
│ │ │ │  00027af0: 2920 2865 2029 2028 2d31 2920 2020 2020  ) (e ) (-1)     
│ │ │ │  00027b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027b10: 2020 7c0a 7c20 2031 2020 2020 3220 2020    |.|  1    2   
│ │ │ │ @@ -10173,16 +10173,16 @@
│ │ │ │  00027bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027bd0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
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│ │ │ │  00027bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027c00: 2020 7c0a 7c20 2020 3133 3a20 2e20 3220    |.|   13: . 2 
│ │ │ │  00027c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027c20: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ -00027c30: 202e 3030 3030 3936 3132 7320 656c 6170   .00009612s elap
│ │ │ │ -00027c40: 7365 6420 2020 2020 2020 2020 2020 2020  sed             
│ │ │ │ +00027c30: 202e 3030 3030 3830 3135 3173 2065 6c61   .000080151s ela
│ │ │ │ +00027c40: 7073 6564 2020 2020 2020 2020 2020 2020  psed            
│ │ │ │  00027c50: 2020 7c0a 7c20 2020 2032 2020 2020 3420    |.|    2    4 
│ │ │ │  00027c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027c70: 2020 2020 2020 2020 2020 7c0a 7c28 6520            |.|(e 
│ │ │ │  00027c80: 2920 2865 2029 2028 3329 2020 2020 2020  ) (e ) (3)      
│ │ │ │  00027c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027ca0: 2020 7c0a 7c20 2031 2020 2020 3220 2020    |.|  1    2   
│ │ │ │  00027cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00027d40: 2020 7c0a 7c20 2020 3131 3a20 3120 3120    |.|   11: 1 1 
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│ │ │ │ -00027d70: 202e 3030 3030 3635 3432 3173 2065 6c61   .000065421s ela
│ │ │ │ -00027d80: 7073 6564 2020 2020 2020 2020 2020 2020  psed            
│ │ │ │ +00027d70: 202e 3030 3030 3730 3273 2065 6c61 7073   .0000702s elaps
│ │ │ │ +00027d80: 6564 2020 2020 2020 2020 2020 2020 2020  ed              
│ │ │ │  00027d90: 2020 7c0a 7c20 2020 2032 2020 2020 2020    |.|    2      
│ │ │ │  00027da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027db0: 2020 2020 2020 2020 2020 7c0a 7c28 6520            |.|(e 
│ │ │ │  00027dc0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  00027dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027de0: 2020 7c0a 7c20 2031 2020 2020 2020 2020    |.|  1        
│ │ │ │  00027df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -10215,16 +10215,16 @@
│ │ │ │  00027e60: 3131 3a20 3220 2e20 2020 2020 2020 2020  11: 2 .         
│ │ │ │  00027e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027e80: 2020 7c0a 7c20 2020 3132 3a20 2e20 2e20    |.|   12: . . 
│ │ │ │  00027e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027ea0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
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│ │ │ │  00027ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027ed0: 2020 7c0a 7c20 2d2d 202e 3030 3030 3737    |.| -- .000077
│ │ │ │ -00027ee0: 3638 3573 2065 6c61 7073 6564 2020 2020  685s elapsed    
│ │ │ │ +00027ed0: 2020 7c0a 7c20 2d2d 202e 3030 3030 3634    |.| -- .000064
│ │ │ │ +00027ee0: 3333 3873 2065 6c61 7073 6564 2020 2020  338s elapsed    
│ │ │ │  00027ef0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00027f00: 2032 2020 2020 3220 2020 2020 2020 2020   2    2         
│ │ │ │  00027f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027f20: 2020 7c0a 7c28 6520 2920 2865 2029 2020    |.|(e ) (e )  
│ │ │ │  00027f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027f40: 2020 2020 2020 2020 2020 7c0a 7c20 2031            |.|  1
│ │ │ │  00027f50: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ @@ -10233,15 +10233,15 @@
│ │ │ │  00027f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027f90: 2020 2020 2020 2020 2020 7c0a 7c74 6f74            |.|tot
│ │ │ │  00027fa0: 616c 3a20 3120 3120 2020 2020 2020 2020  al: 1 1         
│ │ │ │  00027fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027fc0: 2020 7c0a 7c20 2020 3131 3a20 3120 3120    |.|   11: 1 1 
│ │ │ │  00027fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027fe0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ -00027ff0: 202e 3030 3030 3234 3633 3673 2065 6c61   .000024636s ela
│ │ │ │ +00027ff0: 202e 3030 3030 3235 3534 3573 2065 6c61   .000025545s ela
│ │ │ │  00028000: 7073 6564 2020 2020 2020 2020 2020 2020  psed            
│ │ │ │  00028010: 2020 7c0a 7c28 6520 2920 2020 2020 2020    |.|(e )       
│ │ │ │  00028020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028030: 2020 2020 2020 2020 2020 7c0a 7c20 2031            |.|  1
│ │ │ │  00028040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028060: 2020 7c0a 7c20 2020 2020 2020 3020 3120    |.|       0 1 
│ │ │ │ @@ -10250,16 +10250,16 @@
│ │ │ │  00028090: 616c 3a20 3220 3220 2020 2020 2020 2020  al: 2 2         
│ │ │ │  000280a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000280b0: 2020 7c0a 7c20 2020 3132 3a20 3220 2e20    |.|   12: 2 . 
│ │ │ │  000280c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000280d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000280e0: 3133 3a20 2e20 3220 2020 2020 2020 2020  13: . 2         
│ │ │ │  000280f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028100: 2020 7c0a 7c20 2d2d 202e 3030 3030 3736    |.| -- .000076
│ │ │ │ -00028110: 3339 3273 2065 6c61 7073 6564 2020 2020  392s elapsed    
│ │ │ │ +00028100: 2020 7c0a 7c20 2d2d 202e 3030 3030 3639    |.| -- .000069
│ │ │ │ +00028110: 3331 3973 2065 6c61 7073 6564 2020 2020  319s elapsed    
│ │ │ │  00028120: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00028130: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  00028140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028150: 2020 7c0a 7c28 6520 2920 2865 2029 282d    |.|(e ) (e )(-
│ │ │ │  00028160: 3129 2020 2020 2020 2020 2020 2020 2020  1)              
│ │ │ │  00028170: 2020 2020 2020 2020 2020 7c0a 7c20 2031            |.|  1
│ │ │ │  00028180: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ @@ -10268,15 +10268,15 @@
│ │ │ │  000281b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000281c0: 2020 2020 2020 2020 2020 7c0a 7c74 6f74            |.|tot
│ │ │ │  000281d0: 616c 3a20 3120 3120 2020 2020 2020 2020  al: 1 1         
│ │ │ │  000281e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000281f0: 2020 7c0a 7c20 2020 3133 3a20 3120 3120    |.|   13: 1 1 
│ │ │ │  00028200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028210: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ -00028220: 202e 3030 3030 3237 3034 3173 2065 6c61   .000027041s ela
│ │ │ │ +00028220: 202e 3030 3030 3237 3635 3573 2065 6c61   .000027655s ela
│ │ │ │  00028230: 7073 6564 2020 2020 2020 2020 2020 2020  psed            
│ │ │ │  00028240: 2020 7c0a 7c28 6520 2920 2020 2020 2020    |.|(e )       
│ │ │ │  00028250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028260: 2020 2020 2020 2020 2020 7c0a 7c20 2031            |.|  1
│ │ │ │  00028270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028290: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|
│ │ ├── ./usr/share/info/LLLBases.info.gz
│ │ │ ├── LLLBases.info
│ │ │ │ @@ -2531,16 +2531,16 @@
│ │ │ │  00009e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320  ----------+.|i3 
│ │ │ │  00009e40: 3a20 7469 6d65 204c 4c4c 206d 3b20 2020  : time LLL m;   
│ │ │ │  00009e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009e80: 207c 0a7c 202d 2d20 7573 6564 2030 2e30   |.| -- used 0.0
│ │ │ │ -00009e90: 3039 3034 3433 3573 2028 6370 7529 3b20  0904435s (cpu); 
│ │ │ │ -00009ea0: 302e 3030 3930 3430 3031 7320 2874 6872  0.00904001s (thr
│ │ │ │ +00009e90: 3039 3830 3036 3273 2028 6370 7529 3b20  0980062s (cpu); 
│ │ │ │ +00009ea0: 302e 3030 3937 3938 3939 7320 2874 6872  0.00979899s (thr
│ │ │ │  00009eb0: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │  00009ec0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00009ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00009f00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00009f10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ @@ -2557,18 +2557,18 @@
│ │ │ │  00009fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00009fe0: 2d2d 2d2d 2b0a 7c69 3420 3a20 7469 6d65  ----+.|i4 : time
│ │ │ │  00009ff0: 204c 4c4c 286d 2c20 5374 7261 7465 6779   LLL(m, Strategy
│ │ │ │  0000a000: 3d3e 436f 6865 6e45 6e67 696e 6529 3b20  =>CohenEngine); 
│ │ │ │  0000a010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a020: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -0000a030: 2d20 7573 6564 2030 2e30 3237 3338 3836  - used 0.0273886
│ │ │ │ -0000a040: 7320 2863 7075 293b 2030 2e30 3237 3339  s (cpu); 0.02739
│ │ │ │ -0000a050: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ -0000a060: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │ +0000a030: 2d20 7573 6564 2030 2e30 3238 3639 3534  - used 0.0286954
│ │ │ │ +0000a040: 7320 2863 7075 293b 2030 2e30 3238 3730  s (cpu); 0.02870
│ │ │ │ +0000a050: 3231 7320 2874 6872 6561 6429 3b20 3073  21s (thread); 0s
│ │ │ │ +0000a060: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │  0000a070: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0000a080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a0b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0000a0c0: 2020 2020 2020 2020 2020 3530 2020 2020            50    
│ │ │ │  0000a0d0: 2020 2034 3720 2020 2020 2020 2020 2020     47           
│ │ │ │ @@ -2584,17 +2584,17 @@
│ │ │ │  0000a170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  0000a190: 7c69 3520 3a20 7469 6d65 204c 4c4c 286d  |i5 : time LLL(m
│ │ │ │  0000a1a0: 2c20 5374 7261 7465 6779 3d3e 436f 6865  , Strategy=>Cohe
│ │ │ │  0000a1b0: 6e54 6f70 4c65 7665 6c29 3b20 2020 2020  nTopLevel);     
│ │ │ │  0000a1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a1d0: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -0000a1e0: 2030 2e31 3037 3435 3373 2028 6370 7529   0.107453s (cpu)
│ │ │ │ -0000a1f0: 3b20 302e 3130 3734 3373 2028 7468 7265  ; 0.10743s (thre
│ │ │ │ -0000a200: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │ +0000a1e0: 2030 2e31 3232 3035 3573 2028 6370 7529   0.122055s (cpu)
│ │ │ │ +0000a1f0: 3b20 302e 3132 3230 3631 7320 2874 6872  ; 0.122061s (thr
│ │ │ │ +0000a200: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │  0000a210: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0000a220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a260: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0000a270: 2020 2020 3530 2020 2020 2020 2034 3720      50       47 
│ │ │ │ @@ -2610,17 +2610,17 @@
│ │ │ │  0000a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a330: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │  0000a340: 7469 6d65 204c 4c4c 286d 2c20 5374 7261  time LLL(m, Stra
│ │ │ │  0000a350: 7465 6779 3d3e 7b47 6976 656e 732c 5265  tegy=>{Givens,Re
│ │ │ │  0000a360: 616c 4650 7d29 3b20 2020 2020 2020 2020  alFP});         
│ │ │ │  0000a370: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000a380: 0a7c 202d 2d20 7573 6564 2030 2e30 3131  .| -- used 0.011
│ │ │ │ -0000a390: 3836 3973 2028 6370 7529 3b20 302e 3031  869s (cpu); 0.01
│ │ │ │ -0000a3a0: 3138 3639 3373 2028 7468 7265 6164 293b  18693s (thread);
│ │ │ │ +0000a380: 0a7c 202d 2d20 7573 6564 2030 2e30 3133  .| -- used 0.013
│ │ │ │ +0000a390: 3036 3334 7320 2863 7075 293b 2030 2e30  0634s (cpu); 0.0
│ │ │ │ +0000a3a0: 3133 3036 3873 2028 7468 7265 6164 293b  13068s (thread);
│ │ │ │  0000a3b0: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │  0000a3c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0000a3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a400: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0000a410: 2020 2020 2020 2020 2020 2020 2020 3530                50
│ │ │ │ @@ -2637,16 +2637,16 @@
│ │ │ │  0000a4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a4e0: 2d2d 2b0a 7c69 3720 3a20 7469 6d65 204c  --+.|i7 : time L
│ │ │ │  0000a4f0: 4c4c 286d 2c20 5374 7261 7465 6779 3d3e  LL(m, Strategy=>
│ │ │ │  0000a500: 7b47 6976 656e 732c 5265 616c 5150 7d29  {Givens,RealQP})
│ │ │ │  0000a510: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │  0000a520: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ -0000a530: 7573 6564 2030 2e30 3438 3037 3033 7320  used 0.0480703s 
│ │ │ │ -0000a540: 2863 7075 293b 2030 2e30 3438 3037 3432  (cpu); 0.0480742
│ │ │ │ +0000a530: 7573 6564 2030 2e30 3632 3333 3135 7320  used 0.0623315s 
│ │ │ │ +0000a540: 2863 7075 293b 2030 2e30 3632 3136 3532  (cpu); 0.0621652
│ │ │ │  0000a550: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │  0000a560: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │  0000a570: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0000a580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a5b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ @@ -2664,16 +2664,16 @@
│ │ │ │  0000a670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  0000a690: 3820 3a20 7469 6d65 204c 4c4c 286d 2c20  8 : time LLL(m, 
│ │ │ │  0000a6a0: 5374 7261 7465 6779 3d3e 7b47 6976 656e  Strategy=>{Given
│ │ │ │  0000a6b0: 732c 5265 616c 5844 7d29 3b20 2020 2020  s,RealXD});     
│ │ │ │  0000a6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a6d0: 2020 207c 0a7c 202d 2d20 7573 6564 2030     |.| -- used 0
│ │ │ │ -0000a6e0: 2e30 3539 3038 3138 7320 2863 7075 293b  .0590818s (cpu);
│ │ │ │ -0000a6f0: 2030 2e30 3539 3038 3139 7320 2874 6872   0.0590819s (thr
│ │ │ │ +0000a6e0: 2e30 3635 3430 3631 7320 2863 7075 293b  .0654061s (cpu);
│ │ │ │ +0000a6f0: 2030 2e30 3635 3430 3631 7320 2874 6872   0.0654061s (thr
│ │ │ │  0000a700: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │  0000a710: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0000a720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a760: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ @@ -2690,17 +2690,17 @@
│ │ │ │  0000a810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a830: 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20 7469  ------+.|i9 : ti
│ │ │ │  0000a840: 6d65 204c 4c4c 286d 2c20 5374 7261 7465  me LLL(m, Strate
│ │ │ │  0000a850: 6779 3d3e 7b47 6976 656e 732c 5265 616c  gy=>{Givens,Real
│ │ │ │  0000a860: 5252 7d29 3b20 2020 2020 2020 2020 2020  RR});           
│ │ │ │  0000a870: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0000a880: 202d 2d20 7573 6564 2030 2e33 3437 3039   -- used 0.34709
│ │ │ │ -0000a890: 3273 2028 6370 7529 3b20 302e 3334 3730  2s (cpu); 0.3470
│ │ │ │ -0000a8a0: 3932 7320 2874 6872 6561 6429 3b20 3073  92s (thread); 0s
│ │ │ │ +0000a880: 202d 2d20 7573 6564 2030 2e33 3434 3330   -- used 0.34430
│ │ │ │ +0000a890: 3573 2028 6370 7529 3b20 302e 3334 3433  5s (cpu); 0.3443
│ │ │ │ +0000a8a0: 3131 7320 2874 6872 6561 6429 3b20 3073  11s (thread); 0s
│ │ │ │  0000a8b0: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │  0000a8c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0000a8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a900: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0000a910: 2020 2020 2020 2020 2020 2020 3530 2020              50  
│ │ │ │ @@ -2717,16 +2717,16 @@
│ │ │ │  0000a9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a9e0: 2b0a 7c69 3130 203a 2074 696d 6520 4c4c  +.|i10 : time LL
│ │ │ │  0000a9f0: 4c28 6d2c 2053 7472 6174 6567 793d 3e7b  L(m, Strategy=>{
│ │ │ │  0000aa00: 424b 5a2c 4769 7665 6e73 2c52 6561 6c51  BKZ,Givens,RealQ
│ │ │ │  0000aa10: 507d 293b 2020 2020 2020 2020 2020 2020  P});            
│ │ │ │  0000aa20: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -0000aa30: 6564 2030 2e31 3133 3239 3873 2028 6370  ed 0.113298s (cp
│ │ │ │ -0000aa40: 7529 3b20 302e 3131 3333 3031 7320 2874  u); 0.113301s (t
│ │ │ │ +0000aa30: 6564 2030 2e31 3535 3331 3773 2028 6370  ed 0.155317s (cp
│ │ │ │ +0000aa40: 7529 3b20 302e 3135 3533 3232 7320 2874  u); 0.155322s (t
│ │ │ │  0000aa50: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │  0000aa60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  0000aa70: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0000aa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000aa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000aaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000aab0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|
│ │ ├── ./usr/share/info/LatticePolytopes.info.gz
│ │ │ ├── LatticePolytopes.info
│ │ │ │ @@ -1196,32 +1196,32 @@
│ │ │ │  00004ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004ad0: 2d2b 0a7c 6936 203a 2074 696d 6520 6172  -+.|i6 : time ar
│ │ │ │  00004ae0: 6549 736f 6d6f 7270 6869 6328 502c 5029  eIsomorphic(P,P)
│ │ │ │  00004af0: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │  00004b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004b20: 207c 0a7c 202d 2d20 7573 6564 2030 2e35   |.| -- used 0.5
│ │ │ │ -00004b30: 3933 3939 3273 2028 6370 7529 3b20 302e  93992s (cpu); 0.
│ │ │ │ -00004b40: 3434 3132 3135 7320 2874 6872 6561 6429  441215s (thread)
│ │ │ │ -00004b50: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ +00004b20: 207c 0a7c 202d 2d20 7573 6564 2031 2e30   |.| -- used 1.0
│ │ │ │ +00004b30: 3834 3034 7320 2863 7075 293b 2030 2e35  8404s (cpu); 0.5
│ │ │ │ +00004b40: 3338 3736 3373 2028 7468 7265 6164 293b  38763s (thread);
│ │ │ │ +00004b50: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │  00004b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004b70: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00004b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004bc0: 2d2b 0a7c 6937 203a 2074 696d 6520 6172  -+.|i7 : time ar
│ │ │ │  00004bd0: 6549 736f 6d6f 7270 6869 6328 502c 502c  eIsomorphic(P,P,
│ │ │ │  00004be0: 736d 6f6f 7468 5465 7374 3d3e 6661 6c73  smoothTest=>fals
│ │ │ │  00004bf0: 6529 3b20 2020 2020 2020 2020 2020 2020  e);             
│ │ │ │  00004c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00004c10: 207c 0a7c 202d 2d20 7573 6564 2030 2e34   |.| -- used 0.4
│ │ │ │ -00004c20: 3439 3531 3573 2028 6370 7529 3b20 302e  49515s (cpu); 0.
│ │ │ │ -00004c30: 3239 3239 3138 7320 2874 6872 6561 6429  292918s (thread)
│ │ │ │ +00004c10: 207c 0a7c 202d 2d20 7573 6564 2030 2e39   |.| -- used 0.9
│ │ │ │ +00004c20: 3638 3236 3673 2028 6370 7529 3b20 302e  68266s (cpu); 0.
│ │ │ │ +00004c30: 3337 3730 3731 7320 2874 6872 6561 6429  377071s (thread)
│ │ │ │  00004c40: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │  00004c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004c60: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00004c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ ├── ./usr/share/info/LinearTruncations.info.gz
│ │ │ ├── LinearTruncations.info
│ │ │ │ @@ -2197,16 +2197,16 @@
│ │ │ │  00008940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00008950: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a  ---------+.|i6 :
│ │ │ │  00008960: 2065 6c61 7073 6564 5469 6d65 2066 696e   elapsedTime fin
│ │ │ │  00008970: 6452 6567 696f 6e28 7b7b 302c 307d 2c7b  dRegion({{0,0},{
│ │ │ │  00008980: 342c 347d 7d2c 4d2c 6629 2020 2020 2020  4,4}},M,f)      
│ │ │ │  00008990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000089a0: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ -000089b0: 2e31 3239 3639 3973 2065 6c61 7073 6564  .129699s elapsed
│ │ │ │ -000089c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000089b0: 2e30 3831 3933 3939 7320 656c 6170 7365  .0819399s elapse
│ │ │ │ +000089c0: 6420 2020 2020 2020 2020 2020 2020 2020  d               
│ │ │ │  000089d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000089e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000089f0: 2020 2020 2020 2020 207c 0a7c 2020 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                  
│ │ │ │ @@ -2232,15 +2232,15 @@
│ │ │ │  00008b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00008b80: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a  ---------+.|i7 :
│ │ │ │  00008b90: 2065 6c61 7073 6564 5469 6d65 2066 696e   elapsedTime fin
│ │ │ │  00008ba0: 6452 6567 696f 6e28 7b7b 302c 307d 2c7b  dRegion({{0,0},{
│ │ │ │  00008bb0: 342c 347d 7d2c 4d2c 662c 496e 6e65 723d  4,4}},M,f,Inner=
│ │ │ │  00008bc0: 3e7b 7b31 2c32 7d2c 7b33 2c31 7d7d 2c4f  >{{1,2},{3,1}},O
│ │ │ │  00008bd0: 7574 6572 3d3e 7b7b 317c 0a7c 202d 2d20  uter=>{{1|.| -- 
│ │ │ │ -00008be0: 2e30 3132 3137 3436 7320 656c 6170 7365  .0121746s elapse
│ │ │ │ +00008be0: 2e30 3134 3832 3933 7320 656c 6170 7365  .0148293s elapse
│ │ │ │  00008bf0: 6420 2020 2020 2020 2020 2020 2020 2020  d               
│ │ │ │  00008c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008c20: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00008c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4119,16 +4119,16 @@
│ │ │ │  00010160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010190: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │  000101a0: 656c 6170 7365 6454 696d 6520 6c69 6e65  elapsedTime line
│ │ │ │  000101b0: 6172 5472 756e 6361 7469 6f6e 7328 7b7b  arTruncations({{
│ │ │ │  000101c0: 322c 322c 327d 2c7b 342c 342c 347d 7d2c  2,2,2},{4,4,4}},
│ │ │ │ -000101d0: 204d 297c 0a7c 202d 2d20 342e 3136 3035   M)|.| -- 4.1605
│ │ │ │ -000101e0: 3973 2065 6c61 7073 6564 2020 2020 2020  9s elapsed      
│ │ │ │ +000101d0: 204d 297c 0a7c 202d 2d20 332e 3035 3138   M)|.| -- 3.0518
│ │ │ │ +000101e0: 3873 2065 6c61 7073 6564 2020 2020 2020  8s elapsed      
│ │ │ │  000101f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010200: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00010210: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00010220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010240: 2020 2020 2020 2020 207c 0a7c 6f36 203d           |.|o6 =
│ │ │ │  00010250: 207b 7b34 2c20 332c 2033 7d2c 207b 342c   {{4, 3, 3}, {4,
│ │ │ │ @@ -4145,15 +4145,15 @@
│ │ │ │  00010300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010330: 2d2d 2d2d 2d2b 0a7c 6937 203a 2065 6c61  -----+.|i7 : ela
│ │ │ │  00010340: 7073 6564 5469 6d65 206c 696e 6561 7254  psedTime linearT
│ │ │ │  00010350: 7275 6e63 6174 696f 6e73 426f 756e 6420  runcationsBound 
│ │ │ │  00010360: 4d20 2020 2020 2020 2020 2020 2020 2020  M               
│ │ │ │ -00010370: 7c0a 7c20 2d2d 202e 3032 3636 3239 3573  |.| -- .0266295s
│ │ │ │ +00010370: 7c0a 7c20 2d2d 202e 3032 3735 3337 3473  |.| -- .0275374s
│ │ │ │  00010380: 2065 6c61 7073 6564 2020 2020 2020 2020   elapsed        
│ │ │ │  00010390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000103a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000103b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000103c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000103d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000103e0: 2020 2020 2020 7c0a 7c6f 3720 3d20 7b7b        |.|o7 = {{
│ │ ├── ./usr/share/info/LocalRings.info.gz
│ │ │ ├── LocalRings.info
│ │ │ │ @@ -2607,16 +2607,16 @@
│ │ │ │  0000a2e0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  0000a2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  0000a320: 6935 203a 2065 6c61 7073 6564 5469 6d65  i5 : elapsedTime
│ │ │ │  0000a330: 2068 696c 6265 7274 5361 6d75 656c 4675   hilbertSamuelFu
│ │ │ │  0000a340: 6e63 7469 6f6e 284d 2c20 302c 2036 2920  nction(M, 0, 6) 
│ │ │ │ -0000a350: 2020 2020 207c 0a7c 202d 2d20 2e32 3338       |.| -- .238
│ │ │ │ -0000a360: 3635 3173 2065 6c61 7073 6564 2020 2020  651s elapsed    
│ │ │ │ +0000a350: 2020 2020 207c 0a7c 202d 2d20 2e31 3939       |.| -- .199
│ │ │ │ +0000a360: 3534 3773 2065 6c61 7073 6564 2020 2020  547s elapsed    
│ │ │ │  0000a370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a380: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0000a390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a3c0: 2020 2020 207c 0a7c 6f35 203d 207b 312c       |.|o5 = {1,
│ │ │ │  0000a3d0: 2033 2c20 362c 2037 2c20 362c 2033 2c20   3, 6, 7, 6, 3, 
│ │ │ │ @@ -2745,15 +2745,15 @@
│ │ │ │  0000ab80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000ab90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  0000aba0: 7c69 3131 203a 2065 6c61 7073 6564 5469  |i11 : elapsedTi
│ │ │ │  0000abb0: 6d65 2068 696c 6265 7274 5361 6d75 656c  me hilbertSamuel
│ │ │ │  0000abc0: 4675 6e63 7469 6f6e 284e 2c20 302c 2035  Function(N, 0, 5
│ │ │ │  0000abd0: 2920 2d2d 206e 2b31 202d 2d20 302e 3032  ) -- n+1 -- 0.02
│ │ │ │  0000abe0: 2073 6563 6f6e 6473 2020 2020 2020 7c0a   seconds      |.
│ │ │ │ -0000abf0: 7c20 2d2d 202e 3037 3534 3737 3473 2065  | -- .0754774s e
│ │ │ │ +0000abf0: 7c20 2d2d 202e 3031 3636 3838 3373 2065  | -- .0166883s e
│ │ │ │  0000ac00: 6c61 7073 6564 2020 2020 2020 2020 2020  lapsed          
│ │ │ │  0000ac10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ac20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ac30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  0000ac40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0000ac50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ac60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -2780,15 +2780,15 @@
│ │ │ │  0000adb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000adc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  0000add0: 7c69 3132 203a 2065 6c61 7073 6564 5469  |i12 : elapsedTi
│ │ │ │  0000ade0: 6d65 2068 696c 6265 7274 5361 6d75 656c  me hilbertSamuel
│ │ │ │  0000adf0: 4675 6e63 7469 6f6e 2871 2c20 4e2c 2030  Function(q, N, 0
│ │ │ │  0000ae00: 2c20 3529 202d 2d20 3628 6e2b 3129 202d  , 5) -- 6(n+1) -
│ │ │ │  0000ae10: 2d20 302e 3332 2073 6563 6f6e 6473 7c0a  - 0.32 seconds|.
│ │ │ │ -0000ae20: 7c20 2d2d 202e 3338 3134 3032 7320 656c  | -- .381402s el
│ │ │ │ +0000ae20: 7c20 2d2d 202e 3236 3338 3835 7320 656c  | -- .263885s el
│ │ │ │  0000ae30: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │  0000ae40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ae50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ae60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  0000ae70: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0000ae80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ae90: 2020 2020 2020 2020 2020 2020 2020 2020
│ │ ├── ./usr/share/info/Macaulay2Doc.info.gz
│ │ │ ├── Macaulay2Doc.info
│ │ │ │ @@ -4921,16 +4921,16 @@
│ │ │ │  00013380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000133a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3539  ----------+.|i59
│ │ │ │  000133b0: 203a 2074 696d 6520 4320 3d20 7265 736f   : time C = reso
│ │ │ │  000133c0: 6c75 7469 6f6e 204d 2020 2020 2020 2020  lution M        
│ │ │ │  000133d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000133e0: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ -000133f0: 6420 302e 3030 3139 3237 3738 7320 2863  d 0.00192778s (c
│ │ │ │ -00013400: 7075 293b 2030 2e30 3031 3931 3934 3573  pu); 0.00191945s
│ │ │ │ +000133f0: 6420 302e 3030 3139 3530 3531 7320 2863  d 0.00195051s (c
│ │ │ │ +00013400: 7075 293b 2030 2e30 3031 3934 3439 3173  pu); 0.00194491s
│ │ │ │  00013410: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │  00013420: 6329 7c0a 7c20 2020 2020 2020 2020 2020  c)|.|           
│ │ │ │  00013430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013450: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00013460: 7c20 2020 2020 2020 3320 2020 2020 2036  |       3      6
│ │ │ │  00013470: 2020 2020 2020 3135 2020 2020 2020 3138        15      18
│ │ │ │ @@ -24315,17 +24315,17 @@
│ │ │ │  0005efa0: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 2863  ------+.|i3 : (c
│ │ │ │  0005efb0: 203d 2043 6f6d 6d61 6e64 2022 6461 7465   = Command "date
│ │ │ │  0005efc0: 223b 2920 2020 2020 2020 207c 0a2b 2d2d  ";)        |.+--
│ │ │ │  0005efd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005efe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005eff0: 2b0a 7c69 3420 3a20 6320 2020 2020 2020  +.|i4 : c       
│ │ │ │  0005f000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f010: 2020 2020 207c 0a7c 5375 6e20 4465 6320       |.|Sun Dec 
│ │ │ │ -0005f020: 3134 2031 353a 3236 3a35 3620 5554 4320  14 15:26:56 UTC 
│ │ │ │ -0005f030: 3230 3235 2020 2020 2020 7c0a 7c20 2020  2025      |.|   
│ │ │ │ +0005f010: 2020 2020 207c 0a7c 5468 7520 4a61 6e20       |.|Thu Jan 
│ │ │ │ +0005f020: 2031 2031 313a 3032 3a33 3120 5554 4320   1 11:02:31 UTC 
│ │ │ │ +0005f030: 3230 3236 2020 2020 2020 7c0a 7c20 2020  2026      |.|   
│ │ │ │  0005f040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f050: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0005f060: 0a7c 6f34 203d 2030 2020 2020 2020 2020  .|o4 = 0        
│ │ │ │  0005f070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f080: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  0005f090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f0a0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520  ---------+..See 
│ │ │ │ @@ -100453,16 +100453,16 @@
│ │ │ │  00188640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00188650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00188660: 2d2d 2d2d 2d2b 0a7c 6938 203a 2074 696d  -----+.|i8 : tim
│ │ │ │  00188670: 6520 4a20 3d20 7472 756e 6361 7465 2838  e J = truncate(8
│ │ │ │  00188680: 2c20 492c 204d 696e 696d 616c 4765 6e65  , I, MinimalGene
│ │ │ │  00188690: 7261 746f 7273 203d 3e20 6661 6c73 6529  rators => false)
│ │ │ │  001886a0: 3b7c 0a7c 202d 2d20 7573 6564 2030 2e30  ;|.| -- used 0.0
│ │ │ │ -001886b0: 3039 3036 3535 3273 2028 6370 7529 3b20  0906552s (cpu); 
│ │ │ │ -001886c0: 302e 3030 3930 3538 3638 7320 2874 6872  0.00905868s (thr
│ │ │ │ +001886b0: 3035 3238 3239 3673 2028 6370 7529 3b20  0528296s (cpu); 
│ │ │ │ +001886c0: 302e 3030 3532 3737 3736 7320 2874 6872  0.00527776s (thr
│ │ │ │  001886d0: 6561 6429 3b20 3073 2028 6763 297c 0a7c  ead); 0s (gc)|.|
│ │ │ │  001886e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  001886f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00188700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00188710: 2020 2020 2020 2020 207c 0a7c 6f38 203a           |.|o8 :
│ │ │ │  00188720: 2049 6465 616c 206f 6620 5220 2020 2020   Ideal of R     
│ │ │ │  00188730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -100471,17 +100471,17 @@
│ │ │ │  00188760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00188770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00188780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00188790: 2d2b 0a7c 6939 203a 2074 696d 6520 4b20  -+.|i9 : time K 
│ │ │ │  001887a0: 3d20 7472 756e 6361 7465 2838 2c20 492c  = truncate(8, I,
│ │ │ │  001887b0: 204d 696e 696d 616c 4765 6e65 7261 746f   MinimalGenerato
│ │ │ │  001887c0: 7273 203d 3e20 7472 7565 293b 207c 0a7c  rs => true); |.|
│ │ │ │ -001887d0: 202d 2d20 7573 6564 2030 2e30 3739 3436   -- used 0.07946
│ │ │ │ -001887e0: 3634 7320 2863 7075 293b 2030 2e30 3739  64s (cpu); 0.079
│ │ │ │ -001887f0: 3437 3432 7320 2874 6872 6561 6429 3b20  4742s (thread); 
│ │ │ │ +001887d0: 202d 2d20 7573 6564 2030 2e30 3535 3033   -- used 0.05503
│ │ │ │ +001887e0: 3238 7320 2863 7075 293b 2030 2e30 3535  28s (cpu); 0.055
│ │ │ │ +001887f0: 3034 3139 7320 2874 6872 6561 6429 3b20  0419s (thread); 
│ │ │ │  00188800: 3073 2028 6763 2920 207c 0a7c 2020 2020  0s (gc)  |.|    
│ │ │ │  00188810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00188820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00188830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00188840: 2020 2020 207c 0a7c 6f39 203a 2049 6465       |.|o9 : Ide
│ │ │ │  00188850: 616c 206f 6620 5220 2020 2020 2020 2020  al of R         
│ │ │ │  00188860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -101080,15 +101080,15 @@
│ │ │ │  0018ad70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0018ad80: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │  0018ad90: 2d20 636f 6d70 7574 696e 6720 7064 696d  - computing pdim
│ │ │ │  0018ada0: 2720 2020 2020 2020 2020 2020 2020 2020  '               
│ │ │ │  0018adb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0018adc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0018add0: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -0018ade0: 2d20 2e30 3036 3730 3034 3773 2065 6c61  - .00670047s ela
│ │ │ │ +0018ade0: 2d20 2e30 3033 3832 3533 3873 2065 6c61  - .00382538s ela
│ │ │ │  0018adf0: 7073 6564 2020 2020 2020 2020 2020 2020  psed            
│ │ │ │  0018ae00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0018ae10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0018ae20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0018ae30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0018ae40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0018ae50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -101105,15 +101105,15 @@
│ │ │ │  0018af00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0018af10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935  -----------+.|i5
│ │ │ │  0018af20: 203a 2065 6c61 7073 6564 5469 6d65 2070   : elapsedTime p
│ │ │ │  0018af30: 6469 6d27 204d 2020 2020 2020 2020 2020  dim' M          
│ │ │ │  0018af40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0018af50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0018af60: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -0018af70: 2d20 2e30 3030 3030 3135 3133 7320 656c  - .000001513s el
│ │ │ │ +0018af70: 2d20 2e30 3030 3030 3236 3833 7320 656c  - .000002683s el
│ │ │ │  0018af80: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │  0018af90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0018afa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0018afb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0018afc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0018afd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0018afe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -101608,18 +101608,18 @@
│ │ │ │  0018ce70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0018ce80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0018ce90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0018cea0: 2d2d 2d2d 2d2b 0a7c 6932 203a 2074 696d  -----+.|i2 : tim
│ │ │ │  0018ceb0: 6520 6669 6220 3238 2020 2020 2020 2020  e fib 28        
│ │ │ │  0018cec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0018ced0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0018cee0: 207c 0a7c 202d 2d20 7573 6564 2031 2e32   |.| -- used 1.2
│ │ │ │ -0018cef0: 3633 3333 7320 2863 7075 293b 2030 2e37  6333s (cpu); 0.7
│ │ │ │ -0018cf00: 3236 3938 3773 2028 7468 7265 6164 293b  26987s (thread);
│ │ │ │ -0018cf10: 2030 7320 2867 6329 2020 2020 207c 0a7c   0s (gc)     |.|
│ │ │ │ +0018cee0: 207c 0a7c 202d 2d20 7573 6564 2030 2e37   |.| -- used 0.7
│ │ │ │ +0018cef0: 3534 3736 3273 2028 6370 7529 3b20 302e  54762s (cpu); 0.
│ │ │ │ +0018cf00: 3537 3736 3934 7320 2874 6872 6561 6429  577694s (thread)
│ │ │ │ +0018cf10: 3b20 3073 2028 6763 2920 2020 207c 0a7c  ; 0s (gc)    |.|
│ │ │ │  0018cf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0018cf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0018cf40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0018cf50: 2020 2020 2020 2020 207c 0a7c 6f32 203d           |.|o2 =
│ │ │ │  0018cf60: 2035 3134 3232 3920 2020 2020 2020 2020   514229         
│ │ │ │  0018cf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0018cf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -101650,16 +101650,16 @@
│ │ │ │  0018d110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0018d120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0018d130: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a  ---------+.|i4 :
│ │ │ │  0018d140: 2074 696d 6520 6669 6220 3238 2020 2020   time fib 28    
│ │ │ │  0018d150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0018d160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0018d170: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -0018d180: 2037 2e35 3534 3265 2d30 3573 2028 6370   7.5542e-05s (cp
│ │ │ │ -0018d190: 7529 3b20 372e 3438 3031 652d 3035 7320  u); 7.4801e-05s 
│ │ │ │ +0018d180: 2038 2e32 3935 3565 2d30 3573 2028 6370   8.2955e-05s (cp
│ │ │ │ +0018d190: 7529 3b20 382e 3137 3739 652d 3035 7320  u); 8.1779e-05s 
│ │ │ │  0018d1a0: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │  0018d1b0: 297c 0a7c 2020 2020 2020 2020 2020 2020  )|.|            
│ │ │ │  0018d1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0018d1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0018d1e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0018d1f0: 6f34 203d 2035 3134 3232 3920 2020 2020  o4 = 514229     
│ │ │ │  0018d200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -101668,17 +101668,17 @@
│ │ │ │  0018d230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0018d240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0018d250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0018d260: 2d2d 2d2d 2d2b 0a7c 6935 203a 2074 696d  -----+.|i5 : tim
│ │ │ │  0018d270: 6520 6669 6220 3238 2020 2020 2020 2020  e fib 28        
│ │ │ │  0018d280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0018d290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0018d2a0: 207c 0a7c 202d 2d20 7573 6564 2033 2e39   |.| -- used 3.9
│ │ │ │ -0018d2b0: 3837 652d 3036 7320 2863 7075 293b 2033  87e-06s (cpu); 3
│ │ │ │ -0018d2c0: 2e36 3237 652d 3036 7320 2874 6872 6561  .627e-06s (threa
│ │ │ │ +0018d2a0: 207c 0a7c 202d 2d20 7573 6564 2033 2e36   |.| -- used 3.6
│ │ │ │ +0018d2b0: 3936 652d 3036 7320 2863 7075 293b 2033  96e-06s (cpu); 3
│ │ │ │ +0018d2c0: 2e34 3437 652d 3036 7320 2874 6872 6561  .447e-06s (threa
│ │ │ │  0018d2d0: 6429 3b20 3073 2028 6763 2920 207c 0a7c  d); 0s (gc)  |.|
│ │ │ │  0018d2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0018d2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0018d300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0018d310: 2020 2020 2020 2020 207c 0a7c 6f35 203d           |.|o5 =
│ │ │ │  0018d320: 2035 3134 3232 3920 2020 2020 2020 2020   514229         
│ │ │ │  0018d330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -104209,15 +104209,15 @@
│ │ │ │  00197100: 2020 206c 696e 654e 756d 6265 7220 3d3e     lineNumber =>
│ │ │ │  00197110: 2032 2020 2020 2020 2020 2020 2020 7c0a   2            |.
│ │ │ │  00197120: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00197130: 6c6f 6164 4465 7074 6820 3d3e 2033 2020  loadDepth => 3  
│ │ │ │  00197140: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00197150: 2020 2020 2020 2020 2020 2020 206d 6178               max
│ │ │ │  00197160: 416c 6c6f 7761 626c 6554 6872 6561 6473  AllowableThreads
│ │ │ │ -00197170: 203d 3e20 3720 2020 7c0a 7c20 2020 2020   => 7   |.|     
│ │ │ │ +00197170: 203d 3e20 3137 2020 7c0a 7c20 2020 2020   => 17  |.|     
│ │ │ │  00197180: 2020 2020 2020 2020 2020 6d61 7845 7870            maxExp
│ │ │ │  00197190: 6f6e 656e 7420 3d3e 2031 3037 3337 3431  onent => 1073741
│ │ │ │  001971a0: 3832 3320 207c 0a7c 2020 2020 2020 2020  823  |.|        
│ │ │ │  001971b0: 2020 2020 2020 206d 696e 4578 706f 6e65         minExpone
│ │ │ │  001971c0: 6e74 203d 3e20 2d31 3037 3337 3431 3832  nt => -107374182
│ │ │ │  001971d0: 3420 7c0a 7c20 2020 2020 2020 2020 2020  4 |.|           
│ │ │ │  001971e0: 2020 2020 6e75 6d54 4242 5468 7265 6164      numTBBThread
│ │ │ │ @@ -114602,16 +114602,16 @@
│ │ │ │  001bfa90: 636f 6e64 732e 0a2b 2d2d 2d2d 2d2d 2d2d  conds..+--------
│ │ │ │  001bfaa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  001bfab0: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 6265  ------+.|i1 : be
│ │ │ │  001bfac0: 6e63 686d 6172 6b20 2273 7172 7420 3270  nchmark "sqrt 2p
│ │ │ │  001bfad0: 3130 3030 3030 227c 0a7c 2020 2020 2020  100000"|.|      
│ │ │ │  001bfae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  001bfaf0: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20          |.|o1 = 
│ │ │ │ -001bfb00: 2e30 3030 3239 3036 3937 3836 3133 3637  .000290697861367
│ │ │ │ -001bfb10: 3333 3220 2020 2020 207c 0a7c 2020 2020  332      |.|    
│ │ │ │ +001bfb00: 2e30 3030 3334 3738 3634 3038 3031 3434  .000347864080144
│ │ │ │ +001bfb10: 3532 3539 2020 2020 207c 0a7c 2020 2020  5259     |.|    
│ │ │ │  001bfb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  001bfb30: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │  001bfb40: 3a20 5252 2028 6f66 2070 7265 6369 7369  : RR (of precisi
│ │ │ │  001bfb50: 6f6e 2035 3329 2020 2020 207c 0a2b 2d2d  on 53)     |.+--
│ │ │ │  001bfb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  001bfb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 5468  ------------+.Th
│ │ │ │  001bfb80: 6520 736e 6970 7065 7420 6f66 2063 6f64  e snippet of cod
│ │ │ │ @@ -120072,40 +120072,40 @@
│ │ │ │  001d5070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  001d5080: 2020 2020 2020 2020 2020 2020 207d 7c0a               }|.
│ │ │ │  001d5090: 7c20 2020 2020 7b31 3820 3d3e 2028 6d61  |     {18 => (ma
│ │ │ │  001d50a0: 7468 4d4c 2c20 4265 7474 6954 616c 6c79  thML, BettiTally
│ │ │ │  001d50b0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  001d50c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  001d50d0: 2020 207d 7c0a 7c20 2020 2020 7b31 3920     }|.|     {19 
│ │ │ │ -001d50e0: 3d3e 2028 636f 6469 6d2c 2042 6574 7469  => (codim, Betti
│ │ │ │ -001d50f0: 5461 6c6c 7929 2020 2020 2020 2020 2020  Tally)          
│ │ │ │ -001d5100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +001d50e0: 3d3e 2028 7472 756e 6361 7465 2c20 4265  => (truncate, Be
│ │ │ │ +001d50f0: 7474 6954 616c 6c79 2c20 496e 6669 6e69  ttiTally, Infini
│ │ │ │ +001d5100: 7465 4e75 6d62 6572 2c20 5a5a 2920 2020  teNumber, ZZ)   
│ │ │ │  001d5110: 2020 2020 2020 2020 207d 7c0a 7c20 2020           }|.|   
│ │ │ │  001d5120: 2020 7b32 3020 3d3e 2028 7472 756e 6361    {20 => (trunca
│ │ │ │  001d5130: 7465 2c20 4265 7474 6954 616c 6c79 2c20  te, BettiTally, 
│ │ │ │  001d5140: 5a5a 2c20 5a5a 2920 2020 2020 2020 2020  ZZ, ZZ)         
│ │ │ │  001d5150: 2020 2020 2020 2020 2020 2020 2020 207d                 }
│ │ │ │  001d5160: 7c0a 7c20 2020 2020 7b32 3120 3d3e 2028  |.|     {21 => (
│ │ │ │ -001d5170: 6475 616c 2c20 4265 7474 6954 616c 6c79  dual, BettiTally
│ │ │ │ -001d5180: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -001d5190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +001d5170: 7472 756e 6361 7465 2c20 4265 7474 6954  truncate, BettiT
│ │ │ │ +001d5180: 616c 6c79 2c20 5a5a 2c20 496e 6669 6e69  ally, ZZ, Infini
│ │ │ │ +001d5190: 7465 4e75 6d62 6572 2920 2020 2020 2020  teNumber)       
│ │ │ │  001d51a0: 2020 2020 207d 7c0a 7c20 2020 2020 7b32       }|.|     {2
│ │ │ │  001d51b0: 3220 3d3e 2028 7472 756e 6361 7465 2c20  2 => (truncate, 
│ │ │ │  001d51c0: 4265 7474 6954 616c 6c79 2c20 496e 6669  BettiTally, Infi
│ │ │ │ -001d51d0: 6e69 7465 4e75 6d62 6572 2c20 5a5a 2920  niteNumber, ZZ) 
│ │ │ │ -001d51e0: 2020 2020 2020 2020 2020 207d 7c0a 7c20             }|.| 
│ │ │ │ -001d51f0: 2020 2020 7b32 3320 3d3e 2028 7472 756e      {23 => (trun
│ │ │ │ -001d5200: 6361 7465 2c20 4265 7474 6954 616c 6c79  cate, BettiTally
│ │ │ │ -001d5210: 2c20 5a5a 2c20 496e 6669 6e69 7465 4e75  , ZZ, InfiniteNu
│ │ │ │ -001d5220: 6d62 6572 2920 2020 2020 2020 2020 2020  mber)           
│ │ │ │ +001d51d0: 6e69 7465 4e75 6d62 6572 2c20 496e 6669  niteNumber, Infi
│ │ │ │ +001d51e0: 6e69 7465 4e75 6d62 6572 297d 7c0a 7c20  niteNumber)}|.| 
│ │ │ │ +001d51f0: 2020 2020 7b32 3320 3d3e 2028 636f 6469      {23 => (codi
│ │ │ │ +001d5200: 6d2c 2042 6574 7469 5461 6c6c 7929 2020  m, BettiTally)  
│ │ │ │ +001d5210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +001d5220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  001d5230: 207d 7c0a 7c20 2020 2020 7b32 3420 3d3e   }|.|     {24 =>
│ │ │ │ -001d5240: 2028 7472 756e 6361 7465 2c20 4265 7474   (truncate, Bett
│ │ │ │ -001d5250: 6954 616c 6c79 2c20 496e 6669 6e69 7465  iTally, Infinite
│ │ │ │ -001d5260: 4e75 6d62 6572 2c20 496e 6669 6e69 7465  Number, Infinite
│ │ │ │ -001d5270: 4e75 6d62 6572 297d 7c0a 7c20 2020 2020  Number)}|.|     
│ │ │ │ +001d5240: 2028 6475 616c 2c20 4265 7474 6954 616c   (dual, BettiTal
│ │ │ │ +001d5250: 6c79 2920 2020 2020 2020 2020 2020 2020  ly)             
│ │ │ │ +001d5260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +001d5270: 2020 2020 2020 207d 7c0a 7c20 2020 2020         }|.|     
│ │ │ │  001d5280: 7b32 3520 3d3e 2028 5e2c 2052 696e 672c  {25 => (^, Ring,
│ │ │ │  001d5290: 2042 6574 7469 5461 6c6c 7929 2020 2020   BettiTally)    
│ │ │ │  001d52a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  001d52b0: 2020 2020 2020 2020 2020 2020 207d 7c0a               }|.
│ │ │ │  001d52c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  001d52d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  001d52e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -120282,38 +120282,38 @@
│ │ │ │  001d5d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  001d5da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  001d5db0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  001d5dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  001d5dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  001d5de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  001d5df0: 2020 2020 2020 2020 207c 0a7c 6f35 203d           |.|o5 =
│ │ │ │ -001d5e00: 207b 3020 3d3e 2028 636f 6e74 7261 6374   {0 => (contract
│ │ │ │ -001d5e10: 2c20 4d61 7472 6978 2c20 4d61 7472 6978  , Matrix, Matrix
│ │ │ │ -001d5e20: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +001d5e00: 207b 3020 3d3e 2028 2b2c 204d 6174 7269   {0 => (+, Matri
│ │ │ │ +001d5e10: 782c 204d 6174 7269 7829 2020 2020 2020  x, Matrix)      
│ │ │ │ +001d5e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  001d5e30: 2020 2020 2020 2020 2020 2020 207d 7c0a               }|.
│ │ │ │ -001d5e40: 7c20 2020 2020 7b31 203d 3e20 2864 6966  |     {1 => (dif
│ │ │ │ -001d5e50: 662c 204d 6174 7269 782c 204d 6174 7269  f, Matrix, Matri
│ │ │ │ -001d5e60: 7829 2020 2020 2020 2020 2020 2020 2020  x)              
│ │ │ │ +001d5e40: 7c20 2020 2020 7b31 203d 3e20 282d 2c20  |     {1 => (-, 
│ │ │ │ +001d5e50: 4d61 7472 6978 2c20 4d61 7472 6978 2920  Matrix, Matrix) 
│ │ │ │ +001d5e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  001d5e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  001d5e80: 2020 7d7c 0a7c 2020 2020 207b 3220 3d3e    }|.|     {2 =>
│ │ │ │ -001d5e90: 2028 6469 6666 272c 204d 6174 7269 782c   (diff', Matrix,
│ │ │ │ -001d5ea0: 204d 6174 7269 7829 2020 2020 2020 2020   Matrix)        
│ │ │ │ +001d5e90: 2028 636f 6e74 7261 6374 2c20 4d61 7472   (contract, Matr
│ │ │ │ +001d5ea0: 6978 2c20 4d61 7472 6978 2920 2020 2020  ix, Matrix)     
│ │ │ │  001d5eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  001d5ec0: 2020 2020 2020 207d 7c0a 7c20 2020 2020         }|.|     
│ │ │ │ -001d5ed0: 7b33 203d 3e20 282d 2c20 4d61 7472 6978  {3 => (-, Matrix
│ │ │ │ -001d5ee0: 2c20 4d61 7472 6978 2920 2020 2020 2020  , Matrix)       
│ │ │ │ +001d5ed0: 7b33 203d 3e20 2864 6966 662c 204d 6174  {3 => (diff, Mat
│ │ │ │ +001d5ee0: 7269 782c 204d 6174 7269 7829 2020 2020  rix, Matrix)    
│ │ │ │  001d5ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  001d5f00: 2020 2020 2020 2020 2020 2020 7d7c 0a7c              }|.|
│ │ │ │  001d5f10: 2020 2020 207b 3420 3d3e 2028 636f 6e74       {4 => (cont
│ │ │ │  001d5f20: 7261 6374 272c 204d 6174 7269 782c 204d  ract', Matrix, M
│ │ │ │  001d5f30: 6174 7269 7829 2020 2020 2020 2020 2020  atrix)          
│ │ │ │  001d5f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  001d5f50: 207d 7c0a 7c20 2020 2020 7b35 203d 3e20   }|.|     {5 => 
│ │ │ │ -001d5f60: 282b 2c20 4d61 7472 6978 2c20 4d61 7472  (+, Matrix, Matr
│ │ │ │ -001d5f70: 6978 2920 2020 2020 2020 2020 2020 2020  ix)             
│ │ │ │ +001d5f60: 2864 6966 6627 2c20 4d61 7472 6978 2c20  (diff', Matrix, 
│ │ │ │ +001d5f70: 4d61 7472 6978 2920 2020 2020 2020 2020  Matrix)         
│ │ │ │  001d5f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  001d5f90: 2020 2020 2020 7d7c 0a7c 2020 2020 207b        }|.|     {
│ │ │ │  001d5fa0: 3620 3d3e 2028 6d61 726b 6564 4742 2c20  6 => (markedGB, 
│ │ │ │  001d5fb0: 4d61 7472 6978 2c20 4d61 7472 6978 2920  Matrix, Matrix) 
│ │ │ │  001d5fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  001d5fd0: 2020 2020 2020 2020 2020 207d 7c0a 7c20             }|.| 
│ │ │ │  001d5fe0: 2020 2020 7b37 203d 3e20 2848 6f6d 2c20      {7 => (Hom, 
│ │ │ │ @@ -120403,29 +120403,29 @@
│ │ │ │  001d6520: 6978 2920 2020 2020 2020 2020 2020 2020  ix)             
│ │ │ │  001d6530: 2020 2020 2020 2020 2020 2020 2020 207d                 }
│ │ │ │  001d6540: 7c0a 7c20 2020 2020 7b32 3720 3d3e 2028  |.|     {27 => (
│ │ │ │  001d6550: 736f 6c76 652c 204d 6174 7269 782c 204d  solve, Matrix, M
│ │ │ │  001d6560: 6174 7269 7829 2020 2020 2020 2020 2020  atrix)          
│ │ │ │  001d6570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  001d6580: 2020 2020 7d7c 0a7c 2020 2020 207b 3238      }|.|     {28
│ │ │ │ -001d6590: 203d 3e20 2870 756c 6c62 6163 6b2c 204d   => (pullback, M
│ │ │ │ -001d65a0: 6174 7269 782c 204d 6174 7269 7829 2020  atrix, Matrix)  
│ │ │ │ -001d65b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +001d6590: 203d 3e20 2869 6e74 6572 7365 6374 2c20   => (intersect, 
│ │ │ │ +001d65a0: 4d61 7472 6978 2c20 4d61 7472 6978 2c20  Matrix, Matrix, 
│ │ │ │ +001d65b0: 4d61 7472 6978 2c20 4d61 7472 6978 2920  Matrix, Matrix) 
│ │ │ │  001d65c0: 2020 2020 2020 2020 207d 7c0a 7c20 2020           }|.|   
│ │ │ │  001d65d0: 2020 7b32 3920 3d3e 2028 696e 7465 7273    {29 => (inters
│ │ │ │  001d65e0: 6563 742c 204d 6174 7269 782c 204d 6174  ect, Matrix, Mat
│ │ │ │  001d65f0: 7269 7829 2020 2020 2020 2020 2020 2020  rix)            
│ │ │ │  001d6600: 2020 2020 2020 2020 2020 2020 2020 7d7c                }|
│ │ │ │ -001d6610: 0a7c 2020 2020 207b 3330 203d 3e20 2869  .|     {30 => (i
│ │ │ │ -001d6620: 6e74 6572 7365 6374 2c20 4d61 7472 6978  ntersect, Matrix
│ │ │ │ -001d6630: 2c20 4d61 7472 6978 2c20 4d61 7472 6978  , Matrix, Matrix
│ │ │ │ -001d6640: 2c20 4d61 7472 6978 2920 2020 2020 2020  , Matrix)       
│ │ │ │ +001d6610: 0a7c 2020 2020 207b 3330 203d 3e20 2874  .|     {30 => (t
│ │ │ │ +001d6620: 656e 736f 722c 204d 6174 7269 782c 204d  ensor, Matrix, M
│ │ │ │ +001d6630: 6174 7269 7829 2020 2020 2020 2020 2020  atrix)          
│ │ │ │ +001d6640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  001d6650: 2020 207d 7c0a 7c20 2020 2020 7b33 3120     }|.|     {31 
│ │ │ │ -001d6660: 3d3e 2028 7465 6e73 6f72 2c20 4d61 7472  => (tensor, Matr
│ │ │ │ -001d6670: 6978 2c20 4d61 7472 6978 2920 2020 2020  ix, Matrix)     
│ │ │ │ +001d6660: 3d3e 2028 7075 6c6c 6261 636b 2c20 4d61  => (pullback, Ma
│ │ │ │ +001d6670: 7472 6978 2c20 4d61 7472 6978 2920 2020  trix, Matrix)   
│ │ │ │  001d6680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  001d6690: 2020 2020 2020 2020 7d7c 0a7c 2020 2020          }|.|    
│ │ │ │  001d66a0: 207b 3332 203d 3e20 2873 7562 7374 6974   {32 => (substit
│ │ │ │  001d66b0: 7574 652c 204d 6174 7269 782c 204d 6174  ute, Matrix, Mat
│ │ │ │  001d66c0: 7269 7829 2020 2020 2020 2020 2020 2020  rix)            
│ │ │ │  001d66d0: 2020 2020 2020 2020 2020 2020 207d 7c0a               }|.
│ │ │ │  001d66e0: 7c20 2020 2020 7b33 3320 3d3e 2028 796f  |     {33 => (yo
│ │ │ │ @@ -121335,84 +121335,84 @@
│ │ │ │  001d9f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  001d9f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  001d9f80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  001d9f90: 6f32 203d 2023 7275 6e20 2025 7469 6d65  o2 = #run  %time
│ │ │ │  001d9fa0: 2020 2070 6f73 6974 696f 6e20 2020 2020     position     
│ │ │ │  001d9fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  001d9fc0: 2020 2020 7c0a 7c20 2020 2020 3120 2020      |.|     1   
│ │ │ │ -001d9fd0: 2020 3934 2e35 3220 2020 2e2e 2f2e 2e2f    94.52   ../../
│ │ │ │ +001d9fd0: 2020 3933 2e33 3520 2020 2e2e 2f2e 2e2f    93.35   ../../
│ │ │ │  001d9fe0: 6d32 2f6d 6174 7269 7831 2e6d 323a 3237  m2/matrix1.m2:27
│ │ │ │  001d9ff0: 393a 342d 3238 323a 3538 207c 0a7c 2020  9:4-282:58 |.|  
│ │ │ │ -001da000: 2020 2031 2020 2020 2039 322e 3132 2020     1     92.12  
│ │ │ │ +001da000: 2020 2031 2020 2020 2039 302e 3635 2020     1     90.65  
│ │ │ │  001da010: 202e 2e2f 2e2e 2f6d 322f 6d61 7472 6978   ../../m2/matrix
│ │ │ │  001da020: 312e 6d32 3a32 3831 3a32 322d 3238 313a  1.m2:281:22-281:
│ │ │ │  001da030: 3433 7c0a 7c20 2020 2020 3120 2020 2020  43|.|     1     
│ │ │ │ -001da040: 3434 2e31 3620 2020 2e2e 2f2e 2e2f 6d32  44.16   ../../m2
│ │ │ │ +001da040: 3433 2e30 3120 2020 2e2e 2f2e 2e2f 6d32  43.01   ../../m2
│ │ │ │  001da050: 2f6d 6174 7269 7831 2e6d 323a 3139 333a  /matrix1.m2:193:
│ │ │ │  001da060: 3235 2d31 3933 3a35 327c 0a7c 2020 2020  25-193:52|.|    
│ │ │ │ -001da070: 2031 2020 2020 2033 302e 3539 2020 202e   1     30.59   .
│ │ │ │ +001da070: 2031 2020 2020 2032 392e 3934 2020 202e   1     29.94   .
│ │ │ │  001da080: 2e2f 2e2e 2f6d 322f 6d61 7472 6978 312e  ./../m2/matrix1.
│ │ │ │  001da090: 6d32 3a31 3134 3a35 2d31 3536 3a37 3220  m2:114:5-156:72 
│ │ │ │ -001da0a0: 7c0a 7c20 2020 2020 3120 2020 2020 3239  |.|     1     29
│ │ │ │ -001da0b0: 2e34 3720 2020 2e2e 2f2e 2e2f 6d32 2f6d  .47   ../../m2/m
│ │ │ │ +001da0a0: 7c0a 7c20 2020 2020 3120 2020 2020 3238  |.|     1     28
│ │ │ │ +001da0b0: 2e37 3220 2020 2e2e 2f2e 2e2f 6d32 2f6d  .72   ../../m2/m
│ │ │ │  001da0c0: 6174 7269 7831 2e6d 323a 3134 303a 3130  atrix1.m2:140:10
│ │ │ │  001da0d0: 2d31 3535 3a31 367c 0a7c 2020 2020 2031  -155:16|.|     1
│ │ │ │ -001da0e0: 2020 2020 2032 332e 3833 2020 202e 2e2f       23.83   ../
│ │ │ │ +001da0e0: 2020 2020 2032 332e 3436 2020 202e 2e2f       23.46   ../
│ │ │ │  001da0f0: 2e2e 2f6d 322f 6d61 7472 6978 312e 6d32  ../m2/matrix1.m2
│ │ │ │  001da100: 3a31 3831 3a34 2d31 3831 3a34 3220 7c0a  :181:4-181:42 |.
│ │ │ │ -001da110: 7c20 2020 2020 3120 2020 2020 3232 2e35  |     1     22.5
│ │ │ │ -001da120: 3420 2020 2e2e 2f2e 2e2f 6d32 2f73 6574  4   ../../m2/set
│ │ │ │ +001da110: 7c20 2020 2020 3120 2020 2020 3232 2e31  |     1     22.1
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│ │ │ │  001da130: 2e6d 323a 3132 373a 352d 3132 373a 3631  .m2:127:5-127:61
│ │ │ │  001da140: 2020 2020 207c 0a7c 2020 2020 2031 2020       |.|     1  
│ │ │ │ -001da150: 2020 2032 302e 3836 2020 202e 2e2f 2e2e     20.86   ../..
│ │ │ │ +001da150: 2020 2032 302e 3220 2020 202e 2e2f 2e2e     20.2    ../..
│ │ │ │  001da160: 2f6d 322f 6d61 7472 6978 312e 6d32 3a34  /m2/matrix1.m2:4
│ │ │ │  001da170: 353a 3130 2d34 393a 3232 2020 7c0a 7c20  5:10-49:22  |.| 
│ │ │ │ -001da180: 2020 2020 3120 2020 2020 332e 3330 2020      1     3.30  
│ │ │ │ +001da180: 2020 2020 3120 2020 2020 332e 3236 2020      1     3.26  
│ │ │ │  001da190: 2020 2e2e 2f2e 2e2f 6d32 2f6d 6174 7269    ../../m2/matri
│ │ │ │  001da1a0: 7831 2e6d 323a 3131 323a 352d 3131 323a  x1.m2:112:5-112:
│ │ │ │  001da1b0: 3239 207c 0a7c 2020 2020 2031 2020 2020  29 |.|     1    
│ │ │ │ -001da1c0: 2032 2e33 3420 2020 202e 2e2f 2e2e 2f6d   2.34    ../../m
│ │ │ │ +001da1c0: 2032 2e38 2020 2020 202e 2e2f 2e2e 2f6d   2.8     ../../m
│ │ │ │  001da1d0: 322f 6d61 7472 6978 312e 6d32 3a31 3431  2/matrix1.m2:141
│ │ │ │  001da1e0: 3a31 332d 3134 313a 3738 7c0a 7c20 2020  :13-141:78|.|   
│ │ │ │ -001da1f0: 2020 3120 2020 2020 322e 3138 2020 2020    1     2.18    
│ │ │ │ +001da1f0: 2020 3120 2020 2020 322e 3136 2020 2020    1     2.16    
│ │ │ │  001da200: 2e2e 2f2e 2e2f 6d32 2f6d 6174 7269 7831  ../../m2/matrix1
│ │ │ │  001da210: 2e6d 323a 3936 3a35 2d31 3039 3a31 3120  .m2:96:5-109:11 
│ │ │ │  001da220: 207c 0a7c 2020 2020 2031 2020 2020 2031   |.|     1     1
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│ │ │ │  001da250: 2d32 3831 3a31 3620 7c0a 7c20 2020 2020  -281:16 |.|     
│ │ │ │ -001da260: 3120 2020 2020 312e 3239 2020 2020 2e2e  1     1.29    ..
│ │ │ │ +001da260: 3120 2020 2020 312e 3337 2020 2020 2e2e  1     1.37    ..
│ │ │ │  001da270: 2f2e 2e2f 6d32 2f6d 6174 7269 7831 2e6d  /../m2/matrix1.m
│ │ │ │ -001da280: 323a 3134 373a 3230 2d31 3437 3a36 347c  2:147:20-147:64|
│ │ │ │ +001da280: 323a 3237 363a 342d 3237 373a 3733 207c  2:276:4-277:73 |
│ │ │ │  001da290: 0a7c 2020 2020 2031 2020 2020 2031 2e32  .|     1     1.2
│ │ │ │ -001da2a0: 3920 2020 202e 2e2f 2e2e 2f6d 322f 6d61  9    ../../m2/ma
│ │ │ │ -001da2b0: 7472 6978 312e 6d32 3a31 3131 3a35 2d31  trix1.m2:111:5-1
│ │ │ │ -001da2c0: 3131 3a39 3120 7c0a 7c20 2020 2020 3120  11:91 |.|     1 
│ │ │ │ -001da2d0: 2020 2020 312e 3237 2020 2020 2e2e 2f2e      1.27    ../.
│ │ │ │ +001da2a0: 3620 2020 202e 2e2f 2e2e 2f6d 322f 6d61  6    ../../m2/ma
│ │ │ │ +001da2b0: 7472 6978 312e 6d32 3a31 3437 3a32 302d  trix1.m2:147:20-
│ │ │ │ +001da2c0: 3134 373a 3634 7c0a 7c20 2020 2020 3120  147:64|.|     1 
│ │ │ │ +001da2d0: 2020 2020 312e 3134 2020 2020 2e2e 2f2e      1.14    ../.
│ │ │ │  001da2e0: 2e2f 6d32 2f6d 6174 7269 7831 2e6d 323a  ./m2/matrix1.m2:
│ │ │ │ -001da2f0: 3237 363a 342d 3237 373a 3733 207c 0a7c  276:4-277:73 |.|
│ │ │ │ -001da300: 2020 2020 2031 2020 2020 2031 2e30 3220       1     1.02 
│ │ │ │ +001da2f0: 3131 313a 352d 3131 313a 3931 207c 0a7c  111:5-111:91 |.|
│ │ │ │ +001da300: 2020 2020 2031 2020 2020 2031 2e31 3020       1     1.10 
│ │ │ │  001da310: 2020 202e 2e2f 2e2e 2f6d 322f 6d61 7472     ../../m2/matr
│ │ │ │ -001da320: 6978 312e 6d32 3a39 383a 3130 2d39 383a  ix1.m2:98:10-98:
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│ │ │ │ -001da340: 2020 2e39 3720 2020 2020 2e2e 2f2e 2e2f    .97     ../../
│ │ │ │ -001da350: 6d32 2f6d 6174 7269 7831 2e6d 323a 3138  m2/matrix1.m2:18
│ │ │ │ -001da360: 323a 342d 3138 343a 3734 207c 0a7c 2020  2:4-184:74 |.|  
│ │ │ │ -001da370: 2020 2031 2020 2020 202e 3831 2020 2020     1     .81    
│ │ │ │ -001da380: 202e 2e2f 2e2e 2f6d 322f 6d6f 6475 6c65   ../../m2/module
│ │ │ │ -001da390: 732e 6d32 3a32 3739 3a34 2d32 3739 3a35  s.m2:279:4-279:5
│ │ │ │ -001da3a0: 3220 7c0a 7c20 2020 2020 3230 2020 2020  2 |.|     20    
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│ │ │ │ -001da3c0: 2f6d 6174 7269 7831 2e6d 323a 3139 313a  /matrix1.m2:191:
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│ │ │ │ -001da3e0: 2032 3020 2020 202e 3437 2020 2020 202e   20    .47     .
│ │ │ │ +001da320: 6978 312e 6d32 3a31 3832 3a34 2d31 3834  ix1.m2:182:4-184
│ │ │ │ +001da330: 3a37 3420 7c0a 7c20 2020 2020 3120 2020  :74 |.|     1   
│ │ │ │ +001da340: 2020 312e 3038 2020 2020 2e2e 2f2e 2e2f    1.08    ../../
│ │ │ │ +001da350: 6d32 2f6d 6174 7269 7831 2e6d 323a 3938  m2/matrix1.m2:98
│ │ │ │ +001da360: 3a31 302d 3938 3a34 3620 207c 0a7c 2020  :10-98:46  |.|  
│ │ │ │ +001da370: 2020 2032 3020 2020 202e 3935 2020 2020     20    .95    
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│ │ │ │ +001da390: 312e 6d32 3a31 3931 3a31 342d 3139 323a  1.m2:191:14-192:
│ │ │ │ +001da3a0: 3637 7c0a 7c20 2020 2020 3230 2020 2020  67|.|     20    
│ │ │ │ +001da3b0: 2e36 3720 2020 2020 2e2e 2f2e 2e2f 6d32  .67     ../../m2
│ │ │ │ +001da3c0: 2f6d 6174 7269 7831 2e6d 323a 3437 3a34  /matrix1.m2:47:4
│ │ │ │ +001da3d0: 332d 3437 3a37 3120 207c 0a7c 2020 2020  3-47:71  |.|    
│ │ │ │ +001da3e0: 2032 3020 2020 202e 3630 2020 2020 202e   20    .60     .
│ │ │ │  001da3f0: 2e2f 2e2e 2f6d 322f 6d61 7472 6978 312e  ./../m2/matrix1.
│ │ │ │ -001da400: 6d32 3a34 373a 3433 2d34 373a 3731 2020  m2:47:43-47:71  
│ │ │ │ +001da400: 6d32 3a31 3930 3a31 372d 3139 303a 3239  m2:190:17-190:29
│ │ │ │  001da410: 7c0a 7c20 2020 2020 3120 2020 2020 2e30  |.|     1     .0
│ │ │ │ -001da420: 3033 3873 2020 656c 6170 7365 6420 746f  038s  elapsed to
│ │ │ │ +001da420: 3033 3573 2020 656c 6170 7365 6420 746f  035s  elapsed to
│ │ │ │  001da430: 7461 6c20 2020 2020 2020 2020 2020 2020  tal             
│ │ │ │  001da440: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  001da450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  001da460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  001da470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  001da480: 7c69 3320 3a20 636f 7665 7261 6765 5375  |i3 : coverageSu
│ │ │ │  001da490: 6d6d 6172 7920 2020 2020 2020 2020 2020  mmary           
│ │ │ │ @@ -127828,21 +127828,21 @@
│ │ │ │  001f3530: 206f 6620 780a 0a44 6573 6372 6970 7469   of x..Descripti
│ │ │ │  001f3540: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │  001f3550: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  001f3560: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2073  -------+.|i1 : s
│ │ │ │  001f3570: 6572 6961 6c4e 756d 6265 7220 6173 6466  erialNumber asdf
│ │ │ │  001f3580: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  001f3590: 2020 2020 2020 2020 207c 0a7c 6f31 203d           |.|o1 =
│ │ │ │ -001f35a0: 2031 3432 3632 3733 2020 2020 2020 2020   1426273        
│ │ │ │ +001f35a0: 2031 3532 3632 3733 2020 2020 2020 2020   1526273        
│ │ │ │  001f35b0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  001f35c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │  001f35d0: 203a 2073 6572 6961 6c4e 756d 6265 7220   : serialNumber 
│ │ │ │  001f35e0: 666f 6f20 7c0a 7c20 2020 2020 2020 2020  foo |.|         
│ │ │ │  001f35f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -001f3600: 6f32 203d 2031 3432 3632 3735 2020 2020  o2 = 1426275    
│ │ │ │ +001f3600: 6f32 203d 2031 3532 3632 3735 2020 2020  o2 = 1526275    
│ │ │ │  001f3610: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  001f3620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  001f3630: 0a7c 6933 203a 2073 6572 6961 6c4e 756d  .|i3 : serialNum
│ │ │ │  001f3640: 6265 7220 5a5a 2020 7c0a 7c20 2020 2020  ber ZZ  |.|     
│ │ │ │  001f3650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  001f3660: 207c 0a7c 6f33 203d 2031 3030 3030 3530   |.|o3 = 1000050
│ │ │ │  001f3670: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ @@ -131620,16 +131620,16 @@
│ │ │ │  00202230: 656c 6f61 6469 6e67 2046 6972 7374 5061  eloading FirstPa
│ │ │ │  00202240: 636b 6167 653b 2072 6563 7265 6174 6520  ckage; recreate 
│ │ │ │  00202250: 696e 7374 616e 6365 7320 6f66 2074 7970  instances of typ
│ │ │ │  00202260: 6573 2066 726f 6d20 7468 6973 2020 207c  es from this   |
│ │ │ │  00202270: 0a7c 202d 2d20 6361 7074 7572 696e 6720  .| -- capturing 
│ │ │ │  00202280: 6368 6563 6b28 312c 2022 4669 7273 7450  check(1, "FirstP
│ │ │ │  00202290: 6163 6b61 6765 2229 2020 2020 2020 2020  ackage")        
│ │ │ │ -002022a0: 2d2d 202e 3135 3134 3773 2065 6c61 7073  -- .15147s elaps
│ │ │ │ -002022b0: 6564 2020 2020 2020 2020 2020 2020 207c  ed             |
│ │ │ │ +002022a0: 2d2d 202e 3132 3235 3434 7320 656c 6170  -- .122544s elap
│ │ │ │ +002022b0: 7365 6420 2020 2020 2020 2020 2020 207c  sed            |
│ │ │ │  002022c0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  002022d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  002022e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  002022f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00202300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │  00202310: 0a7c 7061 636b 6167 6520 2020 2020 2020  .|package       
│ │ │ │  00202320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -131645,20 +131645,20 @@
│ │ │ │  002023c0: 7374 5061 636b 6167 6520 2020 2020 2020  stPackage       
│ │ │ │  002023d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002023e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002023f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00202400: 0a7c 202d 2d20 6361 7074 7572 696e 6720  .| -- capturing 
│ │ │ │  00202410: 6368 6563 6b28 302c 2022 4669 7273 7450  check(0, "FirstP
│ │ │ │  00202420: 6163 6b61 6765 2229 2020 2020 2020 2020  ackage")        
│ │ │ │ -00202430: 2d2d 202e 3135 3031 3831 7320 656c 6170  -- .150181s elap
│ │ │ │ +00202430: 2d2d 202e 3131 3633 3432 7320 656c 6170  -- .116342s elap
│ │ │ │  00202440: 7365 6420 2020 2020 2020 2020 2020 207c  sed            |
│ │ │ │  00202450: 0a7c 202d 2d20 6361 7074 7572 696e 6720  .| -- capturing 
│ │ │ │  00202460: 6368 6563 6b28 312c 2022 4669 7273 7450  check(1, "FirstP
│ │ │ │  00202470: 6163 6b61 6765 2229 2020 2020 2020 2020  ackage")        
│ │ │ │ -00202480: 2d2d 202e 3135 3039 3635 7320 656c 6170  -- .150965s elap
│ │ │ │ +00202480: 2d2d 202e 3131 3836 3038 7320 656c 6170  -- .118608s elap
│ │ │ │  00202490: 7365 6420 2020 2020 2020 2020 2020 207c  sed            |
│ │ │ │  002024a0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  002024b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  002024c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  002024d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  002024e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  002024f0: 0a0a 416c 7465 726e 6174 6976 656c 792c  ..Alternatively,
│ │ │ │ @@ -131674,32 +131674,32 @@
│ │ │ │  00202590: 2b0a 7c69 3420 3a20 6368 6563 6b5f 3120  +.|i4 : check_1 
│ │ │ │  002025a0: 2246 6972 7374 5061 636b 6167 6522 2020  "FirstPackage"  
│ │ │ │  002025b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002025c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002025d0: 2020 2020 7c0a 7c20 2d2d 2063 6170 7475      |.| -- captu
│ │ │ │  002025e0: 7269 6e67 2063 6865 636b 2831 2c20 2246  ring check(1, "F
│ │ │ │  002025f0: 6972 7374 5061 636b 6167 6522 2920 2020  irstPackage")   
│ │ │ │ -00202600: 2020 2020 202d 2d20 2e31 3532 3537 3973       -- .152579s
│ │ │ │ +00202600: 2020 2020 202d 2d20 2e31 3138 3630 3873       -- .118608s
│ │ │ │  00202610: 2065 6c61 7073 6564 7c0a 2b2d 2d2d 2d2d   elapsed|.+-----
│ │ │ │  00202620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00202630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00202640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00202650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  00202660: 3520 3a20 6368 6563 6b20 2246 6972 7374  5 : check "First
│ │ │ │  00202670: 5061 636b 6167 6522 2020 2020 2020 2020  Package"        
│ │ │ │  00202680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00202690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002026a0: 7c0a 7c20 2d2d 2063 6170 7475 7269 6e67  |.| -- capturing
│ │ │ │  002026b0: 2063 6865 636b 2830 2c20 2246 6972 7374   check(0, "First
│ │ │ │  002026c0: 5061 636b 6167 6522 2920 2020 2020 2020  Package")       
│ │ │ │ -002026d0: 202d 2d20 2e31 3532 3035 3373 2065 6c61   -- .152053s ela
│ │ │ │ +002026d0: 202d 2d20 2e31 3230 3037 3273 2065 6c61   -- .120072s ela
│ │ │ │  002026e0: 7073 6564 7c0a 7c20 2d2d 2063 6170 7475  psed|.| -- captu
│ │ │ │  002026f0: 7269 6e67 2063 6865 636b 2831 2c20 2246  ring check(1, "F
│ │ │ │  00202700: 6972 7374 5061 636b 6167 6522 2920 2020  irstPackage")   
│ │ │ │ -00202710: 2020 2020 202d 2d20 2e31 3531 3836 3773       -- .151867s
│ │ │ │ +00202710: 2020 2020 202d 2d20 2e31 3139 3531 3373       -- .119513s
│ │ │ │  00202720: 2065 6c61 7073 6564 7c0a 2b2d 2d2d 2d2d   elapsed|.+-----
│ │ │ │  00202730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00202740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00202750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00202760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a41  ------------+..A
│ │ │ │  00202770: 202a 6e6f 7465 2054 6573 7449 6e70 7574   *note TestInput
│ │ │ │  00202780: 3a20 7465 7374 732c 206f 626a 6563 7420  : tests, object 
│ │ │ │ @@ -131741,15 +131741,15 @@
│ │ │ │  002029c0: 6937 203a 2063 6865 636b 206f 6f20 2020  i7 : check oo   
│ │ │ │  002029d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002029e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002029f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00202a00: 2020 2020 207c 0a7c 202d 2d20 6361 7074       |.| -- capt
│ │ │ │  00202a10: 7572 696e 6720 6368 6563 6b28 312c 2022  uring check(1, "
│ │ │ │  00202a20: 4669 7273 7450 6163 6b61 6765 2229 2020  FirstPackage")  
│ │ │ │ -00202a30: 2020 2020 2020 2d2d 202e 3135 3330 3833        -- .153083
│ │ │ │ +00202a30: 2020 2020 2020 2d2d 202e 3131 3933 3939        -- .119399
│ │ │ │  00202a40: 7320 656c 6170 7365 6420 2020 207c 0a2b  s elapsed    |.+
│ │ │ │  00202a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00202a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00202a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00202a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00202a90: 2d2d 2d2d 2d2b 0a7c 6938 203a 2074 6573  -----+.|i8 : tes
│ │ │ │  00202aa0: 7473 2022 4669 7273 7450 6163 6b61 6765  ts "FirstPackage
│ │ │ │ @@ -131786,20 +131786,20 @@
│ │ │ │  00202c90: 6939 203a 2063 6865 636b 206f 6f20 2020  i9 : check oo   
│ │ │ │  00202ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00202cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00202cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00202cd0: 2020 2020 207c 0a7c 202d 2d20 6361 7074       |.| -- capt
│ │ │ │  00202ce0: 7572 696e 6720 6368 6563 6b28 302c 2022  uring check(0, "
│ │ │ │  00202cf0: 4669 7273 7450 6163 6b61 6765 2229 2020  FirstPackage")  
│ │ │ │ -00202d00: 2020 2020 2020 2d2d 202e 3135 3237 3033        -- .152703
│ │ │ │ +00202d00: 2020 2020 2020 2d2d 202e 3131 3830 3131        -- .118011
│ │ │ │  00202d10: 7320 656c 6170 7365 6420 2020 207c 0a7c  s elapsed    |.|
│ │ │ │  00202d20: 202d 2d20 6361 7074 7572 696e 6720 6368   -- capturing ch
│ │ │ │  00202d30: 6563 6b28 312c 2022 4669 7273 7450 6163  eck(1, "FirstPac
│ │ │ │  00202d40: 6b61 6765 2229 2020 2020 2020 2020 2d2d  kage")        --
│ │ │ │ -00202d50: 202e 3135 3039 3031 7320 656c 6170 7365   .150901s elapse
│ │ │ │ +00202d50: 202e 3131 3633 3734 7320 656c 6170 7365   .116374s elapse
│ │ │ │  00202d60: 6420 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d  d    |.+--------
│ │ │ │  00202d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00202d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00202d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00202da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │  00202db0: 4966 206f 6e6c 7920 616e 2069 6e74 6567  If only an integ
│ │ │ │  00202dc0: 6572 2069 7320 7061 7373 6564 2061 7320  er is passed as 
│ │ │ │ @@ -131849,15 +131849,15 @@
│ │ │ │  00203080: 636b 2031 2020 2020 2020 2020 2020 2020  ck 1            
│ │ │ │  00203090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002030a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002030b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  002030c0: 202d 2d20 6361 7074 7572 696e 6720 6368   -- capturing ch
│ │ │ │  002030d0: 6563 6b28 312c 2022 4669 7273 7450 6163  eck(1, "FirstPac
│ │ │ │  002030e0: 6b61 6765 2229 2020 2020 2020 2020 2d2d  kage")        --
│ │ │ │ -002030f0: 202e 3135 3133 3436 7320 656c 6170 7365   .151346s elapse
│ │ │ │ +002030f0: 202e 3132 3335 3432 7320 656c 6170 7365   .123542s elapse
│ │ │ │  00203100: 6420 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d  d     |.+-------
│ │ │ │  00203110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00203120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00203130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00203140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  00203150: 0a0a 4361 7665 6174 0a3d 3d3d 3d3d 3d0a  ..Caveat.======.
│ │ │ │  00203160: 0a43 7572 7265 6e74 6c79 2c20 6966 2074  .Currently, if t
│ │ │ │ @@ -135799,15 +135799,15 @@
│ │ │ │  00212760: 2020 2020 2020 2020 7261 7720 646f 6375          raw docu
│ │ │ │  00212770: 6d65 6e74 6174 696f 6e20 3d3e 204d 7574  mentation => Mut
│ │ │ │  00212780: 6162 6c65 4861 7368 5461 626c 657b 7d20  ableHashTable{} 
│ │ │ │  00212790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002127a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  002127b0: 2020 2020 2020 2020 736f 7572 6365 2064          source d
│ │ │ │  002127c0: 6972 6563 746f 7279 203d 3e20 2f74 6d70  irectory => /tmp
│ │ │ │ -002127d0: 2f4d 322d 3130 3139 312d 302f 3931 2d72  /M2-10191-0/91-r
│ │ │ │ +002127d0: 2f4d 322d 3130 3331 312d 302f 3931 2d72  /M2-10311-0/91-r
│ │ │ │  002127e0: 756e 6469 722f 2020 2020 2020 2020 2020  undir/          
│ │ │ │  002127f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00212800: 2020 2020 2020 2020 736f 7572 6365 2066          source f
│ │ │ │  00212810: 696c 6520 3d3e 2073 7464 696f 2020 2020  ile => stdio    
│ │ │ │  00212820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00212830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00212840: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ @@ -141794,26 +141794,26 @@
│ │ │ │  00229e10: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  00229e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  00229e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  00229e50: 7c69 3320 3a20 656c 6170 7365 6454 696d  |i3 : elapsedTim
│ │ │ │  00229e60: 6520 2020 2020 2020 2020 6170 706c 7928  e         apply(
│ │ │ │  00229e70: 312e 2e31 3030 2c20 6e20 2d3e 2073 6f72  1..100, n -> sor
│ │ │ │ -00229e80: 7420 4c29 3b7c 0a7c 202d 2d20 2e36 3430  t L);|.| -- .640
│ │ │ │ -00229e90: 3637 3473 2065 6c61 7073 6564 2020 2020  674s elapsed    
│ │ │ │ +00229e80: 7420 4c29 3b7c 0a7c 202d 2d20 2e36 3831  t L);|.| -- .681
│ │ │ │ +00229e90: 3335 3673 2065 6c61 7073 6564 2020 2020  356s elapsed    
│ │ │ │  00229ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00229eb0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  00229ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00229ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00229ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00229ef0: 2d2d 2d2b 0a7c 6934 203a 2065 6c61 7073  ---+.|i4 : elaps
│ │ │ │  00229f00: 6564 5469 6d65 2070 6172 616c 6c65 6c41  edTime parallelA
│ │ │ │  00229f10: 7070 6c79 2831 2e2e 3130 302c 206e 202d  pply(1..100, n -
│ │ │ │  00229f20: 3e20 736f 7274 204c 293b 7c0a 7c20 2d2d  > sort L);|.| --
│ │ │ │ -00229f30: 202e 3330 3739 3139 7320 656c 6170 7365   .307919s elapse
│ │ │ │ +00229f30: 202e 3138 3230 3638 7320 656c 6170 7365   .182068s elapse
│ │ │ │  00229f40: 6420 2020 2020 2020 2020 2020 2020 2020  d               
│ │ │ │  00229f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00229f60: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00229f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00229f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00229f90: 2d2d 2d2d 2d2d 2d2d 2b0a 0a59 6f75 2077  --------+..You w
│ │ │ │  00229fa0: 696c 6c20 6861 7665 2074 6f20 7472 7920  ill have to try 
│ │ │ │ @@ -141894,15 +141894,15 @@
│ │ │ │  0022a450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0022a460: 2d2b 0a7c 6936 203a 2061 6c6c 6f77 6162  -+.|i6 : allowab
│ │ │ │  0022a470: 6c65 5468 7265 6164 7320 3d20 6d61 7841  leThreads = maxA
│ │ │ │  0022a480: 6c6c 6f77 6162 6c65 5468 7265 6164 737c  llowableThreads|
│ │ │ │  0022a490: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0022a4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0022a4b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0022a4c0: 6f36 203d 2037 2020 2020 2020 2020 2020  o6 = 7          
│ │ │ │ +0022a4c0: 6f36 203d 2031 3720 2020 2020 2020 2020  o6 = 17         
│ │ │ │  0022a4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0022a4e0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  0022a4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0022a500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0022a510: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 546f 2072  ---------+..To r
│ │ │ │  0022a520: 756e 2061 2066 756e 6374 696f 6e20 696e  un a function in
│ │ │ │  0022a530: 2061 6e6f 7468 6572 2074 6872 6561 6420   another thread 
│ │ │ │ @@ -142027,16 +142027,16 @@
│ │ │ │  0022aca0: 6861 7320 636f 6d70 6c65 7465 6420 7468  has completed th
│ │ │ │  0022acb0: 650a 636f 6d70 7574 6174 696f 6e2e 0a0a  e.computation...
│ │ │ │  0022acc0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0022acd0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131 203a  --------+.|i11 :
│ │ │ │  0022ace0: 2074 2020 2020 2020 2020 2020 2020 2020   t              
│ │ │ │  0022acf0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0022ad00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0022ad10: 3131 203d 203c 3c74 6173 6b2c 2072 756e  11 = <>|.|       
│ │ │ │ +0022ad10: 3131 203d 203c 3c74 6173 6b2c 2063 7265  11 = <>|.|       
│ │ │ │  0022ad30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0022ad40: 7c0a 7c6f 3131 203a 2054 6173 6b20 2020  |.|o11 : Task   
│ │ │ │  0022ad50: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  0022ad60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0022ad70: 2d2d 2d2d 2b0a 0a55 7365 202a 6e6f 7465  ----+..Use *note
│ │ │ │  0022ad80: 2069 7352 6561 6479 3a20 6973 5265 6164   isReady: isRead
│ │ │ │  0022ad90: 795f 6c70 4669 6c65 5f72 702c 2074 6f20  y_lpFile_rp, to 
│ │ │ │ @@ -143389,16 +143389,16 @@
│ │ │ │  002301c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  002301d0: 2d2d 2d2d 2b0a 7c69 3520 3a20 6e20 2020  ----+.|i5 : n   
│ │ │ │  002301e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002301f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00230200: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00230210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00230220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00230230: 2020 2020 2020 7c0a 7c6f 3520 3d20 3731        |.|o5 = 71
│ │ │ │ -00230240: 3132 3036 2020 2020 2020 2020 2020 2020  1206            
│ │ │ │ +00230230: 2020 2020 2020 7c0a 7c6f 3520 3d20 3130        |.|o5 = 10
│ │ │ │ +00230240: 3935 3031 3520 2020 2020 2020 2020 2020  95015           
│ │ │ │  00230250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00230260: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  00230270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00230280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00230290: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │  002302a0: 736c 6565 7020 3120 2020 2020 2020 2020  sleep 1         
│ │ │ │  002302b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -143432,16 +143432,16 @@
│ │ │ │  00230470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00230480: 2d2d 2b0a 7c69 3820 3a20 6e20 2020 2020  --+.|i8 : n     
│ │ │ │  00230490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002304a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002304b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  002304c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002304d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002304e0: 2020 2020 7c0a 7c6f 3820 3d20 3134 3533      |.|o8 = 1453
│ │ │ │ -002304f0: 3533 3320 2020 2020 2020 2020 2020 2020  533             
│ │ │ │ +002304e0: 2020 2020 7c0a 7c6f 3820 3d20 3232 3230      |.|o8 = 2220
│ │ │ │ +002304f0: 3831 3420 2020 2020 2020 2020 2020 2020  814             
│ │ │ │  00230500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00230510: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  00230520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00230530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00230540: 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20 6973  ------+.|i9 : is
│ │ │ │  00230550: 5265 6164 7920 7420 2020 2020 2020 2020  Ready t         
│ │ │ │  00230560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -143497,15 +143497,15 @@
│ │ │ │  00230880: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20  -------+.|i13 : 
│ │ │ │  00230890: 6e20 2020 2020 2020 2020 2020 2020 2020  n               
│ │ │ │  002308a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002308b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  002308c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002308d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002308e0: 2020 2020 2020 2020 207c 0a7c 6f31 3320           |.|o13 
│ │ │ │ -002308f0: 3d20 3134 3533 3734 3620 2020 2020 2020  = 1453746       
│ │ │ │ +002308f0: 3d20 3232 3231 3030 3120 2020 2020 2020  = 2221001       
│ │ │ │  00230900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00230910: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  00230920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00230930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00230940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │  00230950: 3420 3a20 736c 6565 7020 3120 2020 2020  4 : sleep 1     
│ │ │ │  00230960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -143521,16 +143521,16 @@
│ │ │ │  00230a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  00230a10: 0a7c 6931 3520 3a20 6e20 2020 2020 2020  .|i15 : n       
│ │ │ │  00230a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00230a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00230a40: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00230a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00230a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00230a70: 207c 0a7c 6f31 3520 3d20 3134 3533 3734   |.|o15 = 145374
│ │ │ │ -00230a80: 3620 2020 2020 2020 2020 2020 2020 2020  6               
│ │ │ │ +00230a70: 207c 0a7c 6f31 3520 3d20 3232 3231 3030   |.|o15 = 222100
│ │ │ │ +00230a80: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │  00230a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00230aa0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00230ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00230ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00230ad0: 2d2d 2d2b 0a7c 6931 3620 3a20 6973 5265  ---+.|i16 : isRe
│ │ │ │  00230ae0: 6164 7920 7420 2020 2020 2020 2020 2020  ady t           
│ │ │ │  00230af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -144288,15 +144288,15 @@
│ │ │ │  002339f0: 7365 740a 0a44 6573 6372 6970 7469 6f6e  set..Description
│ │ │ │  00233a00: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2b2d  .===========..+-
│ │ │ │  00233a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00233a20: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 206d  -------+.|i1 : m
│ │ │ │  00233a30: 6178 416c 6c6f 7761 626c 6554 6872 6561  axAllowableThrea
│ │ │ │  00233a40: 6473 7c0a 7c20 2020 2020 2020 2020 2020  ds|.|           
│ │ │ │  00233a50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00233a60: 6f31 203d 2037 2020 2020 2020 2020 2020  o1 = 7          
│ │ │ │ +00233a60: 6f31 203d 2031 3720 2020 2020 2020 2020  o1 = 17         
│ │ │ │  00233a70: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │  00233a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00233a90: 2d2d 2d2b 0a0a 5365 6520 616c 736f 0a3d  ---+..See also.=
│ │ │ │  00233aa0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f  =======..  * *no
│ │ │ │  00233ab0: 7465 2070 6172 616c 6c65 6c20 7072 6f67  te parallel prog
│ │ │ │  00233ac0: 7261 6d6d 696e 6720 7769 7468 2074 6872  ramming with thr
│ │ │ │  00233ad0: 6561 6473 2061 6e64 2074 6173 6b73 3a20  eads and tasks: 
│ │ │ │ @@ -145520,15 +145520,15 @@
│ │ │ │  002386f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00238700: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00238710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00238720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00238730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00238740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00238750: 207c 0a7c 6f33 203d 202f 746d 702f 4d32   |.|o3 = /tmp/M2
│ │ │ │ -00238760: 2d31 3039 3332 2d30 2f30 2020 2020 2020  -10932-0/0      
│ │ │ │ +00238760: 2d31 3137 3832 2d30 2f30 2020 2020 2020  -11782-0/0      
│ │ │ │  00238770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00238780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00238790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002387a0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  002387b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  002387c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  002387d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -145540,15 +145540,15 @@
│ │ │ │  00238830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00238840: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00238850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00238860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00238870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00238880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00238890: 207c 0a7c 6f34 203d 202f 746d 702f 4d32   |.|o4 = /tmp/M2
│ │ │ │ -002388a0: 2d31 3039 3332 2d30 2f30 2020 2020 2020  -10932-0/0      
│ │ │ │ +002388a0: 2d31 3137 3832 2d30 2f30 2020 2020 2020  -11782-0/0      
│ │ │ │  002388b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002388c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002388d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002388e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  002388f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00238900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00238910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -145700,15 +145700,15 @@
│ │ │ │  00239230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00239240: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00239250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00239260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00239270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00239280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00239290: 207c 0a7c 6f39 203d 202f 746d 702f 4d32   |.|o9 = /tmp/M2
│ │ │ │ -002392a0: 2d31 3039 3332 2d30 2f30 2020 2020 2020  -10932-0/0      
│ │ │ │ +002392a0: 2d31 3137 3832 2d30 2f30 2020 2020 2020  -11782-0/0      
│ │ │ │  002392b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002392c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002392d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002392e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  002392f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00239300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00239310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -145760,15 +145760,15 @@
│ │ │ │  002395f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00239600: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00239610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00239620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00239630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00239640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00239650: 207c 0a7c 6f31 3120 3d20 2f74 6d70 2f4d   |.|o11 = /tmp/M
│ │ │ │ -00239660: 322d 3130 3933 322d 302f 3020 2020 2020  2-10932-0/0     
│ │ │ │ +00239660: 322d 3131 3738 322d 302f 3020 2020 2020  2-11782-0/0     
│ │ │ │  00239670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00239680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00239690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002396a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  002396b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002396c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002396d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -146165,15 +146165,15 @@
│ │ │ │  0023af40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0023af50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0023af60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0023af70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0023af80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0023af90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0023afa0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -0023afb0: 203d 202f 746d 702f 4d32 2d31 3131 3037   = /tmp/M2-11107
│ │ │ │ +0023afb0: 203d 202f 746d 702f 4d32 2d31 3231 3337   = /tmp/M2-12137
│ │ │ │  0023afc0: 2d30 2f30 2020 2020 2020 2020 2020 2020  -0/0            
│ │ │ │  0023afd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0023afe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0023aff0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  0023b000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0023b010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0023b020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -146185,15 +146185,15 @@
│ │ │ │  0023b080: 5e32 2b31 322a 795e 322a 7a5e 322b 785e  ^2+12*y^2*z^2+x^
│ │ │ │  0023b090: 332b 362a 785e 322a 792b 207c 0a7c 2020  3+6*x^2*y+ |.|  
│ │ │ │  0023b0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0023b0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0023b0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0023b0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0023b0e0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -0023b0f0: 203d 202f 746d 702f 4d32 2d31 3131 3037   = /tmp/M2-11107
│ │ │ │ +0023b0f0: 203d 202f 746d 702f 4d32 2d31 3231 3337   = /tmp/M2-12137
│ │ │ │  0023b100: 2d30 2f30 2020 2020 2020 2020 2020 2020  -0/0            
│ │ │ │  0023b110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0023b120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0023b130: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0023b140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0023b150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0023b160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -146401,15 +146401,15 @@
│ │ │ │  0023be00: 7361 6d70 6c65 203d 2032 5e31 3030 7c0a  sample = 2^100|.
│ │ │ │  0023be10: 7c20 2020 2020 7072 696e 7420 7361 6d70  |     print samp
│ │ │ │  0023be20: 6c65 2020 2020 2020 2020 207c 0a7c 2020  le         |.|  
│ │ │ │  0023be30: 2020 2022 203c 3c20 636c 6f73 6520 2020     " << close   
│ │ │ │  0023be40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0023be50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0023be60: 2020 2020 207c 0a7c 6f37 203d 202f 746d       |.|o7 = /tm
│ │ │ │ -0023be70: 702f 4d32 2d31 3131 3037 2d30 2f30 2020  p/M2-11107-0/0  
│ │ │ │ +0023be70: 702f 4d32 2d31 3231 3337 2d30 2f30 2020  p/M2-12137-0/0  
│ │ │ │  0023be80: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0023be90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0023bea0: 0a7c 6f37 203a 2046 696c 6520 2020 2020  .|o7 : File     
│ │ │ │  0023beb0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  0023bec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0023bed0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a4e 6f77 2076  ---------+.Now v
│ │ │ │  0023bee0: 6572 6966 7920 7468 6174 2069 7420 636f  erify that it co
│ │ │ │ @@ -147002,15 +147002,15 @@
│ │ │ │  0023e390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0023e3a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0023e3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0023e3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0023e3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0023e3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0023e3f0: 2020 207c 0a7c 6f35 203d 202f 746d 702f     |.|o5 = /tmp/
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│ │ │ │  0023e420: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ @@ -147022,15 +147022,15 @@
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│ │ │ │  0023e4e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0023e4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0023e500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0023e510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0023e520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0023e530: 2020 207c 0a7c 6f36 203d 202f 746d 702f     |.|o6 = /tmp/
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│ │ │ │  0023e550: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0023e570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0023e580: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0023e590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0023e5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0023e5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -147987,24 +147987,24 @@
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│ │ │ │  00242160: 696c 654e 616d 6528 2920 7c0a 7c20 2020  ileName() |.|   
│ │ │ │  00242170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00242180: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -00242190: 203d 202f 746d 702f 4d32 2d31 3231 3838   = /tmp/M2-12188
│ │ │ │ +00242190: 203d 202f 746d 702f 4d32 2d31 3433 3438   = /tmp/M2-14348
│ │ │ │  002421a0: 2d30 2f30 2020 2020 2020 2020 7c0a 2b2d  -0/0        |.+-
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│ │ │ │  002421c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  002421d0: 6932 203a 2066 6e20 3c3c 2022 6869 2074  i2 : fn << "hi t
│ │ │ │  002421e0: 6865 7265 2220 3c3c 2063 6c6f 7365 7c0a  here" << close|.
│ │ │ │  002421f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00242200: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00242210: 0a7c 6f32 203d 202f 746d 702f 4d32 2d31  .|o2 = /tmp/M2-1
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│ │ │ │ +00242220: 3433 3438 2d30 2f30 2020 2020 2020 2020  4348-0/0        
│ │ │ │  00242230: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00242240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00242250: 207c 0a7c 6f32 203a 2046 696c 6520 2020   |.|o2 : File   
│ │ │ │  00242260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00242270: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
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│ │ │ │  00242290: 2d2d 2d2b 0a7c 6933 203a 2069 7352 6567  ---+.|i3 : isReg
│ │ │ │ @@ -148090,24 +148090,24 @@
│ │ │ │  00242790: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
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│ │ │ │  002427c0: 666e 203d 2074 656d 706f 7261 7279 4669  fn = temporaryFi
│ │ │ │  002427d0: 6c65 4e61 6d65 2829 207c 0a7c 2020 2020  leName() |.|    
│ │ │ │  002427e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002427f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ -00242800: 3d20 2f74 6d70 2f4d 322d 3130 3338 302d  = /tmp/M2-10380-
│ │ │ │ +00242800: 3d20 2f74 6d70 2f4d 322d 3130 3637 302d  = /tmp/M2-10670-
│ │ │ │  00242810: 302f 3020 2020 2020 2020 207c 0a2b 2d2d  0/0        |.+--
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│ │ │ │  00242830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  00242840: 3320 3a20 666e 203c 3c20 2268 6920 7468  3 : fn << "hi th
│ │ │ │  00242850: 6572 6522 203c 3c20 636c 6f73 657c 0a7c  ere" << close|.|
│ │ │ │  00242860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00242870: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00242880: 7c6f 3320 3d20 2f74 6d70 2f4d 322d 3130  |o3 = /tmp/M2-10
│ │ │ │ -00242890: 3338 302d 302f 3020 2020 2020 2020 207c  380-0/0        |
│ │ │ │ +00242890: 3637 302d 302f 3020 2020 2020 2020 207c  670-0/0        |
│ │ │ │  002428a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  002428b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002428c0: 7c0a 7c6f 3320 3a20 4669 6c65 2020 2020  |.|o3 : File    
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│ │ │ │  00242900: 2d2d 2b0a 7c69 3420 3a20 6973 4469 7265  --+.|i4 : isDire
│ │ │ │ @@ -148286,25 +148286,25 @@
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│ │ │ │  002433e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  002433f0: 0a7c 6931 203a 2074 656d 706f 7261 7279  .|i1 : temporary
│ │ │ │  00243400: 4669 6c65 4e61 6d65 2028 2920 7c20 222e  FileName () | ".
│ │ │ │  00243410: 7465 7822 207c 0a7c 2020 2020 2020 2020  tex" |.|        
│ │ │ │  00243420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00243430: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -00243440: 203d 202f 746d 702f 4d32 2d31 3231 3639   = /tmp/M2-12169
│ │ │ │ +00243440: 203d 202f 746d 702f 4d32 2d31 3433 3039   = /tmp/M2-14309
│ │ │ │  00243450: 2d30 2f30 2e74 6578 2020 2020 2020 2020  -0/0.tex        
│ │ │ │  00243460: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00243470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00243480: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2074  -------+.|i2 : t
│ │ │ │  00243490: 656d 706f 7261 7279 4669 6c65 4e61 6d65  emporaryFileName
│ │ │ │  002434a0: 2028 2920 7c20 222e 6874 6d6c 227c 0a7c   () | ".html"|.|
│ │ │ │  002434b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002434c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002434d0: 2020 207c 0a7c 6f32 203d 202f 746d 702f     |.|o2 = /tmp/
│ │ │ │ -002434e0: 4d32 2d31 3231 3639 2d30 2f31 2e68 746d  M2-12169-0/1.htm
│ │ │ │ +002434e0: 4d32 2d31 3433 3039 2d30 2f31 2e68 746d  M2-14309-0/1.htm
│ │ │ │  002434f0: 6c20 2020 2020 2020 207c 0a2b 2d2d 2d2d  l        |.+----
│ │ │ │  00243500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00243510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  00243520: 0a0a 5468 6973 2066 756e 6374 696f 6e20  ..This function 
│ │ │ │  00243530: 7769 6c6c 2077 6f72 6b20 756e 6465 7220  will work under 
│ │ │ │  00243540: 556e 6978 2c20 616e 6420 616c 736f 2075  Unix, and also u
│ │ │ │  00243550: 6e64 6572 2057 696e 646f 7773 2069 6620  nder Windows if 
│ │ │ │ @@ -148468,23 +148468,23 @@
│ │ │ │  00243f30: 3d3d 3d0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  ===..+----------
│ │ │ │  00243f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00243f50: 2d2d 2d2b 0a7c 6931 203a 2066 6e20 3d20  ---+.|i1 : fn = 
│ │ │ │  00243f60: 7465 6d70 6f72 6172 7946 696c 654e 616d  temporaryFileNam
│ │ │ │  00243f70: 6528 297c 0a7c 2020 2020 2020 2020 2020  e()|.|          
│ │ │ │  00243f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00243f90: 2020 207c 0a7c 6f31 203d 202f 746d 702f     |.|o1 = /tmp/
│ │ │ │ -00243fa0: 4d32 2d31 3133 3735 2d30 2f30 2020 2020  M2-11375-0/0    
│ │ │ │ +00243fa0: 4d32 2d31 3236 3835 2d30 2f30 2020 2020  M2-12685-0/0    
│ │ │ │  00243fb0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
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│ │ │ │  00243fd0: 2d2d 2d2b 0a7c 6932 203a 2066 203d 2066  ---+.|i2 : f = f
│ │ │ │  00243fe0: 6e20 3c3c 2022 6869 2074 6865 7265 2220  n << "hi there" 
│ │ │ │  00243ff0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00244000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00244010: 2020 207c 0a7c 6f32 203d 202f 746d 702f     |.|o2 = /tmp/
│ │ │ │ -00244020: 4d32 2d31 3133 3735 2d30 2f30 2020 2020  M2-11375-0/0    
│ │ │ │ +00244020: 4d32 2d31 3236 3835 2d30 2f30 2020 2020  M2-12685-0/0    
│ │ │ │  00244030: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00244040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00244050: 2020 207c 0a7c 6f32 203a 2046 696c 6520     |.|o2 : File 
│ │ │ │  00244060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00244070: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00244080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00244090: 2d2d 2d2b 0a7c 6933 203a 2066 696c 654d  ---+.|i3 : fileM
│ │ │ │ @@ -148496,15 +148496,15 @@
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│ │ │ │  00244110: 2d2d 2d2b 0a7c 6934 203a 2063 6c6f 7365  ---+.|i4 : close
│ │ │ │  00244120: 2066 2020 2020 2020 2020 2020 2020 2020   f              
│ │ │ │  00244130: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00244140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00244150: 2020 207c 0a7c 6f34 203d 202f 746d 702f     |.|o4 = /tmp/
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│ │ │ │ +00244160: 4d32 2d31 3236 3835 2d30 2f30 2020 2020  M2-12685-0/0    
│ │ │ │  00244170: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00244180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00244190: 2020 207c 0a7c 6f34 203a 2046 696c 6520     |.|o4 : File 
│ │ │ │  002441a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  002441d0: 2d2d 2d2b 0a7c 6935 203a 2072 656d 6f76  ---+.|i5 : remov
│ │ │ │ @@ -148562,23 +148562,23 @@
│ │ │ │  00244510: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2b2d 2d2d  ==========..+---
│ │ │ │  00244520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00244530: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120  ----------+.|i1 
│ │ │ │  00244540: 3a20 666e 203d 2074 656d 706f 7261 7279  : fn = temporary
│ │ │ │  00244550: 4669 6c65 4e61 6d65 2829 7c0a 7c20 2020  FileName()|.|   
│ │ │ │  00244560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00244570: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ -00244580: 3d20 2f74 6d70 2f4d 322d 3130 3835 342d  = /tmp/M2-10854-
│ │ │ │ +00244580: 3d20 2f74 6d70 2f4d 322d 3131 3632 342d  = /tmp/M2-11624-
│ │ │ │  00244590: 302f 3020 2020 2020 2020 7c0a 2b2d 2d2d  0/0       |.+---
│ │ │ │  002445a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  002445b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220  ----------+.|i2 
│ │ │ │  002445c0: 3a20 6620 3d20 666e 203c 3c20 2268 6920  : f = fn << "hi 
│ │ │ │  002445d0: 7468 6572 6522 2020 2020 7c0a 7c20 2020  there"    |.|   
│ │ │ │  002445e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002445f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ -00244600: 3d20 2f74 6d70 2f4d 322d 3130 3835 342d  = /tmp/M2-10854-
│ │ │ │ +00244600: 3d20 2f74 6d70 2f4d 322d 3131 3632 342d  = /tmp/M2-11624-
│ │ │ │  00244610: 302f 3020 2020 2020 2020 7c0a 7c20 2020  0/0       |.|   
│ │ │ │  00244620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00244630: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │  00244640: 3a20 4669 6c65 2020 2020 2020 2020 2020  : File          
│ │ │ │  00244650: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  00244660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00244670: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320  ----------+.|i3 
│ │ │ │ @@ -148602,15 +148602,15 @@
│ │ │ │  00244790: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  002447a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  002447b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620  ----------+.|i6 
│ │ │ │  002447c0: 3a20 636c 6f73 6520 6620 2020 2020 2020  : close f       
│ │ │ │  002447d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  002447e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002447f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3620            |.|o6 
│ │ │ │ -00244800: 3d20 2f74 6d70 2f4d 322d 3130 3835 342d  = /tmp/M2-10854-
│ │ │ │ +00244800: 3d20 2f74 6d70 2f4d 322d 3131 3632 342d  = /tmp/M2-11624-
│ │ │ │  00244810: 302f 3020 2020 2020 2020 7c0a 7c20 2020  0/0       |.|   
│ │ │ │  00244820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00244830: 2020 2020 2020 2020 2020 7c0a 7c6f 3620            |.|o6 
│ │ │ │  00244840: 3a20 4669 6c65 2020 2020 2020 2020 2020  : File          
│ │ │ │  00244850: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  00244860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00244870: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720  ----------+.|i7 
│ │ │ │ @@ -148678,24 +148678,24 @@
│ │ │ │  00244c50: 3d3d 3d3d 3d3d 3d0a 0a2b 2d2d 2d2d 2d2d  =======..+------
│ │ │ │  00244c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00244c70: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │  00244c80: 666e 203d 2074 656d 706f 7261 7279 4669  fn = temporaryFi
│ │ │ │  00244c90: 6c65 4e61 6d65 2829 207c 0a7c 2020 2020  leName() |.|    
│ │ │ │  00244ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00244cb0: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ -00244cc0: 3d20 2f74 6d70 2f4d 322d 3130 3938 392d  = /tmp/M2-10989-
│ │ │ │ +00244cc0: 3d20 2f74 6d70 2f4d 322d 3131 3839 392d  = /tmp/M2-11899-
│ │ │ │  00244cd0: 302f 3020 2020 2020 2020 207c 0a2b 2d2d  0/0        |.+--
│ │ │ │  00244ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00244cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  00244d00: 3220 3a20 666e 203c 3c20 2268 6920 7468  2 : fn << "hi th
│ │ │ │  00244d10: 6572 6522 203c 3c20 636c 6f73 657c 0a7c  ere" << close|.|
│ │ │ │  00244d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00244d30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00244d40: 7c6f 3220 3d20 2f74 6d70 2f4d 322d 3130  |o2 = /tmp/M2-10
│ │ │ │ -00244d50: 3938 392d 302f 3020 2020 2020 2020 207c  989-0/0        |
│ │ │ │ +00244d40: 7c6f 3220 3d20 2f74 6d70 2f4d 322d 3131  |o2 = /tmp/M2-11
│ │ │ │ +00244d50: 3839 392d 302f 3020 2020 2020 2020 207c  899-0/0        |
│ │ │ │  00244d60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00244d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00244d80: 7c0a 7c6f 3220 3a20 4669 6c65 2020 2020  |.|o2 : File    
│ │ │ │  00244d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00244da0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00244db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00244dc0: 2d2d 2b0a 7c69 3320 3a20 6669 6c65 4d6f  --+.|i3 : fileMo
│ │ │ │ @@ -148763,23 +148763,23 @@
│ │ │ │  002451a0: 3d0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  =..+------------
│ │ │ │  002451b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  002451c0: 2d2d 2b0a 7c69 3120 3a20 666e 203d 2074  --+.|i1 : fn = t
│ │ │ │  002451d0: 656d 706f 7261 7279 4669 6c65 4e61 6d65  emporaryFileName
│ │ │ │  002451e0: 2829 207c 0a7c 2020 2020 2020 2020 2020  () |.|          
│ │ │ │  002451f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00245200: 2020 2020 7c0a 7c6f 3120 3d20 2f74 6d70      |.|o1 = /tmp
│ │ │ │ -00245210: 2f4d 322d 3131 3937 372d 302f 3020 2020  /M2-11977-0/0   
│ │ │ │ +00245210: 2f4d 322d 3133 3931 372d 302f 3020 2020  /M2-13917-0/0   
│ │ │ │  00245220: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  00245230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00245240: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 666e  ------+.|i2 : fn
│ │ │ │  00245250: 203c 3c20 2268 6920 7468 6572 6522 203c   << "hi there" <
│ │ │ │  00245260: 3c20 636c 6f73 657c 0a7c 2020 2020 2020  < close|.|      
│ │ │ │  00245270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00245280: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20          |.|o2 = 
│ │ │ │ -00245290: 2f74 6d70 2f4d 322d 3131 3937 372d 302f  /tmp/M2-11977-0/
│ │ │ │ +00245290: 2f74 6d70 2f4d 322d 3133 3931 372d 302f  /tmp/M2-13917-0/
│ │ │ │  002452a0: 3020 2020 2020 2020 207c 0a7c 2020 2020  0        |.|    
│ │ │ │  002452b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002452c0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │  002452d0: 3a20 4669 6c65 2020 2020 2020 2020 2020  : File          
│ │ │ │  002452e0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  002452f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00245300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ @@ -148857,15 +148857,15 @@
│ │ │ │  00245780: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00245790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  002457a0: 2b0a 7c69 3120 3a20 666e 203d 2074 656d  +.|i1 : fn = tem
│ │ │ │  002457b0: 706f 7261 7279 4669 6c65 4e61 6d65 2829  poraryFileName()
│ │ │ │  002457c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  002457d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002457e0: 2020 7c0a 7c6f 3120 3d20 2f74 6d70 2f4d    |.|o1 = /tmp/M
│ │ │ │ -002457f0: 322d 3130 3535 382d 302f 3020 2020 2020  2-10558-0/0     
│ │ │ │ +002457f0: 322d 3131 3032 382d 302f 3020 2020 2020  2-11028-0/0     
│ │ │ │  00245800: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00245810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00245820: 2d2d 2d2d 2b0a 7c69 3220 3a20 6669 6c65  ----+.|i2 : file
│ │ │ │  00245830: 4578 6973 7473 2066 6e20 2020 2020 2020  Exists fn       
│ │ │ │  00245840: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00245850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00245860: 2020 2020 2020 7c0a 7c6f 3220 3d20 6661        |.|o2 = fa
│ │ │ │ @@ -148873,15 +148873,15 @@
│ │ │ │  00245880: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  00245890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  002458a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │  002458b0: 666e 203c 3c20 2268 6920 7468 6572 6522  fn << "hi there"
│ │ │ │  002458c0: 203c 3c20 636c 6f73 657c 0a7c 2020 2020   << close|.|    
│ │ │ │  002458d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002458e0: 2020 2020 2020 2020 2020 7c0a 7c6f 3320            |.|o3 
│ │ │ │ -002458f0: 3d20 2f74 6d70 2f4d 322d 3130 3535 382d  = /tmp/M2-10558-
│ │ │ │ +002458f0: 3d20 2f74 6d70 2f4d 322d 3131 3032 382d  = /tmp/M2-11028-
│ │ │ │  00245900: 302f 3020 2020 2020 2020 207c 0a7c 2020  0/0        |.|  
│ │ │ │  00245910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00245920: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │  00245930: 3320 3a20 4669 6c65 2020 2020 2020 2020  3 : File        
│ │ │ │  00245940: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │  00245950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00245960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ @@ -149175,15 +149175,15 @@
│ │ │ │  00246b60: 2e0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │  00246b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00246b80: 2d2d 2d2d 2b0a 7c69 3120 3a20 6375 7272  ----+.|i1 : curr
│ │ │ │  00246b90: 656e 7454 696d 6528 2920 2d20 6669 6c65  entTime() - file
│ │ │ │  00246ba0: 5469 6d65 2022 2e22 7c0a 7c20 2020 2020  Time "."|.|     
│ │ │ │  00246bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00246bc0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00246bd0: 3120 3d20 3631 2020 2020 2020 2020 2020  1 = 61          
│ │ │ │ +00246bd0: 3120 3d20 3436 2020 2020 2020 2020 2020  1 = 46          
│ │ │ │  00246be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00246bf0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00246c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00246c10: 2d2d 2d2d 2b0a 0a53 6565 2061 6c73 6f0a  ----+..See also.
│ │ │ │  00246c20: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │  00246c30: 6f74 6520 6375 7272 656e 7454 696d 653a  ote currentTime:
│ │ │ │  00246c40: 2063 7572 7265 6e74 5469 6d65 2c20 2d2d   currentTime, --
│ │ │ │ @@ -149284,15 +149284,15 @@
│ │ │ │  00247230: 2829 2020 2020 2020 2020 2020 2020 2020  ()              
│ │ │ │  00247240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00247250: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00247260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00247270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00247280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00247290: 2020 2020 2020 207c 0a7c 6f31 203d 202f         |.|o1 = /
│ │ │ │ -002472a0: 746d 702f 4d32 2d31 3039 3730 2d30 2f30  tmp/M2-10970-0/0
│ │ │ │ +002472a0: 746d 702f 4d32 2d31 3138 3630 2d30 2f30  tmp/M2-11860-0/0
│ │ │ │  002472b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002472c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002472d0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  002472e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  002472f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00247300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00247310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ @@ -149301,15 +149301,15 @@
│ │ │ │  00247340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00247350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00247360: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00247370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00247380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00247390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002473a0: 2020 207c 0a7c 6f32 203d 202f 746d 702f     |.|o2 = /tmp/
│ │ │ │ -002473b0: 4d32 2d31 3039 3730 2d30 2f31 2020 2020  M2-10970-0/1    
│ │ │ │ +002473b0: 4d32 2d31 3138 3630 2d30 2f31 2020 2020  M2-11860-0/1    
│ │ │ │  002473c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002473d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002473e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  002473f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00247400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00247410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00247420: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ @@ -149318,15 +149318,15 @@
│ │ │ │  00247450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00247460: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00247470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00247480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00247490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002474a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  002474b0: 0a7c 6f33 203d 202f 746d 702f 4d32 2d31  .|o3 = /tmp/M2-1
│ │ │ │ -002474c0: 3039 3730 2d30 2f30 2020 2020 2020 2020  0970-0/0        
│ │ │ │ +002474c0: 3138 3630 2d30 2f30 2020 2020 2020 2020  1860-0/0        
│ │ │ │  002474d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002474e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002474f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00247500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00247510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00247520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00247530: 2020 2020 207c 0a7c 6f33 203a 2046 696c       |.|o3 : Fil
│ │ │ │ @@ -149339,16 +149339,16 @@
│ │ │ │  002475a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  002475b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │  002475c0: 203a 2063 6f70 7946 696c 6528 7372 632c   : copyFile(src,
│ │ │ │  002475d0: 6473 742c 5665 7262 6f73 653d 3e74 7275  dst,Verbose=>tru
│ │ │ │  002475e0: 6529 2020 2020 2020 2020 2020 2020 2020  e)              
│ │ │ │  002475f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00247600: 7c20 2d2d 2063 6f70 7969 6e67 3a20 2f74  | -- copying: /t
│ │ │ │ -00247610: 6d70 2f4d 322d 3130 3937 302d 302f 3020  mp/M2-10970-0/0 
│ │ │ │ -00247620: 2d3e 202f 746d 702f 4d32 2d31 3039 3730  -> /tmp/M2-10970
│ │ │ │ +00247610: 6d70 2f4d 322d 3131 3836 302d 302f 3020  mp/M2-11860-0/0 
│ │ │ │ +00247620: 2d3e 202f 746d 702f 4d32 2d31 3138 3630  -> /tmp/M2-11860
│ │ │ │  00247630: 2d30 2f31 2020 2020 2020 2020 2020 2020  -0/1            
│ │ │ │  00247640: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00247650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00247660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00247670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00247680: 2d2d 2d2d 2b0a 7c69 3520 3a20 6765 7420  ----+.|i5 : get 
│ │ │ │  00247690: 6473 7420 2020 2020 2020 2020 2020 2020  dst             
│ │ │ │ @@ -149368,32 +149368,32 @@
│ │ │ │  00247770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00247780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00247790: 2b0a 7c69 3620 3a20 636f 7079 4669 6c65  +.|i6 : copyFile
│ │ │ │  002477a0: 2873 7263 2c64 7374 2c56 6572 626f 7365  (src,dst,Verbose
│ │ │ │  002477b0: 3d3e 7472 7565 2c55 7064 6174 654f 6e6c  =>true,UpdateOnl
│ │ │ │  002477c0: 7920 3d3e 2074 7275 6529 2020 2020 2020  y => true)      
│ │ │ │  002477d0: 2020 207c 0a7c 202d 2d20 736b 6970 7069     |.| -- skippi
│ │ │ │ -002477e0: 6e67 3a20 2f74 6d70 2f4d 322d 3130 3937  ng: /tmp/M2-1097
│ │ │ │ +002477e0: 6e67 3a20 2f74 6d70 2f4d 322d 3131 3836  ng: /tmp/M2-1186
│ │ │ │  002477f0: 302d 302f 3020 6e6f 7420 6e65 7765 7220  0-0/0 not newer 
│ │ │ │ -00247800: 7468 616e 202f 746d 702f 4d32 2d31 3039  than /tmp/M2-109
│ │ │ │ -00247810: 3730 2d30 2f31 7c0a 2b2d 2d2d 2d2d 2d2d  70-0/1|.+-------
│ │ │ │ +00247800: 7468 616e 202f 746d 702f 4d32 2d31 3138  than /tmp/M2-118
│ │ │ │ +00247810: 3630 2d30 2f31 7c0a 2b2d 2d2d 2d2d 2d2d  60-0/1|.+-------
│ │ │ │  00247820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00247830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00247840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00247850: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a  ---------+.|i7 :
│ │ │ │  00247860: 2073 7263 203c 3c20 2268 6f20 7468 6572   src << "ho ther
│ │ │ │  00247870: 6522 203c 3c20 636c 6f73 6520 2020 2020  e" << close     
│ │ │ │  00247880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00247890: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  002478a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002478b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002478c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002478d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  002478e0: 0a7c 6f37 203d 202f 746d 702f 4d32 2d31  .|o7 = /tmp/M2-1
│ │ │ │ -002478f0: 3039 3730 2d30 2f30 2020 2020 2020 2020  0970-0/0        
│ │ │ │ +002478f0: 3138 3630 2d30 2f30 2020 2020 2020 2020  1860-0/0        
│ │ │ │  00247900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00247910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00247920: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00247930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00247940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00247950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00247960: 2020 2020 207c 0a7c 6f37 203a 2046 696c       |.|o7 : Fil
│ │ │ │ @@ -149406,17 +149406,17 @@
│ │ │ │  002479d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  002479e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938  -----------+.|i8
│ │ │ │  002479f0: 203a 2063 6f70 7946 696c 6528 7372 632c   : copyFile(src,
│ │ │ │  00247a00: 6473 742c 5665 7262 6f73 653d 3e74 7275  dst,Verbose=>tru
│ │ │ │  00247a10: 652c 5570 6461 7465 4f6e 6c79 203d 3e20  e,UpdateOnly => 
│ │ │ │  00247a20: 7472 7565 2920 2020 2020 2020 2020 7c0a  true)         |.
│ │ │ │  00247a30: 7c20 2d2d 2073 6b69 7070 696e 673a 202f  | -- skipping: /
│ │ │ │ -00247a40: 746d 702f 4d32 2d31 3039 3730 2d30 2f30  tmp/M2-10970-0/0
│ │ │ │ +00247a40: 746d 702f 4d32 2d31 3138 3630 2d30 2f30  tmp/M2-11860-0/0
│ │ │ │  00247a50: 206e 6f74 206e 6577 6572 2074 6861 6e20   not newer than 
│ │ │ │ -00247a60: 2f74 6d70 2f4d 322d 3130 3937 302d 302f  /tmp/M2-10970-0/
│ │ │ │ +00247a60: 2f74 6d70 2f4d 322d 3131 3836 302d 302f  /tmp/M2-11860-0/
│ │ │ │  00247a70: 317c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  1|.+------------
│ │ │ │  00247a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00247a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00247aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00247ab0: 2d2d 2d2d 2b0a 7c69 3920 3a20 6765 7420  ----+.|i9 : get 
│ │ │ │  00247ac0: 6473 7420 2020 2020 2020 2020 2020 2020  dst             
│ │ │ │  00247ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -149554,41 +149554,41 @@
│ │ │ │  00248310: 2d2b 0a7c 6931 203a 2073 7263 203d 2074  -+.|i1 : src = t
│ │ │ │  00248320: 656d 706f 7261 7279 4669 6c65 4e61 6d65  emporaryFileName
│ │ │ │  00248330: 2829 2020 2020 2020 2020 2020 2020 2020  ()              
│ │ │ │  00248340: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00248350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00248360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00248370: 2020 2020 2020 207c 0a7c 6f31 203d 202f         |.|o1 = /
│ │ │ │ -00248380: 746d 702f 4d32 2d31 3036 3135 2d30 2f30  tmp/M2-10615-0/0
│ │ │ │ +00248380: 746d 702f 4d32 2d31 3131 3435 2d30 2f30  tmp/M2-11145-0/0
│ │ │ │  00248390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002483a0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  002483b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  002483c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  002483d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  002483e0: 6932 203a 2064 7374 203d 2074 656d 706f  i2 : dst = tempo
│ │ │ │  002483f0: 7261 7279 4669 6c65 4e61 6d65 2829 2020  raryFileName()  
│ │ │ │  00248400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00248410: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00248420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00248430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00248440: 2020 207c 0a7c 6f32 203d 202f 746d 702f     |.|o2 = /tmp/
│ │ │ │ -00248450: 4d32 2d31 3036 3135 2d30 2f31 2020 2020  M2-10615-0/1    
│ │ │ │ +00248450: 4d32 2d31 3131 3435 2d30 2f31 2020 2020  M2-11145-0/1    
│ │ │ │  00248460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00248470: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  00248480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00248490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  002484a0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │  002484b0: 2073 7263 203c 3c20 2268 6920 7468 6572   src << "hi ther
│ │ │ │  002484c0: 6522 203c 3c20 636c 6f73 6520 2020 2020  e" << close     
│ │ │ │  002484d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  002484e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002484f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00248500: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00248510: 0a7c 6f33 203d 202f 746d 702f 4d32 2d31  .|o3 = /tmp/M2-1
│ │ │ │ -00248520: 3036 3135 2d30 2f30 2020 2020 2020 2020  0615-0/0        
│ │ │ │ +00248520: 3131 3435 2d30 2f30 2020 2020 2020 2020  1145-0/0        
│ │ │ │  00248530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00248540: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00248550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00248560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00248570: 2020 2020 207c 0a7c 6f33 203a 2046 696c       |.|o3 : Fil
│ │ │ │  00248580: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │  00248590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -149596,16 +149596,16 @@
│ │ │ │  002485b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  002485c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  002485d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │  002485e0: 203a 206d 6f76 6546 696c 6528 7372 632c   : moveFile(src,
│ │ │ │  002485f0: 6473 742c 5665 7262 6f73 653d 3e74 7275  dst,Verbose=>tru
│ │ │ │  00248600: 6529 2020 2020 2020 2020 2020 2020 7c0a  e)            |.
│ │ │ │  00248610: 7c2d 2d6d 6f76 696e 673a 202f 746d 702f  |--moving: /tmp/
│ │ │ │ -00248620: 4d32 2d31 3036 3135 2d30 2f30 202d 3e20  M2-10615-0/0 -> 
│ │ │ │ -00248630: 2f74 6d70 2f4d 322d 3130 3631 352d 302f  /tmp/M2-10615-0/
│ │ │ │ +00248620: 4d32 2d31 3131 3435 2d30 2f30 202d 3e20  M2-11145-0/0 -> 
│ │ │ │ +00248630: 2f74 6d70 2f4d 322d 3131 3134 352d 302f  /tmp/M2-11145-0/
│ │ │ │  00248640: 317c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  1|.+------------
│ │ │ │  00248650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00248660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00248670: 2d2d 2d2d 2b0a 7c69 3520 3a20 6765 7420  ----+.|i5 : get 
│ │ │ │  00248680: 6473 7420 2020 2020 2020 2020 2020 2020  dst             
│ │ │ │  00248690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002486a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ @@ -149619,20 +149619,20 @@
│ │ │ │  00248720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00248730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00248740: 2b0a 7c69 3620 3a20 6261 6b20 3d20 6d6f  +.|i6 : bak = mo
│ │ │ │  00248750: 7665 4669 6c65 2864 7374 2c56 6572 626f  veFile(dst,Verbo
│ │ │ │  00248760: 7365 3d3e 7472 7565 2920 2020 2020 2020  se=>true)       
│ │ │ │  00248770: 2020 207c 0a7c 2d2d 6261 636b 7570 2066     |.|--backup f
│ │ │ │  00248780: 696c 6520 6372 6561 7465 643a 202f 746d  ile created: /tm
│ │ │ │ -00248790: 702f 4d32 2d31 3036 3135 2d30 2f31 2e62  p/M2-10615-0/1.b
│ │ │ │ +00248790: 702f 4d32 2d31 3131 3435 2d30 2f31 2e62  p/M2-11145-0/1.b
│ │ │ │  002487a0: 616b 2020 2020 7c0a 7c20 2020 2020 2020  ak    |.|       
│ │ │ │  002487b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002487c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002487d0: 2020 2020 2020 2020 207c 0a7c 6f36 203d           |.|o6 =
│ │ │ │ -002487e0: 202f 746d 702f 4d32 2d31 3036 3135 2d30   /tmp/M2-10615-0
│ │ │ │ +002487e0: 202f 746d 702f 4d32 2d31 3131 3435 2d30   /tmp/M2-11145-0
│ │ │ │  002487f0: 2f31 2e62 616b 2020 2020 2020 2020 2020  /1.bak          
│ │ │ │  00248800: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  00248810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00248820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00248830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  00248840: 0a7c 6937 203a 2072 656d 6f76 6546 696c  .|i7 : removeFil
│ │ │ │  00248850: 6520 6261 6b20 2020 2020 2020 2020 2020  e bak           
│ │ │ │ @@ -149848,15 +149848,15 @@
│ │ │ │  00249570: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00249580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  00249590: 0a7c 6931 203a 2066 6e20 3d20 7465 6d70  .|i1 : fn = temp
│ │ │ │  002495a0: 6f72 6172 7946 696c 654e 616d 6528 297c  oraryFileName()|
│ │ │ │  002495b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  002495c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  002495d0: 0a7c 6f31 203d 202f 746d 702f 4d32 2d31  .|o1 = /tmp/M2-1
│ │ │ │ -002495e0: 3132 3034 2d30 2f30 2020 2020 2020 207c  1204-0/0       |
│ │ │ │ +002495e0: 3233 3334 2d30 2f30 2020 2020 2020 207c  2334-0/0       |
│ │ │ │  002495f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00249600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  00249610: 0a7c 6932 203a 2073 796d 6c69 6e6b 4669  .|i2 : symlinkFi
│ │ │ │  00249620: 6c65 2822 7177 6572 7422 2c20 666e 297c  le("qwert", fn)|
│ │ │ │  00249630: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00249640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  00249650: 0a7c 6933 203a 2066 696c 6545 7869 7374  .|i3 : fileExist
│ │ │ │ @@ -150009,15 +150009,15 @@
│ │ │ │  00249f80: 3d3d 3d3d 3d3d 3d0a 0a2b 2d2d 2d2d 2d2d  =======..+------
│ │ │ │  00249f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00249fa0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2070  -------+.|i1 : p
│ │ │ │  00249fb0: 203d 2074 656d 706f 7261 7279 4669 6c65   = temporaryFile
│ │ │ │  00249fc0: 4e61 6d65 2028 297c 0a7c 2020 2020 2020  Name ()|.|      
│ │ │ │  00249fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00249fe0: 2020 2020 2020 207c 0a7c 6f31 203d 202f         |.|o1 = /
│ │ │ │ -00249ff0: 746d 702f 4d32 2d31 3138 3036 2d30 2f30  tmp/M2-11806-0/0
│ │ │ │ +00249ff0: 746d 702f 4d32 2d31 3335 3536 2d30 2f30  tmp/M2-13556-0/0
│ │ │ │  0024a000: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  0024a010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024a020: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2073  -------+.|i2 : s
│ │ │ │  0024a030: 796d 6c69 6e6b 4669 6c65 2028 2266 6f6f  ymlinkFile ("foo
│ │ │ │  0024a040: 222c 2070 2920 207c 0a2b 2d2d 2d2d 2d2d  ", p)  |.+------
│ │ │ │  0024a050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024a060: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2072  -------+.|i3 : r
│ │ │ │ @@ -150099,66 +150099,66 @@
│ │ │ │  0024a520: 3d3d 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  ==..+-----------
│ │ │ │  0024a530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024a540: 2d2d 2d2d 2b0a 7c69 3120 3a20 7265 616c  ----+.|i1 : real
│ │ │ │  0024a550: 7061 7468 2022 2e22 2020 2020 2020 2020  path "."        
│ │ │ │  0024a560: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0024a570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024a580: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20          |.|o1 = 
│ │ │ │ -0024a590: 2f74 6d70 2f4d 322d 3130 3139 312d 302f  /tmp/M2-10191-0/
│ │ │ │ +0024a590: 2f74 6d70 2f4d 322d 3130 3331 312d 302f  /tmp/M2-10311-0/
│ │ │ │  0024a5a0: 3836 2d72 756e 6469 722f 7c0a 2b2d 2d2d  86-rundir/|.+---
│ │ │ │  0024a5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024a5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  0024a5d0: 3220 3a20 7020 3d20 7465 6d70 6f72 6172  2 : p = temporar
│ │ │ │  0024a5e0: 7946 696c 654e 616d 6528 2920 2020 7c0a  yFileName()   |.
│ │ │ │  0024a5f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0024a600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024a610: 7c0a 7c6f 3220 3d20 2f74 6d70 2f4d 322d  |.|o2 = /tmp/M2-
│ │ │ │ -0024a620: 3131 3832 352d 302f 3020 2020 2020 2020  11825-0/0       
│ │ │ │ +0024a620: 3133 3539 352d 302f 3020 2020 2020 2020  13595-0/0       
│ │ │ │  0024a630: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  0024a640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024a650: 2d2d 2d2d 2b0a 7c69 3320 3a20 7120 3d20  ----+.|i3 : q = 
│ │ │ │  0024a660: 7465 6d70 6f72 6172 7946 696c 654e 616d  temporaryFileNam
│ │ │ │  0024a670: 6528 2920 2020 7c0a 7c20 2020 2020 2020  e()   |.|       
│ │ │ │  0024a680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024a690: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20          |.|o3 = 
│ │ │ │ -0024a6a0: 2f74 6d70 2f4d 322d 3131 3832 352d 302f  /tmp/M2-11825-0/
│ │ │ │ +0024a6a0: 2f74 6d70 2f4d 322d 3133 3539 352d 302f  /tmp/M2-13595-0/
│ │ │ │  0024a6b0: 3120 2020 2020 2020 2020 7c0a 2b2d 2d2d  1         |.+---
│ │ │ │  0024a6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024a6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  0024a6e0: 3420 3a20 7379 6d6c 696e 6b46 696c 6528  4 : symlinkFile(
│ │ │ │  0024a6f0: 702c 7129 2020 2020 2020 2020 2020 7c0a  p,q)          |.
│ │ │ │  0024a700: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0024a710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024a720: 2b0a 7c69 3520 3a20 7020 3c3c 2063 6c6f  +.|i5 : p << clo
│ │ │ │  0024a730: 7365 2020 2020 2020 2020 2020 2020 2020  se              
│ │ │ │  0024a740: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0024a750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024a760: 2020 2020 7c0a 7c6f 3520 3d20 2f74 6d70      |.|o5 = /tmp
│ │ │ │ -0024a770: 2f4d 322d 3131 3832 352d 302f 3020 2020  /M2-11825-0/0   
│ │ │ │ +0024a770: 2f4d 322d 3133 3539 352d 302f 3020 2020  /M2-13595-0/0   
│ │ │ │  0024a780: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0024a790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024a7a0: 2020 2020 2020 2020 7c0a 7c6f 3520 3a20          |.|o5 : 
│ │ │ │  0024a7b0: 4669 6c65 2020 2020 2020 2020 2020 2020  File            
│ │ │ │  0024a7c0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  0024a7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024a7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  0024a7f0: 3620 3a20 7265 6164 6c69 6e6b 2071 2020  6 : readlink q  
│ │ │ │  0024a800: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  0024a810: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0024a820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024a830: 7c0a 7c6f 3620 3d20 2f74 6d70 2f4d 322d  |.|o6 = /tmp/M2-
│ │ │ │ -0024a840: 3131 3832 352d 302f 3020 2020 2020 2020  11825-0/0       
│ │ │ │ +0024a840: 3133 3539 352d 302f 3020 2020 2020 2020  13595-0/0       
│ │ │ │  0024a850: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  0024a860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024a870: 2d2d 2d2d 2b0a 7c69 3720 3a20 7265 616c  ----+.|i7 : real
│ │ │ │  0024a880: 7061 7468 2071 2020 2020 2020 2020 2020  path q          
│ │ │ │  0024a890: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0024a8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024a8b0: 2020 2020 2020 2020 7c0a 7c6f 3720 3d20          |.|o7 = 
│ │ │ │ -0024a8c0: 2f74 6d70 2f4d 322d 3131 3832 352d 302f  /tmp/M2-11825-0/
│ │ │ │ +0024a8c0: 2f74 6d70 2f4d 322d 3133 3539 352d 302f  /tmp/M2-13595-0/
│ │ │ │  0024a8d0: 3020 2020 2020 2020 2020 7c0a 2b2d 2d2d  0         |.+---
│ │ │ │  0024a8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024a8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  0024a900: 3820 3a20 7265 6d6f 7665 4669 6c65 2070  8 : removeFile p
│ │ │ │  0024a910: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  0024a920: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0024a930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -150174,15 +150174,15 @@
│ │ │ │  0024a9d0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0024a9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024a9f0: 2d2d 2b0a 7c69 3130 203a 2072 6561 6c70  --+.|i10 : realp
│ │ │ │  0024aa00: 6174 6820 2222 2020 2020 2020 2020 2020  ath ""          
│ │ │ │  0024aa10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0024aa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024aa30: 2020 2020 2020 2020 7c0a 7c6f 3130 203d          |.|o10 =
│ │ │ │ -0024aa40: 202f 746d 702f 4d32 2d31 3031 3931 2d30   /tmp/M2-10191-0
│ │ │ │ +0024aa40: 202f 746d 702f 4d32 2d31 3033 3131 2d30   /tmp/M2-10311-0
│ │ │ │  0024aa50: 2f38 362d 7275 6e64 6972 2f7c 0a2b 2d2d  /86-rundir/|.+--
│ │ │ │  0024aa60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024aa70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  0024aa80: 0a43 6176 6561 740a 3d3d 3d3d 3d3d 0a0a  .Caveat.======..
│ │ │ │  0024aa90: 4576 6572 7920 636f 6d70 6f6e 656e 7420  Every component 
│ │ │ │  0024aaa0: 6f66 2074 6865 2070 6174 6820 6d75 7374  of the path must
│ │ │ │  0024aab0: 2065 7869 7374 2069 6e20 7468 6520 6669   exist in the fi
│ │ │ │ @@ -150340,43 +150340,43 @@
│ │ │ │  0024b430: 6469 7220 3d20 7465 6d70 6f72 6172 7946  dir = temporaryF
│ │ │ │  0024b440: 696c 654e 616d 6528 2920 2020 2020 2020  ileName()       
│ │ │ │  0024b450: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0024b460: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0024b470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024b480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024b490: 2020 2020 2020 7c0a 7c6f 3120 3d20 2f74        |.|o1 = /t
│ │ │ │ -0024b4a0: 6d70 2f4d 322d 3131 3536 352d 302f 3020  mp/M2-11565-0/0 
│ │ │ │ +0024b4a0: 6d70 2f4d 322d 3133 3037 352d 302f 3020  mp/M2-13075-0/0 
│ │ │ │  0024b4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024b4c0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │  0024b4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024b4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024b4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024b500: 2d2d 2d2d 2b0a 7c69 3220 3a20 6d61 6b65  ----+.|i2 : make
│ │ │ │  0024b510: 4469 7265 6374 6f72 7920 6469 7220 2020  Directory dir   
│ │ │ │  0024b520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024b530: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0024b540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024b550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024b560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024b570: 2020 7c0a 7c6f 3220 3d20 2f74 6d70 2f4d    |.|o2 = /tmp/M
│ │ │ │ -0024b580: 322d 3131 3536 352d 302f 3020 2020 2020  2-11565-0/0     
│ │ │ │ +0024b580: 322d 3133 3037 352d 302f 3020 2020 2020  2-13075-0/0     
│ │ │ │  0024b590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024b5a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  0024b5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024b5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024b5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024b5e0: 2b0a 7c69 3320 3a20 2866 6e20 3d20 6469  +.|i3 : (fn = di
│ │ │ │  0024b5f0: 7220 7c20 222f 2220 7c20 2266 6f6f 2229  r | "/" | "foo")
│ │ │ │  0024b600: 203c 3c20 2268 6920 7468 6572 6522 203c   << "hi there" <
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│ │ │ │  0024b640: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │ +0024b650: 7c6f 3320 3d20 2f74 6d70 2f4d 322d 3133  |o3 = /tmp/M2-13
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│ │ │ │  0024b6b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │  0024b6c0: 3320 3a20 4669 6c65 2020 2020 2020 2020  3 : File        
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│ │ │ │ @@ -150496,24 +150496,24 @@
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│ │ │ │ @@ -150590,15 +150590,15 @@
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│ │ │ │  0024c5b0: 2b0a 7c69 3420 3a20 2866 6e20 3d20 7020  +.|i4 : (fn = p 
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│ │ │ │  0024ccf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024cd00: 2020 2020 2020 207c 0a7c 6f32 203d 202f         |.|o2 = /
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│ │ │ │  0024cd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024cd80: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ -0024cd90: 203d 202f 746d 702f 4d32 2d31 3034 3633   = /tmp/M2-10463
│ │ │ │ +0024cd90: 203d 202f 746d 702f 4d32 2d31 3038 3333   = /tmp/M2-10833
│ │ │ │  0024cda0: 2d30 2f30 2f20 2020 2020 2020 7c0a 2b2d  -0/0/       |.+-
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│ │ │ │  0024ce70: 6375 7272 656e 7420 776f 726b 696e 6720  current working 
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│ │ │ │  0024d690: 7c0a 7c6f 3120 3d20 2f74 6d70 2f4d 322d  |.|o1 = /tmp/M2-
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│ │ │ │  0024d6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024d6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024d6d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │  0024d710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  0024d7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024d7c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0024d7d0: 3220 3d20 2f74 6d70 2f4d 322d 3131 3138  2 = /tmp/M2-1118
│ │ │ │ +0024d7d0: 3220 3d20 2f74 6d70 2f4d 322d 3132 3239  2 = /tmp/M2-1229
│ │ │ │  0024d7e0: 352d 302f 312f 2020 2020 2020 2020 2020  5-0/1/          
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│ │ │ │ @@ -150923,15 +150923,15 @@
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│ │ │ │  0024d8b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0024d8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024d8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024d8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024d8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024d900: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20          |.|o3 = 
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│ │ │ │  0024d940: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0024d980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -150943,15 +150943,15 @@
│ │ │ │  0024d9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024d9f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0024da00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024da10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024da20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024da30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024da40: 2020 2020 7c0a 7c6f 3420 3d20 2f74 6d70      |.|o4 = /tmp
│ │ │ │ -0024da50: 2f4d 322d 3131 3138 352d 302f 302f 622f  /M2-11185-0/0/b/
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│ │ │ │  0024da60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024da70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024da80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024da90: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
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│ │ │ │  0024dab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024dac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -150963,15 +150963,15 @@
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│ │ │ │  0024db30: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0024db40: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0024db70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024db80: 7c0a 7c6f 3520 3d20 2f74 6d70 2f4d 322d  |.|o5 = /tmp/M2-
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│ │ │ │  0024dc00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -150982,15 +150982,15 @@
│ │ │ │  0024dc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0024dc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  0024dca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024dcb0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0024dcc0: 3620 3d20 2f74 6d70 2f4d 322d 3131 3138  6 = /tmp/M2-1118
│ │ │ │ +0024dcc0: 3620 3d20 2f74 6d70 2f4d 322d 3132 3239  6 = /tmp/M2-1229
│ │ │ │  0024dcd0: 352d 302f 302f 612f 6620 2020 2020 2020  5-0/0/a/f       
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│ │ │ │  0024dd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024dd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024dd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -151012,15 +151012,15 @@
│ │ │ │  0024de30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024de40: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0024de50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024de60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024de70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024de80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024de90: 2020 2020 2020 7c0a 7c6f 3720 3d20 2f74        |.|o7 = /t
│ │ │ │ -0024dea0: 6d70 2f4d 322d 3131 3138 352d 302f 302f  mp/M2-11185-0/0/
│ │ │ │ +0024dea0: 6d70 2f4d 322d 3132 3239 352d 302f 302f  mp/M2-12295-0/0/
│ │ │ │  0024deb0: 612f 6720 2020 2020 2020 2020 2020 2020  a/g             
│ │ │ │  0024dec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024ded0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024dee0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0024def0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024df00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024df10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -151042,15 +151042,15 @@
│ │ │ │  0024e010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e020: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0024e030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e070: 7c0a 7c6f 3820 3d20 2f74 6d70 2f4d 322d  |.|o8 = /tmp/M2-
│ │ │ │ -0024e080: 3131 3138 352d 302f 302f 622f 632f 6720  11185-0/0/b/c/g 
│ │ │ │ +0024e080: 3132 3239 352d 302f 302f 622f 632f 6720  12295-0/0/b/c/g 
│ │ │ │  0024e090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e0b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0024e0c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0024e0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -151071,98 +151071,98 @@
│ │ │ │  0024e1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e1f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0024e200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e240: 2020 2020 2020 2020 2020 7c0a 7c6f 3920            |.|o9 
│ │ │ │ -0024e250: 3d20 2f74 6d70 2f4d 322d 3131 3138 352d  = /tmp/M2-11185-
│ │ │ │ +0024e250: 3d20 2f74 6d70 2f4d 322d 3132 3239 352d  = /tmp/M2-12295-
│ │ │ │  0024e260: 302f 302f 2020 2020 2020 2020 2020 2020  0/0/            
│ │ │ │  0024e270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e290: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0024e2a0: 202f 746d 702f 4d32 2d31 3131 3835 2d30   /tmp/M2-11185-0
│ │ │ │ -0024e2b0: 2f30 2f62 2f20 2020 2020 2020 2020 2020  /0/b/           
│ │ │ │ +0024e2a0: 202f 746d 702f 4d32 2d31 3232 3935 2d30   /tmp/M2-12295-0
│ │ │ │ +0024e2b0: 2f30 2f61 2f20 2020 2020 2020 2020 2020  /0/a/           
│ │ │ │  0024e2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e2e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0024e2f0: 2f74 6d70 2f4d 322d 3131 3138 352d 302f  /tmp/M2-11185-0/
│ │ │ │ -0024e300: 302f 622f 632f 2020 2020 2020 2020 2020  0/b/c/          
│ │ │ │ +0024e2f0: 2f74 6d70 2f4d 322d 3132 3239 352d 302f  /tmp/M2-12295-0/
│ │ │ │ +0024e300: 302f 612f 6720 2020 2020 2020 2020 2020  0/a/g           
│ │ │ │  0024e310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e330: 2020 2020 2020 207c 0a7c 2020 2020 202f         |.|     /
│ │ │ │ -0024e340: 746d 702f 4d32 2d31 3131 3835 2d30 2f30  tmp/M2-11185-0/0
│ │ │ │ -0024e350: 2f62 2f63 2f67 2020 2020 2020 2020 2020  /b/c/g          
│ │ │ │ +0024e340: 746d 702f 4d32 2d31 3232 3935 2d30 2f30  tmp/M2-12295-0/0
│ │ │ │ +0024e350: 2f61 2f66 2020 2020 2020 2020 2020 2020  /a/f            
│ │ │ │  0024e360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e380: 2020 2020 2020 7c0a 7c20 2020 2020 2f74        |.|     /t
│ │ │ │ -0024e390: 6d70 2f4d 322d 3131 3138 352d 302f 302f  mp/M2-11185-0/0/
│ │ │ │ -0024e3a0: 612f 2020 2020 2020 2020 2020 2020 2020  a/              
│ │ │ │ +0024e390: 6d70 2f4d 322d 3132 3239 352d 302f 302f  mp/M2-12295-0/0/
│ │ │ │ +0024e3a0: 622f 2020 2020 2020 2020 2020 2020 2020  b/              
│ │ │ │  0024e3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e3d0: 2020 2020 207c 0a7c 2020 2020 202f 746d       |.|     /tm
│ │ │ │ -0024e3e0: 702f 4d32 2d31 3131 3835 2d30 2f30 2f61  p/M2-11185-0/0/a
│ │ │ │ -0024e3f0: 2f67 2020 2020 2020 2020 2020 2020 2020  /g              
│ │ │ │ +0024e3e0: 702f 4d32 2d31 3232 3935 2d30 2f30 2f62  p/M2-12295-0/0/b
│ │ │ │ +0024e3f0: 2f63 2f20 2020 2020 2020 2020 2020 2020  /c/             
│ │ │ │  0024e400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e420: 2020 2020 7c0a 7c20 2020 2020 2f74 6d70      |.|     /tmp
│ │ │ │ -0024e430: 2f4d 322d 3131 3138 352d 302f 302f 612f  /M2-11185-0/0/a/
│ │ │ │ -0024e440: 6620 2020 2020 2020 2020 2020 2020 2020  f               
│ │ │ │ +0024e430: 2f4d 322d 3132 3239 352d 302f 302f 622f  /M2-12295-0/0/b/
│ │ │ │ +0024e440: 632f 6720 2020 2020 2020 2020 2020 2020  c/g             
│ │ │ │  0024e450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e470: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  0024e480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024e490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024e4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024e4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024e4c0: 2d2d 2b0a 7c69 3130 203a 2063 6f70 7944  --+.|i10 : copyD
│ │ │ │  0024e4d0: 6972 6563 746f 7279 2873 7263 2c64 7374  irectory(src,dst
│ │ │ │  0024e4e0: 2c56 6572 626f 7365 3d3e 7472 7565 2920  ,Verbose=>true) 
│ │ │ │  0024e4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e510: 207c 0a7c 202d 2d20 636f 7079 696e 673a   |.| -- copying:
│ │ │ │ -0024e520: 202f 746d 702f 4d32 2d31 3131 3835 2d30   /tmp/M2-11185-0
│ │ │ │ -0024e530: 2f30 2f62 2f63 2f67 202d 3e20 2f74 6d70  /0/b/c/g -> /tmp
│ │ │ │ -0024e540: 2f4d 322d 3131 3138 352d 302f 312f 622f  /M2-11185-0/1/b/
│ │ │ │ -0024e550: 632f 6720 2020 2020 2020 2020 2020 2020  c/g             
│ │ │ │ +0024e520: 202f 746d 702f 4d32 2d31 3232 3935 2d30   /tmp/M2-12295-0
│ │ │ │ +0024e530: 2f30 2f61 2f67 202d 3e20 2f74 6d70 2f4d  /0/a/g -> /tmp/M
│ │ │ │ +0024e540: 322d 3132 3239 352d 302f 312f 612f 6720  2-12295-0/1/a/g 
│ │ │ │ +0024e550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e560: 7c0a 7c20 2d2d 2063 6f70 7969 6e67 3a20  |.| -- copying: 
│ │ │ │ -0024e570: 2f74 6d70 2f4d 322d 3131 3138 352d 302f  /tmp/M2-11185-0/
│ │ │ │ -0024e580: 302f 612f 6720 2d3e 202f 746d 702f 4d32  0/a/g -> /tmp/M2
│ │ │ │ -0024e590: 2d31 3131 3835 2d30 2f31 2f61 2f67 2020  -11185-0/1/a/g  
│ │ │ │ +0024e570: 2f74 6d70 2f4d 322d 3132 3239 352d 302f  /tmp/M2-12295-0/
│ │ │ │ +0024e580: 302f 612f 6620 2d3e 202f 746d 702f 4d32  0/a/f -> /tmp/M2
│ │ │ │ +0024e590: 2d31 3232 3935 2d30 2f31 2f61 2f66 2020  -12295-0/1/a/f  
│ │ │ │  0024e5a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0024e5b0: 0a7c 202d 2d20 636f 7079 696e 673a 202f  .| -- copying: /
│ │ │ │ -0024e5c0: 746d 702f 4d32 2d31 3131 3835 2d30 2f30  tmp/M2-11185-0/0
│ │ │ │ -0024e5d0: 2f61 2f66 202d 3e20 2f74 6d70 2f4d 322d  /a/f -> /tmp/M2-
│ │ │ │ -0024e5e0: 3131 3138 352d 302f 312f 612f 6620 2020  11185-0/1/a/f   
│ │ │ │ -0024e5f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0024e5c0: 746d 702f 4d32 2d31 3232 3935 2d30 2f30  tmp/M2-12295-0/0
│ │ │ │ +0024e5d0: 2f62 2f63 2f67 202d 3e20 2f74 6d70 2f4d  /b/c/g -> /tmp/M
│ │ │ │ +0024e5e0: 322d 3132 3239 352d 302f 312f 622f 632f  2-12295-0/1/b/c/
│ │ │ │ +0024e5f0: 6720 2020 2020 2020 2020 2020 2020 7c0a  g             |.
│ │ │ │  0024e600: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0024e610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024e620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024e630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024e640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  0024e650: 6931 3120 3a20 636f 7079 4469 7265 6374  i11 : copyDirect
│ │ │ │  0024e660: 6f72 7928 7372 632c 6473 742c 5665 7262  ory(src,dst,Verb
│ │ │ │  0024e670: 6f73 653d 3e74 7275 652c 5570 6461 7465  ose=>true,Update
│ │ │ │  0024e680: 4f6e 6c79 203d 3e20 7472 7565 2920 2020  Only => true)   
│ │ │ │  0024e690: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0024e6a0: 2d2d 2073 6b69 7070 696e 673a 202f 746d  -- skipping: /tm
│ │ │ │ -0024e6b0: 702f 4d32 2d31 3131 3835 2d30 2f30 2f62  p/M2-11185-0/0/b
│ │ │ │ -0024e6c0: 2f63 2f67 206e 6f74 206e 6577 6572 2074  /c/g not newer t
│ │ │ │ -0024e6d0: 6861 6e20 2f74 6d70 2f4d 322d 3131 3138  han /tmp/M2-1118
│ │ │ │ -0024e6e0: 352d 302f 312f 622f 632f 677c 0a7c 202d  5-0/1/b/c/g|.| -
│ │ │ │ +0024e6b0: 702f 4d32 2d31 3232 3935 2d30 2f30 2f61  p/M2-12295-0/0/a
│ │ │ │ +0024e6c0: 2f67 206e 6f74 206e 6577 6572 2074 6861  /g not newer tha
│ │ │ │ +0024e6d0: 6e20 2f74 6d70 2f4d 322d 3132 3239 352d  n /tmp/M2-12295-
│ │ │ │ +0024e6e0: 302f 312f 612f 6720 2020 207c 0a7c 202d  0/1/a/g    |.| -
│ │ │ │  0024e6f0: 2d20 736b 6970 7069 6e67 3a20 2f74 6d70  - skipping: /tmp
│ │ │ │ -0024e700: 2f4d 322d 3131 3138 352d 302f 302f 612f  /M2-11185-0/0/a/
│ │ │ │ -0024e710: 6720 6e6f 7420 6e65 7765 7220 7468 616e  g not newer than
│ │ │ │ -0024e720: 202f 746d 702f 4d32 2d31 3131 3835 2d30   /tmp/M2-11185-0
│ │ │ │ -0024e730: 2f31 2f61 2f67 2020 2020 7c0a 7c20 2d2d  /1/a/g    |.| --
│ │ │ │ +0024e700: 2f4d 322d 3132 3239 352d 302f 302f 612f  /M2-12295-0/0/a/
│ │ │ │ +0024e710: 6620 6e6f 7420 6e65 7765 7220 7468 616e  f not newer than
│ │ │ │ +0024e720: 202f 746d 702f 4d32 2d31 3232 3935 2d30   /tmp/M2-12295-0
│ │ │ │ +0024e730: 2f31 2f61 2f66 2020 2020 7c0a 7c20 2d2d  /1/a/f    |.| --
│ │ │ │  0024e740: 2073 6b69 7070 696e 673a 202f 746d 702f   skipping: /tmp/
│ │ │ │ -0024e750: 4d32 2d31 3131 3835 2d30 2f30 2f61 2f66  M2-11185-0/0/a/f
│ │ │ │ -0024e760: 206e 6f74 206e 6577 6572 2074 6861 6e20   not newer than 
│ │ │ │ -0024e770: 2f74 6d70 2f4d 322d 3131 3138 352d 302f  /tmp/M2-11185-0/
│ │ │ │ -0024e780: 312f 612f 6620 2020 207c 0a2b 2d2d 2d2d  1/a/f    |.+----
│ │ │ │ +0024e750: 4d32 2d31 3232 3935 2d30 2f30 2f62 2f63  M2-12295-0/0/b/c
│ │ │ │ +0024e760: 2f67 206e 6f74 206e 6577 6572 2074 6861  /g not newer tha
│ │ │ │ +0024e770: 6e20 2f74 6d70 2f4d 322d 3132 3239 352d  n /tmp/M2-12295-
│ │ │ │ +0024e780: 302f 312f 622f 632f 677c 0a2b 2d2d 2d2d  0/1/b/c/g|.+----
│ │ │ │  0024e790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024e7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024e7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024e7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024e7d0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3132 203a  --------+.|i12 :
│ │ │ │  0024e7e0: 2073 7461 636b 2066 696e 6446 696c 6573   stack findFiles
│ │ │ │  0024e7f0: 2064 7374 2020 2020 2020 2020 2020 2020   dst            
│ │ │ │ @@ -151170,45 +151170,45 @@
│ │ │ │  0024e810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e820: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0024e830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e870: 2020 2020 2020 7c0a 7c6f 3132 203d 202f        |.|o12 = /
│ │ │ │ -0024e880: 746d 702f 4d32 2d31 3131 3835 2d30 2f31  tmp/M2-11185-0/1
│ │ │ │ +0024e880: 746d 702f 4d32 2d31 3232 3935 2d30 2f31  tmp/M2-12295-0/1
│ │ │ │  0024e890: 2f20 2020 2020 2020 2020 2020 2020 2020  /               
│ │ │ │  0024e8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e8c0: 2020 2020 207c 0a7c 2020 2020 2020 2f74       |.|      /t
│ │ │ │ -0024e8d0: 6d70 2f4d 322d 3131 3138 352d 302f 312f  mp/M2-11185-0/1/
│ │ │ │ +0024e8d0: 6d70 2f4d 322d 3132 3239 352d 302f 312f  mp/M2-12295-0/1/
│ │ │ │  0024e8e0: 612f 2020 2020 2020 2020 2020 2020 2020  a/              
│ │ │ │  0024e8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e910: 2020 2020 7c0a 7c20 2020 2020 202f 746d      |.|      /tm
│ │ │ │ -0024e920: 702f 4d32 2d31 3131 3835 2d30 2f31 2f61  p/M2-11185-0/1/a
│ │ │ │ -0024e930: 2f66 2020 2020 2020 2020 2020 2020 2020  /f              
│ │ │ │ +0024e920: 702f 4d32 2d31 3232 3935 2d30 2f31 2f61  p/M2-12295-0/1/a
│ │ │ │ +0024e930: 2f67 2020 2020 2020 2020 2020 2020 2020  /g              
│ │ │ │  0024e940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e960: 2020 207c 0a7c 2020 2020 2020 2f74 6d70     |.|      /tmp
│ │ │ │ -0024e970: 2f4d 322d 3131 3138 352d 302f 312f 612f  /M2-11185-0/1/a/
│ │ │ │ -0024e980: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ +0024e970: 2f4d 322d 3132 3239 352d 302f 312f 612f  /M2-12295-0/1/a/
│ │ │ │ +0024e980: 6620 2020 2020 2020 2020 2020 2020 2020  f               
│ │ │ │  0024e990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e9b0: 2020 7c0a 7c20 2020 2020 202f 746d 702f    |.|      /tmp/
│ │ │ │ -0024e9c0: 4d32 2d31 3131 3835 2d30 2f31 2f62 2f20  M2-11185-0/1/b/ 
│ │ │ │ +0024e9c0: 4d32 2d31 3232 3935 2d30 2f31 2f62 2f20  M2-12295-0/1/b/ 
│ │ │ │  0024e9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024e9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024ea00: 207c 0a7c 2020 2020 2020 2f74 6d70 2f4d   |.|      /tmp/M
│ │ │ │ -0024ea10: 322d 3131 3138 352d 302f 312f 622f 632f  2-11185-0/1/b/c/
│ │ │ │ +0024ea10: 322d 3132 3239 352d 302f 312f 622f 632f  2-12295-0/1/b/c/
│ │ │ │  0024ea20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024ea30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024ea40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024ea50: 7c0a 7c20 2020 2020 202f 746d 702f 4d32  |.|      /tmp/M2
│ │ │ │ -0024ea60: 2d31 3131 3835 2d30 2f31 2f62 2f63 2f67  -11185-0/1/b/c/g
│ │ │ │ +0024ea60: 2d31 3232 3935 2d30 2f31 2f62 2f63 2f67  -12295-0/1/b/c/g
│ │ │ │  0024ea70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024ea80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024ea90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0024eaa0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0024eab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024eac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024ead0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -151349,31 +151349,31 @@
│ │ │ │  0024f340: 3d3d 3d3d 3d3d 0a0a 2b2d 2d2d 2d2d 2d2d  ======..+-------
│ │ │ │  0024f350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024f360: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2064  -------+.|i1 : d
│ │ │ │  0024f370: 6972 203d 2074 656d 706f 7261 7279 4669  ir = temporaryFi
│ │ │ │  0024f380: 6c65 4e61 6d65 2829 7c0a 7c20 2020 2020  leName()|.|     
│ │ │ │  0024f390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024f3a0: 2020 2020 2020 2020 207c 0a7c 6f31 203d           |.|o1 =
│ │ │ │ -0024f3b0: 202f 746d 702f 4d32 2d31 3037 3739 2d30   /tmp/M2-10779-0
│ │ │ │ +0024f3b0: 202f 746d 702f 4d32 2d31 3134 3639 2d30   /tmp/M2-11469-0
│ │ │ │  0024f3c0: 2f30 2020 2020 2020 2020 7c0a 2b2d 2d2d  /0        |.+---
│ │ │ │  0024f3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0024f3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │  0024f3f0: 203a 206d 616b 6544 6972 6563 746f 7279   : makeDirectory
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│ │ │ │  0024f410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024f420: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0024f430: 6f32 203d 202f 746d 702f 4d32 2d31 3037  o2 = /tmp/M2-107
│ │ │ │ -0024f440: 3739 2d30 2f30 2020 2020 2020 2020 7c0a  79-0/0        |.
│ │ │ │ +0024f430: 6f32 203d 202f 746d 702f 4d32 2d31 3134  o2 = /tmp/M2-114
│ │ │ │ +0024f440: 3639 2d30 2f30 2020 2020 2020 2020 7c0a  69-0/0        |.
│ │ │ │  0024f450: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0024f460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0024f470: 0a7c 6933 203a 2072 6561 6444 6972 6563  .|i3 : readDirec
│ │ │ │  0024f480: 746f 7279 2064 6972 2020 2020 2020 2020  tory dir        
│ │ │ │  0024f490: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0024f4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0024f4b0: 207c 0a7c 6f33 203d 207b 2e2c 202e 2e7d   |.|o3 = {., ..}
│ │ │ │ +0024f4b0: 207c 0a7c 6f33 203d 207b 2e2e 2c20 2e7d   |.|o3 = {.., .}
│ │ │ │  0024f4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024f4d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0024f4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024f4f0: 2020 207c 0a7c 6f33 203a 204c 6973 7420     |.|o3 : List 
│ │ │ │  0024f500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024f510: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  0024f520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -151540,30 +151540,30 @@
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│ │ │ │  0024ff60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024ff70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024ff80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0024ff90: 2020 2020 7c0a 7c6f 3120 3d20 2f74 6d70      |.|o1 = /tmp
│ │ │ │ -0024ffa0: 2f4d 322d 3131 3134 372d 302f 302f 2020  /M2-11147-0/0/  
│ │ │ │ +0024ffa0: 2f4d 322d 3132 3231 372d 302f 302f 2020  /M2-12217-0/0/  
│ │ │ │  0024ffb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00250040: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │  00250060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00250070: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00250090: 3220 3d20 2f74 6d70 2f4d 322d 3131 3134  2 = /tmp/M2-1114
│ │ │ │ +00250090: 3220 3d20 2f74 6d70 2f4d 322d 3132 3231  2 = /tmp/M2-1221
│ │ │ │  002500a0: 372d 302f 312f 2020 2020 2020 2020 2020  7-0/1/          
│ │ │ │  002500b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002500c0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
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│ │ │ │ @@ -151571,30 +151571,30 @@
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│ │ │ │  00250130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00250140: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00250150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00250160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00250170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00250180: 2020 2020 7c0a 7c6f 3320 3d20 2f74 6d70      |.|o3 = /tmp
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│ │ │ │  002501a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002501b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002501c0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  002501d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  002501e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  002501f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │  00250210: 6374 6f72 7920 2873 7263 7c22 622f 2229  ctory (src|"b/")
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│ │ │ │  00250230: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00250240: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00250250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00250260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00250270: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00250280: 3420 3d20 2f74 6d70 2f4d 322d 3131 3134  4 = /tmp/M2-1114
│ │ │ │ +00250280: 3420 3d20 2f74 6d70 2f4d 322d 3132 3231  4 = /tmp/M2-1221
│ │ │ │  00250290: 372d 302f 302f 622f 2020 2020 2020 2020  7-0/0/b/        
│ │ │ │  002502a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002502b0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
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│ │ │ │  002502e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  002502f0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20  --------+.|i5 : 
│ │ │ │ @@ -151602,30 +151602,30 @@
│ │ │ │  00250310: 7263 7c22 622f 632f 2229 2020 2020 2020  rc|"b/c/")      
│ │ │ │  00250320: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00250340: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00250370: 2020 2020 7c0a 7c6f 3520 3d20 2f74 6d70      |.|o5 = /tmp
│ │ │ │ -00250380: 2f4d 322d 3131 3134 372d 302f 302f 622f  /M2-11147-0/0/b/
│ │ │ │ +00250380: 2f4d 322d 3132 3231 372d 302f 302f 622f  /M2-12217-0/0/b/
│ │ │ │  00250390: 632f 2020 2020 2020 2020 2020 2020 2020  c/              
│ │ │ │  002503a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002503b0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
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│ │ │ │  002503e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  002503f0: 2b0a 7c69 3620 3a20 7372 637c 2261 2f66  +.|i6 : src|"a/f
│ │ │ │  00250400: 2220 3c3c 2022 6869 2074 6865 7265 2220  " << "hi there" 
│ │ │ │  00250410: 3c3c 2063 6c6f 7365 2020 2020 2020 2020  << close        
│ │ │ │  00250420: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00250430: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00250440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00250450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00250460: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00250470: 3620 3d20 2f74 6d70 2f4d 322d 3131 3134  6 = /tmp/M2-1114
│ │ │ │ +00250470: 3620 3d20 2f74 6d70 2f4d 322d 3132 3231  6 = /tmp/M2-1221
│ │ │ │  00250480: 372d 302f 302f 612f 6620 2020 2020 2020  7-0/0/a/f       
│ │ │ │  00250490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002504a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  002504b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002504c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002504d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002504e0: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20          |.|o6 : 
│ │ │ │ @@ -151641,15 +151641,15 @@
│ │ │ │  00250580: 7265 2220 3c3c 2063 6c6f 7365 2020 2020  re" << close    
│ │ │ │  00250590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002505a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  002505b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002505c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002505d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002505e0: 7c0a 7c6f 3720 3d20 2f74 6d70 2f4d 322d  |.|o7 = /tmp/M2-
│ │ │ │ -002505f0: 3131 3134 372d 302f 302f 612f 6720 2020  11147-0/0/a/g   
│ │ │ │ +002505f0: 3132 3231 372d 302f 302f 612f 6720 2020  12217-0/0/a/g   
│ │ │ │  00250600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00250610: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00250620: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00250630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00250640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00250650: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │  00250660: 3720 3a20 4669 6c65 2020 2020 2020 2020  7 : File        
│ │ │ │ @@ -151664,15 +151664,15 @@
│ │ │ │  002506f0: 686f 2074 6865 7265 2220 3c3c 2063 6c6f  ho there" << clo
│ │ │ │  00250700: 7365 2020 2020 2020 2020 2020 2020 2020  se              
│ │ │ │  00250710: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00250720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00250730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00250740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00250750: 2020 2020 7c0a 7c6f 3820 3d20 2f74 6d70      |.|o8 = /tmp
│ │ │ │ -00250760: 2f4d 322d 3131 3134 372d 302f 302f 622f  /M2-11147-0/0/b/
│ │ │ │ +00250760: 2f4d 322d 3132 3231 372d 302f 302f 622f  /M2-12217-0/0/b/
│ │ │ │  00250770: 632f 6720 2020 2020 2020 2020 2020 2020  c/g             
│ │ │ │  00250780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00250790: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  002507a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002507b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002507c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002507d0: 7c0a 7c6f 3820 3a20 4669 6c65 2020 2020  |.|o8 : File    
│ │ │ │ @@ -151684,25 +151684,25 @@
│ │ │ │  00250830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00250840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  00250850: 3920 3a20 7379 6d6c 696e 6b44 6972 6563  9 : symlinkDirec
│ │ │ │  00250860: 746f 7279 2873 7263 2c64 7374 2c56 6572  tory(src,dst,Ver
│ │ │ │  00250870: 626f 7365 3d3e 7472 7565 2920 2020 2020  bose=>true)     
│ │ │ │  00250880: 2020 2020 2020 2020 2020 7c0a 7c2d 2d73            |.|--s
│ │ │ │  00250890: 796d 6c69 6e6b 696e 673a 202e 2e2f 2e2e  ymlinking: ../..
│ │ │ │ -002508a0: 2f2e 2e2f 302f 622f 632f 6720 2d3e 202f  /../0/b/c/g -> /
│ │ │ │ -002508b0: 746d 702f 4d32 2d31 3131 3437 2d30 2f31  tmp/M2-11147-0/1
│ │ │ │ -002508c0: 2f62 2f63 2f67 2020 7c0a 7c2d 2d73 796d  /b/c/g  |.|--sym
│ │ │ │ +002508a0: 2f30 2f61 2f67 202d 3e20 2f74 6d70 2f4d  /0/a/g -> /tmp/M
│ │ │ │ +002508b0: 322d 3132 3231 372d 302f 312f 612f 6720  2-12217-0/1/a/g 
│ │ │ │ +002508c0: 2020 2020 2020 2020 7c0a 7c2d 2d73 796d          |.|--sym
│ │ │ │  002508d0: 6c69 6e6b 696e 673a 202e 2e2f 2e2e 2f30  linking: ../../0
│ │ │ │ -002508e0: 2f61 2f67 202d 3e20 2f74 6d70 2f4d 322d  /a/g -> /tmp/M2-
│ │ │ │ -002508f0: 3131 3134 372d 302f 312f 612f 6720 2020  11147-0/1/a/g   
│ │ │ │ +002508e0: 2f61 2f66 202d 3e20 2f74 6d70 2f4d 322d  /a/f -> /tmp/M2-
│ │ │ │ +002508f0: 3132 3231 372d 302f 312f 612f 6620 2020  12217-0/1/a/f   
│ │ │ │  00250900: 2020 2020 2020 7c0a 7c2d 2d73 796d 6c69        |.|--symli
│ │ │ │ -00250910: 6e6b 696e 673a 202e 2e2f 2e2e 2f30 2f61  nking: ../../0/a
│ │ │ │ -00250920: 2f66 202d 3e20 2f74 6d70 2f4d 322d 3131  /f -> /tmp/M2-11
│ │ │ │ -00250930: 3134 372d 302f 312f 612f 6620 2020 2020  147-0/1/a/f     
│ │ │ │ -00250940: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00250910: 6e6b 696e 673a 202e 2e2f 2e2e 2f2e 2e2f  nking: ../../../
│ │ │ │ +00250920: 302f 622f 632f 6720 2d3e 202f 746d 702f  0/b/c/g -> /tmp/
│ │ │ │ +00250930: 4d32 2d31 3232 3137 2d30 2f31 2f62 2f63  M2-12217-0/1/b/c
│ │ │ │ +00250940: 2f67 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d  /g  |.+---------
│ │ │ │  00250950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00250960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00250970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00250980: 2d2d 2b0a 7c69 3130 203a 2067 6574 2028  --+.|i10 : get (
│ │ │ │  00250990: 6473 747c 2262 2f63 2f67 2229 2020 2020  dst|"b/c/g")    
│ │ │ │  002509a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002509b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -151719,25 +151719,25 @@
│ │ │ │  00250a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00250a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131  ----------+.|i11
│ │ │ │  00250a80: 203a 2073 796d 6c69 6e6b 4469 7265 6374   : symlinkDirect
│ │ │ │  00250a90: 6f72 7928 7372 632c 6473 742c 5665 7262  ory(src,dst,Verb
│ │ │ │  00250aa0: 6f73 653d 3e74 7275 652c 556e 646f 3d3e  ose=>true,Undo=>
│ │ │ │  00250ab0: 7472 7565 2920 2020 7c0a 7c2d 2d75 6e73  true)   |.|--uns
│ │ │ │  00250ac0: 796d 6c69 6e6b 696e 673a 202e 2e2f 2e2e  ymlinking: ../..
│ │ │ │ -00250ad0: 2f2e 2e2f 302f 622f 632f 6720 2d3e 202f  /../0/b/c/g -> /
│ │ │ │ -00250ae0: 746d 702f 4d32 2d31 3131 3437 2d30 2f31  tmp/M2-11147-0/1
│ │ │ │ -00250af0: 2f62 2f63 2f67 7c0a 7c2d 2d75 6e73 796d  /b/c/g|.|--unsym
│ │ │ │ +00250ad0: 2f30 2f61 2f67 202d 3e20 2f74 6d70 2f4d  /0/a/g -> /tmp/M
│ │ │ │ +00250ae0: 322d 3132 3231 372d 302f 312f 612f 6720  2-12217-0/1/a/g 
│ │ │ │ +00250af0: 2020 2020 2020 7c0a 7c2d 2d75 6e73 796d        |.|--unsym
│ │ │ │  00250b00: 6c69 6e6b 696e 673a 202e 2e2f 2e2e 2f30  linking: ../../0
│ │ │ │ -00250b10: 2f61 2f67 202d 3e20 2f74 6d70 2f4d 322d  /a/g -> /tmp/M2-
│ │ │ │ -00250b20: 3131 3134 372d 302f 312f 612f 6720 2020  11147-0/1/a/g   
│ │ │ │ +00250b10: 2f61 2f66 202d 3e20 2f74 6d70 2f4d 322d  /a/f -> /tmp/M2-
│ │ │ │ +00250b20: 3132 3231 372d 302f 312f 612f 6620 2020  12217-0/1/a/f   
│ │ │ │  00250b30: 2020 2020 7c0a 7c2d 2d75 6e73 796d 6c69      |.|--unsymli
│ │ │ │ -00250b40: 6e6b 696e 673a 202e 2e2f 2e2e 2f30 2f61  nking: ../../0/a
│ │ │ │ -00250b50: 2f66 202d 3e20 2f74 6d70 2f4d 322d 3131  /f -> /tmp/M2-11
│ │ │ │ -00250b60: 3134 372d 302f 312f 612f 6620 2020 2020  147-0/1/a/f     
│ │ │ │ -00250b70: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00250b40: 6e6b 696e 673a 202e 2e2f 2e2e 2f2e 2e2f  nking: ../../../
│ │ │ │ +00250b50: 302f 622f 632f 6720 2d3e 202f 746d 702f  0/b/c/g -> /tmp/
│ │ │ │ +00250b60: 4d32 2d31 3232 3137 2d30 2f31 2f62 2f63  M2-12217-0/1/b/c
│ │ │ │ +00250b70: 2f67 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  /g|.+-----------
│ │ │ │  00250b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00250b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00250ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00250bb0: 2b0a 4e6f 7720 7765 2072 656d 6f76 6520  +.Now we remove 
│ │ │ │  00250bc0: 7468 6520 6669 6c65 7320 616e 6420 6469  the files and di
│ │ │ │  00250bd0: 7265 6374 6f72 6965 7320 7765 2063 7265  rectories we cre
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│ │ │ │ -002521c0: 6572 7420 2020 2020 2020 2020 2020 2020  ert             
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│ │ │ │ +00252240: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ -00252300: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00252380: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ -00252680: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
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│ │ │ │ -002526f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00252700: 2020 7c0a 7c20 2020 2020 696e 7665 7273    |.|     invers
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│ │ │ │ -00253a40: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
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│ │ │ │ -00253a70: 3a20 7275 6e2c 2069 7320 6120 2a6e 6f74  : run, is a *not
│ │ │ │ -00253a80: 6520 636f 6d70 696c 6564 2066 756e 6374  e compiled funct
│ │ │ │ -00253a90: 696f 6e3a 2043 6f6d 7069 6c65 6446 756e  ion: CompiledFun
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│ │ │ │ -00253ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00253ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002534d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002534e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002534f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
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│ │ │ │ +002535c0: 2063 6f6d 6d75 6e69 6361 7469 6e67 2077   communicating w
│ │ │ │ +002535d0: 6974 6820 7072 6f67 7261 6d73 0a0a 7275  ith programs..ru
│ │ │ │ +002535e0: 6e20 2d2d 2072 756e 2061 6e20 6578 7465  n -- run an exte
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│ │ │ │ +00253600: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00253610: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a  ***********..  *
│ │ │ │ +00253620: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ +00253630: 2072 756e 2073 0a20 202a 2049 6e70 7574   run s.  * Input
│ │ │ │ +00253640: 733a 0a20 2020 2020 202a 2073 2c20 6120  s:.      * s, a 
│ │ │ │ +00253650: 2a6e 6f74 6520 7374 7269 6e67 3a20 5374  *note string: St
│ │ │ │ +00253660: 7269 6e67 2c2c 2061 2063 6f6d 6d61 6e64  ring,, a command
│ │ │ │ +00253670: 2075 6e64 6572 7374 616e 6461 626c 6520   understandable 
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│ │ │ │ +002536a0: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ +002536b0: 2020 202a 2061 6e20 2a6e 6f74 6520 696e     * an *note in
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│ │ │ │ +002536d0: 7265 7475 726e 2063 6f64 650a 0a44 6573  return code..Des
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│ │ │ │ +002536f0: 3d3d 3d3d 0a0a 5468 6520 7072 6f63 6573  ====..The proces
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│ │ │ │ +002537b0: 3620 7469 6d65 7320 7468 6520 6578 6974  6 times the exit
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│ │ │ │ +00253990: 2053 6967 6e61 6c0a 6e75 6d62 6572 7320   Signal.numbers 
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│ │ │ │ +00253a70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +00253a80: 5468 6520 6f62 6a65 6374 202a 6e6f 7465  The object *note
│ │ │ │ +00253a90: 2072 756e 3a20 7275 6e2c 2069 7320 6120   run: run, is a 
│ │ │ │ +00253aa0: 2a6e 6f74 6520 636f 6d70 696c 6564 2066  *note compiled f
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│ │ │ │ +00253ac0: 6446 756e 6374 696f 6e2c 2e0a 0a2d 2d2d  dFunction,...---
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│ │ │ │  00253ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00253af0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f  --------..The so
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│ │ │ │ -00253c20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00253c30: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361  *******..  * Usa
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│ │ │ │ -00253c80: 2020 2020 2066 696e 6450 726f 6772 616d       findProgram
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│ │ │ │ -00253ca0: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
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│ │ │ │ +00253c40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -00254870: 7072 6f67 7261 6d50 6174 6873 2322 6766  programPaths#"gf
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│ │ │ │ -00254890: 6766 616e 2f22 2020 2020 2020 2020 2020  gfan/"          
│ │ │ │ -002548a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00254870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +002548b0: 2f74 6f2f 6766 616e 2f22 2020 2020 2020  /to/gfan/"      
│ │ │ │  002548c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002548d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002548d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  002548e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002548f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │ -00254910: 6661 6e2f 2020 2020 2020 2020 2020 2020  fan/            
│ │ │ │ -00254920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00254930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00254940: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -00254950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00254960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002548f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00254900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00254910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00254920: 2020 7c0a 7c6f 3120 3d20 2f70 6174 682f    |.|o1 = /path/
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│ │ │ │ +00254940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00254950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00254960: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │  00254970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00254980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00254990: 2d2d 2d2d 2b0a 7c69 3220 3a20 6766 616e  ----+.|i2 : gfan
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│ │ │ │ -002549e0: 0a7c 202d 2d20 2f70 6174 682f 746f 2f67  .| -- /path/to/g
│ │ │ │ -002549f0: 6661 6e2f 6766 616e 2064 6f65 7320 6e6f  fan/gfan does no
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│ │ │ │ -00254a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00254a20: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ -00254a30: 202f 7573 722f 6c69 6265 7865 632f 4d61   /usr/libexec/Ma
│ │ │ │ -00254a40: 6361 756c 6179 322f 6269 6e2f 6766 616e  caulay2/bin/gfan
│ │ │ │ -00254a50: 2064 6f65 7320 6e6f 7420 6578 6973 7420   does not exist 
│ │ │ │ -00254a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00254a70: 2020 2020 207c 0a7c 202d 2d20 2f75 7372       |.| -- /usr
│ │ │ │ -00254a80: 2f6c 6f63 616c 2f73 6269 6e2f 6766 616e  /local/sbin/gfan
│ │ │ │ -00254a90: 2064 6f65 7320 6e6f 7420 6578 6973 7420   does not exist 
│ │ │ │ -00254aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00254ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00254ac0: 7c0a 7c20 2d2d 202f 7573 722f 6c6f 6361  |.| -- /usr/loca
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│ │ │ │ -00254ae0: 6e6f 7420 6578 6973 7420 2020 2020 2020  not exist       
│ │ │ │ -00254af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00254b00: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -00254b10: 2d20 2f75 7372 2f73 6269 6e2f 6766 616e  - /usr/sbin/gfan
│ │ │ │ -00254b20: 2064 6f65 7320 6e6f 7420 6578 6973 7420   does not exist 
│ │ │ │ -00254b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00254b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00254b50: 2020 2020 2020 7c0a 7c20 2d2d 202f 7573        |.| -- /us
│ │ │ │ -00254b60: 722f 6269 6e2f 6766 616e 2065 7869 7374  r/bin/gfan exist
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│ │ │ │ -00254b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00254ba0: 207c 0a7c 202d 2d20 7275 6e6e 696e 6720   |.| -- running 
│ │ │ │ -00254bb0: 222f 7573 722f 6269 6e2f 6766 616e 205f  "/usr/bin/gfan _
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│ │ │ │ -00254bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00254be0: 2020 2020 2020 2020 2020 2020 7c0a 7c54              |.|T
│ │ │ │ -00254bf0: 6869 7320 7072 6f67 7261 6d20 7772 6974  his program writ
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│ │ │ │ -00254c20: 6520 4766 616e 2069 6e73 7461 6c6c 6174  e Gfan installat
│ │ │ │ -00254c30: 696f 6e2e 2020 207c 0a7c 202d 2d20 7265  ion.   |.| -- re
│ │ │ │ -00254c40: 7475 726e 2076 616c 7565 3a20 3020 2020  turn value: 0   
│ │ │ │ -00254c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00254c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00254c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00254c80: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00254990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002549a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002549b0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20  --------+.|i2 : 
│ │ │ │ +002549c0: 6766 616e 203d 2066 696e 6450 726f 6772  gfan = findProgr
│ │ │ │ +002549d0: 616d 2822 6766 616e 222c 2022 6766 616e  am("gfan", "gfan
│ │ │ │ +002549e0: 205f 7665 7273 696f 6e20 2d2d 6865 6c70   _version --help
│ │ │ │ +002549f0: 222c 2056 6572 626f 7365 203d 3e20 7472  ", Verbose => tr
│ │ │ │ +00254a00: 7565 297c 0a7c 202d 2d20 2f70 6174 682f  ue)|.| -- /path/
│ │ │ │ +00254a10: 746f 2f67 6661 6e2f 6766 616e 2064 6f65  to/gfan/gfan doe
│ │ │ │ +00254a20: 7320 6e6f 7420 6578 6973 7420 2020 2020  s not exist     
│ │ │ │ +00254a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00254a40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00254a50: 7c20 2d2d 202f 7573 722f 6c69 6265 7865  | -- /usr/libexe
│ │ │ │ +00254a60: 632f 4d61 6361 756c 6179 322f 6269 6e2f  c/Macaulay2/bin/
│ │ │ │ +00254a70: 6766 616e 2064 6f65 7320 6e6f 7420 6578  gfan does not ex
│ │ │ │ +00254a80: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +00254a90: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ +00254aa0: 2f75 7372 2f6c 6f63 616c 2f73 6269 6e2f  /usr/local/sbin/
│ │ │ │ +00254ab0: 6766 616e 2064 6f65 7320 6e6f 7420 6578  gfan does not ex
│ │ │ │ +00254ac0: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +00254ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00254ae0: 2020 2020 7c0a 7c20 2d2d 202f 7573 722f      |.| -- /usr/
│ │ │ │ +00254af0: 6c6f 6361 6c2f 6269 6e2f 6766 616e 2064  local/bin/gfan d
│ │ │ │ +00254b00: 6f65 7320 6e6f 7420 6578 6973 7420 2020  oes not exist   
│ │ │ │ +00254b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00254b20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00254b30: 0a7c 202d 2d20 2f75 7372 2f73 6269 6e2f  .| -- /usr/sbin/
│ │ │ │ +00254b40: 6766 616e 2064 6f65 7320 6e6f 7420 6578  gfan does not ex
│ │ │ │ +00254b50: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +00254b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00254b70: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ +00254b80: 202f 7573 722f 6269 6e2f 6766 616e 2065   /usr/bin/gfan e
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│ │ │ │ +00254ba0: 6375 7461 626c 6520 2020 2020 2020 2020  cutable         
│ │ │ │ +00254bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00254bc0: 2020 2020 207c 0a7c 202d 2d20 7275 6e6e       |.| -- runn
│ │ │ │ +00254bd0: 696e 6720 222f 7573 722f 6269 6e2f 6766  ing "/usr/bin/gf
│ │ │ │ +00254be0: 616e 205f 7665 7273 696f 6e20 2d2d 6865  an _version --he
│ │ │ │ +00254bf0: 6c70 223a 2020 2020 2020 2020 2020 2020  lp":            
│ │ │ │ +00254c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00254c10: 7c0a 7c54 6869 7320 7072 6f67 7261 6d20  |.|This program 
│ │ │ │ +00254c20: 7772 6974 6573 206f 7574 2076 6572 7369  writes out versi
│ │ │ │ +00254c30: 6f6e 2069 6e66 6f72 6d61 7469 6f6e 206f  on information o
│ │ │ │ +00254c40: 6620 7468 6520 4766 616e 2069 6e73 7461  f the Gfan insta
│ │ │ │ +00254c50: 6c6c 6174 696f 6e2e 2020 207c 0a7c 202d  llation.   |.| -
│ │ │ │ +00254c60: 2d20 7265 7475 726e 2076 616c 7565 3a20  - return value: 
│ │ │ │ +00254c70: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +00254c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00254c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00254ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00254ca0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00254cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00254cc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00254cd0: 6f32 203d 2067 6661 6e20 2020 2020 2020  o2 = gfan       
│ │ │ │ +00254cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00254cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00254ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00254cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00254cf0: 207c 0a7c 6f32 203d 2067 6661 6e20 2020   |.|o2 = gfan   
│ │ │ │  00254d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00254d10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00254d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00254d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00254d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00254d30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00254d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00254d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00254d60: 2020 207c 0a7c 6f32 203a 2050 726f 6772     |.|o2 : Progr
│ │ │ │ -00254d70: 616d 2020 2020 2020 2020 2020 2020 2020  am              
│ │ │ │ -00254d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00254d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00254da0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00254db0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -00254dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00254dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00254d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00254d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00254d80: 2020 2020 2020 207c 0a7c 6f32 203a 2050         |.|o2 : P
│ │ │ │ +00254d90: 726f 6772 616d 2020 2020 2020 2020 2020  rogram          
│ │ │ │ +00254da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00254db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00254dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00254dd0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00254de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00254df0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4966 2063  ---------+..If c
│ │ │ │ -00254e00: 6d64 2069 7320 6e6f 7420 7072 6f76 6964  md is not provid
│ │ │ │ -00254e10: 6564 2c20 7468 656e 206e 616d 6520 6973  ed, then name is
│ │ │ │ -00254e20: 2072 756e 2077 6974 6820 7468 6520 636f   run with the co
│ │ │ │ -00254e30: 6d6d 6f6e 202d 2d76 6572 7369 6f6e 2063  mmon --version c
│ │ │ │ -00254e40: 6f6d 6d61 6e64 206c 696e 650a 6f70 7469  ommand line.opti
│ │ │ │ -00254e50: 6f6e 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  on...+----------
│ │ │ │ -00254e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00254e70: 2d2b 0a7c 6933 203a 2066 696e 6450 726f  -+.|i3 : findPro
│ │ │ │ -00254e80: 6772 616d 2022 6e6f 726d 616c 697a 227c  gram "normaliz"|
│ │ │ │ -00254e90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00254ea0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00254eb0: 6f33 203d 206e 6f72 6d61 6c69 7a20 2020  o3 = normaliz   
│ │ │ │ -00254ec0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00254ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00254ee0: 2020 2020 2020 2020 207c 0a7c 6f33 203a           |.|o3 :
│ │ │ │ -00254ef0: 2050 726f 6772 616d 2020 2020 2020 2020   Program        
│ │ │ │ -00254f00: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -00254f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00254f20: 2d2d 2d2d 2d2b 0a0a 4f6e 6520 7072 6f67  -----+..One prog
│ │ │ │ -00254f30: 7261 6d20 7468 6174 2069 7320 7368 6970  ram that is ship
│ │ │ │ -00254f40: 7065 6420 7769 7468 2061 2076 6172 6965  ped with a varie
│ │ │ │ -00254f50: 7479 206f 6620 7072 6566 6978 6573 2069  ty of prefixes i
│ │ │ │ -00254f60: 6e20 6469 6666 6572 656e 740a 6469 7374  n different.dist
│ │ │ │ -00254f70: 7269 6275 7469 6f6e 7320 616e 6420 666f  ributions and fo
│ │ │ │ -00254f80: 7220 7768 6963 6820 7468 6520 5072 6566  r which the Pref
│ │ │ │ -00254f90: 6978 206f 7074 696f 6e20 6973 2075 7365  ix option is use
│ │ │ │ -00254fa0: 6675 6c20 6973 2054 4f50 434f 4d3a 0a0a  ful is TOPCOM:..
│ │ │ │ -00254fb0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -00254fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00254fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00254df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00254e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00254e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ +00254e20: 4966 2063 6d64 2069 7320 6e6f 7420 7072  If cmd is not pr
│ │ │ │ +00254e30: 6f76 6964 6564 2c20 7468 656e 206e 616d  ovided, then nam
│ │ │ │ +00254e40: 6520 6973 2072 756e 2077 6974 6820 7468  e is run with th
│ │ │ │ +00254e50: 6520 636f 6d6d 6f6e 202d 2d76 6572 7369  e common --versi
│ │ │ │ +00254e60: 6f6e 2063 6f6d 6d61 6e64 206c 696e 650a  on command line.
│ │ │ │ +00254e70: 6f70 7469 6f6e 2e0a 0a2b 2d2d 2d2d 2d2d  option...+------
│ │ │ │ +00254e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00254e90: 2d2d 2d2d 2d2b 0a7c 6933 203a 2066 696e  -----+.|i3 : fin
│ │ │ │ +00254ea0: 6450 726f 6772 616d 2022 6e6f 726d 616c  dProgram "normal
│ │ │ │ +00254eb0: 697a 227c 0a7c 2020 2020 2020 2020 2020  iz"|.|          
│ │ │ │ +00254ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00254ed0: 207c 0a7c 6f33 203d 206e 6f72 6d61 6c69   |.|o3 = normali
│ │ │ │ +00254ee0: 7a20 2020 2020 2020 2020 2020 2020 207c  z              |
│ │ │ │ +00254ef0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00254f00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00254f10: 6f33 203a 2050 726f 6772 616d 2020 2020  o3 : Program    
│ │ │ │ +00254f20: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00254f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00254f40: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4f6e 6520  ---------+..One 
│ │ │ │ +00254f50: 7072 6f67 7261 6d20 7468 6174 2069 7320  program that is 
│ │ │ │ +00254f60: 7368 6970 7065 6420 7769 7468 2061 2076  shipped with a v
│ │ │ │ +00254f70: 6172 6965 7479 206f 6620 7072 6566 6978  ariety of prefix
│ │ │ │ +00254f80: 6573 2069 6e20 6469 6666 6572 656e 740a  es in different.
│ │ │ │ +00254f90: 6469 7374 7269 6275 7469 6f6e 7320 616e  distributions an
│ │ │ │ +00254fa0: 6420 666f 7220 7768 6963 6820 7468 6520  d for which the 
│ │ │ │ +00254fb0: 5072 6566 6978 206f 7074 696f 6e20 6973  Prefix option is
│ │ │ │ +00254fc0: 2075 7365 6675 6c20 6973 2054 4f50 434f   useful is TOPCO
│ │ │ │ +00254fd0: 4d3a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  M:..+-----------
│ │ │ │  00254fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00254ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00255000: 7c69 3420 3a20 6669 6e64 5072 6f67 7261  |i4 : findProgra
│ │ │ │ -00255010: 6d28 2274 6f70 636f 6d22 2c20 2263 7562  m("topcom", "cub
│ │ │ │ -00255020: 6520 3322 2c20 5665 7262 6f73 6520 3d3e  e 3", Verbose =>
│ │ │ │ -00255030: 2074 7275 652c 2050 7265 6669 7820 3d3e   true, Prefix =>
│ │ │ │ -00255040: 207b 2020 2020 2020 2020 2020 2020 7c0a   {            |.
│ │ │ │ -00255050: 7c20 2020 2020 2020 2822 2e2a 222c 2022  |       (".*", "
│ │ │ │ -00255060: 746f 7063 6f6d 2d22 292c 2020 2020 2020  topcom-"),      
│ │ │ │ -00255070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255090: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -002550a0: 7c20 2020 2020 2020 2822 5e28 6372 6f73  |       ("^(cros
│ │ │ │ -002550b0: 737c 6375 6265 7c63 7963 6c69 637c 6879  s|cube|cyclic|hy
│ │ │ │ -002550c0: 7065 7273 696d 706c 6578 7c6c 6174 7469  persimplex|latti
│ │ │ │ -002550d0: 6365 2924 222c 2022 544f 5043 4f4d 2d22  ce)$", "TOPCOM-"
│ │ │ │ -002550e0: 292c 2020 2020 2020 2020 2020 2020 7c0a  ),            |.
│ │ │ │ -002550f0: 7c20 2020 2020 2020 2822 5e63 7562 6524  |       ("^cube$
│ │ │ │ -00255100: 222c 2022 746f 7063 6f6d 5f22 297d 2920  ", "topcom_")}) 
│ │ │ │ -00255110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255130: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00255140: 7c20 2d2d 202f 7573 722f 6c69 6265 7865  | -- /usr/libexe
│ │ │ │ -00255150: 632f 4d61 6361 756c 6179 322f 6269 6e2f  c/Macaulay2/bin/
│ │ │ │ -00255160: 6375 6265 2064 6f65 7320 6e6f 7420 6578  cube does not ex
│ │ │ │ -00255170: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ -00255180: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00255190: 7c20 2d2d 202f 7573 722f 6c69 6265 7865  | -- /usr/libexe
│ │ │ │ -002551a0: 632f 4d61 6361 756c 6179 322f 6269 6e2f  c/Macaulay2/bin/
│ │ │ │ -002551b0: 746f 7063 6f6d 2d63 7562 6520 646f 6573  topcom-cube does
│ │ │ │ -002551c0: 206e 6f74 2065 7869 7374 2020 2020 2020   not exist      
│ │ │ │ -002551d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -002551e0: 7c20 2d2d 202f 7573 722f 6c69 6265 7865  | -- /usr/libexe
│ │ │ │ -002551f0: 632f 4d61 6361 756c 6179 322f 6269 6e2f  c/Macaulay2/bin/
│ │ │ │ -00255200: 544f 5043 4f4d 2d63 7562 6520 646f 6573  TOPCOM-cube does
│ │ │ │ -00255210: 206e 6f74 2065 7869 7374 2020 2020 2020   not exist      
│ │ │ │ -00255220: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00255230: 7c20 2d2d 202f 7573 722f 6c69 6265 7865  | -- /usr/libexe
│ │ │ │ -00255240: 632f 4d61 6361 756c 6179 322f 6269 6e2f  c/Macaulay2/bin/
│ │ │ │ -00255250: 746f 7063 6f6d 5f63 7562 6520 646f 6573  topcom_cube does
│ │ │ │ -00255260: 206e 6f74 2065 7869 7374 2020 2020 2020   not exist      
│ │ │ │ -00255270: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00255280: 7c20 2d2d 202f 7573 722f 6c6f 6361 6c2f  | -- /usr/local/
│ │ │ │ -00255290: 7362 696e 2f63 7562 6520 646f 6573 206e  sbin/cube does n
│ │ │ │ -002552a0: 6f74 2065 7869 7374 2020 2020 2020 2020  ot exist        
│ │ │ │ -002552b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002552c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -002552d0: 7c20 2d2d 202f 7573 722f 6c6f 6361 6c2f  | -- /usr/local/
│ │ │ │ -002552e0: 7362 696e 2f74 6f70 636f 6d2d 6375 6265  sbin/topcom-cube
│ │ │ │ -002552f0: 2064 6f65 7320 6e6f 7420 6578 6973 7420   does not exist 
│ │ │ │ -00255300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255310: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00255320: 7c20 2d2d 202f 7573 722f 6c6f 6361 6c2f  | -- /usr/local/
│ │ │ │ -00255330: 7362 696e 2f54 4f50 434f 4d2d 6375 6265  sbin/TOPCOM-cube
│ │ │ │ -00255340: 2064 6f65 7320 6e6f 7420 6578 6973 7420   does not exist 
│ │ │ │ -00255350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255360: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00255370: 7c20 2d2d 202f 7573 722f 6c6f 6361 6c2f  | -- /usr/local/
│ │ │ │ -00255380: 7362 696e 2f74 6f70 636f 6d5f 6375 6265  sbin/topcom_cube
│ │ │ │ -00255390: 2064 6f65 7320 6e6f 7420 6578 6973 7420   does not exist 
│ │ │ │ -002553a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002553b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -002553c0: 7c20 2d2d 202f 7573 722f 6c6f 6361 6c2f  | -- /usr/local/
│ │ │ │ -002553d0: 6269 6e2f 6375 6265 2064 6f65 7320 6e6f  bin/cube does no
│ │ │ │ -002553e0: 7420 6578 6973 7420 2020 2020 2020 2020  t exist         
│ │ │ │ -002553f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255400: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00255410: 7c20 2d2d 202f 7573 722f 6c6f 6361 6c2f  | -- /usr/local/
│ │ │ │ -00255420: 6269 6e2f 746f 7063 6f6d 2d63 7562 6520  bin/topcom-cube 
│ │ │ │ -00255430: 646f 6573 206e 6f74 2065 7869 7374 2020  does not exist  
│ │ │ │ -00255440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255450: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00255460: 7c20 2d2d 202f 7573 722f 6c6f 6361 6c2f  | -- /usr/local/
│ │ │ │ -00255470: 6269 6e2f 544f 5043 4f4d 2d63 7562 6520  bin/TOPCOM-cube 
│ │ │ │ -00255480: 646f 6573 206e 6f74 2065 7869 7374 2020  does not exist  
│ │ │ │ -00255490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002554a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -002554b0: 7c20 2d2d 202f 7573 722f 6c6f 6361 6c2f  | -- /usr/local/
│ │ │ │ -002554c0: 6269 6e2f 746f 7063 6f6d 5f63 7562 6520  bin/topcom_cube 
│ │ │ │ -002554d0: 646f 6573 206e 6f74 2065 7869 7374 2020  does not exist  
│ │ │ │ -002554e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002554f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00255500: 7c20 2d2d 202f 7573 722f 7362 696e 2f63  | -- /usr/sbin/c
│ │ │ │ -00255510: 7562 6520 646f 6573 206e 6f74 2065 7869  ube does not exi
│ │ │ │ -00255520: 7374 2020 2020 2020 2020 2020 2020 2020  st              
│ │ │ │ -00255530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255540: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00255550: 7c20 2d2d 202f 7573 722f 7362 696e 2f74  | -- /usr/sbin/t
│ │ │ │ -00255560: 6f70 636f 6d2d 6375 6265 2064 6f65 7320  opcom-cube does 
│ │ │ │ -00255570: 6e6f 7420 6578 6973 7420 2020 2020 2020  not exist       
│ │ │ │ -00255580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255590: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -002555a0: 7c20 2d2d 202f 7573 722f 7362 696e 2f54  | -- /usr/sbin/T
│ │ │ │ -002555b0: 4f50 434f 4d2d 6375 6265 2064 6f65 7320  OPCOM-cube does 
│ │ │ │ -002555c0: 6e6f 7420 6578 6973 7420 2020 2020 2020  not exist       
│ │ │ │ -002555d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002555e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -002555f0: 7c20 2d2d 202f 7573 722f 7362 696e 2f74  | -- /usr/sbin/t
│ │ │ │ -00255600: 6f70 636f 6d5f 6375 6265 2064 6f65 7320  opcom_cube does 
│ │ │ │ -00255610: 6e6f 7420 6578 6973 7420 2020 2020 2020  not exist       
│ │ │ │ -00255620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255630: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00255640: 7c20 2d2d 202f 7573 722f 6269 6e2f 6375  | -- /usr/bin/cu
│ │ │ │ -00255650: 6265 2064 6f65 7320 6e6f 7420 6578 6973  be does not exis
│ │ │ │ -00255660: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ -00255670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255680: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00255690: 7c20 2d2d 202f 7573 722f 6269 6e2f 746f  | -- /usr/bin/to
│ │ │ │ -002556a0: 7063 6f6d 2d63 7562 6520 6578 6973 7473  pcom-cube exists
│ │ │ │ -002556b0: 2061 6e64 2069 7320 6578 6563 7574 6162   and is executab
│ │ │ │ -002556c0: 6c65 2020 2020 2020 2020 2020 2020 2020  le              
│ │ │ │ -002556d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -002556e0: 7c20 2d2d 2072 756e 6e69 6e67 2022 2f75  | -- running "/u
│ │ │ │ -002556f0: 7372 2f62 696e 2f74 6f70 636f 6d2d 6375  sr/bin/topcom-cu
│ │ │ │ -00255700: 6265 2033 223a 2020 2020 2020 2020 2020  be 3":          
│ │ │ │ -00255710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255720: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00255730: 7c5b 5b30 2c30 2c30 2c31 5d2c 5b31 2c30  |[[0,0,0,1],[1,0
│ │ │ │ -00255740: 2c30 2c31 5d2c 5b30 2c31 2c30 2c31 5d2c  ,0,1],[0,1,0,1],
│ │ │ │ -00255750: 5b31 2c31 2c30 2c31 5d2c 5b30 2c30 2c31  [1,1,0,1],[0,0,1
│ │ │ │ -00255760: 2c31 5d2c 5b31 2c30 2c31 2c31 5d2c 5b30  ,1],[1,0,1,1],[0
│ │ │ │ -00255770: 2c31 2c31 2c31 5d2c 5b31 2c31 2c31 7c0a  ,1,1,1],[1,1,1|.
│ │ │ │ -00255780: 7c5b 5b37 2c36 2c35 2c34 2c33 2c32 2c31  |[[7,6,5,4,3,2,1
│ │ │ │ -00255790: 2c30 5d2c 5b36 2c37 2c34 2c35 2c32 2c33  ,0],[6,7,4,5,2,3
│ │ │ │ -002557a0: 2c30 2c31 5d2c 5b34 2c36 2c35 2c37 2c30  ,0,1],[4,6,5,7,0
│ │ │ │ -002557b0: 2c32 2c31 2c33 5d2c 5b30 2c34 2c32 2c36  ,2,1,3],[0,4,2,6
│ │ │ │ -002557c0: 2c31 2c35 2c33 2c37 5d5d 2020 2020 7c0a  ,1,5,3,7]]    |.
│ │ │ │ -002557d0: 7c20 2d2d 2072 6574 7572 6e20 7661 6c75  | -- return valu
│ │ │ │ -002557e0: 653a 2030 2020 2020 2020 2020 2020 2020  e: 0            
│ │ │ │ -002557f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255810: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00255820: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00254ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00255000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00255010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00255020: 2d2d 2b0a 7c69 3420 3a20 6669 6e64 5072  --+.|i4 : findPr
│ │ │ │ +00255030: 6f67 7261 6d28 2274 6f70 636f 6d22 2c20  ogram("topcom", 
│ │ │ │ +00255040: 2263 7562 6520 3322 2c20 5665 7262 6f73  "cube 3", Verbos
│ │ │ │ +00255050: 6520 3d3e 2074 7275 652c 2050 7265 6669  e => true, Prefi
│ │ │ │ +00255060: 7820 3d3e 207b 2020 2020 2020 2020 2020  x => {          
│ │ │ │ +00255070: 2020 7c0a 7c20 2020 2020 2020 2822 2e2a    |.|       (".*
│ │ │ │ +00255080: 222c 2022 746f 7063 6f6d 2d22 292c 2020  ", "topcom-"),  
│ │ │ │ +00255090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002550a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002550b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002550c0: 2020 7c0a 7c20 2020 2020 2020 2822 5e28    |.|       ("^(
│ │ │ │ +002550d0: 6372 6f73 737c 6375 6265 7c63 7963 6c69  cross|cube|cycli
│ │ │ │ +002550e0: 637c 6879 7065 7273 696d 706c 6578 7c6c  c|hypersimplex|l
│ │ │ │ +002550f0: 6174 7469 6365 2924 222c 2022 544f 5043  attice)$", "TOPC
│ │ │ │ +00255100: 4f4d 2d22 292c 2020 2020 2020 2020 2020  OM-"),          
│ │ │ │ +00255110: 2020 7c0a 7c20 2020 2020 2020 2822 5e63    |.|       ("^c
│ │ │ │ +00255120: 7562 6524 222c 2022 746f 7063 6f6d 5f22  ube$", "topcom_"
│ │ │ │ +00255130: 297d 2920 2020 2020 2020 2020 2020 2020  )})             
│ │ │ │ +00255140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255160: 2020 7c0a 7c20 2d2d 202f 7573 722f 6c69    |.| -- /usr/li
│ │ │ │ +00255170: 6265 7865 632f 4d61 6361 756c 6179 322f  bexec/Macaulay2/
│ │ │ │ +00255180: 6269 6e2f 6375 6265 2064 6f65 7320 6e6f  bin/cube does no
│ │ │ │ +00255190: 7420 6578 6973 7420 2020 2020 2020 2020  t exist         
│ │ │ │ +002551a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002551b0: 2020 7c0a 7c20 2d2d 202f 7573 722f 6c69    |.| -- /usr/li
│ │ │ │ +002551c0: 6265 7865 632f 4d61 6361 756c 6179 322f  bexec/Macaulay2/
│ │ │ │ +002551d0: 6269 6e2f 746f 7063 6f6d 2d63 7562 6520  bin/topcom-cube 
│ │ │ │ +002551e0: 646f 6573 206e 6f74 2065 7869 7374 2020  does not exist  
│ │ │ │ +002551f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255200: 2020 7c0a 7c20 2d2d 202f 7573 722f 6c69    |.| -- /usr/li
│ │ │ │ +00255210: 6265 7865 632f 4d61 6361 756c 6179 322f  bexec/Macaulay2/
│ │ │ │ +00255220: 6269 6e2f 544f 5043 4f4d 2d63 7562 6520  bin/TOPCOM-cube 
│ │ │ │ +00255230: 646f 6573 206e 6f74 2065 7869 7374 2020  does not exist  
│ │ │ │ +00255240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255250: 2020 7c0a 7c20 2d2d 202f 7573 722f 6c69    |.| -- /usr/li
│ │ │ │ +00255260: 6265 7865 632f 4d61 6361 756c 6179 322f  bexec/Macaulay2/
│ │ │ │ +00255270: 6269 6e2f 746f 7063 6f6d 5f63 7562 6520  bin/topcom_cube 
│ │ │ │ +00255280: 646f 6573 206e 6f74 2065 7869 7374 2020  does not exist  
│ │ │ │ +00255290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002552a0: 2020 7c0a 7c20 2d2d 202f 7573 722f 6c6f    |.| -- /usr/lo
│ │ │ │ +002552b0: 6361 6c2f 7362 696e 2f63 7562 6520 646f  cal/sbin/cube do
│ │ │ │ +002552c0: 6573 206e 6f74 2065 7869 7374 2020 2020  es not exist    
│ │ │ │ +002552d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002552e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002552f0: 2020 7c0a 7c20 2d2d 202f 7573 722f 6c6f    |.| -- /usr/lo
│ │ │ │ +00255300: 6361 6c2f 7362 696e 2f74 6f70 636f 6d2d  cal/sbin/topcom-
│ │ │ │ +00255310: 6375 6265 2064 6f65 7320 6e6f 7420 6578  cube does not ex
│ │ │ │ +00255320: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +00255330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255340: 2020 7c0a 7c20 2d2d 202f 7573 722f 6c6f    |.| -- /usr/lo
│ │ │ │ +00255350: 6361 6c2f 7362 696e 2f54 4f50 434f 4d2d  cal/sbin/TOPCOM-
│ │ │ │ +00255360: 6375 6265 2064 6f65 7320 6e6f 7420 6578  cube does not ex
│ │ │ │ +00255370: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +00255380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255390: 2020 7c0a 7c20 2d2d 202f 7573 722f 6c6f    |.| -- /usr/lo
│ │ │ │ +002553a0: 6361 6c2f 7362 696e 2f74 6f70 636f 6d5f  cal/sbin/topcom_
│ │ │ │ +002553b0: 6375 6265 2064 6f65 7320 6e6f 7420 6578  cube does not ex
│ │ │ │ +002553c0: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +002553d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002553e0: 2020 7c0a 7c20 2d2d 202f 7573 722f 6c6f    |.| -- /usr/lo
│ │ │ │ +002553f0: 6361 6c2f 6269 6e2f 6375 6265 2064 6f65  cal/bin/cube doe
│ │ │ │ +00255400: 7320 6e6f 7420 6578 6973 7420 2020 2020  s not exist     
│ │ │ │ +00255410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255430: 2020 7c0a 7c20 2d2d 202f 7573 722f 6c6f    |.| -- /usr/lo
│ │ │ │ +00255440: 6361 6c2f 6269 6e2f 746f 7063 6f6d 2d63  cal/bin/topcom-c
│ │ │ │ +00255450: 7562 6520 646f 6573 206e 6f74 2065 7869  ube does not exi
│ │ │ │ +00255460: 7374 2020 2020 2020 2020 2020 2020 2020  st              
│ │ │ │ +00255470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255480: 2020 7c0a 7c20 2d2d 202f 7573 722f 6c6f    |.| -- /usr/lo
│ │ │ │ +00255490: 6361 6c2f 6269 6e2f 544f 5043 4f4d 2d63  cal/bin/TOPCOM-c
│ │ │ │ +002554a0: 7562 6520 646f 6573 206e 6f74 2065 7869  ube does not exi
│ │ │ │ +002554b0: 7374 2020 2020 2020 2020 2020 2020 2020  st              
│ │ │ │ +002554c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002554d0: 2020 7c0a 7c20 2d2d 202f 7573 722f 6c6f    |.| -- /usr/lo
│ │ │ │ +002554e0: 6361 6c2f 6269 6e2f 746f 7063 6f6d 5f63  cal/bin/topcom_c
│ │ │ │ +002554f0: 7562 6520 646f 6573 206e 6f74 2065 7869  ube does not exi
│ │ │ │ +00255500: 7374 2020 2020 2020 2020 2020 2020 2020  st              
│ │ │ │ +00255510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255520: 2020 7c0a 7c20 2d2d 202f 7573 722f 7362    |.| -- /usr/sb
│ │ │ │ +00255530: 696e 2f63 7562 6520 646f 6573 206e 6f74  in/cube does not
│ │ │ │ +00255540: 2065 7869 7374 2020 2020 2020 2020 2020   exist          
│ │ │ │ +00255550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255570: 2020 7c0a 7c20 2d2d 202f 7573 722f 7362    |.| -- /usr/sb
│ │ │ │ +00255580: 696e 2f74 6f70 636f 6d2d 6375 6265 2064  in/topcom-cube d
│ │ │ │ +00255590: 6f65 7320 6e6f 7420 6578 6973 7420 2020  oes not exist   
│ │ │ │ +002555a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002555b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002555c0: 2020 7c0a 7c20 2d2d 202f 7573 722f 7362    |.| -- /usr/sb
│ │ │ │ +002555d0: 696e 2f54 4f50 434f 4d2d 6375 6265 2064  in/TOPCOM-cube d
│ │ │ │ +002555e0: 6f65 7320 6e6f 7420 6578 6973 7420 2020  oes not exist   
│ │ │ │ +002555f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255610: 2020 7c0a 7c20 2d2d 202f 7573 722f 7362    |.| -- /usr/sb
│ │ │ │ +00255620: 696e 2f74 6f70 636f 6d5f 6375 6265 2064  in/topcom_cube d
│ │ │ │ +00255630: 6f65 7320 6e6f 7420 6578 6973 7420 2020  oes not exist   
│ │ │ │ +00255640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255660: 2020 7c0a 7c20 2d2d 202f 7573 722f 6269    |.| -- /usr/bi
│ │ │ │ +00255670: 6e2f 6375 6265 2064 6f65 7320 6e6f 7420  n/cube does not 
│ │ │ │ +00255680: 6578 6973 7420 2020 2020 2020 2020 2020  exist           
│ │ │ │ +00255690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002556a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002556b0: 2020 7c0a 7c20 2d2d 202f 7573 722f 6269    |.| -- /usr/bi
│ │ │ │ +002556c0: 6e2f 746f 7063 6f6d 2d63 7562 6520 6578  n/topcom-cube ex
│ │ │ │ +002556d0: 6973 7473 2061 6e64 2069 7320 6578 6563  ists and is exec
│ │ │ │ +002556e0: 7574 6162 6c65 2020 2020 2020 2020 2020  utable          
│ │ │ │ +002556f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255700: 2020 7c0a 7c20 2d2d 2072 756e 6e69 6e67    |.| -- running
│ │ │ │ +00255710: 2022 2f75 7372 2f62 696e 2f74 6f70 636f   "/usr/bin/topco
│ │ │ │ +00255720: 6d2d 6375 6265 2033 223a 2020 2020 2020  m-cube 3":      
│ │ │ │ +00255730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255750: 2020 7c0a 7c5b 5b30 2c30 2c30 2c31 5d2c    |.|[[0,0,0,1],
│ │ │ │ +00255760: 5b31 2c30 2c30 2c31 5d2c 5b30 2c31 2c30  [1,0,0,1],[0,1,0
│ │ │ │ +00255770: 2c31 5d2c 5b31 2c31 2c30 2c31 5d2c 5b30  ,1],[1,1,0,1],[0
│ │ │ │ +00255780: 2c30 2c31 2c31 5d2c 5b31 2c30 2c31 2c31  ,0,1,1],[1,0,1,1
│ │ │ │ +00255790: 5d2c 5b30 2c31 2c31 2c31 5d2c 5b31 2c31  ],[0,1,1,1],[1,1
│ │ │ │ +002557a0: 2c31 7c0a 7c5b 5b37 2c36 2c35 2c34 2c33  ,1|.|[[7,6,5,4,3
│ │ │ │ +002557b0: 2c32 2c31 2c30 5d2c 5b36 2c37 2c34 2c35  ,2,1,0],[6,7,4,5
│ │ │ │ +002557c0: 2c32 2c33 2c30 2c31 5d2c 5b34 2c36 2c35  ,2,3,0,1],[4,6,5
│ │ │ │ +002557d0: 2c37 2c30 2c32 2c31 2c33 5d2c 5b30 2c34  ,7,0,2,1,3],[0,4
│ │ │ │ +002557e0: 2c32 2c36 2c31 2c35 2c33 2c37 5d5d 2020  ,2,6,1,5,3,7]]  
│ │ │ │ +002557f0: 2020 7c0a 7c20 2d2d 2072 6574 7572 6e20    |.| -- return 
│ │ │ │ +00255800: 7661 6c75 653a 2030 2020 2020 2020 2020  value: 0        
│ │ │ │ +00255810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00255830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255840: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00255850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255860: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00255870: 7c6f 3420 3d20 746f 7063 6f6d 2020 2020  |o4 = topcom    
│ │ │ │ +00255860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00255880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255890: 2020 7c0a 7c6f 3420 3d20 746f 7063 6f6d    |.|o4 = topcom
│ │ │ │  002558a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002558b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -002558c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +002558b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002558c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002558d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002558e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002558e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  002558f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255900: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00255910: 7c6f 3420 3a20 5072 6f67 7261 6d20 2020  |o4 : Program   
│ │ │ │ +00255900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00255920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255950: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00255960: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |---------------
│ │ │ │ -00255970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00255980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00255930: 2020 7c0a 7c6f 3420 3a20 5072 6f67 7261    |.|o4 : Progra
│ │ │ │ +00255940: 6d20 2020 2020 2020 2020 2020 2020 2020  m               
│ │ │ │ +00255950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255980: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.|-----------
│ │ │ │  00255990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -002559a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ -002559b0: 7c2c 315d 5d20 2020 2020 2020 2020 2020  |,1]]           
│ │ │ │ -002559c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002559d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002559a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002559b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002559c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002559d0: 2d2d 7c0a 7c2c 315d 5d20 2020 2020 2020  --|.|,1]]       
│ │ │ │  002559e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002559f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00255a00: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -00255a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00255a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002559f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255a20: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00255a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00255a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00255a50: 0a4e 6f74 6520 7468 6174 2077 6865 6e20  .Note that when 
│ │ │ │ -00255a60: 7573 696e 6720 7468 6520 4d69 6e69 6d75  using the Minimu
│ │ │ │ -00255a70: 6d56 6572 7369 6f6e 206f 7074 696f 6e2c  mVersion option,
│ │ │ │ -00255a80: 2074 6865 2063 6f6d 6d61 6e64 2075 7365   the command use
│ │ │ │ -00255a90: 6420 746f 206f 6274 6169 6e20 7468 650a  d to obtain the.
│ │ │ │ -00255aa0: 6375 7272 656e 7420 7665 7273 696f 6e20  current version 
│ │ │ │ -00255ab0: 6e75 6d62 6572 206d 7573 7420 7265 6d6f  number must remo
│ │ │ │ -00255ac0: 7665 2065 7665 7279 7468 696e 6720 6578  ve everything ex
│ │ │ │ -00255ad0: 6365 7074 2074 6865 2076 6572 7369 6f6e  cept the version
│ │ │ │ -00255ae0: 206e 756d 6265 7220 6974 7365 6c66 0a61   number itself.a
│ │ │ │ -00255af0: 6e64 2061 6e79 206c 6561 6469 6e67 206f  nd any leading o
│ │ │ │ -00255b00: 7220 7472 6169 6c69 6e67 2077 6869 7465  r trailing white
│ │ │ │ -00255b10: 7370 6163 652e 2020 5069 7069 6e67 2077  space.  Piping w
│ │ │ │ -00255b20: 6974 6820 7374 616e 6461 7264 2055 4e49  ith standard UNI
│ │ │ │ -00255b30: 5820 7574 696c 6974 6965 730a 7375 6368  X utilities.such
│ │ │ │ -00255b40: 2061 7320 7365 642c 2068 6561 642c 2074   as sed, head, t
│ │ │ │ -00255b50: 6169 6c2c 2063 7574 2c20 616e 6420 7472  ail, cut, and tr
│ │ │ │ -00255b60: 206d 6179 2062 6520 7573 6566 756c 2e0a   may be useful..
│ │ │ │ -00255b70: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00255b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00255b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00255a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00255a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00255a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00255a70: 2d2d 2b0a 0a4e 6f74 6520 7468 6174 2077  --+..Note that w
│ │ │ │ +00255a80: 6865 6e20 7573 696e 6720 7468 6520 4d69  hen using the Mi
│ │ │ │ +00255a90: 6e69 6d75 6d56 6572 7369 6f6e 206f 7074  nimumVersion opt
│ │ │ │ +00255aa0: 696f 6e2c 2074 6865 2063 6f6d 6d61 6e64  ion, the command
│ │ │ │ +00255ab0: 2075 7365 6420 746f 206f 6274 6169 6e20   used to obtain 
│ │ │ │ +00255ac0: 7468 650a 6375 7272 656e 7420 7665 7273  the.current vers
│ │ │ │ +00255ad0: 696f 6e20 6e75 6d62 6572 206d 7573 7420  ion number must 
│ │ │ │ +00255ae0: 7265 6d6f 7665 2065 7665 7279 7468 696e  remove everythin
│ │ │ │ +00255af0: 6720 6578 6365 7074 2074 6865 2076 6572  g except the ver
│ │ │ │ +00255b00: 7369 6f6e 206e 756d 6265 7220 6974 7365  sion number itse
│ │ │ │ +00255b10: 6c66 0a61 6e64 2061 6e79 206c 6561 6469  lf.and any leadi
│ │ │ │ +00255b20: 6e67 206f 7220 7472 6169 6c69 6e67 2077  ng or trailing w
│ │ │ │ +00255b30: 6869 7465 7370 6163 652e 2020 5069 7069  hitespace.  Pipi
│ │ │ │ +00255b40: 6e67 2077 6974 6820 7374 616e 6461 7264  ng with standard
│ │ │ │ +00255b50: 2055 4e49 5820 7574 696c 6974 6965 730a   UNIX utilities.
│ │ │ │ +00255b60: 7375 6368 2061 7320 7365 642c 2068 6561  such as sed, hea
│ │ │ │ +00255b70: 642c 2074 6169 6c2c 2063 7574 2c20 616e  d, tail, cut, an
│ │ │ │ +00255b80: 6420 7472 206d 6179 2062 6520 7573 6566  d tr may be usef
│ │ │ │ +00255b90: 756c 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  ul...+----------
│ │ │ │  00255ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00255bb0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2066  -------+.|i5 : f
│ │ │ │ -00255bc0: 696e 6450 726f 6772 616d 2822 6766 616e  indProgram("gfan
│ │ │ │ -00255bd0: 222c 2022 6766 616e 205f 7665 7273 696f  ", "gfan _versio
│ │ │ │ -00255be0: 6e20 2d2d 6865 6c70 222c 2056 6572 626f  n --help", Verbo
│ │ │ │ -00255bf0: 7365 203d 3e20 7472 7565 2c20 2020 207c  se => true,    |
│ │ │ │ -00255c00: 0a7c 2020 2020 2020 204d 696e 696d 756d  .|       Minimum
│ │ │ │ -00255c10: 5665 7273 696f 6e20 3d3e 2028 2230 2e35  Version => ("0.5
│ │ │ │ -00255c20: 222c 2020 2020 2020 2020 2020 2020 2020  ",              
│ │ │ │ -00255c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255c40: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00255c50: 2267 6661 6e20 5f76 6572 7369 6f6e 207c  "gfan _version |
│ │ │ │ -00255c60: 2068 6561 6420 2d32 207c 2074 6169 6c20   head -2 | tail 
│ │ │ │ -00255c70: 2d31 207c 2073 6564 2027 732f 6766 616e  -1 | sed 's/gfan
│ │ │ │ -00255c80: 2f2f 2722 2929 2020 2020 2020 2020 207c  //'"))         |
│ │ │ │ -00255c90: 0a7c 202d 2d20 2f70 6174 682f 746f 2f67  .| -- /path/to/g
│ │ │ │ -00255ca0: 6661 6e2f 6766 616e 2064 6f65 7320 6e6f  fan/gfan does no
│ │ │ │ -00255cb0: 7420 6578 6973 7420 2020 2020 2020 2020  t exist         
│ │ │ │ -00255cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255cd0: 2020 2020 2020 207c 0a7c 202d 2d20 2f75         |.| -- /u
│ │ │ │ -00255ce0: 7372 2f6c 6962 6578 6563 2f4d 6163 6175  sr/libexec/Macau
│ │ │ │ -00255cf0: 6c61 7932 2f62 696e 2f67 6661 6e20 646f  lay2/bin/gfan do
│ │ │ │ -00255d00: 6573 206e 6f74 2065 7869 7374 2020 2020  es not exist    
│ │ │ │ -00255d10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00255d20: 0a7c 202d 2d20 2f75 7372 2f6c 6f63 616c  .| -- /usr/local
│ │ │ │ -00255d30: 2f73 6269 6e2f 6766 616e 2064 6f65 7320  /sbin/gfan does 
│ │ │ │ -00255d40: 6e6f 7420 6578 6973 7420 2020 2020 2020  not exist       
│ │ │ │ -00255d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255d60: 2020 2020 2020 207c 0a7c 202d 2d20 2f75         |.| -- /u
│ │ │ │ -00255d70: 7372 2f6c 6f63 616c 2f62 696e 2f67 6661  sr/local/bin/gfa
│ │ │ │ -00255d80: 6e20 646f 6573 206e 6f74 2065 7869 7374  n does not exist
│ │ │ │ -00255d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255da0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00255db0: 0a7c 202d 2d20 2f75 7372 2f73 6269 6e2f  .| -- /usr/sbin/
│ │ │ │ -00255dc0: 6766 616e 2064 6f65 7320 6e6f 7420 6578  gfan does not ex
│ │ │ │ -00255dd0: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ -00255de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255df0: 2020 2020 2020 207c 0a7c 202d 2d20 2f75         |.| -- /u
│ │ │ │ -00255e00: 7372 2f62 696e 2f67 6661 6e20 6578 6973  sr/bin/gfan exis
│ │ │ │ -00255e10: 7473 2061 6e64 2069 7320 6578 6563 7574  ts and is execut
│ │ │ │ -00255e20: 6162 6c65 2020 2020 2020 2020 2020 2020  able            
│ │ │ │ -00255e30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00255e40: 0a7c 202d 2d20 7275 6e6e 696e 6720 222f  .| -- running "/
│ │ │ │ -00255e50: 7573 722f 6269 6e2f 6766 616e 205f 7665  usr/bin/gfan _ve
│ │ │ │ -00255e60: 7273 696f 6e20 2d2d 6865 6c70 223a 2020  rsion --help":  
│ │ │ │ -00255e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255e80: 2020 2020 2020 207c 0a7c 5468 6973 2070         |.|This p
│ │ │ │ -00255e90: 726f 6772 616d 2077 7269 7465 7320 6f75  rogram writes ou
│ │ │ │ -00255ea0: 7420 7665 7273 696f 6e20 696e 666f 726d  t version inform
│ │ │ │ -00255eb0: 6174 696f 6e20 6f66 2074 6865 2047 6661  ation of the Gfa
│ │ │ │ -00255ec0: 6e20 696e 7374 616c 6c61 7469 6f6e 2e7c  n installation.|
│ │ │ │ -00255ed0: 0a7c 202d 2d20 7265 7475 726e 2076 616c  .| -- return val
│ │ │ │ -00255ee0: 7565 3a20 3020 2020 2020 2020 2020 2020  ue: 0           
│ │ │ │ -00255ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255f10: 2020 2020 2020 207c 0a7c 202d 2d20 666f         |.| -- fo
│ │ │ │ -00255f20: 756e 6420 7665 7273 696f 6e20 302e 3720  und version 0.7 
│ │ │ │ -00255f30: 3e3d 2030 2e35 2020 2020 2020 2020 2020  >= 0.5          
│ │ │ │ -00255f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255f50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00255f60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00255bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00255bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00255bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935  -----------+.|i5
│ │ │ │ +00255be0: 203a 2066 696e 6450 726f 6772 616d 2822   : findProgram("
│ │ │ │ +00255bf0: 6766 616e 222c 2022 6766 616e 205f 7665  gfan", "gfan _ve
│ │ │ │ +00255c00: 7273 696f 6e20 2d2d 6865 6c70 222c 2056  rsion --help", V
│ │ │ │ +00255c10: 6572 626f 7365 203d 3e20 7472 7565 2c20  erbose => true, 
│ │ │ │ +00255c20: 2020 207c 0a7c 2020 2020 2020 204d 696e     |.|       Min
│ │ │ │ +00255c30: 696d 756d 5665 7273 696f 6e20 3d3e 2028  imumVersion => (
│ │ │ │ +00255c40: 2230 2e35 222c 2020 2020 2020 2020 2020  "0.5",          
│ │ │ │ +00255c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255c60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00255c70: 2020 2020 2267 6661 6e20 5f76 6572 7369      "gfan _versi
│ │ │ │ +00255c80: 6f6e 207c 2068 6561 6420 2d32 207c 2074  on | head -2 | t
│ │ │ │ +00255c90: 6169 6c20 2d31 207c 2073 6564 2027 732f  ail -1 | sed 's/
│ │ │ │ +00255ca0: 6766 616e 2f2f 2722 2929 2020 2020 2020  gfan//'"))      
│ │ │ │ +00255cb0: 2020 207c 0a7c 202d 2d20 2f70 6174 682f     |.| -- /path/
│ │ │ │ +00255cc0: 746f 2f67 6661 6e2f 6766 616e 2064 6f65  to/gfan/gfan doe
│ │ │ │ +00255cd0: 7320 6e6f 7420 6578 6973 7420 2020 2020  s not exist     
│ │ │ │ +00255ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255cf0: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ +00255d00: 2d20 2f75 7372 2f6c 6962 6578 6563 2f4d  - /usr/libexec/M
│ │ │ │ +00255d10: 6163 6175 6c61 7932 2f62 696e 2f67 6661  acaulay2/bin/gfa
│ │ │ │ +00255d20: 6e20 646f 6573 206e 6f74 2065 7869 7374  n does not exist
│ │ │ │ +00255d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255d40: 2020 207c 0a7c 202d 2d20 2f75 7372 2f6c     |.| -- /usr/l
│ │ │ │ +00255d50: 6f63 616c 2f73 6269 6e2f 6766 616e 2064  ocal/sbin/gfan d
│ │ │ │ +00255d60: 6f65 7320 6e6f 7420 6578 6973 7420 2020  oes not exist   
│ │ │ │ +00255d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255d80: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ +00255d90: 2d20 2f75 7372 2f6c 6f63 616c 2f62 696e  - /usr/local/bin
│ │ │ │ +00255da0: 2f67 6661 6e20 646f 6573 206e 6f74 2065  /gfan does not e
│ │ │ │ +00255db0: 7869 7374 2020 2020 2020 2020 2020 2020  xist            
│ │ │ │ +00255dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255dd0: 2020 207c 0a7c 202d 2d20 2f75 7372 2f73     |.| -- /usr/s
│ │ │ │ +00255de0: 6269 6e2f 6766 616e 2064 6f65 7320 6e6f  bin/gfan does no
│ │ │ │ +00255df0: 7420 6578 6973 7420 2020 2020 2020 2020  t exist         
│ │ │ │ +00255e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255e10: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ +00255e20: 2d20 2f75 7372 2f62 696e 2f67 6661 6e20  - /usr/bin/gfan 
│ │ │ │ +00255e30: 6578 6973 7473 2061 6e64 2069 7320 6578  exists and is ex
│ │ │ │ +00255e40: 6563 7574 6162 6c65 2020 2020 2020 2020  ecutable        
│ │ │ │ +00255e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255e60: 2020 207c 0a7c 202d 2d20 7275 6e6e 696e     |.| -- runnin
│ │ │ │ +00255e70: 6720 222f 7573 722f 6269 6e2f 6766 616e  g "/usr/bin/gfan
│ │ │ │ +00255e80: 205f 7665 7273 696f 6e20 2d2d 6865 6c70   _version --help
│ │ │ │ +00255e90: 223a 2020 2020 2020 2020 2020 2020 2020  ":              
│ │ │ │ +00255ea0: 2020 2020 2020 2020 2020 207c 0a7c 5468             |.|Th
│ │ │ │ +00255eb0: 6973 2070 726f 6772 616d 2077 7269 7465  is program write
│ │ │ │ +00255ec0: 7320 6f75 7420 7665 7273 696f 6e20 696e  s out version in
│ │ │ │ +00255ed0: 666f 726d 6174 696f 6e20 6f66 2074 6865  formation of the
│ │ │ │ +00255ee0: 2047 6661 6e20 696e 7374 616c 6c61 7469   Gfan installati
│ │ │ │ +00255ef0: 6f6e 2e7c 0a7c 202d 2d20 7265 7475 726e  on.|.| -- return
│ │ │ │ +00255f00: 2076 616c 7565 3a20 3020 2020 2020 2020   value: 0       
│ │ │ │ +00255f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255f30: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ +00255f40: 2d20 666f 756e 6420 7665 7273 696f 6e20  - found version 
│ │ │ │ +00255f50: 302e 3720 3e3d 2030 2e35 2020 2020 2020  0.7 >= 0.5      
│ │ │ │ +00255f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00255f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255f80: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00255f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255fa0: 2020 2020 2020 207c 0a7c 6f35 203d 2067         |.|o5 = g
│ │ │ │ -00255fb0: 6661 6e20 2020 2020 2020 2020 2020 2020  fan             
│ │ │ │ -00255fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00255fe0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00255ff0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00255fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255fc0: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ +00255fd0: 203d 2067 6661 6e20 2020 2020 2020 2020   = gfan         
│ │ │ │ +00255fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00255ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00256000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00256010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00256010: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00256020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00256030: 2020 2020 2020 207c 0a7c 6f35 203a 2050         |.|o5 : P
│ │ │ │ -00256040: 726f 6772 616d 2020 2020 2020 2020 2020  rogram          
│ │ │ │ -00256050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00256060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00256070: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00256080: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00256090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -002560a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00256030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00256040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00256050: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ +00256060: 203a 2050 726f 6772 616d 2020 2020 2020   : Program      
│ │ │ │ +00256070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00256080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00256090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002560a0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  002560b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -002560c0: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c  -------+..See al
│ │ │ │ -002560d0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  so.========..  *
│ │ │ │ -002560e0: 202a 6e6f 7465 2072 756e 5072 6f67 7261   *note runProgra
│ │ │ │ -002560f0: 6d3a 2072 756e 5072 6f67 7261 6d2c 202d  m: runProgram, -
│ │ │ │ -00256100: 2d20 7275 6e20 616e 2065 7874 6572 6e61  - run an externa
│ │ │ │ -00256110: 6c20 7072 6f67 7261 6d0a 2020 2a20 2a6e  l program.  * *n
│ │ │ │ -00256120: 6f74 6520 7365 6172 6368 5061 7468 3a20  ote searchPath: 
│ │ │ │ -00256130: 7365 6172 6368 5061 7468 5f6c 704c 6973  searchPath_lpLis
│ │ │ │ -00256140: 745f 636d 5374 7269 6e67 5f72 702c 202d  t_cmString_rp, -
│ │ │ │ -00256150: 2d20 7365 6172 6368 2061 2070 6174 6820  - search a path 
│ │ │ │ -00256160: 666f 7220 610a 2020 2020 6669 6c65 0a2a  for a.    file.*
│ │ │ │ -00256170: 204d 656e 753a 0a0a 2a20 5072 6f67 7261   Menu:..* Progra
│ │ │ │ -00256180: 6d3a 3a20 2020 2020 2020 2020 2020 2020  m::             
│ │ │ │ -00256190: 2020 2020 2020 2020 6578 7465 726e 616c          external
│ │ │ │ -002561a0: 2070 726f 6772 616d 206f 626a 6563 740a   program object.
│ │ │ │ -002561b0: 2a20 7072 6f67 7261 6d50 6174 6873 3a3a  * programPaths::
│ │ │ │ -002561c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002561d0: 7573 6572 2d64 6566 696e 6564 2065 7874  user-defined ext
│ │ │ │ -002561e0: 6572 6e61 6c20 7072 6f67 7261 6d20 7061  ernal program pa
│ │ │ │ -002561f0: 7468 730a 0a57 6179 7320 746f 2075 7365  ths..Ways to use
│ │ │ │ -00256200: 2066 696e 6450 726f 6772 616d 3a0a 3d3d   findProgram:.==
│ │ │ │ -00256210: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00256220: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2266 696e  ======..  * "fin
│ │ │ │ -00256230: 6450 726f 6772 616d 2853 7472 696e 6729  dProgram(String)
│ │ │ │ -00256240: 220a 2020 2a20 2266 696e 6450 726f 6772  ".  * "findProgr
│ │ │ │ -00256250: 616d 2853 7472 696e 672c 4c69 7374 2922  am(String,List)"
│ │ │ │ -00256260: 0a20 202a 2022 6669 6e64 5072 6f67 7261  .  * "findProgra
│ │ │ │ -00256270: 6d28 5374 7269 6e67 2c53 7472 696e 6729  m(String,String)
│ │ │ │ -00256280: 220a 0a46 6f72 2074 6865 2070 726f 6772  "..For the progr
│ │ │ │ -00256290: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d  ammer.==========
│ │ │ │ -002562a0: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62  ========..The ob
│ │ │ │ -002562b0: 6a65 6374 202a 6e6f 7465 2066 696e 6450  ject *note findP
│ │ │ │ -002562c0: 726f 6772 616d 3a20 6669 6e64 5072 6f67  rogram: findProg
│ │ │ │ -002562d0: 7261 6d2c 2069 7320 6120 2a6e 6f74 6520  ram, is a *note 
│ │ │ │ -002562e0: 6d65 7468 6f64 2066 756e 6374 696f 6e20  method function 
│ │ │ │ -002562f0: 7769 7468 0a6f 7074 696f 6e73 3a20 4d65  with.options: Me
│ │ │ │ -00256300: 7468 6f64 4675 6e63 7469 6f6e 5769 7468  thodFunctionWith
│ │ │ │ -00256310: 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d  Options,...-----
│ │ │ │ -00256320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00256330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002560c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002560d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002560e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365  -----------+..Se
│ │ │ │ +002560f0: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a  e also.========.
│ │ │ │ +00256100: 0a20 202a 202a 6e6f 7465 2072 756e 5072  .  * *note runPr
│ │ │ │ +00256110: 6f67 7261 6d3a 2072 756e 5072 6f67 7261  ogram: runProgra
│ │ │ │ +00256120: 6d2c 202d 2d20 7275 6e20 616e 2065 7874  m, -- run an ext
│ │ │ │ +00256130: 6572 6e61 6c20 7072 6f67 7261 6d0a 2020  ernal program.  
│ │ │ │ +00256140: 2a20 2a6e 6f74 6520 7365 6172 6368 5061  * *note searchPa
│ │ │ │ +00256150: 7468 3a20 7365 6172 6368 5061 7468 5f6c  th: searchPath_l
│ │ │ │ +00256160: 704c 6973 745f 636d 5374 7269 6e67 5f72  pList_cmString_r
│ │ │ │ +00256170: 702c 202d 2d20 7365 6172 6368 2061 2070  p, -- search a p
│ │ │ │ +00256180: 6174 6820 666f 7220 610a 2020 2020 6669  ath for a.    fi
│ │ │ │ +00256190: 6c65 0a2a 204d 656e 753a 0a0a 2a20 5072  le.* Menu:..* Pr
│ │ │ │ +002561a0: 6f67 7261 6d3a 3a20 2020 2020 2020 2020  ogram::         
│ │ │ │ +002561b0: 2020 2020 2020 2020 2020 2020 6578 7465              exte
│ │ │ │ +002561c0: 726e 616c 2070 726f 6772 616d 206f 626a  rnal program obj
│ │ │ │ +002561d0: 6563 740a 2a20 7072 6f67 7261 6d50 6174  ect.* programPat
│ │ │ │ +002561e0: 6873 3a3a 2020 2020 2020 2020 2020 2020  hs::            
│ │ │ │ +002561f0: 2020 2020 7573 6572 2d64 6566 696e 6564      user-defined
│ │ │ │ +00256200: 2065 7874 6572 6e61 6c20 7072 6f67 7261   external progra
│ │ │ │ +00256210: 6d20 7061 7468 730a 0a57 6179 7320 746f  m paths..Ways to
│ │ │ │ +00256220: 2075 7365 2066 696e 6450 726f 6772 616d   use findProgram
│ │ │ │ +00256230: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ +00256240: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ +00256250: 2266 696e 6450 726f 6772 616d 2853 7472  "findProgram(Str
│ │ │ │ +00256260: 696e 6729 220a 2020 2a20 2266 696e 6450  ing)".  * "findP
│ │ │ │ +00256270: 726f 6772 616d 2853 7472 696e 672c 4c69  rogram(String,Li
│ │ │ │ +00256280: 7374 2922 0a20 202a 2022 6669 6e64 5072  st)".  * "findPr
│ │ │ │ +00256290: 6f67 7261 6d28 5374 7269 6e67 2c53 7472  ogram(String,Str
│ │ │ │ +002562a0: 696e 6729 220a 0a46 6f72 2074 6865 2070  ing)"..For the p
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│ │ │ │ -00256970: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ -002579c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
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│ │ │ │ +002575f0: 7574 7075 742e 0a20 2020 2020 202a 2052  utput..      * R
│ │ │ │ +00257600: 6169 7365 4572 726f 7220 3d3e 2061 202a  aiseError => a *
│ │ │ │ +00257610: 6e6f 7465 2042 6f6f 6c65 616e 2076 616c  note Boolean val
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│ │ │ │ +00257630: 6661 756c 7420 7661 6c75 6520 7472 7565  fault value true
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│ │ │ │ +00257690: 2020 2a20 5275 6e44 6972 6563 746f 7279    * RunDirectory
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│ │ │ │ +002576c0: 6175 6c74 2076 616c 7565 206e 756c 6c2c  ault value null,
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│ │ │ │ +00257740: 6620 2a6e 6f74 6520 6e75 6c6c 3a20 6e75  f *note null: nu
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│ │ │ │ +002577c0: 2020 6375 7272 656e 7444 6972 6563 746f    currentDirecto
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│ │ │ │ +002577e0: 7262 6f73 6520 3d3e 2061 202a 6e6f 7465  rbose => a *note
│ │ │ │ +002577f0: 2042 6f6f 6c65 616e 2076 616c 7565 3a20   Boolean value: 
│ │ │ │ +00257800: 426f 6f6c 6561 6e2c 2c20 6465 6661 756c  Boolean,, defaul
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│ │ │ │ +00257820: 2020 2020 2020 2077 6865 7468 6572 2074         whether t
│ │ │ │ +00257830: 6f20 7072 696e 7420 7468 6520 636f 6d6d  o print the comm
│ │ │ │ +00257840: 616e 6420 6c69 6e65 2069 6e70 7574 2061  and line input a
│ │ │ │ +00257850: 6e64 2074 6865 2070 726f 6772 616d 2773  nd the program's
│ │ │ │ +00257860: 206f 7574 7075 742e 0a0a 4465 7363 7269   output...Descri
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│ │ │ │ +00257880: 3d0a 0a54 6869 7320 6d65 7468 6f64 2072  =..This method r
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│ │ │ │ +002578a0: 7072 6f67 7261 6d20 7768 6963 6820 6861  program which ha
│ │ │ │ +002578b0: 7320 616c 7265 6164 7920 6265 656e 206c  s already been l
│ │ │ │ +002578c0: 6f61 6465 6420 7573 696e 6720 2a6e 6f74  oaded using *not
│ │ │ │ +002578d0: 650a 6669 6e64 5072 6f67 7261 6d3a 2066  e.findProgram: f
│ │ │ │ +002578e0: 696e 6450 726f 6772 616d 2c2e 2020 5468  indProgram,.  Th
│ │ │ │ +002578f0: 6520 7265 7375 6c74 7320 6f66 2074 6869  e results of thi
│ │ │ │ +00257900: 7320 7275 6e20 6172 6520 6176 6169 6c61  s run are availa
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│ │ │ │ +00257920: 726f 6772 616d 5275 6e3a 2050 726f 6772  rogramRun: Progr
│ │ │ │ +00257930: 616d 5275 6e2c 206f 626a 6563 742e 0a0a  amRun, object...
│ │ │ │ +00257940: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00257950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00257960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00257970: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120  ----------+.|i1 
│ │ │ │ +00257980: 3a20 6766 616e 203d 2066 696e 6450 726f  : gfan = findPro
│ │ │ │ +00257990: 6772 616d 2822 6766 616e 222c 2022 6766  gram("gfan", "gf
│ │ │ │ +002579a0: 616e 202d 2d68 656c 7022 2920 2020 2020  an --help")     
│ │ │ │ +002579b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +002579c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002579d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002579e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002579f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257a00: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +002579f0: 2020 7c0a 7c6f 3120 3d20 6766 616e 2020    |.|o1 = gfan  
│ │ │ │ +00257a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00257a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257a40: 2020 2020 2020 7c0a 7c6f 3120 3a20 5072        |.|o1 : Pr
│ │ │ │ -00257a50: 6f67 7261 6d20 2020 2020 2020 2020 2020  ogram           
│ │ │ │ -00257a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257a80: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -00257a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00257aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00257ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00257ac0: 7c69 3220 3a20 7275 6e50 726f 6772 616d  |i2 : runProgram
│ │ │ │ -00257ad0: 2867 6661 6e2c 2022 5f76 6572 7369 6f6e  (gfan, "_version
│ │ │ │ -00257ae0: 2229 2020 2020 2020 2020 2020 2020 2020  ")              
│ │ │ │ -00257af0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00257b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257b30: 2020 2020 2020 7c0a 7c6f 3220 3d20 3020        |.|o2 = 0 
│ │ │ │ +00257a20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00257a30: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00257a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00257a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00257a60: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ +00257a70: 3a20 5072 6f67 7261 6d20 2020 2020 2020  : Program       
│ │ │ │ +00257a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00257a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00257aa0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00257ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00257ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00257ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00257ae0: 2d2d 2b0a 7c69 3220 3a20 7275 6e50 726f  --+.|i2 : runPro
│ │ │ │ +00257af0: 6772 616d 2867 6661 6e2c 2022 5f76 6572  gram(gfan, "_ver
│ │ │ │ +00257b00: 7369 6f6e 2229 2020 2020 2020 2020 2020  sion")          
│ │ │ │ +00257b10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00257b20: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00257b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00257b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257b70: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00257b50: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ +00257b60: 3d20 3020 2020 2020 2020 2020 2020 2020  = 0             
│ │ │ │ +00257b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00257b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257ba0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00257bb0: 7c6f 3220 3a20 5072 6f67 7261 6d52 756e  |o2 : ProgramRun
│ │ │ │ +00257b90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00257ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00257bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00257bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257be0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -00257bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00257c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00257c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00257c20: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 6f6f  ------+.|i3 : oo
│ │ │ │ -00257c30: 2322 6f75 7470 7574 2220 2020 2020 2020  #"output"       
│ │ │ │ -00257c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257c60: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00257bd0: 2020 7c0a 7c6f 3220 3a20 5072 6f67 7261    |.|o2 : Progra
│ │ │ │ +00257be0: 6d52 756e 2020 2020 2020 2020 2020 2020  mRun            
│ │ │ │ +00257bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00257c00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00257c10: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00257c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00257c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00257c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320  ----------+.|i3 
│ │ │ │ +00257c50: 3a20 6f6f 2322 6f75 7470 7574 2220 2020  : oo#"output"   
│ │ │ │ +00257c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00257c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257c90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00257ca0: 7c6f 3320 3d20 4766 616e 2076 6572 7369  |o3 = Gfan versi
│ │ │ │ -00257cb0: 6f6e 3a20 2020 2020 2020 2020 2020 2020  on:             
│ │ │ │ -00257cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257cd0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00257ce0: 2020 6766 616e 302e 3720 2020 2020 2020    gfan0.7       
│ │ │ │ -00257cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257d10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00257c80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00257c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00257ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00257cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00257cc0: 2020 7c0a 7c6f 3320 3d20 4766 616e 2076    |.|o3 = Gfan v
│ │ │ │ +00257cd0: 6572 7369 6f6e 3a20 2020 2020 2020 2020  ersion:         
│ │ │ │ +00257ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00257cf0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00257d00: 7c20 2020 2020 6766 616e 302e 3720 2020  |     gfan0.7   
│ │ │ │ +00257d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00257d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00257d30: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00257d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257d50: 2020 7c0a 7c20 2020 2020 466f 726b 6564    |.|     Forked
│ │ │ │ -00257d60: 2066 726f 6d20 736f 7572 6365 2074 7265   from source tre
│ │ │ │ -00257d70: 6520 6f6e 3a20 2020 2020 2020 2020 2020  e on:           
│ │ │ │ -00257d80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00257d90: 7c20 2020 2020 3137 3233 3437 3834 3135  |     1723478415
│ │ │ │ -00257da0: 204d 6f6e 2041 7567 2031 3220 3138 3a30   Mon Aug 12 18:0
│ │ │ │ -00257db0: 303a 3135 2032 3032 3420 2020 2020 2020  0:15 2024       
│ │ │ │ -00257dc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00257dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257e00: 2020 2020 2020 7c0a 7c20 2020 2020 4c69        |.|     Li
│ │ │ │ -00257e10: 6e6b 6564 206c 6962 7261 7269 6573 3a20  nked libraries: 
│ │ │ │ -00257e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257e40: 2020 7c0a 7c20 2020 2020 474d 5020 362e    |.|     GMP 6.
│ │ │ │ -00257e50: 332e 3020 2020 2020 2020 2020 2020 2020  3.0             
│ │ │ │ -00257e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257e70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00257e80: 7c20 2020 2020 4364 646c 6962 2020 2020  |     Cddlib    
│ │ │ │ -00257e90: 2020 2059 4553 2020 2020 2020 2020 2020     YES          
│ │ │ │ -00257ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257eb0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00257ec0: 2020 536f 506c 6578 2020 2020 2020 2020    SoPlex        
│ │ │ │ -00257ed0: 4e4f 2020 2020 2020 2020 2020 2020 2020  NO              
│ │ │ │ -00257ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257ef0: 2020 2020 2020 7c0a 7c20 2020 2020 5369        |.|     Si
│ │ │ │ -00257f00: 6e67 756c 6172 2020 2020 2020 4e4f 2020  ngular      NO  
│ │ │ │ -00257f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257f30: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -00257f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00257f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00257f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00257f70: 7c69 3420 3a20 7275 6e50 726f 6772 616d  |i4 : runProgram
│ │ │ │ -00257f80: 2867 6661 6e2c 2022 5f66 6f6f 222c 2052  (gfan, "_foo", R
│ │ │ │ -00257f90: 6169 7365 4572 726f 7220 3d3e 2066 616c  aiseError => fal
│ │ │ │ -00257fa0: 7365 2920 2020 2020 2020 7c0a 7c20 2020  se)       |.|   
│ │ │ │ -00257fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00257fe0: 2020 2020 2020 7c0a 7c6f 3420 3d20 3235        |.|o4 = 25
│ │ │ │ -00257ff0: 3620 2020 2020 2020 2020 2020 2020 2020  6               
│ │ │ │ -00258000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258020: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00257d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00257d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00257d70: 2020 2020 2020 7c0a 7c20 2020 2020 466f        |.|     Fo
│ │ │ │ +00257d80: 726b 6564 2066 726f 6d20 736f 7572 6365  rked from source
│ │ │ │ +00257d90: 2074 7265 6520 6f6e 3a20 2020 2020 2020   tree on:       
│ │ │ │ +00257da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00257db0: 2020 7c0a 7c20 2020 2020 3137 3233 3437    |.|     172347
│ │ │ │ +00257dc0: 3834 3135 204d 6f6e 2041 7567 2031 3220  8415 Mon Aug 12 
│ │ │ │ +00257dd0: 3138 3a30 303a 3135 2032 3032 3420 2020  18:00:15 2024   
│ │ │ │ +00257de0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00257df0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00257e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00257e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00257e20: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00257e30: 2020 4c69 6e6b 6564 206c 6962 7261 7269    Linked librari
│ │ │ │ +00257e40: 6573 3a20 2020 2020 2020 2020 2020 2020  es:             
│ │ │ │ +00257e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00257e60: 2020 2020 2020 7c0a 7c20 2020 2020 474d        |.|     GM
│ │ │ │ +00257e70: 5020 362e 332e 3020 2020 2020 2020 2020  P 6.3.0         
│ │ │ │ +00257e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00257e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00257ea0: 2020 7c0a 7c20 2020 2020 4364 646c 6962    |.|     Cddlib
│ │ │ │ +00257eb0: 2020 2020 2020 2059 4553 2020 2020 2020         YES      
│ │ │ │ +00257ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00257ed0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00257ee0: 7c20 2020 2020 536f 506c 6578 2020 2020  |     SoPlex    
│ │ │ │ +00257ef0: 2020 2020 4e4f 2020 2020 2020 2020 2020      NO          
│ │ │ │ +00257f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00257f10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00257f20: 2020 5369 6e67 756c 6172 2020 2020 2020    Singular      
│ │ │ │ +00257f30: 4e4f 2020 2020 2020 2020 2020 2020 2020  NO              
│ │ │ │ +00257f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00257f50: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00257f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00257f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00257f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00257f90: 2d2d 2b0a 7c69 3420 3a20 7275 6e50 726f  --+.|i4 : runPro
│ │ │ │ +00257fa0: 6772 616d 2867 6661 6e2c 2022 5f66 6f6f  gram(gfan, "_foo
│ │ │ │ +00257fb0: 222c 2052 6169 7365 4572 726f 7220 3d3e  ", RaiseError =>
│ │ │ │ +00257fc0: 2066 616c 7365 2920 2020 2020 2020 7c0a   false)       |.
│ │ │ │ +00257fd0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00257fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00257ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258000: 2020 2020 2020 2020 2020 7c0a 7c6f 3420            |.|o4 
│ │ │ │ +00258010: 3d20 3235 3620 2020 2020 2020 2020 2020  = 256           
│ │ │ │ +00258020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00258030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258050: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00258060: 7c6f 3420 3a20 5072 6f67 7261 6d52 756e  |o4 : ProgramRun
│ │ │ │ +00258040: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00258050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00258070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258090: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -002580a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -002580b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -002580c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -002580d0: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 6f6f  ------+.|i5 : oo
│ │ │ │ -002580e0: 2322 6572 726f 7222 2020 2020 2020 2020  #"error"        
│ │ │ │ -002580f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258110: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00258080: 2020 7c0a 7c6f 3420 3a20 5072 6f67 7261    |.|o4 : Progra
│ │ │ │ +00258090: 6d52 756e 2020 2020 2020 2020 2020 2020  mRun            
│ │ │ │ +002580a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002580b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +002580c0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +002580d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002580e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002580f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520  ----------+.|i5 
│ │ │ │ +00258100: 3a20 6f6f 2322 6572 726f 7222 2020 2020  : oo#"error"    
│ │ │ │ +00258110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00258120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258140: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00258150: 7c6f 3520 3d20 554e 4b4e 4f57 4e20 4f50  |o5 = UNKNOWN OP
│ │ │ │ -00258160: 5449 4f4e 3a20 5f66 6f6f 2e20 2020 2020  TION: _foo.     
│ │ │ │ -00258170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258180: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00258190: 2020 5553 4520 2d2d 6865 6c70 2041 5320    USE --help AS 
│ │ │ │ -002581a0: 4120 5349 4e47 4c45 204f 5054 494f 4e20  A SINGLE OPTION 
│ │ │ │ -002581b0: 544f 2056 4945 5720 5448 4520 4845 4c50  TO VIEW THE HELP
│ │ │ │ -002581c0: 2054 4558 542e 7c0a 2b2d 2d2d 2d2d 2d2d   TEXT.|.+-------
│ │ │ │ -002581d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -002581e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00258130: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00258140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258170: 2020 7c0a 7c6f 3520 3d20 554e 4b4e 4f57    |.|o5 = UNKNOW
│ │ │ │ +00258180: 4e20 4f50 5449 4f4e 3a20 5f66 6f6f 2e20  N OPTION: _foo. 
│ │ │ │ +00258190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002581a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +002581b0: 7c20 2020 2020 5553 4520 2d2d 6865 6c70  |     USE --help
│ │ │ │ +002581c0: 2041 5320 4120 5349 4e47 4c45 204f 5054   AS A SINGLE OPT
│ │ │ │ +002581d0: 494f 4e20 544f 2056 4945 5720 5448 4520  ION TO VIEW THE 
│ │ │ │ +002581e0: 4845 4c50 2054 4558 542e 7c0a 2b2d 2d2d  HELP TEXT.|.+---
│ │ │ │  002581f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00258200: 2d2d 2b0a 0a54 6865 2076 616c 7565 2063  --+..The value c
│ │ │ │ -00258210: 6f72 7265 7370 6f6e 6469 6e67 2074 6f20  orresponding to 
│ │ │ │ -00258220: 7468 6520 226f 7574 7075 7422 206b 6579  the "output" key
│ │ │ │ -00258230: 206d 6179 2061 6c73 6f20 6265 206f 6274   may also be obt
│ │ │ │ -00258240: 6169 6e65 6420 7573 696e 6720 2a6e 6f74  ained using *not
│ │ │ │ -00258250: 650a 746f 5374 7269 6e67 3a20 746f 5374  e.toString: toSt
│ │ │ │ -00258260: 7269 6e67 2c2e 0a0a 2b2d 2d2d 2d2d 2d2d  ring,...+-------
│ │ │ │ -00258270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00258280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00258290: 2d2d 2d2b 0a7c 6936 203a 2074 6f53 7472  ---+.|i6 : toStr
│ │ │ │ -002582a0: 696e 6720 7275 6e50 726f 6772 616d 2867  ing runProgram(g
│ │ │ │ -002582b0: 6661 6e2c 2022 5f76 6572 7369 6f6e 2229  fan, "_version")
│ │ │ │ -002582c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -002582d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002582e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -002582f0: 6f36 203d 2047 6661 6e20 7665 7273 696f  o6 = Gfan versio
│ │ │ │ -00258300: 6e3a 2020 2020 2020 2020 2020 2020 2020  n:              
│ │ │ │ -00258310: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00258320: 2020 6766 616e 302e 3720 2020 2020 2020    gfan0.7       
│ │ │ │ -00258330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258340: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00258200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00258210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00258220: 2d2d 2d2d 2d2d 2b0a 0a54 6865 2076 616c  ------+..The val
│ │ │ │ +00258230: 7565 2063 6f72 7265 7370 6f6e 6469 6e67  ue corresponding
│ │ │ │ +00258240: 2074 6f20 7468 6520 226f 7574 7075 7422   to the "output"
│ │ │ │ +00258250: 206b 6579 206d 6179 2061 6c73 6f20 6265   key may also be
│ │ │ │ +00258260: 206f 6274 6169 6e65 6420 7573 696e 6720   obtained using 
│ │ │ │ +00258270: 2a6e 6f74 650a 746f 5374 7269 6e67 3a20  *note.toString: 
│ │ │ │ +00258280: 746f 5374 7269 6e67 2c2e 0a0a 2b2d 2d2d  toString,...+---
│ │ │ │ +00258290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002582a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002582b0: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2074  -------+.|i6 : t
│ │ │ │ +002582c0: 6f53 7472 696e 6720 7275 6e50 726f 6772  oString runProgr
│ │ │ │ +002582d0: 616d 2867 6661 6e2c 2022 5f76 6572 7369  am(gfan, "_versi
│ │ │ │ +002582e0: 6f6e 2229 7c0a 7c20 2020 2020 2020 2020  on")|.|         
│ │ │ │ +002582f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258310: 207c 0a7c 6f36 203d 2047 6661 6e20 7665   |.|o6 = Gfan ve
│ │ │ │ +00258320: 7273 696f 6e3a 2020 2020 2020 2020 2020  rsion:          
│ │ │ │ +00258330: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00258340: 7c20 2020 2020 6766 616e 302e 3720 2020  |     gfan0.7   
│ │ │ │  00258350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258370: 2020 2020 7c0a 7c20 2020 2020 466f 726b      |.|     Fork
│ │ │ │ -00258380: 6564 2066 726f 6d20 736f 7572 6365 2074  ed from source t
│ │ │ │ -00258390: 7265 6520 6f6e 3a20 2020 2020 2020 2020  ree on:         
│ │ │ │ -002583a0: 207c 0a7c 2020 2020 2031 3732 3334 3738   |.|     1723478
│ │ │ │ -002583b0: 3431 3520 4d6f 6e20 4175 6720 3132 2031  415 Mon Aug 12 1
│ │ │ │ -002583c0: 383a 3030 3a31 3520 3230 3234 2020 7c0a  8:00:15 2024  |.
│ │ │ │ -002583d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -002583e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002583f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00258400: 2020 204c 696e 6b65 6420 6c69 6272 6172     Linked librar
│ │ │ │ -00258410: 6965 733a 2020 2020 2020 2020 2020 2020  ies:            
│ │ │ │ -00258420: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00258430: 474d 5020 362e 332e 3020 2020 2020 2020  GMP 6.3.0       
│ │ │ │ -00258440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258450: 2020 2020 207c 0a7c 2020 2020 2043 6464       |.|     Cdd
│ │ │ │ -00258460: 6c69 6220 2020 2020 2020 5945 5320 2020  lib       YES   
│ │ │ │ -00258470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258480: 2020 7c0a 7c20 2020 2020 536f 506c 6578    |.|     SoPlex
│ │ │ │ -00258490: 2020 2020 2020 2020 4e4f 2020 2020 2020          NO      
│ │ │ │ -002584a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -002584b0: 0a7c 2020 2020 2053 696e 6775 6c61 7220  .|     Singular 
│ │ │ │ -002584c0: 2020 2020 204e 4f20 2020 2020 2020 2020       NO         
│ │ │ │ -002584d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -002584e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -002584f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00258500: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4974 2069  ---------+..It i
│ │ │ │ -00258510: 7320 616c 736f 2070 6f73 7369 626c 6520  s also possible 
│ │ │ │ -00258520: 746f 2073 6b69 7020 2a6e 6f74 6520 6669  to skip *note fi
│ │ │ │ -00258530: 6e64 5072 6f67 7261 6d3a 2066 696e 6450  ndProgram: findP
│ │ │ │ -00258540: 726f 6772 616d 2c20 616e 6420 6a75 7374  rogram, and just
│ │ │ │ -00258550: 2070 726f 7669 6465 0a74 776f 2073 7472   provide.two str
│ │ │ │ -00258560: 696e 6773 3a20 7468 6520 6e61 6d65 206f  ings: the name o
│ │ │ │ -00258570: 6620 7468 6520 7072 6f67 7261 6d20 616e  f the program an
│ │ │ │ -00258580: 6420 7468 6520 636f 6d6d 616e 6420 6c69  d the command li
│ │ │ │ -00258590: 6e65 2061 7267 756d 656e 7473 2e0a 0a2b  ne arguments...+
│ │ │ │ -002585a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -002585b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -002585c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00258360: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00258370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258390: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +002583a0: 466f 726b 6564 2066 726f 6d20 736f 7572  Forked from sour
│ │ │ │ +002583b0: 6365 2074 7265 6520 6f6e 3a20 2020 2020  ce tree on:     
│ │ │ │ +002583c0: 2020 2020 207c 0a7c 2020 2020 2031 3732       |.|     172
│ │ │ │ +002583d0: 3334 3738 3431 3520 4d6f 6e20 4175 6720  3478415 Mon Aug 
│ │ │ │ +002583e0: 3132 2031 383a 3030 3a31 3520 3230 3234  12 18:00:15 2024
│ │ │ │ +002583f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00258400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258410: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00258420: 0a7c 2020 2020 204c 696e 6b65 6420 6c69  .|     Linked li
│ │ │ │ +00258430: 6272 6172 6965 733a 2020 2020 2020 2020  braries:        
│ │ │ │ +00258440: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00258450: 2020 2020 474d 5020 362e 332e 3020 2020      GMP 6.3.0   
│ │ │ │ +00258460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258470: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00258480: 2043 6464 6c69 6220 2020 2020 2020 5945   Cddlib       YE
│ │ │ │ +00258490: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ +002584a0: 2020 2020 2020 7c0a 7c20 2020 2020 536f        |.|     So
│ │ │ │ +002584b0: 506c 6578 2020 2020 2020 2020 4e4f 2020  Plex        NO  
│ │ │ │ +002584c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002584d0: 2020 207c 0a7c 2020 2020 2053 696e 6775     |.|     Singu
│ │ │ │ +002584e0: 6c61 7220 2020 2020 204e 4f20 2020 2020  lar      NO     
│ │ │ │ +002584f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258500: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00258510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00258520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ +00258530: 4974 2069 7320 616c 736f 2070 6f73 7369  It is also possi
│ │ │ │ +00258540: 626c 6520 746f 2073 6b69 7020 2a6e 6f74  ble to skip *not
│ │ │ │ +00258550: 6520 6669 6e64 5072 6f67 7261 6d3a 2066  e findProgram: f
│ │ │ │ +00258560: 696e 6450 726f 6772 616d 2c20 616e 6420  indProgram, and 
│ │ │ │ +00258570: 6a75 7374 2070 726f 7669 6465 0a74 776f  just provide.two
│ │ │ │ +00258580: 2073 7472 696e 6773 3a20 7468 6520 6e61   strings: the na
│ │ │ │ +00258590: 6d65 206f 6620 7468 6520 7072 6f67 7261  me of the progra
│ │ │ │ +002585a0: 6d20 616e 6420 7468 6520 636f 6d6d 616e  m and the comman
│ │ │ │ +002585b0: 6420 6c69 6e65 2061 7267 756d 656e 7473  d line arguments
│ │ │ │ +002585c0: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...+------------
│ │ │ │  002585d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -002585e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -002585f0: 6937 203a 2072 756e 5072 6f67 7261 6d28  i7 : runProgram(
│ │ │ │ -00258600: 226e 6f72 6d61 6c69 7a22 2c20 222d 2d76  "normaliz", "--v
│ │ │ │ -00258610: 6572 7369 6f6e 2229 2020 2020 2020 2020  ersion")        
│ │ │ │ -00258620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258630: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +002585e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002585f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00258600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00258610: 2d2b 0a7c 6937 203a 2072 756e 5072 6f67  -+.|i7 : runProg
│ │ │ │ +00258620: 7261 6d28 226e 6f72 6d61 6c69 7a22 2c20  ram("normaliz", 
│ │ │ │ +00258630: 222d 2d76 6572 7369 6f6e 2229 2020 2020  "--version")    
│ │ │ │  00258640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00258650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258660: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00258670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258680: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00258690: 6f37 203d 2030 2020 2020 2020 2020 2020  o7 = 0          
│ │ │ │ +00258680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002586a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002586b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002586b0: 207c 0a7c 6f37 203d 2030 2020 2020 2020   |.|o7 = 0      
│ │ │ │  002586c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002586d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +002586d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002586e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002586f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258700: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00258710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258720: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00258730: 6f37 203a 2050 726f 6772 616d 5275 6e20  o7 : ProgramRun 
│ │ │ │ +00258720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00258740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258770: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00258780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00258790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -002587a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00258750: 207c 0a7c 6f37 203a 2050 726f 6772 616d   |.|o7 : Program
│ │ │ │ +00258760: 5275 6e20 2020 2020 2020 2020 2020 2020  Run             
│ │ │ │ +00258770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002587a0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  002587b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -002587c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -002587d0: 6938 203a 2070 6565 6b20 6f6f 2020 2020  i8 : peek oo    
│ │ │ │ -002587e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002587f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002587c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002587d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002587e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002587f0: 2d2b 0a7c 6938 203a 2070 6565 6b20 6f6f  -+.|i8 : peek oo
│ │ │ │  00258800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258810: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00258810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00258820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00258830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258840: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00258850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258860: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00258870: 6f38 203d 2050 726f 6772 616d 5275 6e7b  o8 = ProgramRun{
│ │ │ │ -00258880: 636f 6d6d 616e 6420 3d3e 202f 7573 722f  command => /usr/
│ │ │ │ -00258890: 6269 6e2f 6e6f 726d 616c 697a 202d 2d76  bin/normaliz --v
│ │ │ │ -002588a0: 6572 7369 6f6e 2020 2020 2020 2020 2020  ersion          
│ │ │ │ -002588b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -002588c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002588d0: 6572 726f 7220 3d3e 2020 2020 2020 2020  error =>        
│ │ │ │ -002588e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002588f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258900: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00258860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258890: 207c 0a7c 6f38 203d 2050 726f 6772 616d   |.|o8 = Program
│ │ │ │ +002588a0: 5275 6e7b 636f 6d6d 616e 6420 3d3e 202f  Run{command => /
│ │ │ │ +002588b0: 7573 722f 6269 6e2f 6e6f 726d 616c 697a  usr/bin/normaliz
│ │ │ │ +002588c0: 202d 2d76 6572 7369 6f6e 2020 2020 2020   --version      
│ │ │ │ +002588d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002588e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +002588f0: 2020 2020 6572 726f 7220 3d3e 2020 2020      error =>    
│ │ │ │ +00258900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00258910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258920: 6f75 7470 7574 203d 3e20 4e6f 726d 616c  output => Normal
│ │ │ │ -00258930: 697a 2033 2e31 312e 3020 2020 2020 2020  iz 3.11.0       
│ │ │ │ -00258940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258950: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00258920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258930: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00258940: 2020 2020 6f75 7470 7574 203d 3e20 4e6f      output => No
│ │ │ │ +00258950: 726d 616c 697a 2033 2e31 312e 3020 2020  rmaliz 3.11.0   
│ │ │ │  00258960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258970: 2020 2020 2020 2020 2020 2d2d 2d2d 2d2d            ------
│ │ │ │ -00258980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00258990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -002589a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ -002589b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002589c0: 2020 2020 2020 2020 2020 7769 7468 2070            with p
│ │ │ │ -002589d0: 6163 6b61 6765 2873 2920 466c 696e 7420  ackage(s) Flint 
│ │ │ │ -002589e0: 652d 616e 7469 6320 6e61 7574 7920 2020  e-antic nauty   
│ │ │ │ -002589f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00258a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258a10: 2020 2020 2020 2020 2020 436f 7079 7269            Copyri
│ │ │ │ -00258a20: 6768 7420 2843 2920 3230 3037 2d32 3032  ght (C) 2007-202
│ │ │ │ -00258a30: 3520 2054 6865 204e 6f72 6d61 6c69 7a20  5  The Normaliz 
│ │ │ │ -00258a40: 5465 616d 2c20 556e 6976 6572 737c 0a7c  Team, Univers|.|
│ │ │ │ -00258a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258a60: 2020 2020 2020 2020 2020 5468 6973 2070            This p
│ │ │ │ -00258a70: 726f 6772 616d 2063 6f6d 6573 2077 6974  rogram comes wit
│ │ │ │ -00258a80: 6820 4142 534f 4c55 5445 4c59 204e 4f20  h ABSOLUTELY NO 
│ │ │ │ -00258a90: 5741 5252 414e 5459 3b20 5468 697c 0a7c  WARRANTY; Thi|.|
│ │ │ │ -00258aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258ab0: 2020 2020 2020 2020 2020 616e 6420 796f            and yo
│ │ │ │ -00258ac0: 7520 6172 6520 7765 6c63 6f6d 6520 746f  u are welcome to
│ │ │ │ -00258ad0: 2072 6564 6973 7472 6962 7574 6520 6974   redistribute it
│ │ │ │ -00258ae0: 2075 6e64 6572 2063 6572 7461 697c 0a7c   under certai|.|
│ │ │ │ -00258af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258b00: 2020 2020 2020 2020 2020 5365 6520 434f            See CO
│ │ │ │ -00258b10: 5059 494e 4720 666f 7220 6465 7461 696c  PYING for detail
│ │ │ │ -00258b20: 732e 2020 2020 2020 2020 2020 2020 2020  s.              
│ │ │ │ -00258b30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00258b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258980: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00258990: 2020 2020 2020 2020 2020 2020 2020 2d2d                --
│ │ │ │ +002589a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002589b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002589c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002589d0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +002589e0: 2020 2020 2020 2020 2020 2020 2020 7769                wi
│ │ │ │ +002589f0: 7468 2070 6163 6b61 6765 2873 2920 466c  th package(s) Fl
│ │ │ │ +00258a00: 696e 7420 652d 616e 7469 6320 6e61 7574  int e-antic naut
│ │ │ │ +00258a10: 7920 2020 2020 2020 2020 2020 2020 2020  y               
│ │ │ │ +00258a20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00258a30: 2020 2020 2020 2020 2020 2020 2020 436f                Co
│ │ │ │ +00258a40: 7079 7269 6768 7420 2843 2920 3230 3037  pyright (C) 2007
│ │ │ │ +00258a50: 2d32 3032 3520 2054 6865 204e 6f72 6d61  -2025  The Norma
│ │ │ │ +00258a60: 6c69 7a20 5465 616d 2c20 556e 6976 6572  liz Team, Univer
│ │ │ │ +00258a70: 737c 0a7c 2020 2020 2020 2020 2020 2020  s|.|            
│ │ │ │ +00258a80: 2020 2020 2020 2020 2020 2020 2020 5468                Th
│ │ │ │ +00258a90: 6973 2070 726f 6772 616d 2063 6f6d 6573  is program comes
│ │ │ │ +00258aa0: 2077 6974 6820 4142 534f 4c55 5445 4c59   with ABSOLUTELY
│ │ │ │ +00258ab0: 204e 4f20 5741 5252 414e 5459 3b20 5468   NO WARRANTY; Th
│ │ │ │ +00258ac0: 697c 0a7c 2020 2020 2020 2020 2020 2020  i|.|            
│ │ │ │ +00258ad0: 2020 2020 2020 2020 2020 2020 2020 616e                an
│ │ │ │ +00258ae0: 6420 796f 7520 6172 6520 7765 6c63 6f6d  d you are welcom
│ │ │ │ +00258af0: 6520 746f 2072 6564 6973 7472 6962 7574  e to redistribut
│ │ │ │ +00258b00: 6520 6974 2075 6e64 6572 2063 6572 7461  e it under certa
│ │ │ │ +00258b10: 697c 0a7c 2020 2020 2020 2020 2020 2020  i|.|            
│ │ │ │ +00258b20: 2020 2020 2020 2020 2020 2020 2020 5365                Se
│ │ │ │ +00258b30: 6520 434f 5059 494e 4720 666f 7220 6465  e COPYING for de
│ │ │ │ +00258b40: 7461 696c 732e 2020 2020 2020 2020 2020  tails.          
│ │ │ │  00258b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258b60: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00258b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258b80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00258b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00258b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258ba0: 7265 7475 726e 2076 616c 7565 203d 3e20  return value => 
│ │ │ │ -00258bb0: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ -00258bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258bd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00258be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00258bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00258c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00258ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258bb0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00258bc0: 2020 2020 7265 7475 726e 2076 616c 7565      return value
│ │ │ │ +00258bd0: 203d 3e20 3020 2020 2020 2020 2020 2020   => 0           
│ │ │ │ +00258be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258c00: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00258c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00258c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ -00258c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258c40: 2020 207d 2020 2020 2020 2020 2020 2020     }            
│ │ │ │ -00258c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258c70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00258c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00258c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00258c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00258c50: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +00258c60: 2020 2020 2020 207d 2020 2020 2020 2020         }        
│ │ │ │ +00258c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00258c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00258c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258ca0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00258cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258cc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00258cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00258cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00258ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258cf0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00258d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258d10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00258d20: 2d2d 2d2d 2d2d 2d2d 2d20 2020 2020 2020  ---------       
│ │ │ │ +00258d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00258d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258d40: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d20 2020   |.|---------   
│ │ │ │  00258d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258d60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00258d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00258d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00258d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258d90: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00258da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258db0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00258dc0: 6974 7920 6f66 204f 736e 6162 7275 6563  ity of Osnabruec
│ │ │ │ -00258dd0: 6b2e 2020 2020 2020 2020 2020 2020 2020  k.              
│ │ │ │ -00258de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258e00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00258e10: 7320 6973 2066 7265 6520 736f 6674 7761  s is free softwa
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│ │ │ │ -00258e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00258e50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00258e60: 6e20 636f 6e64 6974 696f 6e73 3b20 2020  n conditions;   
│ │ │ │ +00258db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00258de0: 207c 0a7c 6974 7920 6f66 204f 736e 6162   |.|ity of Osnab
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│ │ │ │ +00258e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00258e30: 207c 0a7c 7320 6973 2066 7265 6520 736f   |.|s is free so
│ │ │ │ +00258e40: 6674 7761 7265 2c20 2020 2020 2020 2020  ftware,         
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│ │ │ │  00258e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00258ea0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00258eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00258f40: 7465 7261 6374 2077 6974 680a 7072 6f67  teract with.prog
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│ │ │ │ -00259040: 6374 0a20 202a 202a 6e6f 7465 2066 696e  ct.  * *note fin
│ │ │ │ -00259050: 6450 726f 6772 616d 3a20 6669 6e64 5072  dProgram: findPr
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│ │ │ │ -00259070: 7874 6572 6e61 6c20 7072 6f67 7261 6d0a  xternal program.
│ │ │ │ -00259080: 2020 2a20 2a6e 6f74 6520 7374 6174 7573    * *note status
│ │ │ │ -00259090: 2850 726f 6772 616d 5275 6e29 3a20 7374  (ProgramRun): st
│ │ │ │ -002590a0: 6174 7573 5f6c 7050 726f 6772 616d 5275  atus_lpProgramRu
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│ │ │ │ -002590c0: 2072 6574 7572 6e20 7374 6174 7573 0a20   return status. 
│ │ │ │ -002590d0: 2020 206f 6620 6120 7072 6f67 7261 6d20     of a program 
│ │ │ │ -002590e0: 7275 6e0a 2020 2a20 2a6e 6f74 6520 7275  run.  * *note ru
│ │ │ │ -002590f0: 6e3a 2072 756e 2c20 2d2d 2072 756e 2061  n: run, -- run a
│ │ │ │ -00259100: 6e20 6578 7465 726e 616c 2063 6f6d 6d61  n external comma
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│ │ │ │ -00259120: 3a20 6765 742c 202d 2d20 6765 7420 7468  : get, -- get th
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│ │ │ │ -00259140: 6669 6c65 0a2a 204d 656e 753a 0a0a 2a20  file.* Menu:..* 
│ │ │ │ -00259150: 5072 6f67 7261 6d52 756e 3a3a 2020 2020  ProgramRun::    
│ │ │ │ -00259160: 2020 2020 2020 2020 2020 2020 2020 7265                re
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│ │ │ │ -002591a0: 2072 756e 5072 6f67 7261 6d3a 0a3d 3d3d   runProgram:.===
│ │ │ │ -002591b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -002591c0: 3d3d 3d3d 0a0a 2020 2a20 2272 756e 5072  ====..  * "runPr
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│ │ │ │ -00259200: 7472 696e 672c 5374 7269 6e67 2922 0a20  tring,String)". 
│ │ │ │ -00259210: 202a 2022 7275 6e50 726f 6772 616d 2853   * "runProgram(S
│ │ │ │ -00259220: 7472 696e 672c 5374 7269 6e67 2922 0a0a  tring,String)"..
│ │ │ │ -00259230: 466f 7220 7468 6520 7072 6f67 7261 6d6d  For the programm
│ │ │ │ -00259240: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  er.=============
│ │ │ │ -00259250: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563  =====..The objec
│ │ │ │ -00259260: 7420 2a6e 6f74 6520 7275 6e50 726f 6772  t *note runProgr
│ │ │ │ -00259270: 616d 3a20 7275 6e50 726f 6772 616d 2c20  am: runProgram, 
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│ │ │ │ -002592a0: 6f70 7469 6f6e 733a 204d 6574 686f 6446  options: MethodF
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│ │ │ │ -002592d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -002592e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00258ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00258f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00258f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00258f20: 2d2b 0a0a 496e 7465 726e 616c 6c79 2c20  -+..Internally, 
│ │ │ │ +00258f30: 7468 6973 2072 6f75 7469 6e65 2075 7365  this routine use
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│ │ │ │ +00258f50: 2c2e 2041 6e6f 7468 6572 2077 6179 2074  ,. Another way t
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│ │ │ │ +00258f70: 7072 6f67 7261 6d73 2069 7320 746f 2070  programs is to p
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│ │ │ │ +00258fa0: 746f 202a 6e6f 7465 2067 6574 3a20 6765  to *note get: ge
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│ │ │ │ +00258fd0: 675f 7270 2c2c 202a 6e6f 7465 206f 7065  g_rp,, *note ope
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│ │ │ │ +00258ff0: 5374 7269 6e67 5f72 702c 2c20 6f72 202a  String_rp,, or *
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│ │ │ │ +00259050: 7874 6572 6e61 6c20 7072 6f67 7261 6d20  xternal program 
│ │ │ │ +00259060: 6f62 6a65 6374 0a20 202a 202a 6e6f 7465  object.  * *note
│ │ │ │ +00259070: 2066 696e 6450 726f 6772 616d 3a20 6669   findProgram: fi
│ │ │ │ +00259080: 6e64 5072 6f67 7261 6d2c 202d 2d20 6c6f  ndProgram, -- lo
│ │ │ │ +00259090: 6164 2065 7874 6572 6e61 6c20 7072 6f67  ad external prog
│ │ │ │ +002590a0: 7261 6d0a 2020 2a20 2a6e 6f74 6520 7374  ram.  * *note st
│ │ │ │ +002590b0: 6174 7573 2850 726f 6772 616d 5275 6e29  atus(ProgramRun)
│ │ │ │ +002590c0: 3a20 7374 6174 7573 5f6c 7050 726f 6772  : status_lpProgr
│ │ │ │ +002590d0: 616d 5275 6e5f 7270 2c20 2d2d 2067 6574  amRun_rp, -- get
│ │ │ │ +002590e0: 2074 6865 2072 6574 7572 6e20 7374 6174   the return stat
│ │ │ │ +002590f0: 7573 0a20 2020 206f 6620 6120 7072 6f67  us.    of a prog
│ │ │ │ +00259100: 7261 6d20 7275 6e0a 2020 2a20 2a6e 6f74  ram run.  * *not
│ │ │ │ +00259110: 6520 7275 6e3a 2072 756e 2c20 2d2d 2072  e run: run, -- r
│ │ │ │ +00259120: 756e 2061 6e20 6578 7465 726e 616c 2063  un an external c
│ │ │ │ +00259130: 6f6d 6d61 6e64 0a20 202a 202a 6e6f 7465  ommand.  * *note
│ │ │ │ +00259140: 2067 6574 3a20 6765 742c 202d 2d20 6765   get: get, -- ge
│ │ │ │ +00259150: 7420 7468 6520 636f 6e74 656e 7473 206f  t the contents o
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│ │ │ │ +00259170: 0a0a 2a20 5072 6f67 7261 6d52 756e 3a3a  ..* ProgramRun::
│ │ │ │ +00259180: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +002591c0: 2075 7365 2072 756e 5072 6f67 7261 6d3a   use runProgram:
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│ │ │ │ +002591e0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2272  ========..  * "r
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│ │ │ │ +00259250: 2922 0a0a 466f 7220 7468 6520 7072 6f67  )"..For the prog
│ │ │ │ +00259260: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d  rammer.=========
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│ │ │ │ +002592a0: 616d 2c20 6973 2061 202a 6e6f 7465 206d  am, is a *note m
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│ │ │ │ +002592d0: 686f 6446 756e 6374 696f 6e57 6974 684f  hodFunctionWithO
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│ │ │ │ -00259420: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00259430: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00259440: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00259450: 2a2a 0a0a 4465 7363 7269 7074 696f 6e0a  **..Description.
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│ │ │ │ -00259470: 6173 6820 7461 626c 6520 7265 7475 726e  ash table return
│ │ │ │ -00259480: 6564 2062 7920 2a6e 6f74 6520 7275 6e50  ed by *note runP
│ │ │ │ -00259490: 726f 6772 616d 3a20 7275 6e50 726f 6772  rogram: runProgr
│ │ │ │ -002594a0: 616d 2c20 7769 7468 2074 6865 2066 6f6c  am, with the fol
│ │ │ │ -002594b0: 6c6f 7769 6e67 0a73 7472 696e 6773 2061  lowing.strings a
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│ │ │ │ -002597e0: 203c 3c20 5468 696e 672c 202d 2d20 7275   << Thing, -- ru
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│ │ │ │ -002598f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00259900: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00259310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +002593b0: 756e 6374 696f 6e73 2f72 756e 5072 6f67  unctions/runProg
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│ │ │ │ +00259410: 726f 6772 616d 5275 6e20 2d2d 2072 6573  rogramRun -- res
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│ │ │ │ +00259430: 6e20 6578 7465 726e 616c 2070 726f 6772  n external progr
│ │ │ │ +00259440: 616d 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  am.*************
│ │ │ │ +00259450: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00259460: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00259470: 2a2a 2a2a 2a2a 0a0a 4465 7363 7269 7074  ******..Descript
│ │ │ │ +00259480: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ion.===========.
│ │ │ │ +00259490: 0a41 2068 6173 6820 7461 626c 6520 7265  .A hash table re
│ │ │ │ +002594a0: 7475 726e 6564 2062 7920 2a6e 6f74 6520  turned by *note 
│ │ │ │ +002594b0: 7275 6e50 726f 6772 616d 3a20 7275 6e50  runProgram: runP
│ │ │ │ +002594c0: 726f 6772 616d 2c20 7769 7468 2074 6865  rogram, with the
│ │ │ │ +002594d0: 2066 6f6c 6c6f 7769 6e67 0a73 7472 696e   following.strin
│ │ │ │ +002594e0: 6773 2061 7320 6b65 7973 3a0a 0a20 202a  gs as keys:..  *
│ │ │ │ +002594f0: 2022 636f 6d6d 616e 6422 2c20 7468 6520   "command", the 
│ │ │ │ +00259500: 636f 6d6d 616e 6420 7468 6174 2077 6173  command that was
│ │ │ │ +00259510: 2075 7365 6420 746f 2072 756e 2074 6865   used to run the
│ │ │ │ +00259520: 2070 726f 6772 616d 2e0a 2020 2a20 226f   program..  * "o
│ │ │ │ +00259530: 7574 7075 7422 2c20 7468 6520 6f75 7470  utput", the outp
│ │ │ │ +00259540: 7574 206f 6620 7468 6520 7072 6f67 7261  ut of the progra
│ │ │ │ +00259550: 6d20 746f 2073 7464 6f75 742e 0a20 202a  m to stdout..  *
│ │ │ │ +00259560: 2022 6572 726f 7222 2c20 7468 6520 6f75   "error", the ou
│ │ │ │ +00259570: 7470 7574 206f 6620 7468 6520 7072 6f67  tput of the prog
│ │ │ │ +00259580: 7261 6d20 746f 2073 7464 6572 722e 0a20  ram to stderr.. 
│ │ │ │ +00259590: 202a 2022 7265 7475 726e 2076 616c 7565   * "return value
│ │ │ │ +002595a0: 222c 2074 6865 2072 6574 7572 6e20 7661  ", the return va
│ │ │ │ +002595b0: 6c75 6520 6f66 2074 6865 2070 726f 6772  lue of the progr
│ │ │ │ +002595c0: 616d 2c20 706f 7373 6962 6c79 206d 756c  am, possibly mul
│ │ │ │ +002595d0: 7469 706c 6965 6420 6279 2032 3536 0a20  tiplied by 256. 
│ │ │ │ +002595e0: 2020 2028 7365 6520 2a6e 6f74 6520 7275     (see *note ru
│ │ │ │ +002595f0: 6e3a 2072 756e 2c29 2e20 204e 6f74 6520  n: run,).  Note 
│ │ │ │ +00259600: 7468 6174 2074 6869 7320 6973 2077 6861  that this is wha
│ │ │ │ +00259610: 7420 6973 2064 6973 706c 6179 6564 2077  t is displayed w
│ │ │ │ +00259620: 6865 6e20 7072 696e 7469 6e67 2061 0a20  hen printing a. 
│ │ │ │ +00259630: 2020 2050 726f 6772 616d 5275 6e20 6f62     ProgramRun ob
│ │ │ │ +00259640: 6a65 6374 2e0a 0a49 6e20 6164 6469 7469  ject...In additi
│ │ │ │ +00259650: 6f6e 2c20 6966 202a 6e6f 7465 2072 756e  on, if *note run
│ │ │ │ +00259660: 5072 6f67 7261 6d3a 2072 756e 5072 6f67  Program: runProg
│ │ │ │ +00259670: 7261 6d2c 2069 7320 6361 6c6c 6564 2077  ram, is called w
│ │ │ │ +00259680: 6974 6820 7468 6520 4b65 6570 4669 6c65  ith the KeepFile
│ │ │ │ +00259690: 730a 6f70 7469 6f6e 2073 6574 2074 6f20  s.option set to 
│ │ │ │ +002596a0: 2a6e 6f74 6520 7472 7565 3a20 7472 7565  *note true: true
│ │ │ │ +002596b0: 2c2c 2074 6865 6e20 7468 6520 666f 6c6c  ,, then the foll
│ │ │ │ +002596c0: 6f77 696e 6720 6b65 7973 2077 696c 6c20  owing keys will 
│ │ │ │ +002596d0: 6265 2070 7265 7365 6e74 3a0a 0a20 202a  be present:..  *
│ │ │ │ +002596e0: 2022 6f75 7470 7574 2066 696c 6522 2c20   "output file", 
│ │ │ │ +002596f0: 7468 6520 7061 7468 2074 6f20 6120 6669  the path to a fi
│ │ │ │ +00259700: 6c65 2063 6f6e 7461 696e 696e 6720 7468  le containing th
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│ │ │ │ +00259720: 7072 6f67 7261 6d20 746f 0a20 2020 2073  program to.    s
│ │ │ │ +00259730: 7464 6f75 742e 0a20 202a 2022 6572 726f  tdout..  * "erro
│ │ │ │ +00259740: 7220 6669 6c65 222c 2074 6865 2070 6174  r file", the pat
│ │ │ │ +00259750: 6820 746f 2061 2066 696c 6520 636f 6e74  h to a file cont
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│ │ │ │ +00259770: 7420 6f66 2074 6865 2070 726f 6772 616d  t of the program
│ │ │ │ +00259780: 2074 6f0a 2020 2020 7374 6465 7272 2e0a   to.    stderr..
│ │ │ │ +00259790: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ +002597a0: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 7275  ==..  * *note ru
│ │ │ │ +002597b0: 6e50 726f 6772 616d 3a20 7275 6e50 726f  nProgram: runPro
│ │ │ │ +002597c0: 6772 616d 2c20 2d2d 2072 756e 2061 6e20  gram, -- run an 
│ │ │ │ +002597d0: 6578 7465 726e 616c 2070 726f 6772 616d  external program
│ │ │ │ +002597e0: 0a20 202a 202a 6e6f 7465 2050 726f 6772  .  * *note Progr
│ │ │ │ +002597f0: 616d 203c 3c20 5468 696e 673a 2050 726f  am << Thing: Pro
│ │ │ │ +00259800: 6772 616d 203c 3c20 5468 696e 672c 202d  gram << Thing, -
│ │ │ │ +00259810: 2d20 7275 6e20 7072 6f67 7261 6d20 7769  - run program wi
│ │ │ │ +00259820: 7468 2069 6e70 7574 0a20 2020 2072 6564  th input.    red
│ │ │ │ +00259830: 6972 6563 7469 6f6e 0a20 202a 202a 6e6f  irection.  * *no
│ │ │ │ +00259840: 7465 2050 726f 6772 616d 3a20 5072 6f67  te Program: Prog
│ │ │ │ +00259850: 7261 6d2c 202d 2d20 6578 7465 726e 616c  ram, -- external
│ │ │ │ +00259860: 2070 726f 6772 616d 206f 626a 6563 740a   program object.
│ │ │ │ +00259870: 2a20 4d65 6e75 3a0a 0a2a 2073 7461 7475  * Menu:..* statu
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│ │ │ │ +00259890: 7461 7475 735f 6c70 5072 6f67 7261 6d52  tatus_lpProgramR
│ │ │ │ +002598a0: 756e 5f72 702e 2020 6765 7420 7468 6520  un_rp.  get the 
│ │ │ │ +002598b0: 7265 7475 726e 2073 7461 7475 7320 6f66  return status of
│ │ │ │ +002598c0: 2061 2070 726f 6772 616d 2072 756e 0a0a   a program run..
│ │ │ │ +002598d0: 4675 6e63 7469 6f6e 7320 616e 6420 6d65  Functions and me
│ │ │ │ +002598e0: 7468 6f64 7320 7265 7475 726e 696e 6720  thods returning 
│ │ │ │ +002598f0: 616e 206f 626a 6563 7420 6f66 2063 6c61  an object of cla
│ │ │ │ +00259900: 7373 2050 726f 6772 616d 5275 6e3a 0a3d  ss ProgramRun:.=
│ │ │ │  00259910: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00259920: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  =========..  * *
│ │ │ │ -00259930: 6e6f 7465 2072 756e 5072 6f67 7261 6d3a  note runProgram:
│ │ │ │ -00259940: 2072 756e 5072 6f67 7261 6d2c 202d 2d20   runProgram, -- 
│ │ │ │ -00259950: 7275 6e20 616e 2065 7874 6572 6e61 6c20  run an external 
│ │ │ │ -00259960: 7072 6f67 7261 6d0a 0a4d 6574 686f 6473  program..Methods
│ │ │ │ -00259970: 2074 6861 7420 7573 6520 616e 206f 626a   that use an obj
│ │ │ │ -00259980: 6563 7420 6f66 2063 6c61 7373 2050 726f  ect of class Pro
│ │ │ │ -00259990: 6772 616d 5275 6e3a 0a3d 3d3d 3d3d 3d3d  gramRun:.=======
│ │ │ │ -002599a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -002599b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -002599c0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ -002599d0: 6f74 6520 7374 6174 7573 2850 726f 6772  ote status(Progr
│ │ │ │ -002599e0: 616d 5275 6e29 3a20 7374 6174 7573 5f6c  amRun): status_l
│ │ │ │ -002599f0: 7050 726f 6772 616d 5275 6e5f 7270 2c20  pProgramRun_rp, 
│ │ │ │ -00259a00: 2d2d 2067 6574 2074 6865 2072 6574 7572  -- get the retur
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│ │ │ │ -00259a20: 6120 7072 6f67 7261 6d20 7275 6e0a 0a46  a program run..F
│ │ │ │ -00259a30: 6f72 2074 6865 2070 726f 6772 616d 6d65  or the programme
│ │ │ │ -00259a40: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r.==============
│ │ │ │ -00259a50: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ -00259a60: 202a 6e6f 7465 2050 726f 6772 616d 5275   *note ProgramRu
│ │ │ │ -00259a70: 6e3a 2050 726f 6772 616d 5275 6e2c 2069  n: ProgramRun, i
│ │ │ │ -00259a80: 7320 6120 2a6e 6f74 6520 7479 7065 3a20  s a *note type: 
│ │ │ │ -00259a90: 5479 7065 2c2c 2077 6974 6820 616e 6365  Type,, with ance
│ │ │ │ -00259aa0: 7374 6f72 0a63 6c61 7373 6573 202a 6e6f  stor.classes *no
│ │ │ │ -00259ab0: 7465 2048 6173 6854 6162 6c65 3a20 4861  te HashTable: Ha
│ │ │ │ -00259ac0: 7368 5461 626c 652c 203c 202a 6e6f 7465  shTable, < *note
│ │ │ │ -00259ad0: 2054 6869 6e67 3a20 5468 696e 672c 2e0a   Thing: Thing,..
│ │ │ │ -00259ae0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .---------------
│ │ │ │ -00259af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00259b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00259920: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00259930: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00259940: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ +00259950: 202a 202a 6e6f 7465 2072 756e 5072 6f67   * *note runProg
│ │ │ │ +00259960: 7261 6d3a 2072 756e 5072 6f67 7261 6d2c  ram: runProgram,
│ │ │ │ +00259970: 202d 2d20 7275 6e20 616e 2065 7874 6572   -- run an exter
│ │ │ │ +00259980: 6e61 6c20 7072 6f67 7261 6d0a 0a4d 6574  nal program..Met
│ │ │ │ +00259990: 686f 6473 2074 6861 7420 7573 6520 616e  hods that use an
│ │ │ │ +002599a0: 206f 626a 6563 7420 6f66 2063 6c61 7373   object of class
│ │ │ │ +002599b0: 2050 726f 6772 616d 5275 6e3a 0a3d 3d3d   ProgramRun:.===
│ │ │ │ +002599c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +002599d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +002599e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
│ │ │ │ +002599f0: 2a20 2a6e 6f74 6520 7374 6174 7573 2850  * *note status(P
│ │ │ │ +00259a00: 726f 6772 616d 5275 6e29 3a20 7374 6174  rogramRun): stat
│ │ │ │ +00259a10: 7573 5f6c 7050 726f 6772 616d 5275 6e5f  us_lpProgramRun_
│ │ │ │ +00259a20: 7270 2c20 2d2d 2067 6574 2074 6865 2072  rp, -- get the r
│ │ │ │ +00259a30: 6574 7572 6e20 7374 6174 7573 0a20 2020  eturn status.   
│ │ │ │ +00259a40: 206f 6620 6120 7072 6f67 7261 6d20 7275   of a program ru
│ │ │ │ +00259a50: 6e0a 0a46 6f72 2074 6865 2070 726f 6772  n..For the progr
│ │ │ │ +00259a60: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d  ammer.==========
│ │ │ │ +00259a70: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62  ========..The ob
│ │ │ │ +00259a80: 6a65 6374 202a 6e6f 7465 2050 726f 6772  ject *note Progr
│ │ │ │ +00259a90: 616d 5275 6e3a 2050 726f 6772 616d 5275  amRun: ProgramRu
│ │ │ │ +00259aa0: 6e2c 2069 7320 6120 2a6e 6f74 6520 7479  n, is a *note ty
│ │ │ │ +00259ab0: 7065 3a20 5479 7065 2c2c 2077 6974 6820  pe: Type,, with 
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│ │ │ │ +00259ad0: 202a 6e6f 7465 2048 6173 6854 6162 6c65   *note HashTable
│ │ │ │ +00259ae0: 3a20 4861 7368 5461 626c 652c 203c 202a  : HashTable, < *
│ │ │ │ +00259af0: 6e6f 7465 2054 6869 6e67 3a20 5468 696e  note Thing: Thin
│ │ │ │ +00259b00: 672c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d  g,...-----------
│ │ │ │  00259b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00259b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00259b30: 0a0a 5468 6520 736f 7572 6365 206f 6620  ..The source of 
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│ │ │ │ -00259b80: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70  s/M2/Macaulay2/p
│ │ │ │ -00259b90: 6163 6b61 6765 732f 0a4d 6163 6175 6c61  ackages/.Macaula
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│ │ │ │ -00259bc0: 323a 3335 3a30 2e0a 1f0a 4669 6c65 3a20  2:35:0....File: 
│ │ │ │ -00259bd0: 4d61 6361 756c 6179 3244 6f63 2e69 6e66  Macaulay2Doc.inf
│ │ │ │ -00259be0: 6f2c 204e 6f64 653a 2073 7461 7475 735f  o, Node: status_
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│ │ │ │ -00259c00: 2055 703a 2050 726f 6772 616d 5275 6e0a   Up: ProgramRun.
│ │ │ │ -00259c10: 0a73 7461 7475 7328 5072 6f67 7261 6d52  .status(ProgramR
│ │ │ │ -00259c20: 756e 2920 2d2d 2067 6574 2074 6865 2072  un) -- get the r
│ │ │ │ -00259c30: 6574 7572 6e20 7374 6174 7573 206f 6620  eturn status of 
│ │ │ │ -00259c40: 6120 7072 6f67 7261 6d20 7275 6e0a 2a2a  a program run.**
│ │ │ │ -00259c50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00259c60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00259c70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00259c80: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ -00259c90: 4675 6e63 7469 6f6e 3a20 2a6e 6f74 6520  Function: *note 
│ │ │ │ -00259ca0: 7374 6174 7573 3a20 7374 6174 7573 2c0a  status: status,.
│ │ │ │ -00259cb0: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ -00259cc0: 2020 2020 7374 6174 7573 2070 720a 2020      status pr.  
│ │ │ │ -00259cd0: 2a20 496e 7075 7473 3a0a 2020 2020 2020  * Inputs:.      
│ │ │ │ -00259ce0: 2a20 7072 2c20 616e 2069 6e73 7461 6e63  * pr, an instanc
│ │ │ │ -00259cf0: 6520 6f66 2074 6865 2074 7970 6520 2a6e  e of the type *n
│ │ │ │ -00259d00: 6f74 6520 5072 6f67 7261 6d52 756e 3a20  ote ProgramRun: 
│ │ │ │ -00259d10: 5072 6f67 7261 6d52 756e 2c2c 200a 2020  ProgramRun,, .  
│ │ │ │ -00259d20: 2a20 2a6e 6f74 6520 4f70 7469 6f6e 616c  * *note Optional
│ │ │ │ -00259d30: 2069 6e70 7574 733a 2075 7369 6e67 2066   inputs: using f
│ │ │ │ -00259d40: 756e 6374 696f 6e73 2077 6974 6820 6f70  unctions with op
│ │ │ │ -00259d50: 7469 6f6e 616c 2069 6e70 7574 732c 3a0a  tional inputs,:.
│ │ │ │ -00259d60: 2020 2020 2020 2a20 2a6e 6f74 6520 4d6f        * *note Mo
│ │ │ │ -00259d70: 6e6f 6d69 616c 733a 2028 4f6c 6443 6861  nomials: (OldCha
│ │ │ │ -00259d80: 696e 436f 6d70 6c65 7865 7329 7374 6174  inComplexes)stat
│ │ │ │ -00259d90: 7573 5f6c 7043 6861 696e 436f 6d70 6c65  us_lpChainComple
│ │ │ │ -00259da0: 785f 7270 2c20 3d3e 202e 2e2e 2c0a 2020  x_rp, => ...,.  
│ │ │ │ -00259db0: 2020 2020 2020 6465 6661 756c 7420 7661        default va
│ │ │ │ -00259dc0: 6c75 6520 6661 6c73 652c 0a20 2020 2020  lue false,.     
│ │ │ │ -00259dd0: 202a 202a 6e6f 7465 2050 6169 7273 5265   * *note PairsRe
│ │ │ │ -00259de0: 6d61 696e 696e 673a 2028 4f6c 6443 6861  maining: (OldCha
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│ │ │ │ -00259e00: 7573 5f6c 7043 6861 696e 436f 6d70 6c65  us_lpChainComple
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│ │ │ │ -00259fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0025aaa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0025aa40: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
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│ │ │ │ +0025aa70: 2054 6869 6e67 2c20 2d2d 2072 756e 2070   Thing, -- run p
│ │ │ │ +0025aa80: 726f 6772 616d 2077 6974 6820 696e 7075  rogram with inpu
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│ │ │ │  0025aab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0025aac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
│ │ │ │ -0025aad0: 6865 2073 6f75 7263 6520 6f66 2074 6869  he source of thi
│ │ │ │ -0025aae0: 7320 646f 6375 6d65 6e74 2069 7320 696e  s document is in
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│ │ │ │ -0025ab40: 6f63 2f66 756e 6374 696f 6e73 2f72 756e  oc/functions/run
│ │ │ │ -0025ab50: 5072 6f67 7261 6d2d 646f 632e 6d32 3a31  Program-doc.m2:1
│ │ │ │ -0025ab60: 3537 3a30 2e0a 1f0a 4669 6c65 3a20 4d61  57:0....File: Ma
│ │ │ │ -0025ab70: 6361 756c 6179 3244 6f63 2e69 6e66 6f2c  caulay2Doc.info,
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│ │ │ │ -0025abb0: 6963 6174 696e 6720 7769 7468 2070 726f  icating with pro
│ │ │ │ -0025abc0: 6772 616d 732c 2055 703a 2073 7973 7465  grams, Up: syste
│ │ │ │ -0025abd0: 6d20 6661 6369 6c69 7469 6573 0a0a 7573  m facilities..us
│ │ │ │ -0025abe0: 696e 6720 736f 636b 6574 730a 2a2a 2a2a  ing sockets.****
│ │ │ │ -0025abf0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a49 7427 7320  *********..It's 
│ │ │ │ -0025ac00: 6561 7379 2074 6f20 7573 6520 736f 636b  easy to use sock
│ │ │ │ -0025ac10: 6574 7320 6173 2074 686f 7567 6820 7468  ets as though th
│ │ │ │ -0025ac20: 6579 2077 6572 6520 6669 6c65 732e 2020  ey were files.  
│ │ │ │ -0025ac30: 5369 6d70 6c79 2072 6570 6c61 6365 2074  Simply replace t
│ │ │ │ -0025ac40: 6865 2066 696c 650a 6e61 6d65 2062 7920  he file.name by 
│ │ │ │ -0025ac50: 6120 7374 7269 6e67 206f 6620 7468 6520  a string of the 
│ │ │ │ -0025ac60: 666f 726d 2024 686f 7374 3a73 6572 7669  form $host:servi
│ │ │ │ -0025ac70: 6365 2077 6865 7265 2068 6f73 7420 6973  ce where host is
│ │ │ │ -0025ac80: 2074 6865 206e 616d 6520 6f66 2049 5020   the name of IP 
│ │ │ │ -0025ac90: 6e75 6d62 6572 0a6f 6620 686f 7374 2074  number.of host t
│ │ │ │ -0025aca0: 6f20 636f 6e74 6163 742c 2061 6e64 2073  o contact, and s
│ │ │ │ -0025acb0: 6572 7669 6365 2069 7320 7468 6520 706f  ervice is the po
│ │ │ │ -0025acc0: 7274 206e 756d 6265 7220 6f72 206e 616d  rt number or nam
│ │ │ │ -0025acd0: 6520 746f 2075 7365 2e20 2049 6620 7365  e to use.  If se
│ │ │ │ -0025ace0: 7276 6963 650a 6973 206f 6d69 7474 6564  rvice.is omitted
│ │ │ │ -0025acf0: 2c20 7468 656e 2070 6f72 7420 3235 3030  , then port 2500
│ │ │ │ -0025ad00: 2069 7320 7573 6564 2e20 2049 6620 686f   is used.  If ho
│ │ │ │ -0025ad10: 7374 2069 7320 6f6d 6974 7465 642c 2074  st is omitted, t
│ │ │ │ -0025ad20: 6865 6e20 616e 2069 6e63 6f6d 696e 670a  hen an incoming.
│ │ │ │ -0025ad30: 636f 6e6e 6563 7469 6f6e 2077 696c 6c20  connection will 
│ │ │ │ -0025ad40: 6265 206c 6973 7465 6e65 6420 666f 722e  be listened for.
│ │ │ │ -0025ad50: 0a0a 5468 6520 666f 6c6c 6f77 696e 6720  ..The following 
│ │ │ │ -0025ad60: 636f 6465 2077 696c 6c20 696c 6c75 7374  code will illust
│ │ │ │ -0025ad70: 7261 7465 2074 776f 2d77 6179 2063 6f6d  rate two-way com
│ │ │ │ -0025ad80: 6d75 6e69 6361 7469 6f6e 2075 7369 6e67  munication using
│ │ │ │ -0025ad90: 2073 6f63 6b65 7473 2073 696d 696c 6172   sockets similar
│ │ │ │ -0025ada0: 0a74 6f20 7468 6520 696e 7465 7261 6374  .to the interact
│ │ │ │ -0025adb0: 696f 6e20 7573 6564 2062 7920 7765 6220  ion used by web 
│ │ │ │ -0025adc0: 7365 7276 6572 732c 2061 6e64 2079 6f75  servers, and you
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│ │ │ │ -0025ade0: 6f6e 2079 6f75 7220 6d61 6368 696e 652c  on your machine,
│ │ │ │ -0025adf0: 0a75 6e6c 6573 7320 6120 6669 7265 7761  .unless a firewa
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│ │ │ │ -0025ae10: 0a69 6620 2870 6964 203d 2066 6f72 6b28  .if (pid = fork(
│ │ │ │ -0025ae20: 2929 203d 3d20 3020 7468 656e 2028 0a20  )) == 0 then (. 
│ │ │ │ -0025ae30: 2020 2020 7472 7920 2224 3a37 3530 3022      try "$:7500"
│ │ │ │ -0025ae40: 203c 3c20 2268 6920 7468 6572 6522 203c   << "hi there" <
│ │ │ │ -0025ae50: 3c20 636c 6f73 653b 0a20 2020 2020 6578  < close;.     ex
│ │ │ │ -0025ae60: 6974 2030 3b0a 2020 2020 2029 0a73 6c65  it 0;.     ).sle
│ │ │ │ -0025ae70: 6570 2032 0a67 6574 2022 246c 6f63 616c  ep 2.get "$local
│ │ │ │ -0025ae80: 686f 7374 3a37 3530 3022 0a77 6169 7420  host:7500".wait 
│ │ │ │ -0025ae90: 7069 640a 0a54 6865 2063 6f64 6520 7573  pid..The code us
│ │ │ │ -0025aea0: 6573 202a 6e6f 7465 2066 6f72 6b3a 2066  es *note fork: f
│ │ │ │ -0025aeb0: 6f72 6b2c 2074 6f20 6372 6561 7465 2061  ork, to create a
│ │ │ │ -0025aec0: 2073 6570 6172 6174 6520 7072 6f63 6573   separate proces
│ │ │ │ -0025aed0: 7320 7468 6174 2077 696c 6c20 6c69 7374  s that will list
│ │ │ │ -0025aee0: 656e 0a66 6f72 2061 2063 6f6e 6e65 6374  en.for a connect
│ │ │ │ -0025aef0: 696f 6e20 6f6e 2070 6f72 7420 3735 3030  ion on port 7500
│ │ │ │ -0025af00: 2061 6e64 2074 6865 6e20 7365 6e64 2075   and then send u
│ │ │ │ -0025af10: 7320 6120 6d65 7373 6167 652e 2020 5468  s a message.  Th
│ │ │ │ -0025af20: 6520 2a6e 6f74 6520 736c 6565 703a 0a73  e *note sleep:.s
│ │ │ │ -0025af30: 6c65 6570 2c20 636f 6d6d 616e 6420 7061  leep, command pa
│ │ │ │ -0025af40: 7573 6573 2066 6f72 2061 2077 6869 6c65  uses for a while
│ │ │ │ -0025af50: 2074 6f20 6d61 6b65 2073 7572 6520 7468   to make sure th
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│ │ │ │ -0025af70: 6861 7320 6861 6420 7469 6d65 0a74 6f20  has had time.to 
│ │ │ │ -0025af80: 7374 6172 7420 6c69 7374 656e 696e 672e  start listening.
│ │ │ │ -0025af90: 2054 6865 6e20 7765 2075 7365 2061 6e20   Then we use an 
│ │ │ │ -0025afa0: 6f72 6469 6e61 7279 2069 6e70 7574 2063  ordinary input c
│ │ │ │ -0025afb0: 6f6d 6d61 6e64 2c20 6e61 6d65 6c79 202a  ommand, namely *
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│ │ │ │ -0025afd0: 746f 206f 6274 6169 6e20 7468 6520 6d65  to obtain the me
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│ │ │ │ -0025aff0: 7765 202a 6e6f 7465 2077 6169 743a 2077  we *note wait: w
│ │ │ │ -0025b000: 6169 742c 2066 6f72 2074 6865 2063 6869  ait, for the chi
│ │ │ │ -0025b010: 6c64 0a70 726f 6365 7373 2074 6f20 6669  ld.process to fi
│ │ │ │ -0025b020: 6e69 7368 2c20 6173 2077 6520 7368 6f75  nish, as we shou
│ │ │ │ -0025b030: 6c64 2e0a 0a53 6565 2061 6c73 6f0a 3d3d  ld...See also.==
│ │ │ │ -0025b040: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74  ======..  * *not
│ │ │ │ -0025b050: 6520 6765 7457 5757 3a20 6765 7457 5757  e getWWW: getWWW
│ │ │ │ -0025b060: 2c20 2d2d 2067 6574 2061 2077 6562 2070  , -- get a web p
│ │ │ │ -0025b070: 6167 650a 2a20 4d65 6e75 3a0a 0a0a 4372  age.* Menu:...Cr
│ │ │ │ -0025b080: 6561 7469 6e67 2061 2073 6f63 6b65 740a  eating a socket.
│ │ │ │ -0025b090: 2a20 6f70 656e 4c69 7374 656e 6572 3a20  * openListener: 
│ │ │ │ -0025b0a0: 6f70 656e 4c69 7374 656e 6572 5f6c 7053  openListener_lpS
│ │ │ │ -0025b0b0: 7472 696e 675f 7270 2e20 206f 7065 6e20  tring_rp.  open 
│ │ │ │ -0025b0c0: 6120 706f 7274 2066 6f72 206c 6973 7465  a port for liste
│ │ │ │ -0025b0d0: 6e69 6e67 0a2a 206f 7065 6e49 6e3a 206f  ning.* openIn: o
│ │ │ │ -0025b0e0: 7065 6e49 6e5f 6c70 5374 7269 6e67 5f72  penIn_lpString_r
│ │ │ │ -0025b0f0: 702e 2020 206f 7065 6e20 616e 2069 6e70  p.   open an inp
│ │ │ │ -0025b100: 7574 2066 696c 650a 2a20 6f70 656e 496e  ut file.* openIn
│ │ │ │ -0025b110: 4f75 743a 3a20 2020 2020 2020 2020 2020  Out::           
│ │ │ │ -0025b120: 2020 2020 2020 2020 6f70 656e 2061 6e20          open an 
│ │ │ │ -0025b130: 696e 7075 7420 6f75 7470 7574 2066 696c  input output fil
│ │ │ │ -0025b140: 650a 2a20 6f70 656e 4f75 743a 206f 7065  e.* openOut: ope
│ │ │ │ -0025b150: 6e4f 7574 5f6c 7053 7472 696e 675f 7270  nOut_lpString_rp
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│ │ │ │ -0025b170: 7420 6669 6c65 0a2a 206f 7065 6e4f 7574  t file.* openOut
│ │ │ │ -0025b180: 4170 7065 6e64 3a20 6f70 656e 4f75 7441  Append: openOutA
│ │ │ │ -0025b190: 7070 656e 645f 6c70 5374 7269 6e67 5f72  ppend_lpString_r
│ │ │ │ -0025b1a0: 702e 2020 6f70 656e 2061 6e20 6f75 7470  p.  open an outp
│ │ │ │ -0025b1b0: 7574 2066 696c 6520 666f 7220 6170 7065  ut file for appe
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│ │ │ │ -0025b1e0: 6e64 6c3a 3a20 2020 2020 2020 2020 2020  ndl::           
│ │ │ │ -0025b1f0: 2020 2020 2020 2020 2020 2020 2065 6e64               end
│ │ │ │ -0025b200: 2061 6e20 6f75 7470 7574 206c 696e 650a   an output line.
│ │ │ │ -0025b210: 2a20 666c 7573 683a 3a20 2020 2020 2020  * flush::       
│ │ │ │ -0025b220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0025b230: 666c 7573 6820 6f75 7470 7574 2074 6f20  flush output to 
│ │ │ │ -0025b240: 6669 6c65 0a2a 2063 6c6f 7365 3a3a 2020  file.* close::  
│ │ │ │ -0025b250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0025b260: 2020 2020 2063 6c6f 7365 2061 2066 696c       close a fil
│ │ │ │ -0025b270: 650a 2a20 636c 6f73 6549 6e3a 3a20 2020  e.* closeIn::   
│ │ │ │ -0025b280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0025b290: 2020 636c 6f73 6520 616e 2069 6e70 7574    close an input
│ │ │ │ -0025b2a0: 2066 696c 650a 2a20 636c 6f73 654f 7574   file.* closeOut
│ │ │ │ -0025b2b0: 3a3a 2020 2020 2020 2020 2020 2020 2020  ::              
│ │ │ │ -0025b2c0: 2020 2020 2020 636c 6f73 6520 616e 206f        close an o
│ │ │ │ -0025b2d0: 7574 7075 7420 6669 6c65 0a0a 496e 7465  utput file..Inte
│ │ │ │ -0025b2e0: 7261 6374 696e 6720 6120 736f 636b 6574  racting a socket
│ │ │ │ -0025b2f0: 0a2a 2067 6574 3a3a 2020 2020 2020 2020  .* get::        
│ │ │ │ -0025b300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0025b310: 2067 6574 2074 6865 2063 6f6e 7465 6e74   get the content
│ │ │ │ -0025b320: 7320 6f66 2061 2066 696c 650a 2a20 6765  s of a file.* ge
│ │ │ │ -0025b330: 7463 3a3a 2020 2020 2020 2020 2020 2020  tc::            
│ │ │ │ -0025b340: 2020 2020 2020 2020 2020 2020 6765 7420              get 
│ │ │ │ -0025b350: 6120 6279 7465 0a2a 2072 6561 643a 3a20  a byte.* read:: 
│ │ │ │ +0025aac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0025aad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0025aae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0025aaf0: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66  -..The source of
│ │ │ │ +0025ab00: 2074 6869 7320 646f 6375 6d65 6e74 2069   this document i
│ │ │ │ +0025ab10: 7320 696e 0a2f 6275 696c 642f 7265 7072  s in./build/repr
│ │ │ │ +0025ab20: 6f64 7563 6962 6c65 2d70 6174 682f 6d61  oducible-path/ma
│ │ │ │ +0025ab30: 6361 756c 6179 322d 312e 3235 2e31 312b  caulay2-1.25.11+
│ │ │ │ +0025ab40: 6473 2f4d 322f 4d61 6361 756c 6179 322f  ds/M2/Macaulay2/
│ │ │ │ +0025ab50: 7061 636b 6167 6573 2f0a 4d61 6361 756c  packages/.Macaul
│ │ │ │ +0025ab60: 6179 3244 6f63 2f66 756e 6374 696f 6e73  ay2Doc/functions
│ │ │ │ +0025ab70: 2f72 756e 5072 6f67 7261 6d2d 646f 632e  /runProgram-doc.
│ │ │ │ +0025ab80: 6d32 3a31 3537 3a30 2e0a 1f0a 4669 6c65  m2:157:0....File
│ │ │ │ +0025ab90: 3a20 4d61 6361 756c 6179 3244 6f63 2e69  : Macaulay2Doc.i
│ │ │ │ +0025aba0: 6e66 6f2c 204e 6f64 653a 2075 7369 6e67  nfo, Node: using
│ │ │ │ +0025abb0: 2073 6f63 6b65 7473 2c20 4e65 7874 3a20   sockets, Next: 
│ │ │ │ +0025abc0: 6e6f 7469 6679 2c20 5072 6576 3a20 636f  notify, Prev: co
│ │ │ │ +0025abd0: 6d6d 756e 6963 6174 696e 6720 7769 7468  mmunicating with
│ │ │ │ +0025abe0: 2070 726f 6772 616d 732c 2055 703a 2073   programs, Up: s
│ │ │ │ +0025abf0: 7973 7465 6d20 6661 6369 6c69 7469 6573  ystem facilities
│ │ │ │ +0025ac00: 0a0a 7573 696e 6720 736f 636b 6574 730a  ..using sockets.
│ │ │ │ +0025ac10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a49  *************..I
│ │ │ │ +0025ac20: 7427 7320 6561 7379 2074 6f20 7573 6520  t's easy to use 
│ │ │ │ +0025ac30: 736f 636b 6574 7320 6173 2074 686f 7567  sockets as thoug
│ │ │ │ +0025ac40: 6820 7468 6579 2077 6572 6520 6669 6c65  h they were file
│ │ │ │ +0025ac50: 732e 2020 5369 6d70 6c79 2072 6570 6c61  s.  Simply repla
│ │ │ │ +0025ac60: 6365 2074 6865 2066 696c 650a 6e61 6d65  ce the file.name
│ │ │ │ +0025ac70: 2062 7920 6120 7374 7269 6e67 206f 6620   by a string of 
│ │ │ │ +0025ac80: 7468 6520 666f 726d 2024 686f 7374 3a73  the form $host:s
│ │ │ │ +0025ac90: 6572 7669 6365 2077 6865 7265 2068 6f73  ervice where hos
│ │ │ │ +0025aca0: 7420 6973 2074 6865 206e 616d 6520 6f66  t is the name of
│ │ │ │ +0025acb0: 2049 5020 6e75 6d62 6572 0a6f 6620 686f   IP number.of ho
│ │ │ │ +0025acc0: 7374 2074 6f20 636f 6e74 6163 742c 2061  st to contact, a
│ │ │ │ +0025acd0: 6e64 2073 6572 7669 6365 2069 7320 7468  nd service is th
│ │ │ │ +0025ace0: 6520 706f 7274 206e 756d 6265 7220 6f72  e port number or
│ │ │ │ +0025acf0: 206e 616d 6520 746f 2075 7365 2e20 2049   name to use.  I
│ │ │ │ +0025ad00: 6620 7365 7276 6963 650a 6973 206f 6d69  f service.is omi
│ │ │ │ +0025ad10: 7474 6564 2c20 7468 656e 2070 6f72 7420  tted, then port 
│ │ │ │ +0025ad20: 3235 3030 2069 7320 7573 6564 2e20 2049  2500 is used.  I
│ │ │ │ +0025ad30: 6620 686f 7374 2069 7320 6f6d 6974 7465  f host is omitte
│ │ │ │ +0025ad40: 642c 2074 6865 6e20 616e 2069 6e63 6f6d  d, then an incom
│ │ │ │ +0025ad50: 696e 670a 636f 6e6e 6563 7469 6f6e 2077  ing.connection w
│ │ │ │ +0025ad60: 696c 6c20 6265 206c 6973 7465 6e65 6420  ill be listened 
│ │ │ │ +0025ad70: 666f 722e 0a0a 5468 6520 666f 6c6c 6f77  for...The follow
│ │ │ │ +0025ad80: 696e 6720 636f 6465 2077 696c 6c20 696c  ing code will il
│ │ │ │ +0025ad90: 6c75 7374 7261 7465 2074 776f 2d77 6179  lustrate two-way
│ │ │ │ +0025ada0: 2063 6f6d 6d75 6e69 6361 7469 6f6e 2075   communication u
│ │ │ │ +0025adb0: 7369 6e67 2073 6f63 6b65 7473 2073 696d  sing sockets sim
│ │ │ │ +0025adc0: 696c 6172 0a74 6f20 7468 6520 696e 7465  ilar.to the inte
│ │ │ │ +0025add0: 7261 6374 696f 6e20 7573 6564 2062 7920  raction used by 
│ │ │ │ +0025ade0: 7765 6220 7365 7276 6572 732c 2061 6e64  web servers, and
│ │ │ │ +0025adf0: 2079 6f75 206d 6179 2074 7279 2069 7420   you may try it 
│ │ │ │ +0025ae00: 6f75 7420 6f6e 2079 6f75 7220 6d61 6368  out on your mach
│ │ │ │ +0025ae10: 696e 652c 0a75 6e6c 6573 7320 6120 6669  ine,.unless a fi
│ │ │ │ +0025ae20: 7265 7761 6c6c 2070 7265 7665 6e74 7320  rewall prevents 
│ │ │ │ +0025ae30: 6974 2e0a 0a69 6620 2870 6964 203d 2066  it...if (pid = f
│ │ │ │ +0025ae40: 6f72 6b28 2929 203d 3d20 3020 7468 656e  ork()) == 0 then
│ │ │ │ +0025ae50: 2028 0a20 2020 2020 7472 7920 2224 3a37   (.     try "$:7
│ │ │ │ +0025ae60: 3530 3022 203c 3c20 2268 6920 7468 6572  500" << "hi ther
│ │ │ │ +0025ae70: 6522 203c 3c20 636c 6f73 653b 0a20 2020  e" << close;.   
│ │ │ │ +0025ae80: 2020 6578 6974 2030 3b0a 2020 2020 2029    exit 0;.     )
│ │ │ │ +0025ae90: 0a73 6c65 6570 2032 0a67 6574 2022 246c  .sleep 2.get "$l
│ │ │ │ +0025aea0: 6f63 616c 686f 7374 3a37 3530 3022 0a77  ocalhost:7500".w
│ │ │ │ +0025aeb0: 6169 7420 7069 640a 0a54 6865 2063 6f64  ait pid..The cod
│ │ │ │ +0025aec0: 6520 7573 6573 202a 6e6f 7465 2066 6f72  e uses *note for
│ │ │ │ +0025aed0: 6b3a 2066 6f72 6b2c 2074 6f20 6372 6561  k: fork, to crea
│ │ │ │ +0025aee0: 7465 2061 2073 6570 6172 6174 6520 7072  te a separate pr
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│ │ │ │ -0025b910: 6e67 3a20 5374 7269 6e67 2c2c 206f 6620  ng: String,, of 
│ │ │ │ -0025b920: 7468 6520 666f 726d 2022 2469 6e74 6572  the form "$inter
│ │ │ │ -0025b930: 6661 6365 3a70 6f72 7422 2e20 2042 6f74  face:port".  Bot
│ │ │ │ -0025b940: 6820 7061 7274 730a 2020 2020 2020 2020  h parts.        
│ │ │ │ -0025b950: 6172 6520 6f70 7469 6f6e 616c 2e20 2049  are optional.  I
│ │ │ │ -0025b960: 6620 7468 6520 706f 7274 2069 7320 6f6d  f the port is om
│ │ │ │ -0025b970: 6974 7465 642c 2074 6865 2063 6f6c 6f6e  itted, the colon
│ │ │ │ -0025b980: 2069 7320 6f70 7469 6f6e 616c 2e0a 2020   is optional..  
│ │ │ │ -0025b990: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ -0025b9a0: 202a 2066 2c20 6120 2a6e 6f74 6520 6669   * f, a *note fi
│ │ │ │ -0025b9b0: 6c65 3a20 4669 6c65 2c2c 2061 6e20 6f70  le: File,, an op
│ │ │ │ -0025b9c0: 656e 206c 6973 7465 6e65 7220 6f6e 2074  en listener on t
│ │ │ │ -0025b9d0: 6865 2073 7065 6369 6669 6564 2069 6e74  he specified int
│ │ │ │ -0025b9e0: 6572 6661 6365 206f 660a 2020 2020 2020  erface of.      
│ │ │ │ -0025b9f0: 2020 7468 6520 6c6f 6361 6c20 686f 7374    the local host
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│ │ │ │ -0025ba10: 6420 7365 7276 6963 6520 706f 7274 2e20  d service port. 
│ │ │ │ -0025ba20: 4966 2074 6865 2070 6f72 7420 6973 206f  If the port is o
│ │ │ │ -0025ba30: 6d69 7474 6564 2c0a 2020 2020 2020 2020  mitted,.        
│ │ │ │ -0025ba40: 6974 2069 7320 7461 6b65 6e20 746f 2062  it is taken to b
│ │ │ │ -0025ba50: 6520 706f 7274 2032 3530 302e 2020 4966  e port 2500.  If
│ │ │ │ -0025ba60: 2074 6865 2069 6e74 6572 6661 6365 2069   the interface i
│ │ │ │ -0025ba70: 7320 6f6d 6974 7465 642c 2074 6865 206c  s omitted, the l
│ │ │ │ -0025ba80: 6973 7465 6e65 720a 2020 2020 2020 2020  istener.        
│ │ │ │ -0025ba90: 6163 6365 7074 7320 636f 6e6e 6563 7469  accepts connecti
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│ │ │ │ -0025bab0: 6661 6365 732e 0a0a 4465 7363 7269 7074  faces...Descript
│ │ │ │ -0025bac0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ion.===========.
│ │ │ │ -0025bad0: 0a55 7365 206f 7065 6e49 6e4f 7574 2066  .Use openInOut f
│ │ │ │ -0025bae0: 2074 6f20 6163 6365 7074 2061 6e20 696e   to accept an in
│ │ │ │ -0025baf0: 636f 6d69 6e67 2063 6f6e 6e65 6374 696f  coming connectio
│ │ │ │ -0025bb00: 6e20 6f6e 2074 6865 206c 6973 7465 6e65  n on the listene
│ │ │ │ -0025bb10: 722c 2072 6574 7572 6e69 6e67 2061 0a6e  r, returning a.n
│ │ │ │ -0025bb20: 6577 2069 6e70 7574 206f 7574 7075 7420  ew input output 
│ │ │ │ -0025bb30: 6669 6c65 2074 6861 7420 7365 7276 6573  file that serves
│ │ │ │ -0025bb40: 2061 7320 7468 6520 636f 6e6e 6563 7469   as the connecti
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│ │ │ │ -0025bb60: 6e20 2a6e 6f74 650a 6973 5265 6164 793a  n *note.isReady:
│ │ │ │ -0025bb70: 2069 7352 6561 6479 5f6c 7046 696c 655f   isReady_lpFile_
│ │ │ │ -0025bb80: 7270 2c20 6361 6e20 6265 2075 7365 6420  rp, can be used 
│ │ │ │ -0025bb90: 746f 2064 6574 6572 6d69 6e65 2077 6865  to determine whe
│ │ │ │ -0025bba0: 7468 6572 2061 6e20 696e 636f 6d69 6e67  ther an incoming
│ │ │ │ -0025bbb0: 0a63 6f6e 6e65 6374 696f 6e20 6861 7320  .connection has 
│ │ │ │ -0025bbc0: 6172 7269 7665 642c 2077 6974 686f 7574  arrived, without
│ │ │ │ -0025bbd0: 2062 6c6f 636b 696e 672e 0a0a 5365 6520   blocking...See 
│ │ │ │ -0025bbe0: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  also.========.. 
│ │ │ │ -0025bbf0: 202a 202a 6e6f 7465 206f 7065 6e49 6e4f   * *note openInO
│ │ │ │ -0025bc00: 7574 3a20 6f70 656e 496e 4f75 742c 202d  ut: openInOut, -
│ │ │ │ -0025bc10: 2d20 6f70 656e 2061 6e20 696e 7075 7420  - open an input 
│ │ │ │ -0025bc20: 6f75 7470 7574 2066 696c 650a 2020 2a20  output file.  * 
│ │ │ │ -0025bc30: 2a6e 6f74 6520 6973 5265 6164 793a 2069  *note isReady: i
│ │ │ │ -0025bc40: 7352 6561 6479 5f6c 7046 696c 655f 7270  sReady_lpFile_rp
│ │ │ │ -0025bc50: 2c20 2d2d 2077 6865 7468 6572 2061 2066  , -- whether a f
│ │ │ │ -0025bc60: 696c 6520 6861 7320 6461 7461 2061 7661  ile has data ava
│ │ │ │ -0025bc70: 696c 6162 6c65 2066 6f72 0a20 2020 2072  ilable for.    r
│ │ │ │ -0025bc80: 6561 6469 6e67 0a0a 5761 7973 2074 6f20  eading..Ways to 
│ │ │ │ -0025bc90: 7573 6520 7468 6973 206d 6574 686f 643a  use this method:
│ │ │ │ -0025bca0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -0025bcb0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  =========..  * *
│ │ │ │ -0025bcc0: 6e6f 7465 206f 7065 6e4c 6973 7465 6e65  note openListene
│ │ │ │ -0025bcd0: 7228 5374 7269 6e67 293a 206f 7065 6e4c  r(String): openL
│ │ │ │ -0025bce0: 6973 7465 6e65 725f 6c70 5374 7269 6e67  istener_lpString
│ │ │ │ -0025bcf0: 5f72 702c 202d 2d20 6f70 656e 2061 2070  _rp, -- open a p
│ │ │ │ -0025bd00: 6f72 7420 666f 720a 2020 2020 6c69 7374  ort for.    list
│ │ │ │ -0025bd10: 656e 696e 670a 2d2d 2d2d 2d2d 2d2d 2d2d  ening.----------
│ │ │ │ -0025bd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0025bd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0025b730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0025b740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0025b750: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f  --------..The so
│ │ │ │ +0025b760: 7572 6365 206f 6620 7468 6973 2064 6f63  urce of this doc
│ │ │ │ +0025b770: 756d 656e 7420 6973 2069 6e0a 2f62 7569  ument is in./bui
│ │ │ │ +0025b780: 6c64 2f72 6570 726f 6475 6369 626c 652d  ld/reproducible-
│ │ │ │ +0025b790: 7061 7468 2f6d 6163 6175 6c61 7932 2d31  path/macaulay2-1
│ │ │ │ +0025b7a0: 2e32 352e 3131 2b64 732f 4d32 2f4d 6163  .25.11+ds/M2/Mac
│ │ │ │ +0025b7b0: 6175 6c61 7932 2f70 6163 6b61 6765 732f  aulay2/packages/
│ │ │ │ +0025b7c0: 0a4d 6163 6175 6c61 7932 446f 632f 6f76  .Macaulay2Doc/ov
│ │ │ │ +0025b7d0: 5f66 696c 6573 2e6d 323a 3530 393a 302e  _files.m2:509:0.
│ │ │ │ +0025b7e0: 0a1f 0a46 696c 653a 204d 6163 6175 6c61  ...File: Macaula
│ │ │ │ +0025b7f0: 7932 446f 632e 696e 666f 2c20 4e6f 6465  y2Doc.info, Node
│ │ │ │ +0025b800: 3a20 6f70 656e 4c69 7374 656e 6572 5f6c  : openListener_l
│ │ │ │ +0025b810: 7053 7472 696e 675f 7270 2c20 4e65 7874  pString_rp, Next
│ │ │ │ +0025b820: 3a20 6f70 656e 496e 5f6c 7053 7472 696e  : openIn_lpStrin
│ │ │ │ +0025b830: 675f 7270 2c20 5570 3a20 7573 696e 6720  g_rp, Up: using 
│ │ │ │ +0025b840: 736f 636b 6574 730a 0a6f 7065 6e4c 6973  sockets..openLis
│ │ │ │ +0025b850: 7465 6e65 7228 5374 7269 6e67 2920 2d2d  tener(String) --
│ │ │ │ +0025b860: 206f 7065 6e20 6120 706f 7274 2066 6f72   open a port for
│ │ │ │ +0025b870: 206c 6973 7465 6e69 6e67 0a2a 2a2a 2a2a   listening.*****
│ │ │ │ +0025b880: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0025b890: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0025b8a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020  ************..  
│ │ │ │ +0025b8b0: 2a20 4675 6e63 7469 6f6e 3a20 2a6e 6f74  * Function: *not
│ │ │ │ +0025b8c0: 6520 6f70 656e 4c69 7374 656e 6572 3a20  e openListener: 
│ │ │ │ +0025b8d0: 6f70 656e 4c69 7374 656e 6572 5f6c 7053  openListener_lpS
│ │ │ │ +0025b8e0: 7472 696e 675f 7270 2c0a 2020 2a20 5573  tring_rp,.  * Us
│ │ │ │ +0025b8f0: 6167 653a 200a 2020 2020 2020 2020 6620  age: .        f 
│ │ │ │ +0025b900: 3d20 6f70 656e 4c69 7374 656e 6572 2073  = openListener s
│ │ │ │ +0025b910: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ +0025b920: 2020 202a 2073 2c20 6120 2a6e 6f74 6520     * s, a *note 
│ │ │ │ +0025b930: 7374 7269 6e67 3a20 5374 7269 6e67 2c2c  string: String,,
│ │ │ │ +0025b940: 206f 6620 7468 6520 666f 726d 2022 2469   of the form "$i
│ │ │ │ +0025b950: 6e74 6572 6661 6365 3a70 6f72 7422 2e20  nterface:port". 
│ │ │ │ +0025b960: 2042 6f74 6820 7061 7274 730a 2020 2020   Both parts.    
│ │ │ │ +0025b970: 2020 2020 6172 6520 6f70 7469 6f6e 616c      are optional
│ │ │ │ +0025b980: 2e20 2049 6620 7468 6520 706f 7274 2069  .  If the port i
│ │ │ │ +0025b990: 7320 6f6d 6974 7465 642c 2074 6865 2063  s omitted, the c
│ │ │ │ +0025b9a0: 6f6c 6f6e 2069 7320 6f70 7469 6f6e 616c  olon is optional
│ │ │ │ +0025b9b0: 2e0a 2020 2a20 4f75 7470 7574 733a 0a20  ..  * Outputs:. 
│ │ │ │ +0025b9c0: 2020 2020 202a 2066 2c20 6120 2a6e 6f74       * f, a *not
│ │ │ │ +0025b9d0: 6520 6669 6c65 3a20 4669 6c65 2c2c 2061  e file: File,, a
│ │ │ │ +0025b9e0: 6e20 6f70 656e 206c 6973 7465 6e65 7220  n open listener 
│ │ │ │ +0025b9f0: 6f6e 2074 6865 2073 7065 6369 6669 6564  on the specified
│ │ │ │ +0025ba00: 2069 6e74 6572 6661 6365 206f 660a 2020   interface of.  
│ │ │ │ +0025ba10: 2020 2020 2020 7468 6520 6c6f 6361 6c20        the local 
│ │ │ │ +0025ba20: 686f 7374 2061 7420 7468 6520 7370 6563  host at the spec
│ │ │ │ +0025ba30: 6966 6965 6420 7365 7276 6963 6520 706f  ified service po
│ │ │ │ +0025ba40: 7274 2e20 4966 2074 6865 2070 6f72 7420  rt. If the port 
│ │ │ │ +0025ba50: 6973 206f 6d69 7474 6564 2c0a 2020 2020  is omitted,.    
│ │ │ │ +0025ba60: 2020 2020 6974 2069 7320 7461 6b65 6e20      it is taken 
│ │ │ │ +0025ba70: 746f 2062 6520 706f 7274 2032 3530 302e  to be port 2500.
│ │ │ │ +0025ba80: 2020 4966 2074 6865 2069 6e74 6572 6661    If the interfa
│ │ │ │ +0025ba90: 6365 2069 7320 6f6d 6974 7465 642c 2074  ce is omitted, t
│ │ │ │ +0025baa0: 6865 206c 6973 7465 6e65 720a 2020 2020  he listener.    
│ │ │ │ +0025bab0: 2020 2020 6163 6365 7074 7320 636f 6e6e      accepts conn
│ │ │ │ +0025bac0: 6563 7469 6f6e 7320 6f6e 2061 6c6c 2069  ections on all i
│ │ │ │ +0025bad0: 6e74 6572 6661 6365 732e 0a0a 4465 7363  nterfaces...Desc
│ │ │ │ +0025bae0: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ +0025baf0: 3d3d 3d0a 0a55 7365 206f 7065 6e49 6e4f  ===..Use openInO
│ │ │ │ +0025bb00: 7574 2066 2074 6f20 6163 6365 7074 2061  ut f to accept a
│ │ │ │ +0025bb10: 6e20 696e 636f 6d69 6e67 2063 6f6e 6e65  n incoming conne
│ │ │ │ +0025bb20: 6374 696f 6e20 6f6e 2074 6865 206c 6973  ction on the lis
│ │ │ │ +0025bb30: 7465 6e65 722c 2072 6574 7572 6e69 6e67  tener, returning
│ │ │ │ +0025bb40: 2061 0a6e 6577 2069 6e70 7574 206f 7574   a.new input out
│ │ │ │ +0025bb50: 7075 7420 6669 6c65 2074 6861 7420 7365  put file that se
│ │ │ │ +0025bb60: 7276 6573 2061 7320 7468 6520 636f 6e6e  rves as the conn
│ │ │ │ +0025bb70: 6563 7469 6f6e 2e20 2054 6865 2066 756e  ection.  The fun
│ │ │ │ +0025bb80: 6374 696f 6e20 2a6e 6f74 650a 6973 5265  ction *note.isRe
│ │ │ │ +0025bb90: 6164 793a 2069 7352 6561 6479 5f6c 7046  ady: isReady_lpF
│ │ │ │ +0025bba0: 696c 655f 7270 2c20 6361 6e20 6265 2075  ile_rp, can be u
│ │ │ │ +0025bbb0: 7365 6420 746f 2064 6574 6572 6d69 6e65  sed to determine
│ │ │ │ +0025bbc0: 2077 6865 7468 6572 2061 6e20 696e 636f   whether an inco
│ │ │ │ +0025bbd0: 6d69 6e67 0a63 6f6e 6e65 6374 696f 6e20  ming.connection 
│ │ │ │ +0025bbe0: 6861 7320 6172 7269 7665 642c 2077 6974  has arrived, wit
│ │ │ │ +0025bbf0: 686f 7574 2062 6c6f 636b 696e 672e 0a0a  hout blocking...
│ │ │ │ +0025bc00: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d  See also.=======
│ │ │ │ +0025bc10: 3d0a 0a20 202a 202a 6e6f 7465 206f 7065  =..  * *note ope
│ │ │ │ +0025bc20: 6e49 6e4f 7574 3a20 6f70 656e 496e 4f75  nInOut: openInOu
│ │ │ │ +0025bc30: 742c 202d 2d20 6f70 656e 2061 6e20 696e  t, -- open an in
│ │ │ │ +0025bc40: 7075 7420 6f75 7470 7574 2066 696c 650a  put output file.
│ │ │ │ +0025bc50: 2020 2a20 2a6e 6f74 6520 6973 5265 6164    * *note isRead
│ │ │ │ +0025bc60: 793a 2069 7352 6561 6479 5f6c 7046 696c  y: isReady_lpFil
│ │ │ │ +0025bc70: 655f 7270 2c20 2d2d 2077 6865 7468 6572  e_rp, -- whether
│ │ │ │ +0025bc80: 2061 2066 696c 6520 6861 7320 6461 7461   a file has data
│ │ │ │ +0025bc90: 2061 7661 696c 6162 6c65 2066 6f72 0a20   available for. 
│ │ │ │ +0025bca0: 2020 2072 6561 6469 6e67 0a0a 5761 7973     reading..Ways
│ │ │ │ +0025bcb0: 2074 6f20 7573 6520 7468 6973 206d 6574   to use this met
│ │ │ │ +0025bcc0: 686f 643a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  hod:.===========
│ │ │ │ +0025bcd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ +0025bce0: 202a 202a 6e6f 7465 206f 7065 6e4c 6973   * *note openLis
│ │ │ │ +0025bcf0: 7465 6e65 7228 5374 7269 6e67 293a 206f  tener(String): o
│ │ │ │ +0025bd00: 7065 6e4c 6973 7465 6e65 725f 6c70 5374  penListener_lpSt
│ │ │ │ +0025bd10: 7269 6e67 5f72 702c 202d 2d20 6f70 656e  ring_rp, -- open
│ │ │ │ +0025bd20: 2061 2070 6f72 7420 666f 720a 2020 2020   a port for.    
│ │ │ │ +0025bd30: 6c69 7374 656e 696e 670a 2d2d 2d2d 2d2d  listening.------
│ │ │ │  0025bd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0025bd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0025bd60: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263  -----..The sourc
│ │ │ │ -0025bd70: 6520 6f66 2074 6869 7320 646f 6375 6d65  e of this docume
│ │ │ │ -0025bd80: 6e74 2069 7320 696e 0a2f 6275 696c 642f  nt is in./build/
│ │ │ │ -0025bd90: 7265 7072 6f64 7563 6962 6c65 2d70 6174  reproducible-pat
│ │ │ │ -0025bda0: 682f 6d61 6361 756c 6179 322d 312e 3235  h/macaulay2-1.25
│ │ │ │ -0025bdb0: 2e31 312b 6473 2f4d 322f 4d61 6361 756c  .11+ds/M2/Macaul
│ │ │ │ -0025bdc0: 6179 322f 7061 636b 6167 6573 2f0a 4d61  ay2/packages/.Ma
│ │ │ │ -0025bdd0: 6361 756c 6179 3244 6f63 2f6f 765f 7379  caulay2Doc/ov_sy
│ │ │ │ -0025bde0: 7374 656d 2e6d 323a 3432 323a 302e 0a1f  stem.m2:422:0...
│ │ │ │ -0025bdf0: 0a46 696c 653a 204d 6163 6175 6c61 7932  .File: Macaulay2
│ │ │ │ -0025be00: 446f 632e 696e 666f 2c20 4e6f 6465 3a20  Doc.info, Node: 
│ │ │ │ -0025be10: 6f70 656e 496e 5f6c 7053 7472 696e 675f  openIn_lpString_
│ │ │ │ -0025be20: 7270 2c20 4e65 7874 3a20 6f70 656e 496e  rp, Next: openIn
│ │ │ │ -0025be30: 4f75 742c 2050 7265 763a 206f 7065 6e4c  Out, Prev: openL
│ │ │ │ -0025be40: 6973 7465 6e65 725f 6c70 5374 7269 6e67  istener_lpString
│ │ │ │ -0025be50: 5f72 702c 2055 703a 2075 7369 6e67 2073  _rp, Up: using s
│ │ │ │ -0025be60: 6f63 6b65 7473 0a0a 6f70 656e 496e 2853  ockets..openIn(S
│ │ │ │ -0025be70: 7472 696e 6729 202d 2d20 6f70 656e 2061  tring) -- open a
│ │ │ │ -0025be80: 6e20 696e 7075 7420 6669 6c65 0a2a 2a2a  n input file.***
│ │ │ │ -0025be90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0025bea0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0025beb0: 2a0a 0a20 202a 2046 756e 6374 696f 6e3a  *..  * Function:
│ │ │ │ -0025bec0: 202a 6e6f 7465 206f 7065 6e49 6e3a 206f   *note openIn: o
│ │ │ │ -0025bed0: 7065 6e49 6e5f 6c70 5374 7269 6e67 5f72  penIn_lpString_r
│ │ │ │ -0025bee0: 702c 0a20 202a 2055 7361 6765 3a20 0a20  p,.  * Usage: . 
│ │ │ │ -0025bef0: 2020 2020 2020 206f 7065 6e49 6e20 666e         openIn fn
│ │ │ │ -0025bf00: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ -0025bf10: 2020 202a 2066 6e2c 2061 202a 6e6f 7465     * fn, a *note
│ │ │ │ -0025bf20: 2073 7472 696e 673a 2053 7472 696e 672c   string: String,
│ │ │ │ -0025bf30: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ -0025bf40: 2020 2020 2a20 6120 2a6e 6f74 6520 6669      * a *note fi
│ │ │ │ -0025bf50: 6c65 3a20 4669 6c65 2c2c 2061 6e20 6f70  le: File,, an op
│ │ │ │ -0025bf60: 656e 2069 6e70 7574 2066 696c 6520 7768  en input file wh
│ │ │ │ -0025bf70: 6f73 6520 6669 6c65 6e61 6d65 2069 7320  ose filename is 
│ │ │ │ -0025bf80: 666e 2e20 4669 6c65 6e61 6d65 730a 2020  fn. Filenames.  
│ │ │ │ -0025bf90: 2020 2020 2020 7374 6172 7469 6e67 2077        starting w
│ │ │ │ -0025bfa0: 6974 6820 2120 6f72 2077 6974 6820 2420  ith ! or with $ 
│ │ │ │ -0025bfb0: 6172 6520 7472 6561 7465 6420 7370 6563  are treated spec
│ │ │ │ -0025bfc0: 6961 6c6c 792c 2073 6565 202a 6e6f 7465  ially, see *note
│ │ │ │ -0025bfd0: 206f 7065 6e49 6e4f 7574 3a0a 2020 2020   openInOut:.    
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│ │ │ │ +0025ca10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ +0025d220: 6e64 6172 6420 6f75 7470 7574 206f 6620  ndard output of 
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│ │ │ │ +0025d290: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f  =======..  * *no
│ │ │ │ +0025d2a0: 7465 206f 7065 6e49 6e3a 206f 7065 6e49  te openIn: openI
│ │ │ │ +0025d2b0: 6e5f 6c70 5374 7269 6e67 5f72 702c 202d  n_lpString_rp, -
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│ │ │ │ +0025d2e0: 7065 6e4f 7574 3a20 6f70 656e 4f75 745f  penOut: openOut_
│ │ │ │ +0025d2f0: 6c70 5374 7269 6e67 5f72 702c 202d 2d20  lpString_rp, -- 
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│ │ │ │ +0025d320: 656e 4c69 7374 656e 6572 3a20 6f70 656e  enListener: open
│ │ │ │ +0025d330: 4c69 7374 656e 6572 5f6c 7053 7472 696e  Listener_lpStrin
│ │ │ │ +0025d340: 675f 7270 2c20 2d2d 206f 7065 6e20 6120  g_rp, -- open a 
│ │ │ │ +0025d350: 706f 7274 2066 6f72 206c 6973 7465 6e69  port for listeni
│ │ │ │ +0025d360: 6e67 0a0a 5761 7973 2074 6f20 7573 6520  ng..Ways to use 
│ │ │ │ +0025d370: 6f70 656e 496e 4f75 743a 0a3d 3d3d 3d3d  openInOut:.=====
│ │ │ │ +0025d380: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0025d390: 3d0a 0a20 202a 2022 6f70 656e 496e 4f75  =..  * "openInOu
│ │ │ │ +0025d3a0: 7428 4669 6c65 2922 0a20 202a 2022 6f70  t(File)".  * "op
│ │ │ │ +0025d3b0: 656e 496e 4f75 7428 5374 7269 6e67 2922  enInOut(String)"
│ │ │ │ +0025d3c0: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
│ │ │ │ +0025d3d0: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
│ │ │ │ +0025d3e0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
│ │ │ │ +0025d3f0: 6563 7420 2a6e 6f74 6520 6f70 656e 496e  ect *note openIn
│ │ │ │ +0025d400: 4f75 743a 206f 7065 6e49 6e4f 7574 2c20  Out: openInOut, 
│ │ │ │ +0025d410: 6973 2061 202a 6e6f 7465 2063 6f6d 7069  is a *note compi
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│ │ │ │ +0025d440: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..--------------
│ │ │ │  0025d450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0025d460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
│ │ │ │ -0025d470: 6865 2073 6f75 7263 6520 6f66 2074 6869  he source of thi
│ │ │ │ -0025d480: 7320 646f 6375 6d65 6e74 2069 7320 696e  s document is in
│ │ │ │ -0025d490: 0a2f 6275 696c 642f 7265 7072 6f64 7563  ./build/reproduc
│ │ │ │ -0025d4a0: 6962 6c65 2d70 6174 682f 6d61 6361 756c  ible-path/macaul
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│ │ │ │ -0025d4c0: 322f 4d61 6361 756c 6179 322f 7061 636b  2/Macaulay2/pack
│ │ │ │ -0025d4d0: 6167 6573 2f0a 4d61 6361 756c 6179 3244  ages/.Macaulay2D
│ │ │ │ -0025d4e0: 6f63 2f6f 765f 7379 7374 656d 2e6d 323a  oc/ov_system.m2:
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│ │ │ │ -0025d510: 2c20 4e6f 6465 3a20 6f70 656e 4f75 745f  , Node: openOut_
│ │ │ │ -0025d520: 6c70 5374 7269 6e67 5f72 702c 204e 6578  lpString_rp, Nex
│ │ │ │ -0025d530: 743a 206f 7065 6e4f 7574 4170 7065 6e64  t: openOutAppend
│ │ │ │ -0025d540: 5f6c 7053 7472 696e 675f 7270 2c20 5072  _lpString_rp, Pr
│ │ │ │ -0025d550: 6576 3a20 6f70 656e 496e 4f75 742c 2055  ev: openInOut, U
│ │ │ │ -0025d560: 703a 2075 7369 6e67 2073 6f63 6b65 7473  p: using sockets
│ │ │ │ -0025d570: 0a0a 6f70 656e 4f75 7428 5374 7269 6e67  ..openOut(String
│ │ │ │ -0025d580: 2920 2d2d 206f 7065 6e20 616e 206f 7574  ) -- open an out
│ │ │ │ -0025d590: 7075 7420 6669 6c65 0a2a 2a2a 2a2a 2a2a  put file.*******
│ │ │ │ -0025d5a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0025d5b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ -0025d5c0: 0a20 202a 2046 756e 6374 696f 6e3a 202a  .  * Function: *
│ │ │ │ -0025d5d0: 6e6f 7465 206f 7065 6e4f 7574 3a20 6f70  note openOut: op
│ │ │ │ -0025d5e0: 656e 4f75 745f 6c70 5374 7269 6e67 5f72  enOut_lpString_r
│ │ │ │ -0025d5f0: 702c 0a20 202a 2055 7361 6765 3a20 0a20  p,.  * Usage: . 
│ │ │ │ -0025d600: 2020 2020 2020 206f 7065 6e4f 7574 2066         openOut f
│ │ │ │ -0025d610: 6e0a 2020 2a20 496e 7075 7473 3a0a 2020  n.  * Inputs:.  
│ │ │ │ -0025d620: 2020 2020 2a20 666e 2c20 6120 2a6e 6f74      * fn, a *not
│ │ │ │ -0025d630: 6520 7374 7269 6e67 3a20 5374 7269 6e67  e string: String
│ │ │ │ -0025d640: 2c0a 2020 2a20 4f75 7470 7574 733a 0a20  ,.  * Outputs:. 
│ │ │ │ -0025d650: 2020 2020 202a 2061 202a 6e6f 7465 2066       * a *note f
│ │ │ │ -0025d660: 696c 653a 2046 696c 652c 2c20 616e 206f  ile: File,, an o
│ │ │ │ -0025d670: 7065 6e20 6f75 7470 7574 2066 696c 6520  pen output file 
│ │ │ │ -0025d680: 7768 6f73 6520 6669 6c65 6e61 6d65 2069  whose filename i
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│ │ │ │ -0025d6a0: 6c65 6e61 6d65 7320 7374 6172 7469 6e67  lenames starting
│ │ │ │ -0025d6b0: 2077 6974 6820 2120 6f72 2077 6974 6820   with ! or with 
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│ │ │ │ -0025d6d0: 6563 6961 6c6c 792c 2073 6565 202a 6e6f  ecially, see *no
│ │ │ │ -0025d6e0: 7465 0a20 2020 2020 2020 206f 7065 6e49  te.        openI
│ │ │ │ -0025d6f0: 6e4f 7574 3a20 6f70 656e 496e 4f75 742c  nOut: openInOut,
│ │ │ │ -0025d700: 2e0a 0a44 6573 6372 6970 7469 6f6e 0a3d  ...Description.=
│ │ │ │ -0025d710: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2b2d 2d2d  ==========..+---
│ │ │ │ -0025d720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0025d730: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
│ │ │ │ -0025d740: 2067 203d 206f 7065 6e4f 7574 2022 7465   g = openOut "te
│ │ │ │ -0025d750: 7374 2d66 696c 6522 7c0a 7c20 2020 2020  st-file"|.|     
│ │ │ │ -0025d760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0025d770: 2020 2020 2020 207c 0a7c 6f31 203d 2074         |.|o1 = t
│ │ │ │ -0025d780: 6573 742d 6669 6c65 2020 2020 2020 2020  est-file        
│ │ │ │ -0025d790: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0025d7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0025d7b0: 2020 2020 207c 0a7c 6f31 203a 2046 696c       |.|o1 : Fil
│ │ │ │ -0025d7c0: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ -0025d7d0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -0025d7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0025d7f0: 2d2d 2d2b 0a7c 6932 203a 2067 203c 3c20  ---+.|i2 : g << 
│ │ │ │ -0025d800: 2268 6920 7468 6572 6522 2020 2020 2020  "hi there"      
│ │ │ │ -0025d810: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0025d820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0025d830: 207c 0a7c 6f32 203d 2074 6573 742d 6669   |.|o2 = test-fi
│ │ │ │ -0025d840: 6c65 2020 2020 2020 2020 2020 2020 2020  le              
│ │ │ │ -0025d850: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0025d860: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0025d870: 0a7c 6f32 203a 2046 696c 6520 2020 2020  .|o2 : File     
│ │ │ │ -0025d880: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0025d890: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -0025d8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -0025d8b0: 6933 203a 2067 203c 3c20 636c 6f73 6520  i3 : g << close 
│ │ │ │ -0025d8c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0025d8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0025d8e0: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ -0025d8f0: 203d 2074 6573 742d 6669 6c65 2020 2020   = test-file    
│ │ │ │ -0025d900: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0025d910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0025d920: 2020 2020 2020 2020 207c 0a7c 6f33 203a           |.|o3 :
│ │ │ │ -0025d930: 2046 696c 6520 2020 2020 2020 2020 2020   File           
│ │ │ │ -0025d940: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -0025d950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0025d960: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2067  -------+.|i4 : g
│ │ │ │ -0025d970: 6574 2022 7465 7374 2d66 696c 6522 2020  et "test-file"  
│ │ │ │ -0025d980: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0025d990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0025d9a0: 2020 2020 207c 0a7c 6f34 203d 2068 6920       |.|o4 = hi 
│ │ │ │ -0025d9b0: 7468 6572 6520 2020 2020 2020 2020 2020  there           
│ │ │ │ -0025d9c0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -0025d9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0025d9e0: 2d2d 2d2b 0a7c 6935 203a 2072 656d 6f76  ---+.|i5 : remov
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│ │ │ │ -0025da00: 2220 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  " |.+-----------
│ │ │ │ -0025da10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0025da20: 2d2b 0a0a 4120 6669 6c65 6e61 6d65 2073  -+..A filename s
│ │ │ │ -0025da30: 7461 7274 696e 6720 7769 7468 207e 2f20  tarting with ~/ 
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│ │ │ │ -0025da80: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  also.========.. 
│ │ │ │ -0025da90: 202a 202a 6e6f 7465 206f 7065 6e49 6e3a   * *note openIn:
│ │ │ │ -0025daa0: 206f 7065 6e49 6e5f 6c70 5374 7269 6e67   openIn_lpString
│ │ │ │ -0025dab0: 5f72 702c 202d 2d20 6f70 656e 2061 6e20  _rp, -- open an 
│ │ │ │ -0025dac0: 696e 7075 7420 6669 6c65 0a20 202a 202a  input file.  * *
│ │ │ │ -0025dad0: 6e6f 7465 206f 7065 6e49 6e4f 7574 3a20  note openInOut: 
│ │ │ │ -0025dae0: 6f70 656e 496e 4f75 742c 202d 2d20 6f70  openInOut, -- op
│ │ │ │ -0025daf0: 656e 2061 6e20 696e 7075 7420 6f75 7470  en an input outp
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│ │ │ │ -0025db20: 206f 7065 6e4f 7574 4170 7065 6e64 5f6c   openOutAppend_l
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│ │ │ │ -0025db60: 6469 6e67 0a20 202a 202a 6e6f 7465 2067  ding.  * *note g
│ │ │ │ -0025db70: 6574 3a20 6765 742c 202d 2d20 6765 7420  et: get, -- get 
│ │ │ │ -0025db80: 7468 6520 636f 6e74 656e 7473 206f 6620  the contents of 
│ │ │ │ -0025db90: 6120 6669 6c65 0a20 202a 202a 6e6f 7465  a file.  * *note
│ │ │ │ -0025dba0: 2072 656d 6f76 6546 696c 653a 2072 656d   removeFile: rem
│ │ │ │ -0025dbb0: 6f76 6546 696c 652c 202d 2d20 7265 6d6f  oveFile, -- remo
│ │ │ │ -0025dbc0: 7665 2061 2066 696c 650a 2020 2a20 2a6e  ve a file.  * *n
│ │ │ │ -0025dbd0: 6f74 6520 636c 6f73 653a 2063 6c6f 7365  ote close: close
│ │ │ │ -0025dbe0: 2c20 2d2d 2063 6c6f 7365 2061 2066 696c  , -- close a fil
│ │ │ │ -0025dbf0: 650a 2020 2a20 2a6e 6f74 6520 4669 6c65  e.  * *note File
│ │ │ │ -0025dc00: 203c 3c20 5468 696e 673a 2070 7269 6e74   << Thing: print
│ │ │ │ -0025dc10: 696e 6720 746f 2061 2066 696c 652c 202d  ing to a file, -
│ │ │ │ -0025dc20: 2d20 7072 696e 7420 746f 2061 2066 696c  - print to a fil
│ │ │ │ -0025dc30: 650a 0a57 6179 7320 746f 2075 7365 2074  e..Ways to use t
│ │ │ │ -0025dc40: 6869 7320 6d65 7468 6f64 3a0a 3d3d 3d3d  his method:.====
│ │ │ │ -0025dc50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0025dc60: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ -0025dc70: 6f70 656e 4f75 7428 5374 7269 6e67 293a  openOut(String):
│ │ │ │ -0025dc80: 206f 7065 6e4f 7574 5f6c 7053 7472 696e   openOut_lpStrin
│ │ │ │ -0025dc90: 675f 7270 2c20 2d2d 206f 7065 6e20 616e  g_rp, -- open an
│ │ │ │ -0025dca0: 206f 7574 7075 7420 6669 6c65 0a2d 2d2d   output file.---
│ │ │ │ -0025dcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0025dcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0025dcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0025d460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0025d470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0025d480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0025d490: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66  -..The source of
│ │ │ │ +0025d4a0: 2074 6869 7320 646f 6375 6d65 6e74 2069   this document i
│ │ │ │ +0025d4b0: 7320 696e 0a2f 6275 696c 642f 7265 7072  s in./build/repr
│ │ │ │ +0025d4c0: 6f64 7563 6962 6c65 2d70 6174 682f 6d61  oducible-path/ma
│ │ │ │ +0025d4d0: 6361 756c 6179 322d 312e 3235 2e31 312b  caulay2-1.25.11+
│ │ │ │ +0025d4e0: 6473 2f4d 322f 4d61 6361 756c 6179 322f  ds/M2/Macaulay2/
│ │ │ │ +0025d4f0: 7061 636b 6167 6573 2f0a 4d61 6361 756c  packages/.Macaul
│ │ │ │ +0025d500: 6179 3244 6f63 2f6f 765f 7379 7374 656d  ay2Doc/ov_system
│ │ │ │ +0025d510: 2e6d 323a 3533 373a 302e 0a1f 0a46 696c  .m2:537:0....Fil
│ │ │ │ +0025d520: 653a 204d 6163 6175 6c61 7932 446f 632e  e: Macaulay2Doc.
│ │ │ │ +0025d530: 696e 666f 2c20 4e6f 6465 3a20 6f70 656e  info, Node: open
│ │ │ │ +0025d540: 4f75 745f 6c70 5374 7269 6e67 5f72 702c  Out_lpString_rp,
│ │ │ │ +0025d550: 204e 6578 743a 206f 7065 6e4f 7574 4170   Next: openOutAp
│ │ │ │ +0025d560: 7065 6e64 5f6c 7053 7472 696e 675f 7270  pend_lpString_rp
│ │ │ │ +0025d570: 2c20 5072 6576 3a20 6f70 656e 496e 4f75  , Prev: openInOu
│ │ │ │ +0025d580: 742c 2055 703a 2075 7369 6e67 2073 6f63  t, Up: using soc
│ │ │ │ +0025d590: 6b65 7473 0a0a 6f70 656e 4f75 7428 5374  kets..openOut(St
│ │ │ │ +0025d5a0: 7269 6e67 2920 2d2d 206f 7065 6e20 616e  ring) -- open an
│ │ │ │ +0025d5b0: 206f 7574 7075 7420 6669 6c65 0a2a 2a2a   output file.***
│ │ │ │ +0025d5c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0025d5d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0025d5e0: 2a2a 2a0a 0a20 202a 2046 756e 6374 696f  ***..  * Functio
│ │ │ │ +0025d5f0: 6e3a 202a 6e6f 7465 206f 7065 6e4f 7574  n: *note openOut
│ │ │ │ +0025d600: 3a20 6f70 656e 4f75 745f 6c70 5374 7269  : openOut_lpStri
│ │ │ │ +0025d610: 6e67 5f72 702c 0a20 202a 2055 7361 6765  ng_rp,.  * Usage
│ │ │ │ +0025d620: 3a20 0a20 2020 2020 2020 206f 7065 6e4f  : .        openO
│ │ │ │ +0025d630: 7574 2066 6e0a 2020 2a20 496e 7075 7473  ut fn.  * Inputs
│ │ │ │ +0025d640: 3a0a 2020 2020 2020 2a20 666e 2c20 6120  :.      * fn, a 
│ │ │ │ +0025d650: 2a6e 6f74 6520 7374 7269 6e67 3a20 5374  *note string: St
│ │ │ │ +0025d660: 7269 6e67 2c0a 2020 2a20 4f75 7470 7574  ring,.  * Output
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│ │ │ │ +0025d780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0025d790: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
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│ │ │ │ +0025d8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0025d900: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0025d910: 0a7c 6f33 203d 2074 6573 742d 6669 6c65  .|o3 = test-file
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│ │ │ │ +0025d930: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
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│ │ │ │ +0025d9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0025d9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0025dac0: 6e49 6e3a 206f 7065 6e49 6e5f 6c70 5374  nIn: openIn_lpSt
│ │ │ │ +0025dad0: 7269 6e67 5f72 702c 202d 2d20 6f70 656e  ring_rp, -- open
│ │ │ │ +0025dae0: 2061 6e20 696e 7075 7420 6669 6c65 0a20   an input file. 
│ │ │ │ +0025daf0: 202a 202a 6e6f 7465 206f 7065 6e49 6e4f   * *note openInO
│ │ │ │ +0025db00: 7574 3a20 6f70 656e 496e 4f75 742c 202d  ut: openInOut, -
│ │ │ │ +0025db10: 2d20 6f70 656e 2061 6e20 696e 7075 7420  - open an input 
│ │ │ │ +0025db20: 6f75 7470 7574 2066 696c 650a 2020 2a20  output file.  * 
│ │ │ │ +0025db30: 2a6e 6f74 6520 6f70 656e 4f75 7441 7070  *note openOutApp
│ │ │ │ +0025db40: 656e 643a 206f 7065 6e4f 7574 4170 7065  end: openOutAppe
│ │ │ │ +0025db50: 6e64 5f6c 7053 7472 696e 675f 7270 2c20  nd_lpString_rp, 
│ │ │ │ +0025db60: 2d2d 206f 7065 6e20 616e 206f 7574 7075  -- open an outpu
│ │ │ │ +0025db70: 7420 6669 6c65 2066 6f72 0a20 2020 2061  t file for.    a
│ │ │ │ +0025db80: 7070 656e 6469 6e67 0a20 202a 202a 6e6f  ppending.  * *no
│ │ │ │ +0025db90: 7465 2067 6574 3a20 6765 742c 202d 2d20  te get: get, -- 
│ │ │ │ +0025dba0: 6765 7420 7468 6520 636f 6e74 656e 7473  get the contents
│ │ │ │ +0025dbb0: 206f 6620 6120 6669 6c65 0a20 202a 202a   of a file.  * *
│ │ │ │ +0025dbc0: 6e6f 7465 2072 656d 6f76 6546 696c 653a  note removeFile:
│ │ │ │ +0025dbd0: 2072 656d 6f76 6546 696c 652c 202d 2d20   removeFile, -- 
│ │ │ │ +0025dbe0: 7265 6d6f 7665 2061 2066 696c 650a 2020  remove a file.  
│ │ │ │ +0025dbf0: 2a20 2a6e 6f74 6520 636c 6f73 653a 2063  * *note close: c
│ │ │ │ +0025dc00: 6c6f 7365 2c20 2d2d 2063 6c6f 7365 2061  lose, -- close a
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│ │ │ │ +0025dc20: 4669 6c65 203c 3c20 5468 696e 673a 2070  File << Thing: p
│ │ │ │ +0025dc30: 7269 6e74 696e 6720 746f 2061 2066 696c  rinting to a fil
│ │ │ │ +0025dc40: 652c 202d 2d20 7072 696e 7420 746f 2061  e, -- print to a
│ │ │ │ +0025dc50: 2066 696c 650a 0a57 6179 7320 746f 2075   file..Ways to u
│ │ │ │ +0025dc60: 7365 2074 6869 7320 6d65 7468 6f64 3a0a  se this method:.
│ │ │ │ +0025dc70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0025dc80: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ +0025dc90: 6f74 6520 6f70 656e 4f75 7428 5374 7269  ote openOut(Stri
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│ │ │ │ +0025dcb0: 7472 696e 675f 7270 2c20 2d2d 206f 7065  tring_rp, -- ope
│ │ │ │ +0025dcc0: 6e20 616e 206f 7574 7075 7420 6669 6c65  n an output file
│ │ │ │ +0025dcd0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .---------------
│ │ │ │  0025dce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0025dcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468  ------------..Th
│ │ │ │ -0025dd00: 6520 736f 7572 6365 206f 6620 7468 6973  e source of this
│ │ │ │ -0025dd10: 2064 6f63 756d 656e 7420 6973 2069 6e0a   document is in.
│ │ │ │ -0025dd20: 2f62 7569 6c64 2f72 6570 726f 6475 6369  /build/reproduci
│ │ │ │ -0025dd30: 626c 652d 7061 7468 2f6d 6163 6175 6c61  ble-path/macaula
│ │ │ │ -0025dd40: 7932 2d31 2e32 352e 3131 2b64 732f 4d32  y2-1.25.11+ds/M2
│ │ │ │ -0025dd50: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61  /Macaulay2/packa
│ │ │ │ -0025dd60: 6765 732f 0a4d 6163 6175 6c61 7932 446f  ges/.Macaulay2Do
│ │ │ │ -0025dd70: 632f 6f76 5f73 7973 7465 6d2e 6d32 3a34  c/ov_system.m2:4
│ │ │ │ -0025dd80: 3633 3a30 2e0a 1f0a 4669 6c65 3a20 4d61  63:0....File: Ma
│ │ │ │ -0025dd90: 6361 756c 6179 3244 6f63 2e69 6e66 6f2c  caulay2Doc.info,
│ │ │ │ -0025dda0: 204e 6f64 653a 206f 7065 6e4f 7574 4170   Node: openOutAp
│ │ │ │ -0025ddb0: 7065 6e64 5f6c 7053 7472 696e 675f 7270  pend_lpString_rp
│ │ │ │ -0025ddc0: 2c20 4e65 7874 3a20 656e 646c 2c20 5072  , Next: endl, Pr
│ │ │ │ -0025ddd0: 6576 3a20 6f70 656e 4f75 745f 6c70 5374  ev: openOut_lpSt
│ │ │ │ -0025dde0: 7269 6e67 5f72 702c 2055 703a 2075 7369  ring_rp, Up: usi
│ │ │ │ -0025ddf0: 6e67 2073 6f63 6b65 7473 0a0a 6f70 656e  ng sockets..open
│ │ │ │ -0025de00: 4f75 7441 7070 656e 6428 5374 7269 6e67  OutAppend(String
│ │ │ │ -0025de10: 2920 2d2d 206f 7065 6e20 616e 206f 7574  ) -- open an out
│ │ │ │ -0025de20: 7075 7420 6669 6c65 2066 6f72 2061 7070  put file for app
│ │ │ │ -0025de30: 656e 6469 6e67 0a2a 2a2a 2a2a 2a2a 2a2a  ending.*********
│ │ │ │ -0025de40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0025de50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0025dcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0025dd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0025dd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0025dd20: 0a0a 5468 6520 736f 7572 6365 206f 6620  ..The source of 
│ │ │ │ +0025dd30: 7468 6973 2064 6f63 756d 656e 7420 6973  this document is
│ │ │ │ +0025dd40: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f   in./build/repro
│ │ │ │ +0025dd50: 6475 6369 626c 652d 7061 7468 2f6d 6163  ducible-path/mac
│ │ │ │ +0025dd60: 6175 6c61 7932 2d31 2e32 352e 3131 2b64  aulay2-1.25.11+d
│ │ │ │ +0025dd70: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70  s/M2/Macaulay2/p
│ │ │ │ +0025dd80: 6163 6b61 6765 732f 0a4d 6163 6175 6c61  ackages/.Macaula
│ │ │ │ +0025dd90: 7932 446f 632f 6f76 5f73 7973 7465 6d2e  y2Doc/ov_system.
│ │ │ │ +0025dda0: 6d32 3a34 3633 3a30 2e0a 1f0a 4669 6c65  m2:463:0....File
│ │ │ │ +0025ddb0: 3a20 4d61 6361 756c 6179 3244 6f63 2e69  : Macaulay2Doc.i
│ │ │ │ +0025ddc0: 6e66 6f2c 204e 6f64 653a 206f 7065 6e4f  nfo, Node: openO
│ │ │ │ +0025ddd0: 7574 4170 7065 6e64 5f6c 7053 7472 696e  utAppend_lpStrin
│ │ │ │ +0025dde0: 675f 7270 2c20 4e65 7874 3a20 656e 646c  g_rp, Next: endl
│ │ │ │ +0025ddf0: 2c20 5072 6576 3a20 6f70 656e 4f75 745f  , Prev: openOut_
│ │ │ │ +0025de00: 6c70 5374 7269 6e67 5f72 702c 2055 703a  lpString_rp, Up:
│ │ │ │ +0025de10: 2075 7369 6e67 2073 6f63 6b65 7473 0a0a   using sockets..
│ │ │ │ +0025de20: 6f70 656e 4f75 7441 7070 656e 6428 5374  openOutAppend(St
│ │ │ │ +0025de30: 7269 6e67 2920 2d2d 206f 7065 6e20 616e  ring) -- open an
│ │ │ │ +0025de40: 206f 7574 7075 7420 6669 6c65 2066 6f72   output file for
│ │ │ │ +0025de50: 2061 7070 656e 6469 6e67 0a2a 2a2a 2a2a   appending.*****
│ │ │ │  0025de60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0025de70: 2a0a 0a20 202a 2046 756e 6374 696f 6e3a  *..  * Function:
│ │ │ │ -0025de80: 202a 6e6f 7465 206f 7065 6e4f 7574 4170   *note openOutAp
│ │ │ │ -0025de90: 7065 6e64 3a20 6f70 656e 4f75 7441 7070  pend: openOutApp
│ │ │ │ -0025dea0: 656e 645f 6c70 5374 7269 6e67 5f72 702c  end_lpString_rp,
│ │ │ │ -0025deb0: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
│ │ │ │ -0025dec0: 2020 2020 206f 7065 6e4f 7574 4170 7065       openOutAppe
│ │ │ │ -0025ded0: 6e64 2066 6e0a 2020 2a20 496e 7075 7473  nd fn.  * Inputs
│ │ │ │ -0025dee0: 3a0a 2020 2020 2020 2a20 666e 2c20 6120  :.      * fn, a 
│ │ │ │ -0025def0: 2a6e 6f74 6520 7374 7269 6e67 3a20 5374  *note string: St
│ │ │ │ -0025df00: 7269 6e67 2c0a 2020 2a20 4f75 7470 7574  ring,.  * Output
│ │ │ │ -0025df10: 733a 0a20 2020 2020 202a 2061 202a 6e6f  s:.      * a *no
│ │ │ │ -0025df20: 7465 2066 696c 653a 2046 696c 652c 2c20  te file: File,, 
│ │ │ │ -0025df30: 616e 206f 7065 6e20 6f75 7470 7574 2066  an open output f
│ │ │ │ -0025df40: 696c 6520 7768 6f73 6520 6669 6c65 6e61  ile whose filena
│ │ │ │ -0025df50: 6d65 2069 7320 666e 0a0a 4465 7363 7269  me is fn..Descri
│ │ │ │ -0025df60: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ -0025df70: 3d0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  =..+------------
│ │ │ │ -0025df80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0025df90: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
│ │ │ │ -0025dfa0: 2067 203d 206f 7065 6e4f 7574 2022 7465   g = openOut "te
│ │ │ │ -0025dfb0: 7374 2d66 696c 6522 2020 2020 2020 2020  st-file"        
│ │ │ │ -0025dfc0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0025dfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0025dfe0: 2020 2020 2020 2020 207c 0a7c 6f31 203d           |.|o1 =
│ │ │ │ -0025dff0: 2074 6573 742d 6669 6c65 2020 2020 2020   test-file      
│ │ │ │ -0025e000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0025e010: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0025de70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0025de80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0025de90: 2a2a 2a2a 2a0a 0a20 202a 2046 756e 6374  *****..  * Funct
│ │ │ │ +0025dea0: 696f 6e3a 202a 6e6f 7465 206f 7065 6e4f  ion: *note openO
│ │ │ │ +0025deb0: 7574 4170 7065 6e64 3a20 6f70 656e 4f75  utAppend: openOu
│ │ │ │ +0025dec0: 7441 7070 656e 645f 6c70 5374 7269 6e67  tAppend_lpString
│ │ │ │ +0025ded0: 5f72 702c 0a20 202a 2055 7361 6765 3a20  _rp,.  * Usage: 
│ │ │ │ +0025dee0: 0a20 2020 2020 2020 206f 7065 6e4f 7574  .        openOut
│ │ │ │ +0025def0: 4170 7065 6e64 2066 6e0a 2020 2a20 496e  Append fn.  * In
│ │ │ │ +0025df00: 7075 7473 3a0a 2020 2020 2020 2a20 666e  puts:.      * fn
│ │ │ │ +0025df10: 2c20 6120 2a6e 6f74 6520 7374 7269 6e67  , a *note string
│ │ │ │ +0025df20: 3a20 5374 7269 6e67 2c0a 2020 2a20 4f75  : String,.  * Ou
│ │ │ │ +0025df30: 7470 7574 733a 0a20 2020 2020 202a 2061  tputs:.      * a
│ │ │ │ +0025df40: 202a 6e6f 7465 2066 696c 653a 2046 696c   *note file: Fil
│ │ │ │ +0025df50: 652c 2c20 616e 206f 7065 6e20 6f75 7470  e,, an open outp
│ │ │ │ +0025df60: 7574 2066 696c 6520 7768 6f73 6520 6669  ut file whose fi
│ │ │ │ +0025df70: 6c65 6e61 6d65 2069 7320 666e 0a0a 4465  lename is fn..De
│ │ │ │ +0025df80: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ +0025df90: 3d3d 3d3d 3d0a 0a2b 2d2d 2d2d 2d2d 2d2d  =====..+--------
│ │ │ │ +0025dfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0025dfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0025dfc0: 6931 203a 2067 203d 206f 7065 6e4f 7574  i1 : g = openOut
│ │ │ │ +0025dfd0: 2022 7465 7374 2d66 696c 6522 2020 2020   "test-file"    
│ │ │ │ +0025dfe0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0025dff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0025e000: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0025e010: 6f31 203d 2074 6573 742d 6669 6c65 2020  o1 = test-file  
│ │ │ │  0025e020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0025e030: 2020 2020 2020 2020 207c 0a7c 6f31 203a           |.|o1 :
│ │ │ │ -0025e040: 2046 696c 6520 2020 2020 2020 2020 2020   File           
│ │ │ │ -0025e050: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0025e390: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0025e3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0025e410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0025e450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0025e460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0025e470: 2d2b 0a0a 4120 6669 6c65 6e61 6d65 2073  -+..A filename s
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│ │ │ │ -0025e690: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
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│ │ │ │ -0025e6c0: 206f 7065 6e4f 7574 4170 7065 6e64 5f6c   openOutAppend_l
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│ │ │ │ -0025e720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0025e300: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0025e310: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0025e360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0025e370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
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│ │ │ │ +0025e3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0025e420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0025e430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0025e620: 202a 6e6f 7465 2072 656d 6f76 6546 696c   *note removeFil
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│ │ │ │ +0025e680: 7072 696e 7420 746f 2061 2066 696c 650a  print to a file.
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│ │ │ │ +0025e6b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │  0025e730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0025e740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0025e870: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a44 6573  ***********..Des
│ │ │ │ -0025e880: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ -0025e890: 3d3d 3d3d 0a0a 6620 3c3c 2065 6e64 6c20  ====..f << endl 
│ │ │ │ -0025e8a0: 2d2d 2065 6e64 7320 7468 6520 6c69 6e65  -- ends the line
│ │ │ │ -0025e8b0: 2063 7572 7265 6e74 6c79 2062 6569 6e67   currently being
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│ │ │ │ -0025e8d0: 6669 6c65 2066 2e0a 0a0a 4974 2069 7320  file f....It is 
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│ │ │ │ -0025e900: 6720 7072 6163 7469 6365 2074 6f20 7573  g practice to us
│ │ │ │ -0025e910: 6520 656e 646c 2061 6c77 6179 732c 2066  e endl always, f
│ │ │ │ -0025e920: 6f72 0a77 7269 7469 6e67 206e 6577 6c69  or.writing newli
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│ │ │ │ -0025e940: 6565 202a 6e6f 7465 206e 6577 6c69 6e65  ee *note newline
│ │ │ │ -0025e950: 3a20 6e65 776c 696e 652c 2920 746f 2061  : newline,) to a
│ │ │ │ -0025e960: 2066 696c 6520 7769 6c6c 206e 6f74 0a74   file will not.t
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│ │ │ │ -0025e980: 636f 6e74 6169 6e69 6e67 206e 6574 7320  containing nets 
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│ │ │ │ -0025e9a0: 2077 696c 6c20 6e6f 7420 666c 7573 6820   will not flush 
│ │ │ │ -0025e9b0: 7468 6520 6f75 7470 7574 0a62 7566 6665  the output.buffe
│ │ │ │ -0025e9c0: 722e 0a0a 5761 7973 2074 6f20 7573 6520  r...Ways to use 
│ │ │ │ -0025e9d0: 656e 646c 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d  endl:.==========
│ │ │ │ -0025e9e0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 4d61  =======..  * "Ma
│ │ │ │ -0025e9f0: 6e69 7075 6c61 746f 7220 4461 7461 6261  nipulator Databa
│ │ │ │ -0025ea00: 7365 2220 2d2d 2073 6565 202a 6e6f 7465  se" -- see *note
│ │ │ │ -0025ea10: 204d 616e 6970 756c 6174 6f72 3a20 4d61   Manipulator: Ma
│ │ │ │ -0025ea20: 6e69 7075 6c61 746f 722c 202d 2d20 7468  nipulator, -- th
│ │ │ │ -0025ea30: 6520 636c 6173 730a 2020 2020 6f66 2061  e class.    of a
│ │ │ │ -0025ea40: 6c6c 2066 696c 6520 6d61 6e69 7075 6c61  ll file manipula
│ │ │ │ -0025ea50: 746f 7273 0a20 202a 2022 4d61 6e69 7075  tors.  * "Manipu
│ │ │ │ -0025ea60: 6c61 746f 7220 4669 6c65 2220 2d2d 2073  lator File" -- s
│ │ │ │ -0025ea70: 6565 202a 6e6f 7465 204d 616e 6970 756c  ee *note Manipul
│ │ │ │ -0025ea80: 6174 6f72 3a20 4d61 6e69 7075 6c61 746f  ator: Manipulato
│ │ │ │ -0025ea90: 722c 202d 2d20 7468 6520 636c 6173 7320  r, -- the class 
│ │ │ │ -0025eaa0: 6f66 0a20 2020 2061 6c6c 2066 696c 6520  of.    all file 
│ │ │ │ -0025eab0: 6d61 6e69 7075 6c61 746f 7273 0a20 202a  manipulators.  *
│ │ │ │ -0025eac0: 2022 4d61 6e69 7075 6c61 746f 7220 4e6f   "Manipulator No
│ │ │ │ -0025ead0: 7468 696e 6722 202d 2d20 7365 6520 2a6e  thing" -- see *n
│ │ │ │ -0025eae0: 6f74 6520 4d61 6e69 7075 6c61 746f 723a  ote Manipulator:
│ │ │ │ -0025eaf0: 204d 616e 6970 756c 6174 6f72 2c20 2d2d   Manipulator, --
│ │ │ │ -0025eb00: 2074 6865 2063 6c61 7373 0a20 2020 206f   the class.    o
│ │ │ │ -0025eb10: 6620 616c 6c20 6669 6c65 206d 616e 6970  f all file manip
│ │ │ │ -0025eb20: 756c 6174 6f72 730a 2020 2a20 226e 6577  ulators.  * "new
│ │ │ │ -0025eb30: 204d 616e 6970 756c 6174 6f72 2066 726f   Manipulator fro
│ │ │ │ -0025eb40: 6d20 4675 6e63 7469 6f6e 2220 2d2d 2073  m Function" -- s
│ │ │ │ -0025eb50: 6565 202a 6e6f 7465 204d 616e 6970 756c  ee *note Manipul
│ │ │ │ -0025eb60: 6174 6f72 3a20 4d61 6e69 7075 6c61 746f  ator: Manipulato
│ │ │ │ -0025eb70: 722c 202d 2d0a 2020 2020 7468 6520 636c  r, --.    the cl
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│ │ │ │ -0025eb90: 6d61 6e69 7075 6c61 746f 7273 0a20 202a  manipulators.  *
│ │ │ │ -0025eba0: 2022 6d65 7468 6f64 7328 4d61 6e69 7075   "methods(Manipu
│ │ │ │ -0025ebb0: 6c61 746f 7229 2220 2d2d 2073 6565 202a  lator)" -- see *
│ │ │ │ -0025ebc0: 6e6f 7465 206d 6574 686f 6473 3a20 6d65  note methods: me
│ │ │ │ -0025ebd0: 7468 6f64 732c 202d 2d20 6c69 7374 206d  thods, -- list m
│ │ │ │ -0025ebe0: 6574 686f 6473 0a20 202a 2022 4d61 6e69  ethods.  * "Mani
│ │ │ │ -0025ebf0: 7075 6c61 746f 7220 4e65 7446 696c 6522  pulator NetFile"
│ │ │ │ -0025ec00: 202d 2d20 7365 6520 2a6e 6f74 6520 4e65   -- see *note Ne
│ │ │ │ -0025ec10: 7446 696c 653a 204e 6574 4669 6c65 2c20  tFile: NetFile, 
│ │ │ │ -0025ec20: 2d2d 2074 6865 2063 6c61 7373 206f 6620  -- the class of 
│ │ │ │ -0025ec30: 616c 6c0a 2020 2020 6e65 7420 6669 6c65  all.    net file
│ │ │ │ -0025ec40: 730a 2020 2a20 224e 6574 4669 6c65 203c  s.  * "NetFile <
│ │ │ │ -0025ec50: 3c20 4d61 6e69 7075 6c61 746f 7222 202d  < Manipulator" -
│ │ │ │ -0025ec60: 2d20 7365 6520 2a6e 6f74 6520 4e65 7446  - see *note NetF
│ │ │ │ -0025ec70: 696c 653a 204e 6574 4669 6c65 2c20 2d2d  ile: NetFile, --
│ │ │ │ -0025ec80: 2074 6865 2063 6c61 7373 206f 6620 616c   the class of al
│ │ │ │ -0025ec90: 6c0a 2020 2020 6e65 7420 6669 6c65 730a  l.    net files.
│ │ │ │ -0025eca0: 2020 2a20 2246 696c 6520 3c3c 204d 616e    * "File << Man
│ │ │ │ -0025ecb0: 6970 756c 6174 6f72 2220 2d2d 2073 6565  ipulator" -- see
│ │ │ │ -0025ecc0: 202a 6e6f 7465 2070 7269 6e74 696e 6720   *note printing 
│ │ │ │ -0025ecd0: 746f 2061 2066 696c 653a 2070 7269 6e74  to a file: print
│ │ │ │ -0025ece0: 696e 6720 746f 2061 2066 696c 652c 0a20  ing to a file,. 
│ │ │ │ -0025ecf0: 2020 202d 2d20 7072 696e 7420 746f 2061     -- print to a
│ │ │ │ -0025ed00: 2066 696c 650a 2020 2a20 224e 6f74 6869   file.  * "Nothi
│ │ │ │ -0025ed10: 6e67 203c 3c20 4d61 6e69 7075 6c61 746f  ng << Manipulato
│ │ │ │ -0025ed20: 7222 202d 2d20 7365 6520 2a6e 6f74 6520  r" -- see *note 
│ │ │ │ -0025ed30: 7072 696e 7469 6e67 2074 6f20 6120 6669  printing to a fi
│ │ │ │ -0025ed40: 6c65 3a20 7072 696e 7469 6e67 2074 6f20  le: printing to 
│ │ │ │ -0025ed50: 610a 2020 2020 6669 6c65 2c20 2d2d 2070  a.    file, -- p
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│ │ │ │ -0025ed70: 466f 7220 7468 6520 7072 6f67 7261 6d6d  For the programm
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│ │ │ │ -0025ed90: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563  =====..The objec
│ │ │ │ -0025eda0: 7420 2a6e 6f74 6520 656e 646c 3a20 656e  t *note endl: en
│ │ │ │ -0025edb0: 646c 2c20 6973 2061 202a 6e6f 7465 206d  dl, is a *note m
│ │ │ │ -0025edc0: 616e 6970 756c 6174 6f72 3a20 4d61 6e69  anipulator: Mani
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│ │ │ │ -0025ede0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0025edf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0025e750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0025e760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0025e770: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f  --------..The so
│ │ │ │ +0025e780: 7572 6365 206f 6620 7468 6973 2064 6f63  urce of this doc
│ │ │ │ +0025e790: 756d 656e 7420 6973 2069 6e0a 2f62 7569  ument is in./bui
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│ │ │ │ +0025e7c0: 2e32 352e 3131 2b64 732f 4d32 2f4d 6163  .25.11+ds/M2/Mac
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│ │ │ │ +0025e7e0: 0a4d 6163 6175 6c61 7932 446f 632f 6f76  .Macaulay2Doc/ov
│ │ │ │ +0025e7f0: 5f73 7973 7465 6d2e 6d32 3a34 3832 3a30  _system.m2:482:0
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│ │ │ │ +0025e840: 4f75 7441 7070 656e 645f 6c70 5374 7269  OutAppend_lpStri
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│ │ │ │ +0025e880: 6c69 6e65 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a  line.***********
│ │ │ │ +0025e890: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ +0025e8a0: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d  .Description.===
│ │ │ │ +0025e8b0: 3d3d 3d3d 3d3d 3d3d 0a0a 6620 3c3c 2065  ========..f << e
│ │ │ │ +0025e8c0: 6e64 6c20 2d2d 2065 6e64 7320 7468 6520  ndl -- ends the 
│ │ │ │ +0025e8d0: 6c69 6e65 2063 7572 7265 6e74 6c79 2062  line currently b
│ │ │ │ +0025e8e0: 6569 6e67 2070 7574 206f 7574 2074 6f20  eing put out to 
│ │ │ │ +0025e8f0: 7468 6520 6669 6c65 2066 2e0a 0a0a 4974  the file f....It
│ │ │ │ +0025e900: 2069 7320 616e 2065 7373 656e 7469 616c   is an essential
│ │ │ │ +0025e910: 2070 6f72 7461 626c 6520 7072 6f67 7261   portable progra
│ │ │ │ +0025e920: 6d6d 696e 6720 7072 6163 7469 6365 2074  mming practice t
│ │ │ │ +0025e930: 6f20 7573 6520 656e 646c 2061 6c77 6179  o use endl alway
│ │ │ │ +0025e940: 732c 2066 6f72 0a77 7269 7469 6e67 206e  s, for.writing n
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│ │ │ │ +0025e960: 7320 2873 6565 202a 6e6f 7465 206e 6577  s (see *note new
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│ │ │ │ +0025e980: 746f 2061 2066 696c 6520 7769 6c6c 206e  to a file will n
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│ │ │ │ +0025e9d0: 7573 6820 7468 6520 6f75 7470 7574 0a62  ush the output.b
│ │ │ │ +0025e9e0: 7566 6665 722e 0a0a 5761 7973 2074 6f20  uffer...Ways to 
│ │ │ │ +0025e9f0: 7573 6520 656e 646c 3a0a 3d3d 3d3d 3d3d  use endl:.======
│ │ │ │ +0025ea00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ +0025ea10: 2022 4d61 6e69 7075 6c61 746f 7220 4461   "Manipulator Da
│ │ │ │ +0025ea20: 7461 6261 7365 2220 2d2d 2073 6565 202a  tabase" -- see *
│ │ │ │ +0025ea30: 6e6f 7465 204d 616e 6970 756c 6174 6f72  note Manipulator
│ │ │ │ +0025ea40: 3a20 4d61 6e69 7075 6c61 746f 722c 202d  : Manipulator, -
│ │ │ │ +0025ea50: 2d20 7468 6520 636c 6173 730a 2020 2020  - the class.    
│ │ │ │ +0025ea60: 6f66 2061 6c6c 2066 696c 6520 6d61 6e69  of all file mani
│ │ │ │ +0025ea70: 7075 6c61 746f 7273 0a20 202a 2022 4d61  pulators.  * "Ma
│ │ │ │ +0025ea80: 6e69 7075 6c61 746f 7220 4669 6c65 2220  nipulator File" 
│ │ │ │ +0025ea90: 2d2d 2073 6565 202a 6e6f 7465 204d 616e  -- see *note Man
│ │ │ │ +0025eaa0: 6970 756c 6174 6f72 3a20 4d61 6e69 7075  ipulator: Manipu
│ │ │ │ +0025eab0: 6c61 746f 722c 202d 2d20 7468 6520 636c  lator, -- the cl
│ │ │ │ +0025eac0: 6173 7320 6f66 0a20 2020 2061 6c6c 2066  ass of.    all f
│ │ │ │ +0025ead0: 696c 6520 6d61 6e69 7075 6c61 746f 7273  ile manipulators
│ │ │ │ +0025eae0: 0a20 202a 2022 4d61 6e69 7075 6c61 746f  .  * "Manipulato
│ │ │ │ +0025eaf0: 7220 4e6f 7468 696e 6722 202d 2d20 7365  r Nothing" -- se
│ │ │ │ +0025eb00: 6520 2a6e 6f74 6520 4d61 6e69 7075 6c61  e *note Manipula
│ │ │ │ +0025eb10: 746f 723a 204d 616e 6970 756c 6174 6f72  tor: Manipulator
│ │ │ │ +0025eb20: 2c20 2d2d 2074 6865 2063 6c61 7373 0a20  , -- the class. 
│ │ │ │ +0025eb30: 2020 206f 6620 616c 6c20 6669 6c65 206d     of all file m
│ │ │ │ +0025eb40: 616e 6970 756c 6174 6f72 730a 2020 2a20  anipulators.  * 
│ │ │ │ +0025eb50: 226e 6577 204d 616e 6970 756c 6174 6f72  "new Manipulator
│ │ │ │ +0025eb60: 2066 726f 6d20 4675 6e63 7469 6f6e 2220   from Function" 
│ │ │ │ +0025eb70: 2d2d 2073 6565 202a 6e6f 7465 204d 616e  -- see *note Man
│ │ │ │ +0025eb80: 6970 756c 6174 6f72 3a20 4d61 6e69 7075  ipulator: Manipu
│ │ │ │ +0025eb90: 6c61 746f 722c 202d 2d0a 2020 2020 7468  lator, --.    th
│ │ │ │ +0025eba0: 6520 636c 6173 7320 6f66 2061 6c6c 2066  e class of all f
│ │ │ │ +0025ebb0: 696c 6520 6d61 6e69 7075 6c61 746f 7273  ile manipulators
│ │ │ │ +0025ebc0: 0a20 202a 2022 6d65 7468 6f64 7328 4d61  .  * "methods(Ma
│ │ │ │ +0025ebd0: 6e69 7075 6c61 746f 7229 2220 2d2d 2073  nipulator)" -- s
│ │ │ │ +0025ebe0: 6565 202a 6e6f 7465 206d 6574 686f 6473  ee *note methods
│ │ │ │ +0025ebf0: 3a20 6d65 7468 6f64 732c 202d 2d20 6c69  : methods, -- li
│ │ │ │ +0025ec00: 7374 206d 6574 686f 6473 0a20 202a 2022  st methods.  * "
│ │ │ │ +0025ec10: 4d61 6e69 7075 6c61 746f 7220 4e65 7446  Manipulator NetF
│ │ │ │ +0025ec20: 696c 6522 202d 2d20 7365 6520 2a6e 6f74  ile" -- see *not
│ │ │ │ +0025ec30: 6520 4e65 7446 696c 653a 204e 6574 4669  e NetFile: NetFi
│ │ │ │ +0025ec40: 6c65 2c20 2d2d 2074 6865 2063 6c61 7373  le, -- the class
│ │ │ │ +0025ec50: 206f 6620 616c 6c0a 2020 2020 6e65 7420   of all.    net 
│ │ │ │ +0025ec60: 6669 6c65 730a 2020 2a20 224e 6574 4669  files.  * "NetFi
│ │ │ │ +0025ec70: 6c65 203c 3c20 4d61 6e69 7075 6c61 746f  le << Manipulato
│ │ │ │ +0025ec80: 7222 202d 2d20 7365 6520 2a6e 6f74 6520  r" -- see *note 
│ │ │ │ +0025ec90: 4e65 7446 696c 653a 204e 6574 4669 6c65  NetFile: NetFile
│ │ │ │ +0025eca0: 2c20 2d2d 2074 6865 2063 6c61 7373 206f  , -- the class o
│ │ │ │ +0025ecb0: 6620 616c 6c0a 2020 2020 6e65 7420 6669  f all.    net fi
│ │ │ │ +0025ecc0: 6c65 730a 2020 2a20 2246 696c 6520 3c3c  les.  * "File <<
│ │ │ │ +0025ecd0: 204d 616e 6970 756c 6174 6f72 2220 2d2d   Manipulator" --
│ │ │ │ +0025ece0: 2073 6565 202a 6e6f 7465 2070 7269 6e74   see *note print
│ │ │ │ +0025ecf0: 696e 6720 746f 2061 2066 696c 653a 2070  ing to a file: p
│ │ │ │ +0025ed00: 7269 6e74 696e 6720 746f 2061 2066 696c  rinting to a fil
│ │ │ │ +0025ed10: 652c 0a20 2020 202d 2d20 7072 696e 7420  e,.    -- print 
│ │ │ │ +0025ed20: 746f 2061 2066 696c 650a 2020 2a20 224e  to a file.  * "N
│ │ │ │ +0025ed30: 6f74 6869 6e67 203c 3c20 4d61 6e69 7075  othing << Manipu
│ │ │ │ +0025ed40: 6c61 746f 7222 202d 2d20 7365 6520 2a6e  lator" -- see *n
│ │ │ │ +0025ed50: 6f74 6520 7072 696e 7469 6e67 2074 6f20  ote printing to 
│ │ │ │ +0025ed60: 6120 6669 6c65 3a20 7072 696e 7469 6e67  a file: printing
│ │ │ │ +0025ed70: 2074 6f20 610a 2020 2020 6669 6c65 2c20   to a.    file, 
│ │ │ │ +0025ed80: 2d2d 2070 7269 6e74 2074 6f20 6120 6669  -- print to a fi
│ │ │ │ +0025ed90: 6c65 0a0a 466f 7220 7468 6520 7072 6f67  le..For the prog
│ │ │ │ +0025eda0: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d  rammer.=========
│ │ │ │ +0025edb0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f  =========..The o
│ │ │ │ +0025edc0: 626a 6563 7420 2a6e 6f74 6520 656e 646c  bject *note endl
│ │ │ │ +0025edd0: 3a20 656e 646c 2c20 6973 2061 202a 6e6f  : endl, is a *no
│ │ │ │ +0025ede0: 7465 206d 616e 6970 756c 6174 6f72 3a20  te manipulator: 
│ │ │ │ +0025edf0: 4d61 6e69 7075 6c61 746f 722c 2e0a 0a2d  Manipulator,...-
│ │ │ │  0025ee00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0025ee10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0025ee20: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520  ----------..The 
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│ │ │ │ -0025ee60: 652d 7061 7468 2f6d 6163 6175 6c61 7932  e-path/macaulay2
│ │ │ │ -0025ee70: 2d31 2e32 352e 3131 2b64 732f 4d32 2f4d  -1.25.11+ds/M2/M
│ │ │ │ -0025ee80: 6163 6175 6c61 7932 2f70 6163 6b61 6765  acaulay2/package
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│ │ │ │ -0025eea0: 6f76 5f73 7973 7465 6d2e 6d32 3a31 3535  ov_system.m2:155
│ │ │ │ -0025eeb0: 333a 302e 0a1f 0a46 696c 653a 204d 6163  3:0....File: Mac
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│ │ │ │ -0025eed0: 4e6f 6465 3a20 666c 7573 682c 204e 6578  Node: flush, Nex
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│ │ │ │ -0025eef0: 656e 646c 2c20 5570 3a20 7573 696e 6720  endl, Up: using 
│ │ │ │ -0025ef00: 736f 636b 6574 730a 0a66 6c75 7368 202d  sockets..flush -
│ │ │ │ -0025ef10: 2d20 666c 7573 6820 6f75 7470 7574 2074  - flush output t
│ │ │ │ -0025ef20: 6f20 6669 6c65 0a2a 2a2a 2a2a 2a2a 2a2a  o file.*********
│ │ │ │ -0025ef30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0025ef40: 2a2a 2a2a 0a0a 4465 7363 7269 7074 696f  ****..Descriptio
│ │ │ │ -0025ef50: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a66  n.===========..f
│ │ │ │ -0025ef60: 203c 3c20 666c 7573 6820 2d2d 2077 7269   << flush -- wri
│ │ │ │ -0025ef70: 7465 7320 6f75 7420 616e 7920 6275 6666  tes out any buff
│ │ │ │ -0025ef80: 6572 6564 206f 7574 7075 7420 666f 7220  ered output for 
│ │ │ │ -0025ef90: 7468 6520 6f75 7470 7574 2066 696c 6520  the output file 
│ │ │ │ -0025efa0: 662e 0a0a 5761 7973 2074 6f20 7573 6520  f...Ways to use 
│ │ │ │ -0025efb0: 666c 7573 683a 0a3d 3d3d 3d3d 3d3d 3d3d  flush:.=========
│ │ │ │ -0025efc0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022  =========..  * "
│ │ │ │ -0025efd0: 4d61 6e69 7075 6c61 746f 7220 4461 7461  Manipulator Data
│ │ │ │ -0025efe0: 6261 7365 2220 2d2d 2073 6565 202a 6e6f  base" -- see *no
│ │ │ │ -0025eff0: 7465 204d 616e 6970 756c 6174 6f72 3a20  te Manipulator: 
│ │ │ │ -0025f000: 4d61 6e69 7075 6c61 746f 722c 202d 2d20  Manipulator, -- 
│ │ │ │ -0025f010: 7468 6520 636c 6173 730a 2020 2020 6f66  the class.    of
│ │ │ │ -0025f020: 2061 6c6c 2066 696c 6520 6d61 6e69 7075   all file manipu
│ │ │ │ -0025f030: 6c61 746f 7273 0a20 202a 2022 4d61 6e69  lators.  * "Mani
│ │ │ │ -0025f040: 7075 6c61 746f 7220 4669 6c65 2220 2d2d  pulator File" --
│ │ │ │ -0025f050: 2073 6565 202a 6e6f 7465 204d 616e 6970   see *note Manip
│ │ │ │ -0025f060: 756c 6174 6f72 3a20 4d61 6e69 7075 6c61  ulator: Manipula
│ │ │ │ -0025f070: 746f 722c 202d 2d20 7468 6520 636c 6173  tor, -- the clas
│ │ │ │ -0025f080: 7320 6f66 0a20 2020 2061 6c6c 2066 696c  s of.    all fil
│ │ │ │ -0025f090: 6520 6d61 6e69 7075 6c61 746f 7273 0a20  e manipulators. 
│ │ │ │ -0025f0a0: 202a 2022 4d61 6e69 7075 6c61 746f 7220   * "Manipulator 
│ │ │ │ -0025f0b0: 4e6f 7468 696e 6722 202d 2d20 7365 6520  Nothing" -- see 
│ │ │ │ -0025f0c0: 2a6e 6f74 6520 4d61 6e69 7075 6c61 746f  *note Manipulato
│ │ │ │ -0025f0d0: 723a 204d 616e 6970 756c 6174 6f72 2c20  r: Manipulator, 
│ │ │ │ -0025f0e0: 2d2d 2074 6865 2063 6c61 7373 0a20 2020  -- the class.   
│ │ │ │ -0025f0f0: 206f 6620 616c 6c20 6669 6c65 206d 616e   of all file man
│ │ │ │ -0025f100: 6970 756c 6174 6f72 730a 2020 2a20 226e  ipulators.  * "n
│ │ │ │ -0025f110: 6577 204d 616e 6970 756c 6174 6f72 2066  ew Manipulator f
│ │ │ │ -0025f120: 726f 6d20 4675 6e63 7469 6f6e 2220 2d2d  rom Function" --
│ │ │ │ -0025f130: 2073 6565 202a 6e6f 7465 204d 616e 6970   see *note Manip
│ │ │ │ -0025f140: 756c 6174 6f72 3a20 4d61 6e69 7075 6c61  ulator: Manipula
│ │ │ │ -0025f150: 746f 722c 202d 2d0a 2020 2020 7468 6520  tor, --.    the 
│ │ │ │ -0025f160: 636c 6173 7320 6f66 2061 6c6c 2066 696c  class of all fil
│ │ │ │ -0025f170: 6520 6d61 6e69 7075 6c61 746f 7273 0a20  e manipulators. 
│ │ │ │ -0025f180: 202a 2022 6d65 7468 6f64 7328 4d61 6e69   * "methods(Mani
│ │ │ │ -0025f190: 7075 6c61 746f 7229 2220 2d2d 2073 6565  pulator)" -- see
│ │ │ │ -0025f1a0: 202a 6e6f 7465 206d 6574 686f 6473 3a20   *note methods: 
│ │ │ │ -0025f1b0: 6d65 7468 6f64 732c 202d 2d20 6c69 7374  methods, -- list
│ │ │ │ -0025f1c0: 206d 6574 686f 6473 0a20 202a 2022 4d61   methods.  * "Ma
│ │ │ │ -0025f1d0: 6e69 7075 6c61 746f 7220 4e65 7446 696c  nipulator NetFil
│ │ │ │ -0025f1e0: 6522 202d 2d20 7365 6520 2a6e 6f74 6520  e" -- see *note 
│ │ │ │ -0025f1f0: 4e65 7446 696c 653a 204e 6574 4669 6c65  NetFile: NetFile
│ │ │ │ -0025f200: 2c20 2d2d 2074 6865 2063 6c61 7373 206f  , -- the class o
│ │ │ │ -0025f210: 6620 616c 6c0a 2020 2020 6e65 7420 6669  f all.    net fi
│ │ │ │ -0025f220: 6c65 730a 2020 2a20 224e 6574 4669 6c65  les.  * "NetFile
│ │ │ │ -0025f230: 203c 3c20 4d61 6e69 7075 6c61 746f 7222   << Manipulator"
│ │ │ │ -0025f240: 202d 2d20 7365 6520 2a6e 6f74 6520 4e65   -- see *note Ne
│ │ │ │ -0025f250: 7446 696c 653a 204e 6574 4669 6c65 2c20  tFile: NetFile, 
│ │ │ │ -0025f260: 2d2d 2074 6865 2063 6c61 7373 206f 6620  -- the class of 
│ │ │ │ -0025f270: 616c 6c0a 2020 2020 6e65 7420 6669 6c65  all.    net file
│ │ │ │ -0025f280: 730a 2020 2a20 2246 696c 6520 3c3c 204d  s.  * "File << M
│ │ │ │ -0025f290: 616e 6970 756c 6174 6f72 2220 2d2d 2073  anipulator" -- s
│ │ │ │ -0025f2a0: 6565 202a 6e6f 7465 2070 7269 6e74 696e  ee *note printin
│ │ │ │ -0025f2b0: 6720 746f 2061 2066 696c 653a 2070 7269  g to a file: pri
│ │ │ │ -0025f2c0: 6e74 696e 6720 746f 2061 2066 696c 652c  nting to a file,
│ │ │ │ -0025f2d0: 0a20 2020 202d 2d20 7072 696e 7420 746f  .    -- print to
│ │ │ │ -0025f2e0: 2061 2066 696c 650a 2020 2a20 224e 6f74   a file.  * "Not
│ │ │ │ -0025f2f0: 6869 6e67 203c 3c20 4d61 6e69 7075 6c61  hing << Manipula
│ │ │ │ -0025f300: 746f 7222 202d 2d20 7365 6520 2a6e 6f74  tor" -- see *not
│ │ │ │ -0025f310: 6520 7072 696e 7469 6e67 2074 6f20 6120  e printing to a 
│ │ │ │ -0025f320: 6669 6c65 3a20 7072 696e 7469 6e67 2074  file: printing t
│ │ │ │ -0025f330: 6f20 610a 2020 2020 6669 6c65 2c20 2d2d  o a.    file, --
│ │ │ │ -0025f340: 2070 7269 6e74 2074 6f20 6120 6669 6c65   print to a file
│ │ │ │ -0025f350: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
│ │ │ │ -0025f360: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
│ │ │ │ -0025f370: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
│ │ │ │ -0025f380: 6563 7420 2a6e 6f74 6520 666c 7573 683a  ect *note flush:
│ │ │ │ -0025f390: 2066 6c75 7368 2c20 6973 2061 202a 6e6f   flush, is a *no
│ │ │ │ -0025f3a0: 7465 206d 616e 6970 756c 6174 6f72 3a20  te manipulator: 
│ │ │ │ -0025f3b0: 4d61 6e69 7075 6c61 746f 722c 2e0a 0a2d  Manipulator,...-
│ │ │ │ -0025f3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0025f3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0025f3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0025ee20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0025ee30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0025ee40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ +0025ee50: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
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│ │ │ │ +0025ee70: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475  n./build/reprodu
│ │ │ │ +0025ee80: 6369 626c 652d 7061 7468 2f6d 6163 6175  cible-path/macau
│ │ │ │ +0025ee90: 6c61 7932 2d31 2e32 352e 3131 2b64 732f  lay2-1.25.11+ds/
│ │ │ │ +0025eea0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163  M2/Macaulay2/pac
│ │ │ │ +0025eeb0: 6b61 6765 732f 0a4d 6163 6175 6c61 7932  kages/.Macaulay2
│ │ │ │ +0025eec0: 446f 632f 6f76 5f73 7973 7465 6d2e 6d32  Doc/ov_system.m2
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│ │ │ │ +0025eee0: 204d 6163 6175 6c61 7932 446f 632e 696e   Macaulay2Doc.in
│ │ │ │ +0025eef0: 666f 2c20 4e6f 6465 3a20 666c 7573 682c  fo, Node: flush,
│ │ │ │ +0025ef00: 204e 6578 743a 2063 6c6f 7365 2c20 5072   Next: close, Pr
│ │ │ │ +0025ef10: 6576 3a20 656e 646c 2c20 5570 3a20 7573  ev: endl, Up: us
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│ │ │ │ +0025ef30: 7368 202d 2d20 666c 7573 6820 6f75 7470  sh -- flush outp
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│ │ │ │ +0025ef50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ +0025ef80: 3d0a 0a66 203c 3c20 666c 7573 6820 2d2d  =..f << flush --
│ │ │ │ +0025ef90: 2077 7269 7465 7320 6f75 7420 616e 7920   writes out any 
│ │ │ │ +0025efa0: 6275 6666 6572 6564 206f 7574 7075 7420  buffered output 
│ │ │ │ +0025efb0: 666f 7220 7468 6520 6f75 7470 7574 2066  for the output f
│ │ │ │ +0025efc0: 696c 6520 662e 0a0a 5761 7973 2074 6f20  ile f...Ways to 
│ │ │ │ +0025efd0: 7573 6520 666c 7573 683a 0a3d 3d3d 3d3d  use flush:.=====
│ │ │ │ +0025efe0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ +0025eff0: 202a 2022 4d61 6e69 7075 6c61 746f 7220   * "Manipulator 
│ │ │ │ +0025f000: 4461 7461 6261 7365 2220 2d2d 2073 6565  Database" -- see
│ │ │ │ +0025f010: 202a 6e6f 7465 204d 616e 6970 756c 6174   *note Manipulat
│ │ │ │ +0025f020: 6f72 3a20 4d61 6e69 7075 6c61 746f 722c  or: Manipulator,
│ │ │ │ +0025f030: 202d 2d20 7468 6520 636c 6173 730a 2020   -- the class.  
│ │ │ │ +0025f040: 2020 6f66 2061 6c6c 2066 696c 6520 6d61    of all file ma
│ │ │ │ +0025f050: 6e69 7075 6c61 746f 7273 0a20 202a 2022  nipulators.  * "
│ │ │ │ +0025f060: 4d61 6e69 7075 6c61 746f 7220 4669 6c65  Manipulator File
│ │ │ │ +0025f070: 2220 2d2d 2073 6565 202a 6e6f 7465 204d  " -- see *note M
│ │ │ │ +0025f080: 616e 6970 756c 6174 6f72 3a20 4d61 6e69  anipulator: Mani
│ │ │ │ +0025f090: 7075 6c61 746f 722c 202d 2d20 7468 6520  pulator, -- the 
│ │ │ │ +0025f0a0: 636c 6173 7320 6f66 0a20 2020 2061 6c6c  class of.    all
│ │ │ │ +0025f0b0: 2066 696c 6520 6d61 6e69 7075 6c61 746f   file manipulato
│ │ │ │ +0025f0c0: 7273 0a20 202a 2022 4d61 6e69 7075 6c61  rs.  * "Manipula
│ │ │ │ +0025f0d0: 746f 7220 4e6f 7468 696e 6722 202d 2d20  tor Nothing" -- 
│ │ │ │ +0025f0e0: 7365 6520 2a6e 6f74 6520 4d61 6e69 7075  see *note Manipu
│ │ │ │ +0025f0f0: 6c61 746f 723a 204d 616e 6970 756c 6174  lator: Manipulat
│ │ │ │ +0025f100: 6f72 2c20 2d2d 2074 6865 2063 6c61 7373  or, -- the class
│ │ │ │ +0025f110: 0a20 2020 206f 6620 616c 6c20 6669 6c65  .    of all file
│ │ │ │ +0025f120: 206d 616e 6970 756c 6174 6f72 730a 2020   manipulators.  
│ │ │ │ +0025f130: 2a20 226e 6577 204d 616e 6970 756c 6174  * "new Manipulat
│ │ │ │ +0025f140: 6f72 2066 726f 6d20 4675 6e63 7469 6f6e  or from Function
│ │ │ │ +0025f150: 2220 2d2d 2073 6565 202a 6e6f 7465 204d  " -- see *note M
│ │ │ │ +0025f160: 616e 6970 756c 6174 6f72 3a20 4d61 6e69  anipulator: Mani
│ │ │ │ +0025f170: 7075 6c61 746f 722c 202d 2d0a 2020 2020  pulator, --.    
│ │ │ │ +0025f180: 7468 6520 636c 6173 7320 6f66 2061 6c6c  the class of all
│ │ │ │ +0025f190: 2066 696c 6520 6d61 6e69 7075 6c61 746f   file manipulato
│ │ │ │ +0025f1a0: 7273 0a20 202a 2022 6d65 7468 6f64 7328  rs.  * "methods(
│ │ │ │ +0025f1b0: 4d61 6e69 7075 6c61 746f 7229 2220 2d2d  Manipulator)" --
│ │ │ │ +0025f1c0: 2073 6565 202a 6e6f 7465 206d 6574 686f   see *note metho
│ │ │ │ +0025f1d0: 6473 3a20 6d65 7468 6f64 732c 202d 2d20  ds: methods, -- 
│ │ │ │ +0025f1e0: 6c69 7374 206d 6574 686f 6473 0a20 202a  list methods.  *
│ │ │ │ +0025f1f0: 2022 4d61 6e69 7075 6c61 746f 7220 4e65   "Manipulator Ne
│ │ │ │ +0025f200: 7446 696c 6522 202d 2d20 7365 6520 2a6e  tFile" -- see *n
│ │ │ │ +0025f210: 6f74 6520 4e65 7446 696c 653a 204e 6574  ote NetFile: Net
│ │ │ │ +0025f220: 4669 6c65 2c20 2d2d 2074 6865 2063 6c61  File, -- the cla
│ │ │ │ +0025f230: 7373 206f 6620 616c 6c0a 2020 2020 6e65  ss of all.    ne
│ │ │ │ +0025f240: 7420 6669 6c65 730a 2020 2a20 224e 6574  t files.  * "Net
│ │ │ │ +0025f250: 4669 6c65 203c 3c20 4d61 6e69 7075 6c61  File << Manipula
│ │ │ │ +0025f260: 746f 7222 202d 2d20 7365 6520 2a6e 6f74  tor" -- see *not
│ │ │ │ +0025f270: 6520 4e65 7446 696c 653a 204e 6574 4669  e NetFile: NetFi
│ │ │ │ +0025f280: 6c65 2c20 2d2d 2074 6865 2063 6c61 7373  le, -- the class
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│ │ │ │ +0025f2a0: 6669 6c65 730a 2020 2a20 2246 696c 6520  files.  * "File 
│ │ │ │ +0025f2b0: 3c3c 204d 616e 6970 756c 6174 6f72 2220  << Manipulator" 
│ │ │ │ +0025f2c0: 2d2d 2073 6565 202a 6e6f 7465 2070 7269  -- see *note pri
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│ │ │ │ +0025f2e0: 2070 7269 6e74 696e 6720 746f 2061 2066   printing to a f
│ │ │ │ +0025f2f0: 696c 652c 0a20 2020 202d 2d20 7072 696e  ile,.    -- prin
│ │ │ │ +0025f300: 7420 746f 2061 2066 696c 650a 2020 2a20  t to a file.  * 
│ │ │ │ +0025f310: 224e 6f74 6869 6e67 203c 3c20 4d61 6e69  "Nothing << Mani
│ │ │ │ +0025f320: 7075 6c61 746f 7222 202d 2d20 7365 6520  pulator" -- see 
│ │ │ │ +0025f330: 2a6e 6f74 6520 7072 696e 7469 6e67 2074  *note printing t
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│ │ │ │ +0025f350: 6e67 2074 6f20 610a 2020 2020 6669 6c65  ng to a.    file
│ │ │ │ +0025f360: 2c20 2d2d 2070 7269 6e74 2074 6f20 6120  , -- print to a 
│ │ │ │ +0025f370: 6669 6c65 0a0a 466f 7220 7468 6520 7072  file..For the pr
│ │ │ │ +0025f380: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ +0025f390: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ +0025f3a0: 206f 626a 6563 7420 2a6e 6f74 6520 666c   object *note fl
│ │ │ │ +0025f3b0: 7573 683a 2066 6c75 7368 2c20 6973 2061  ush: flush, is a
│ │ │ │ +0025f3c0: 202a 6e6f 7465 206d 616e 6970 756c 6174   *note manipulat
│ │ │ │ +0025f3d0: 6f72 3a20 4d61 6e69 7075 6c61 746f 722c  or: Manipulator,
│ │ │ │ +0025f3e0: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...-------------
│ │ │ │  0025f3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0025f400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ -0025f410: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
│ │ │ │ -0025f420: 6973 2064 6f63 756d 656e 7420 6973 2069  is document is i
│ │ │ │ -0025f430: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475  n./build/reprodu
│ │ │ │ -0025f440: 6369 626c 652d 7061 7468 2f6d 6163 6175  cible-path/macau
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│ │ │ │ -0025f860: 3d3d 3d3d 0a0a 2020 2a20 224d 616e 6970  ====..  * "Manip
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│ │ │ │ -0025f8c0: 6669 6c65 206d 616e 6970 756c 6174 6f72  file manipulator
│ │ │ │ -0025f8d0: 730a 2020 2a20 224d 616e 6970 756c 6174  s.  * "Manipulat
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│ │ │ │ -0025f8f0: 2a6e 6f74 6520 4d61 6e69 7075 6c61 746f  *note Manipulato
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│ │ │ │ -0025f930: 6970 756c 6174 6f72 730a 2020 2a20 224d  ipulators.  * "M
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│ │ │ │ -0025f960: 204d 616e 6970 756c 6174 6f72 3a20 4d61   Manipulator: Ma
│ │ │ │ -0025f970: 6e69 7075 6c61 746f 722c 202d 2d20 7468  nipulator, -- th
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│ │ │ │ -0025fac0: 202a 2022 4e65 7446 696c 6520 3c3c 204d   * "NetFile << M
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│ │ │ │ +0025f870: 6c6f 7365 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d  lose:.==========
│ │ │ │ +0025f880: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 224d  ========..  * "M
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│ │ │ │ +0025f900: 756c 6174 6f72 2046 696c 6522 202d 2d20  ulator File" -- 
│ │ │ │ +0025f910: 7365 6520 2a6e 6f74 6520 4d61 6e69 7075  see *note Manipu
│ │ │ │ +0025f920: 6c61 746f 723a 204d 616e 6970 756c 6174  lator: Manipulat
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│ │ │ │ +0025f950: 206d 616e 6970 756c 6174 6f72 730a 2020   manipulators.  
│ │ │ │ +0025f960: 2a20 224d 616e 6970 756c 6174 6f72 204e  * "Manipulator N
│ │ │ │ +0025f970: 6f74 6869 6e67 2220 2d2d 2073 6565 202a  othing" -- see *
│ │ │ │ +0025f980: 6e6f 7465 204d 616e 6970 756c 6174 6f72  note Manipulator
│ │ │ │ +0025f990: 3a20 4d61 6e69 7075 6c61 746f 722c 202d  : Manipulator, -
│ │ │ │ +0025f9a0: 2d20 7468 6520 636c 6173 730a 2020 2020  - the class.    
│ │ │ │ +0025f9b0: 6f66 2061 6c6c 2066 696c 6520 6d61 6e69  of all file mani
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│ │ │ │ +0025f9d0: 7720 4d61 6e69 7075 6c61 746f 7220 6672  w Manipulator fr
│ │ │ │ +0025f9e0: 6f6d 2046 756e 6374 696f 6e22 202d 2d20  om Function" -- 
│ │ │ │ +0025f9f0: 7365 6520 2a6e 6f74 6520 4d61 6e69 7075  see *note Manipu
│ │ │ │ +0025fa00: 6c61 746f 723a 204d 616e 6970 756c 6174  lator: Manipulat
│ │ │ │ +0025fa10: 6f72 2c20 2d2d 0a20 2020 2074 6865 2063  or, --.    the c
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│ │ │ │ +0025fa40: 2a20 226d 6574 686f 6473 284d 616e 6970  * "methods(Manip
│ │ │ │ +0025fa50: 756c 6174 6f72 2922 202d 2d20 7365 6520  ulator)" -- see 
│ │ │ │ +0025fa60: 2a6e 6f74 6520 6d65 7468 6f64 733a 206d  *note methods: m
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│ │ │ │ +0025fa80: 6d65 7468 6f64 730a 2020 2a20 224d 616e  methods.  * "Man
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│ │ │ │ +0025faa0: 2220 2d2d 2073 6565 202a 6e6f 7465 204e  " -- see *note N
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│ │ │ │ +0025fac0: 202d 2d20 7468 6520 636c 6173 7320 6f66   -- the class of
│ │ │ │ +0025fad0: 2061 6c6c 0a20 2020 206e 6574 2066 696c   all.    net fil
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│ │ │ │ +0025faf0: 3c3c 204d 616e 6970 756c 6174 6f72 2220  << Manipulator" 
│ │ │ │ +0025fb00: 2d2d 2073 6565 202a 6e6f 7465 204e 6574  -- see *note Net
│ │ │ │ +0025fb10: 4669 6c65 3a20 4e65 7446 696c 652c 202d  File: NetFile, -
│ │ │ │ +0025fb20: 2d20 7468 6520 636c 6173 7320 6f66 2061  - the class of a
│ │ │ │ +0025fb30: 6c6c 0a20 2020 206e 6574 2066 696c 6573  ll.    net files
│ │ │ │ +0025fb40: 0a20 202a 2022 4669 6c65 203c 3c20 4d61  .  * "File << Ma
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│ │ │ │ +0025fb60: 6520 2a6e 6f74 6520 7072 696e 7469 6e67  e *note printing
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│ │ │ │ +0025fb90: 2020 2020 2d2d 2070 7269 6e74 2074 6f20      -- print to 
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│ │ │ │ +0025fc00: 7072 696e 7420 746f 2061 2066 696c 650a  print to a file.
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│ │ │ │  0025fc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0025fc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0025fdb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -0025fe60: 202a 6e6f 7465 2063 6c6f 7365 3a20 636c   *note close: cl
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│ │ │ │ -0025fe80: 6f20 7573 650a 616e 6420 6861 7320 7468  o use.and has th
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│ │ │ │ -0025ff10: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 4d61  =======..  * "Ma
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│ │ │ │ -0025ff40: 204d 616e 6970 756c 6174 6f72 3a20 4d61   Manipulator: Ma
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│ │ │ │ -0025ffa0: 6565 202a 6e6f 7465 204d 616e 6970 756c  ee *note Manipul
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│ │ │ │ -0025fff0: 2022 4d61 6e69 7075 6c61 746f 7220 4e6f   "Manipulator No
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│ │ │ │ -00260020: 204d 616e 6970 756c 6174 6f72 2c20 2d2d   Manipulator, --
│ │ │ │ -00260030: 2074 6865 2063 6c61 7373 0a20 2020 206f   the class.    o
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│ │ │ │ -00260060: 204d 616e 6970 756c 6174 6f72 2066 726f   Manipulator fro
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│ │ │ │ -002600a0: 722c 202d 2d0a 2020 2020 7468 6520 636c  r, --.    the cl
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│ │ │ │ -002600d0: 2022 6d65 7468 6f64 7328 4d61 6e69 7075   "methods(Manipu
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│ │ │ │ +0025ff10: 0a0a 5761 7973 2074 6f20 7573 6520 636c  ..Ways to use cl
│ │ │ │ +0025ff20: 6f73 6549 6e3a 0a3d 3d3d 3d3d 3d3d 3d3d  oseIn:.=========
│ │ │ │ +0025ff30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ +0025ff40: 2022 4d61 6e69 7075 6c61 746f 7220 4461   "Manipulator Da
│ │ │ │ +0025ff50: 7461 6261 7365 2220 2d2d 2073 6565 202a  tabase" -- see *
│ │ │ │ +0025ff60: 6e6f 7465 204d 616e 6970 756c 6174 6f72  note Manipulator
│ │ │ │ +0025ff70: 3a20 4d61 6e69 7075 6c61 746f 722c 202d  : Manipulator, -
│ │ │ │ +0025ff80: 2d20 7468 6520 636c 6173 730a 2020 2020  - the class.    
│ │ │ │ +0025ff90: 6f66 2061 6c6c 2066 696c 6520 6d61 6e69  of all file mani
│ │ │ │ +0025ffa0: 7075 6c61 746f 7273 0a20 202a 2022 4d61  pulators.  * "Ma
│ │ │ │ +0025ffb0: 6e69 7075 6c61 746f 7220 4669 6c65 2220  nipulator File" 
│ │ │ │ +0025ffc0: 2d2d 2073 6565 202a 6e6f 7465 204d 616e  -- see *note Man
│ │ │ │ +0025ffd0: 6970 756c 6174 6f72 3a20 4d61 6e69 7075  ipulator: Manipu
│ │ │ │ +0025ffe0: 6c61 746f 722c 202d 2d20 7468 6520 636c  lator, -- the cl
│ │ │ │ +0025fff0: 6173 7320 6f66 0a20 2020 2061 6c6c 2066  ass of.    all f
│ │ │ │ +00260000: 696c 6520 6d61 6e69 7075 6c61 746f 7273  ile manipulators
│ │ │ │ +00260010: 0a20 202a 2022 4d61 6e69 7075 6c61 746f  .  * "Manipulato
│ │ │ │ +00260020: 7220 4e6f 7468 696e 6722 202d 2d20 7365  r Nothing" -- se
│ │ │ │ +00260030: 6520 2a6e 6f74 6520 4d61 6e69 7075 6c61  e *note Manipula
│ │ │ │ +00260040: 746f 723a 204d 616e 6970 756c 6174 6f72  tor: Manipulator
│ │ │ │ +00260050: 2c20 2d2d 2074 6865 2063 6c61 7373 0a20  , -- the class. 
│ │ │ │ +00260060: 2020 206f 6620 616c 6c20 6669 6c65 206d     of all file m
│ │ │ │ +00260070: 616e 6970 756c 6174 6f72 730a 2020 2a20  anipulators.  * 
│ │ │ │ +00260080: 226e 6577 204d 616e 6970 756c 6174 6f72  "new Manipulator
│ │ │ │ +00260090: 2066 726f 6d20 4675 6e63 7469 6f6e 2220   from Function" 
│ │ │ │ +002600a0: 2d2d 2073 6565 202a 6e6f 7465 204d 616e  -- see *note Man
│ │ │ │ +002600b0: 6970 756c 6174 6f72 3a20 4d61 6e69 7075  ipulator: Manipu
│ │ │ │ +002600c0: 6c61 746f 722c 202d 2d0a 2020 2020 7468  lator, --.    th
│ │ │ │ +002600d0: 6520 636c 6173 7320 6f66 2061 6c6c 2066  e class of all f
│ │ │ │ +002600e0: 696c 6520 6d61 6e69 7075 6c61 746f 7273  ile manipulators
│ │ │ │ +002600f0: 0a20 202a 2022 6d65 7468 6f64 7328 4d61  .  * "methods(Ma
│ │ │ │ +00260100: 6e69 7075 6c61 746f 7229 2220 2d2d 2073  nipulator)" -- s
│ │ │ │ +00260110: 6565 202a 6e6f 7465 206d 6574 686f 6473  ee *note methods
│ │ │ │ +00260120: 3a20 6d65 7468 6f64 732c 202d 2d20 6c69  : methods, -- li
│ │ │ │ +00260130: 7374 206d 6574 686f 6473 0a20 202a 2022  st methods.  * "
│ │ │ │ +00260140: 4d61 6e69 7075 6c61 746f 7220 4e65 7446  Manipulator NetF
│ │ │ │ +00260150: 696c 6522 202d 2d20 7365 6520 2a6e 6f74  ile" -- see *not
│ │ │ │ +00260160: 6520 4e65 7446 696c 653a 204e 6574 4669  e NetFile: NetFi
│ │ │ │ +00260170: 6c65 2c20 2d2d 2074 6865 2063 6c61 7373  le, -- the class
│ │ │ │ +00260180: 206f 6620 616c 6c0a 2020 2020 6e65 7420   of all.    net 
│ │ │ │ +00260190: 6669 6c65 730a 2020 2a20 224e 6574 4669  files.  * "NetFi
│ │ │ │ +002601a0: 6c65 203c 3c20 4d61 6e69 7075 6c61 746f  le << Manipulato
│ │ │ │ +002601b0: 7222 202d 2d20 7365 6520 2a6e 6f74 6520  r" -- see *note 
│ │ │ │ +002601c0: 4e65 7446 696c 653a 204e 6574 4669 6c65  NetFile: NetFile
│ │ │ │ +002601d0: 2c20 2d2d 2074 6865 2063 6c61 7373 206f  , -- the class o
│ │ │ │ +002601e0: 6620 616c 6c0a 2020 2020 6e65 7420 6669  f all.    net fi
│ │ │ │ +002601f0: 6c65 730a 2020 2a20 2246 696c 6520 3c3c  les.  * "File <<
│ │ │ │ +00260200: 204d 616e 6970 756c 6174 6f72 2220 2d2d   Manipulator" --
│ │ │ │ +00260210: 2073 6565 202a 6e6f 7465 2070 7269 6e74   see *note print
│ │ │ │ +00260220: 696e 6720 746f 2061 2066 696c 653a 2070  ing to a file: p
│ │ │ │ +00260230: 7269 6e74 696e 6720 746f 2061 2066 696c  rinting to a fil
│ │ │ │ +00260240: 652c 0a20 2020 202d 2d20 7072 696e 7420  e,.    -- print 
│ │ │ │ +00260250: 746f 2061 2066 696c 650a 2020 2a20 224e  to a file.  * "N
│ │ │ │ +00260260: 6f74 6869 6e67 203c 3c20 4d61 6e69 7075  othing << Manipu
│ │ │ │ +00260270: 6c61 746f 7222 202d 2d20 7365 6520 2a6e  lator" -- see *n
│ │ │ │ +00260280: 6f74 6520 7072 696e 7469 6e67 2074 6f20  ote printing to 
│ │ │ │ +00260290: 6120 6669 6c65 3a20 7072 696e 7469 6e67  a file: printing
│ │ │ │ +002602a0: 2074 6f20 610a 2020 2020 6669 6c65 2c20   to a.    file, 
│ │ │ │ +002602b0: 2d2d 2070 7269 6e74 2074 6f20 6120 6669  -- print to a fi
│ │ │ │ +002602c0: 6c65 0a0a 466f 7220 7468 6520 7072 6f67  le..For the prog
│ │ │ │ +002602d0: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d  rammer.=========
│ │ │ │ +002602e0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f  =========..The o
│ │ │ │ +002602f0: 626a 6563 7420 2a6e 6f74 6520 636c 6f73  bject *note clos
│ │ │ │ +00260300: 6549 6e3a 2063 6c6f 7365 496e 2c20 6973  eIn: closeIn, is
│ │ │ │ +00260310: 2061 202a 6e6f 7465 206d 616e 6970 756c   a *note manipul
│ │ │ │ +00260320: 6174 6f72 3a20 4d61 6e69 7075 6c61 746f  ator: Manipulato
│ │ │ │ +00260330: 722c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d  r,...-----------
│ │ │ │  00260340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00260350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00260360: 0a0a 5468 6520 736f 7572 6365 206f 6620  ..The source of 
│ │ │ │ -00260370: 7468 6973 2064 6f63 756d 656e 7420 6973  this document is
│ │ │ │ -00260380: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f   in./build/repro
│ │ │ │ -00260390: 6475 6369 626c 652d 7061 7468 2f6d 6163  ducible-path/mac
│ │ │ │ -002603a0: 6175 6c61 7932 2d31 2e32 352e 3131 2b64  aulay2-1.25.11+d
│ │ │ │ -002603b0: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70  s/M2/Macaulay2/p
│ │ │ │ -002603c0: 6163 6b61 6765 732f 0a4d 6163 6175 6c61  ackages/.Macaula
│ │ │ │ -002603d0: 7932 446f 632f 6f76 5f73 7973 7465 6d2e  y2Doc/ov_system.
│ │ │ │ -002603e0: 6d32 3a31 3531 363a 302e 0a1f 0a46 696c  m2:1516:0....Fil
│ │ │ │ -002603f0: 653a 204d 6163 6175 6c61 7932 446f 632e  e: Macaulay2Doc.
│ │ │ │ -00260400: 696e 666f 2c20 4e6f 6465 3a20 636c 6f73  info, Node: clos
│ │ │ │ -00260410: 654f 7574 2c20 4e65 7874 3a20 6765 742c  eOut, Next: get,
│ │ │ │ -00260420: 2050 7265 763a 2063 6c6f 7365 496e 2c20   Prev: closeIn, 
│ │ │ │ -00260430: 5570 3a20 7573 696e 6720 736f 636b 6574  Up: using socket
│ │ │ │ -00260440: 730a 0a63 6c6f 7365 4f75 7420 2d2d 2063  s..closeOut -- c
│ │ │ │ -00260450: 6c6f 7365 2061 6e20 6f75 7470 7574 2066  lose an output f
│ │ │ │ -00260460: 696c 650a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ile.************
│ │ │ │ -00260470: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00260480: 2a2a 2a2a 0a0a 4465 7363 7269 7074 696f  ****..Descriptio
│ │ │ │ -00260490: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a66  n.===========..f
│ │ │ │ -002604a0: 203c 3c20 636c 6f73 654f 7574 202d 2d20   << closeOut -- 
│ │ │ │ -002604b0: 636c 6f73 6573 2074 6865 206f 7574 7075  closes the outpu
│ │ │ │ -002604c0: 7420 6669 6c65 2066 2e0a 636c 6f73 654f  t file f..closeO
│ │ │ │ -002604d0: 7574 2066 202d 2d20 636c 6f73 6573 2074  ut f -- closes t
│ │ │ │ -002604e0: 6865 206f 7574 7075 7420 6669 6c65 2066  he output file f
│ │ │ │ -002604f0: 2e0a 0a0a 416e 7920 6275 6666 6572 6564  ....Any buffered
│ │ │ │ -00260500: 206f 7574 7075 7420 6973 2066 6972 7374   output is first
│ │ │ │ -00260510: 2077 7269 7474 656e 2074 6f20 7468 6520   written to the 
│ │ │ │ -00260520: 6669 6c65 2c20 616e 6420 7468 6520 7265  file, and the re
│ │ │ │ -00260530: 7475 726e 2076 616c 7565 2069 7320 616e  turn value is an
│ │ │ │ -00260540: 0a69 6e74 6567 6572 2c20 6e6f 726d 616c  .integer, normal
│ │ │ │ -00260550: 6c79 2030 2c20 6f72 202d 3120 6f6e 2065  ly 0, or -1 on e
│ │ │ │ -00260560: 7272 6f72 2c20 6f72 2074 6865 2072 6574  rror, or the ret
│ │ │ │ -00260570: 7572 6e20 7374 6174 7573 206f 6620 7468  urn status of th
│ │ │ │ -00260580: 6520 6368 696c 6420 7072 6f63 6573 730a  e child process.
│ │ │ │ -00260590: 696e 2063 6173 6520 7468 6520 6669 6c65  in case the file
│ │ │ │ -002605a0: 2077 6173 2061 2070 6970 652e 0a0a 0a49   was a pipe....I
│ │ │ │ -002605b0: 6620 7468 6520 6669 6c65 2077 6173 206f  f the file was o
│ │ │ │ -002605c0: 7065 6e20 6f6e 6c79 2066 6f72 206f 7574  pen only for out
│ │ │ │ -002605d0: 7075 742c 2074 6865 6e20 2a6e 6f74 6520  put, then *note 
│ │ │ │ -002605e0: 636c 6f73 653a 2063 6c6f 7365 2c20 6973  close: close, is
│ │ │ │ -002605f0: 2065 6173 6965 7220 746f 2075 7365 0a61   easier to use.a
│ │ │ │ -00260600: 6e64 2068 6173 2074 6865 2073 616d 6520  nd has the same 
│ │ │ │ -00260610: 6566 6665 6374 2e0a 0a0a 4966 2074 6865  effect....If the
│ │ │ │ -00260620: 2066 696c 6520 7761 7320 6f70 656e 2066   file was open f
│ │ │ │ -00260630: 6f72 2062 6f74 6820 696e 7075 7420 616e  or both input an
│ │ │ │ -00260640: 6420 6f75 7470 7574 2c20 6974 2072 656d  d output, it rem
│ │ │ │ -00260650: 6169 6e73 206f 7065 6e20 666f 7220 696e  ains open for in
│ │ │ │ -00260660: 7075 742e 0a0a 5761 7973 2074 6f20 7573  put...Ways to us
│ │ │ │ -00260670: 6520 636c 6f73 654f 7574 3a0a 3d3d 3d3d  e closeOut:.====
│ │ │ │ -00260680: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00260690: 3d0a 0a20 202a 2022 4d61 6e69 7075 6c61  =..  * "Manipula
│ │ │ │ -002606a0: 746f 7220 4461 7461 6261 7365 2220 2d2d  tor Database" --
│ │ │ │ -002606b0: 2073 6565 202a 6e6f 7465 204d 616e 6970   see *note Manip
│ │ │ │ -002606c0: 756c 6174 6f72 3a20 4d61 6e69 7075 6c61  ulator: Manipula
│ │ │ │ -002606d0: 746f 722c 202d 2d20 7468 6520 636c 6173  tor, -- the clas
│ │ │ │ -002606e0: 730a 2020 2020 6f66 2061 6c6c 2066 696c  s.    of all fil
│ │ │ │ -002606f0: 6520 6d61 6e69 7075 6c61 746f 7273 0a20  e manipulators. 
│ │ │ │ -00260700: 202a 2022 4d61 6e69 7075 6c61 746f 7220   * "Manipulator 
│ │ │ │ -00260710: 4669 6c65 2220 2d2d 2073 6565 202a 6e6f  File" -- see *no
│ │ │ │ -00260720: 7465 204d 616e 6970 756c 6174 6f72 3a20  te Manipulator: 
│ │ │ │ -00260730: 4d61 6e69 7075 6c61 746f 722c 202d 2d20  Manipulator, -- 
│ │ │ │ -00260740: 7468 6520 636c 6173 7320 6f66 0a20 2020  the class of.   
│ │ │ │ -00260750: 2061 6c6c 2066 696c 6520 6d61 6e69 7075   all file manipu
│ │ │ │ -00260760: 6c61 746f 7273 0a20 202a 2022 4d61 6e69  lators.  * "Mani
│ │ │ │ -00260770: 7075 6c61 746f 7220 4e6f 7468 696e 6722  pulator Nothing"
│ │ │ │ -00260780: 202d 2d20 7365 6520 2a6e 6f74 6520 4d61   -- see *note Ma
│ │ │ │ -00260790: 6e69 7075 6c61 746f 723a 204d 616e 6970  nipulator: Manip
│ │ │ │ -002607a0: 756c 6174 6f72 2c20 2d2d 2074 6865 2063  ulator, -- the c
│ │ │ │ -002607b0: 6c61 7373 0a20 2020 206f 6620 616c 6c20  lass.    of all 
│ │ │ │ -002607c0: 6669 6c65 206d 616e 6970 756c 6174 6f72  file manipulator
│ │ │ │ -002607d0: 730a 2020 2a20 226e 6577 204d 616e 6970  s.  * "new Manip
│ │ │ │ -002607e0: 756c 6174 6f72 2066 726f 6d20 4675 6e63  ulator from Func
│ │ │ │ -002607f0: 7469 6f6e 2220 2d2d 2073 6565 202a 6e6f  tion" -- see *no
│ │ │ │ -00260800: 7465 204d 616e 6970 756c 6174 6f72 3a20  te Manipulator: 
│ │ │ │ -00260810: 4d61 6e69 7075 6c61 746f 722c 202d 2d0a  Manipulator, --.
│ │ │ │ -00260820: 2020 2020 7468 6520 636c 6173 7320 6f66      the class of
│ │ │ │ -00260830: 2061 6c6c 2066 696c 6520 6d61 6e69 7075   all file manipu
│ │ │ │ -00260840: 6c61 746f 7273 0a20 202a 2022 6d65 7468  lators.  * "meth
│ │ │ │ -00260850: 6f64 7328 4d61 6e69 7075 6c61 746f 7229  ods(Manipulator)
│ │ │ │ -00260860: 2220 2d2d 2073 6565 202a 6e6f 7465 206d  " -- see *note m
│ │ │ │ -00260870: 6574 686f 6473 3a20 6d65 7468 6f64 732c  ethods: methods,
│ │ │ │ -00260880: 202d 2d20 6c69 7374 206d 6574 686f 6473   -- list methods
│ │ │ │ -00260890: 0a20 202a 2022 4d61 6e69 7075 6c61 746f  .  * "Manipulato
│ │ │ │ -002608a0: 7220 4e65 7446 696c 6522 202d 2d20 7365  r NetFile" -- se
│ │ │ │ -002608b0: 6520 2a6e 6f74 6520 4e65 7446 696c 653a  e *note NetFile:
│ │ │ │ -002608c0: 204e 6574 4669 6c65 2c20 2d2d 2074 6865   NetFile, -- the
│ │ │ │ -002608d0: 2063 6c61 7373 206f 6620 616c 6c0a 2020   class of all.  
│ │ │ │ -002608e0: 2020 6e65 7420 6669 6c65 730a 2020 2a20    net files.  * 
│ │ │ │ -002608f0: 224e 6574 4669 6c65 203c 3c20 4d61 6e69  "NetFile << Mani
│ │ │ │ -00260900: 7075 6c61 746f 7222 202d 2d20 7365 6520  pulator" -- see 
│ │ │ │ -00260910: 2a6e 6f74 6520 4e65 7446 696c 653a 204e  *note NetFile: N
│ │ │ │ -00260920: 6574 4669 6c65 2c20 2d2d 2074 6865 2063  etFile, -- the c
│ │ │ │ -00260930: 6c61 7373 206f 6620 616c 6c0a 2020 2020  lass of all.    
│ │ │ │ -00260940: 6e65 7420 6669 6c65 730a 2020 2a20 2246  net files.  * "F
│ │ │ │ -00260950: 696c 6520 3c3c 204d 616e 6970 756c 6174  ile << Manipulat
│ │ │ │ -00260960: 6f72 2220 2d2d 2073 6565 202a 6e6f 7465  or" -- see *note
│ │ │ │ -00260970: 2070 7269 6e74 696e 6720 746f 2061 2066   printing to a f
│ │ │ │ -00260980: 696c 653a 2070 7269 6e74 696e 6720 746f  ile: printing to
│ │ │ │ -00260990: 2061 2066 696c 652c 0a20 2020 202d 2d20   a file,.    -- 
│ │ │ │ -002609a0: 7072 696e 7420 746f 2061 2066 696c 650a  print to a file.
│ │ │ │ -002609b0: 2020 2a20 224e 6f74 6869 6e67 203c 3c20    * "Nothing << 
│ │ │ │ -002609c0: 4d61 6e69 7075 6c61 746f 7222 202d 2d20  Manipulator" -- 
│ │ │ │ -002609d0: 7365 6520 2a6e 6f74 6520 7072 696e 7469  see *note printi
│ │ │ │ -002609e0: 6e67 2074 6f20 6120 6669 6c65 3a20 7072  ng to a file: pr
│ │ │ │ -002609f0: 696e 7469 6e67 2074 6f20 610a 2020 2020  inting to a.    
│ │ │ │ -00260a00: 6669 6c65 2c20 2d2d 2070 7269 6e74 2074  file, -- print t
│ │ │ │ -00260a10: 6f20 6120 6669 6c65 0a0a 466f 7220 7468  o a file..For th
│ │ │ │ -00260a20: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ -00260a30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -00260a40: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ -00260a50: 6520 636c 6f73 654f 7574 3a20 636c 6f73  e closeOut: clos
│ │ │ │ -00260a60: 654f 7574 2c20 6973 2061 202a 6e6f 7465  eOut, is a *note
│ │ │ │ -00260a70: 206d 616e 6970 756c 6174 6f72 3a20 4d61   manipulator: Ma
│ │ │ │ -00260a80: 6e69 7075 6c61 746f 722c 2e0a 0a2d 2d2d  nipulator,...---
│ │ │ │ -00260a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00260aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00260ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00260360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00260370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00260380: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365  ----..The source
│ │ │ │ +00260390: 206f 6620 7468 6973 2064 6f63 756d 656e   of this documen
│ │ │ │ +002603a0: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72  t is in./build/r
│ │ │ │ +002603b0: 6570 726f 6475 6369 626c 652d 7061 7468  eproducible-path
│ │ │ │ +002603c0: 2f6d 6163 6175 6c61 7932 2d31 2e32 352e  /macaulay2-1.25.
│ │ │ │ +002603d0: 3131 2b64 732f 4d32 2f4d 6163 6175 6c61  11+ds/M2/Macaula
│ │ │ │ +002603e0: 7932 2f70 6163 6b61 6765 732f 0a4d 6163  y2/packages/.Mac
│ │ │ │ +002603f0: 6175 6c61 7932 446f 632f 6f76 5f73 7973  aulay2Doc/ov_sys
│ │ │ │ +00260400: 7465 6d2e 6d32 3a31 3531 363a 302e 0a1f  tem.m2:1516:0...
│ │ │ │ +00260410: 0a46 696c 653a 204d 6163 6175 6c61 7932  .File: Macaulay2
│ │ │ │ +00260420: 446f 632e 696e 666f 2c20 4e6f 6465 3a20  Doc.info, Node: 
│ │ │ │ +00260430: 636c 6f73 654f 7574 2c20 4e65 7874 3a20  closeOut, Next: 
│ │ │ │ +00260440: 6765 742c 2050 7265 763a 2063 6c6f 7365  get, Prev: close
│ │ │ │ +00260450: 496e 2c20 5570 3a20 7573 696e 6720 736f  In, Up: using so
│ │ │ │ +00260460: 636b 6574 730a 0a63 6c6f 7365 4f75 7420  ckets..closeOut 
│ │ │ │ +00260470: 2d2d 2063 6c6f 7365 2061 6e20 6f75 7470  -- close an outp
│ │ │ │ +00260480: 7574 2066 696c 650a 2a2a 2a2a 2a2a 2a2a  ut file.********
│ │ │ │ +00260490: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +002604a0: 2a2a 2a2a 2a2a 2a2a 0a0a 4465 7363 7269  ********..Descri
│ │ │ │ +002604b0: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ +002604c0: 3d0a 0a66 203c 3c20 636c 6f73 654f 7574  =..f << closeOut
│ │ │ │ +002604d0: 202d 2d20 636c 6f73 6573 2074 6865 206f   -- closes the o
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│ │ │ │ +002604f0: 6f73 654f 7574 2066 202d 2d20 636c 6f73  oseOut f -- clos
│ │ │ │ +00260500: 6573 2074 6865 206f 7574 7075 7420 6669  es the output fi
│ │ │ │ +00260510: 6c65 2066 2e0a 0a0a 416e 7920 6275 6666  le f....Any buff
│ │ │ │ +00260520: 6572 6564 206f 7574 7075 7420 6973 2066  ered output is f
│ │ │ │ +00260530: 6972 7374 2077 7269 7474 656e 2074 6f20  irst written to 
│ │ │ │ +00260540: 7468 6520 6669 6c65 2c20 616e 6420 7468  the file, and th
│ │ │ │ +00260550: 6520 7265 7475 726e 2076 616c 7565 2069  e return value i
│ │ │ │ +00260560: 7320 616e 0a69 6e74 6567 6572 2c20 6e6f  s an.integer, no
│ │ │ │ +00260570: 726d 616c 6c79 2030 2c20 6f72 202d 3120  rmally 0, or -1 
│ │ │ │ +00260580: 6f6e 2065 7272 6f72 2c20 6f72 2074 6865  on error, or the
│ │ │ │ +00260590: 2072 6574 7572 6e20 7374 6174 7573 206f   return status o
│ │ │ │ +002605a0: 6620 7468 6520 6368 696c 6420 7072 6f63  f the child proc
│ │ │ │ +002605b0: 6573 730a 696e 2063 6173 6520 7468 6520  ess.in case the 
│ │ │ │ +002605c0: 6669 6c65 2077 6173 2061 2070 6970 652e  file was a pipe.
│ │ │ │ +002605d0: 0a0a 0a49 6620 7468 6520 6669 6c65 2077  ...If the file w
│ │ │ │ +002605e0: 6173 206f 7065 6e20 6f6e 6c79 2066 6f72  as open only for
│ │ │ │ +002605f0: 206f 7574 7075 742c 2074 6865 6e20 2a6e   output, then *n
│ │ │ │ +00260600: 6f74 6520 636c 6f73 653a 2063 6c6f 7365  ote close: close
│ │ │ │ +00260610: 2c20 6973 2065 6173 6965 7220 746f 2075  , is easier to u
│ │ │ │ +00260620: 7365 0a61 6e64 2068 6173 2074 6865 2073  se.and has the s
│ │ │ │ +00260630: 616d 6520 6566 6665 6374 2e0a 0a0a 4966  ame effect....If
│ │ │ │ +00260640: 2074 6865 2066 696c 6520 7761 7320 6f70   the file was op
│ │ │ │ +00260650: 656e 2066 6f72 2062 6f74 6820 696e 7075  en for both inpu
│ │ │ │ +00260660: 7420 616e 6420 6f75 7470 7574 2c20 6974  t and output, it
│ │ │ │ +00260670: 2072 656d 6169 6e73 206f 7065 6e20 666f   remains open fo
│ │ │ │ +00260680: 7220 696e 7075 742e 0a0a 5761 7973 2074  r input...Ways t
│ │ │ │ +00260690: 6f20 7573 6520 636c 6f73 654f 7574 3a0a  o use closeOut:.
│ │ │ │ +002606a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +002606b0: 3d3d 3d3d 3d0a 0a20 202a 2022 4d61 6e69  =====..  * "Mani
│ │ │ │ +002606c0: 7075 6c61 746f 7220 4461 7461 6261 7365  pulator Database
│ │ │ │ +002606d0: 2220 2d2d 2073 6565 202a 6e6f 7465 204d  " -- see *note M
│ │ │ │ +002606e0: 616e 6970 756c 6174 6f72 3a20 4d61 6e69  anipulator: Mani
│ │ │ │ +002606f0: 7075 6c61 746f 722c 202d 2d20 7468 6520  pulator, -- the 
│ │ │ │ +00260700: 636c 6173 730a 2020 2020 6f66 2061 6c6c  class.    of all
│ │ │ │ +00260710: 2066 696c 6520 6d61 6e69 7075 6c61 746f   file manipulato
│ │ │ │ +00260720: 7273 0a20 202a 2022 4d61 6e69 7075 6c61  rs.  * "Manipula
│ │ │ │ +00260730: 746f 7220 4669 6c65 2220 2d2d 2073 6565  tor File" -- see
│ │ │ │ +00260740: 202a 6e6f 7465 204d 616e 6970 756c 6174   *note Manipulat
│ │ │ │ +00260750: 6f72 3a20 4d61 6e69 7075 6c61 746f 722c  or: Manipulator,
│ │ │ │ +00260760: 202d 2d20 7468 6520 636c 6173 7320 6f66   -- the class of
│ │ │ │ +00260770: 0a20 2020 2061 6c6c 2066 696c 6520 6d61  .    all file ma
│ │ │ │ +00260780: 6e69 7075 6c61 746f 7273 0a20 202a 2022  nipulators.  * "
│ │ │ │ +00260790: 4d61 6e69 7075 6c61 746f 7220 4e6f 7468  Manipulator Noth
│ │ │ │ +002607a0: 696e 6722 202d 2d20 7365 6520 2a6e 6f74  ing" -- see *not
│ │ │ │ +002607b0: 6520 4d61 6e69 7075 6c61 746f 723a 204d  e Manipulator: M
│ │ │ │ +002607c0: 616e 6970 756c 6174 6f72 2c20 2d2d 2074  anipulator, -- t
│ │ │ │ +002607d0: 6865 2063 6c61 7373 0a20 2020 206f 6620  he class.    of 
│ │ │ │ +002607e0: 616c 6c20 6669 6c65 206d 616e 6970 756c  all file manipul
│ │ │ │ +002607f0: 6174 6f72 730a 2020 2a20 226e 6577 204d  ators.  * "new M
│ │ │ │ +00260800: 616e 6970 756c 6174 6f72 2066 726f 6d20  anipulator from 
│ │ │ │ +00260810: 4675 6e63 7469 6f6e 2220 2d2d 2073 6565  Function" -- see
│ │ │ │ +00260820: 202a 6e6f 7465 204d 616e 6970 756c 6174   *note Manipulat
│ │ │ │ +00260830: 6f72 3a20 4d61 6e69 7075 6c61 746f 722c  or: Manipulator,
│ │ │ │ +00260840: 202d 2d0a 2020 2020 7468 6520 636c 6173   --.    the clas
│ │ │ │ +00260850: 7320 6f66 2061 6c6c 2066 696c 6520 6d61  s of all file ma
│ │ │ │ +00260860: 6e69 7075 6c61 746f 7273 0a20 202a 2022  nipulators.  * "
│ │ │ │ +00260870: 6d65 7468 6f64 7328 4d61 6e69 7075 6c61  methods(Manipula
│ │ │ │ +00260880: 746f 7229 2220 2d2d 2073 6565 202a 6e6f  tor)" -- see *no
│ │ │ │ +00260890: 7465 206d 6574 686f 6473 3a20 6d65 7468  te methods: meth
│ │ │ │ +002608a0: 6f64 732c 202d 2d20 6c69 7374 206d 6574  ods, -- list met
│ │ │ │ +002608b0: 686f 6473 0a20 202a 2022 4d61 6e69 7075  hods.  * "Manipu
│ │ │ │ +002608c0: 6c61 746f 7220 4e65 7446 696c 6522 202d  lator NetFile" -
│ │ │ │ +002608d0: 2d20 7365 6520 2a6e 6f74 6520 4e65 7446  - see *note NetF
│ │ │ │ +002608e0: 696c 653a 204e 6574 4669 6c65 2c20 2d2d  ile: NetFile, --
│ │ │ │ +002608f0: 2074 6865 2063 6c61 7373 206f 6620 616c   the class of al
│ │ │ │ +00260900: 6c0a 2020 2020 6e65 7420 6669 6c65 730a  l.    net files.
│ │ │ │ +00260910: 2020 2a20 224e 6574 4669 6c65 203c 3c20    * "NetFile << 
│ │ │ │ +00260920: 4d61 6e69 7075 6c61 746f 7222 202d 2d20  Manipulator" -- 
│ │ │ │ +00260930: 7365 6520 2a6e 6f74 6520 4e65 7446 696c  see *note NetFil
│ │ │ │ +00260940: 653a 204e 6574 4669 6c65 2c20 2d2d 2074  e: NetFile, -- t
│ │ │ │ +00260950: 6865 2063 6c61 7373 206f 6620 616c 6c0a  he class of all.
│ │ │ │ +00260960: 2020 2020 6e65 7420 6669 6c65 730a 2020      net files.  
│ │ │ │ +00260970: 2a20 2246 696c 6520 3c3c 204d 616e 6970  * "File << Manip
│ │ │ │ +00260980: 756c 6174 6f72 2220 2d2d 2073 6565 202a  ulator" -- see *
│ │ │ │ +00260990: 6e6f 7465 2070 7269 6e74 696e 6720 746f  note printing to
│ │ │ │ +002609a0: 2061 2066 696c 653a 2070 7269 6e74 696e   a file: printin
│ │ │ │ +002609b0: 6720 746f 2061 2066 696c 652c 0a20 2020  g to a file,.   
│ │ │ │ +002609c0: 202d 2d20 7072 696e 7420 746f 2061 2066   -- print to a f
│ │ │ │ +002609d0: 696c 650a 2020 2a20 224e 6f74 6869 6e67  ile.  * "Nothing
│ │ │ │ +002609e0: 203c 3c20 4d61 6e69 7075 6c61 746f 7222   << Manipulator"
│ │ │ │ +002609f0: 202d 2d20 7365 6520 2a6e 6f74 6520 7072   -- see *note pr
│ │ │ │ +00260a00: 696e 7469 6e67 2074 6f20 6120 6669 6c65  inting to a file
│ │ │ │ +00260a10: 3a20 7072 696e 7469 6e67 2074 6f20 610a  : printing to a.
│ │ │ │ +00260a20: 2020 2020 6669 6c65 2c20 2d2d 2070 7269      file, -- pri
│ │ │ │ +00260a30: 6e74 2074 6f20 6120 6669 6c65 0a0a 466f  nt to a file..Fo
│ │ │ │ +00260a40: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ +00260a50: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ +00260a60: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ +00260a70: 2a6e 6f74 6520 636c 6f73 654f 7574 3a20  *note closeOut: 
│ │ │ │ +00260a80: 636c 6f73 654f 7574 2c20 6973 2061 202a  closeOut, is a *
│ │ │ │ +00260a90: 6e6f 7465 206d 616e 6970 756c 6174 6f72  note manipulator
│ │ │ │ +00260aa0: 3a20 4d61 6e69 7075 6c61 746f 722c 2e0a  : Manipulator,..
│ │ │ │ +00260ab0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .---------------
│ │ │ │  00260ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00260ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468  ------------..Th
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│ │ │ │ -00260b50: 632f 6f76 5f73 7973 7465 6d2e 6d32 3a31  c/ov_system.m2:1
│ │ │ │ -00260b60: 3533 353a 302e 0a1f 0a46 696c 653a 204d  535:0....File: M
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│ │ │ │ -00260b80: 2c20 4e6f 6465 3a20 6765 742c 204e 6578  , Node: get, Nex
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│ │ │ │ -00260ba0: 6c6f 7365 4f75 742c 2055 703a 2075 7369  loseOut, Up: usi
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│ │ │ │ -00260bc0: 2d2d 2067 6574 2074 6865 2063 6f6e 7465  -- get the conte
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│ │ │ │ -00260be0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00260bf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ -00260c00: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
│ │ │ │ -00260c10: 2020 2020 2067 6574 2066 0a20 202a 2049       get f.  * I
│ │ │ │ -00260c20: 6e70 7574 733a 0a20 2020 2020 202a 2066  nputs:.      * f
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│ │ │ │ -00260c40: 4669 6c65 2c20 6f72 2061 202a 6e6f 7465  File, or a *note
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│ │ │ │ -00260c70: 696e 672c 0a20 2020 2020 2020 2074 6865  ing,.        the
│ │ │ │ -00260c80: 6e20 6974 2069 7320 6f70 656e 6564 2c20  n it is opened, 
│ │ │ │ -00260c90: 6173 2077 6974 6820 2a6e 6f74 6520 6f70  as with *note op
│ │ │ │ -00260ca0: 656e 496e 3a20 6f70 656e 496e 5f6c 7053  enIn: openIn_lpS
│ │ │ │ -00260cb0: 7472 696e 675f 7270 2c2e 0a20 2020 2020  tring_rp,..     
│ │ │ │ -00260cc0: 2020 2046 696c 656e 616d 6573 2073 7461     Filenames sta
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│ │ │ │ -00260ce0: 7769 7468 2024 2061 7265 2074 7265 6174  with $ are treat
│ │ │ │ -00260cf0: 6564 2073 7065 6369 616c 6c79 2c20 7365  ed specially, se
│ │ │ │ -00260d00: 6520 2a6e 6f74 650a 2020 2020 2020 2020  e *note.        
│ │ │ │ -00260d10: 6f70 656e 496e 4f75 743a 206f 7065 6e49  openInOut: openI
│ │ │ │ -00260d20: 6e4f 7574 2c2e 0a20 202a 204f 7574 7075  nOut,..  * Outpu
│ │ │ │ -00260d30: 7473 3a0a 2020 2020 2020 2a20 6120 7374  ts:.      * a st
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│ │ │ │ -00260d50: 7468 6520 636f 6e74 656e 7473 206f 6620  the contents of 
│ │ │ │ -00260d60: 7468 6520 6669 6c65 2e20 2049 6620 7468  the file.  If th
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│ │ │ │ -00260dc0: 7320 6f66 2074 6865 2066 696c 6520 6172  s of the file ar
│ │ │ │ -00260dd0: 650a 2020 2020 2020 2020 7265 7475 726e  e.        return
│ │ │ │ -00260de0: 6564 2e0a 2020 2a20 436f 6e73 6571 7565  ed..  * Conseque
│ │ │ │ -00260df0: 6e63 6573 3a0a 2020 2020 2020 2a20 496e  nces:.      * In
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│ │ │ │ -00260e10: 6620 6973 2061 2066 696c 652c 2069 7473  f is a file, its
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│ │ │ │ -00260e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00260e70: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
│ │ │ │ -00260e80: 2022 7465 7374 2d66 696c 6522 203c 3c20   "test-file" << 
│ │ │ │ -00260e90: 2268 6920 7468 6572 6522 203c 3c20 636c  "hi there" << cl
│ │ │ │ -00260ea0: 6f73 657c 0a7c 2020 2020 2020 2020 2020  ose|.|          
│ │ │ │ -00260eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00260ec0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00260ed0: 6f31 203d 2074 6573 742d 6669 6c65 2020  o1 = test-file  
│ │ │ │ +00260ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00260ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00260af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00260b00: 0a0a 5468 6520 736f 7572 6365 206f 6620  ..The source of 
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│ │ │ │ +00260b30: 6475 6369 626c 652d 7061 7468 2f6d 6163  ducible-path/mac
│ │ │ │ +00260b40: 6175 6c61 7932 2d31 2e32 352e 3131 2b64  aulay2-1.25.11+d
│ │ │ │ +00260b50: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70  s/M2/Macaulay2/p
│ │ │ │ +00260b60: 6163 6b61 6765 732f 0a4d 6163 6175 6c61  ackages/.Macaula
│ │ │ │ +00260b70: 7932 446f 632f 6f76 5f73 7973 7465 6d2e  y2Doc/ov_system.
│ │ │ │ +00260b80: 6d32 3a31 3533 353a 302e 0a1f 0a46 696c  m2:1535:0....Fil
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│ │ │ │ +00260ba0: 696e 666f 2c20 4e6f 6465 3a20 6765 742c  info, Node: get,
│ │ │ │ +00260bb0: 204e 6578 743a 2067 6574 632c 2050 7265   Next: getc, Pre
│ │ │ │ +00260bc0: 763a 2063 6c6f 7365 4f75 742c 2055 703a  v: closeOut, Up:
│ │ │ │ +00260bd0: 2075 7369 6e67 2073 6f63 6b65 7473 0a0a   using sockets..
│ │ │ │ +00260be0: 6765 7420 2d2d 2067 6574 2074 6865 2063  get -- get the c
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│ │ │ │ +00260c10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ +00260c30: 0a20 2020 2020 2020 2067 6574 2066 0a20  .        get f. 
│ │ │ │ +00260c40: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ +00260c50: 202a 2066 2c20 6120 2a6e 6f74 6520 6669   * f, a *note fi
│ │ │ │ +00260c60: 6c65 3a20 4669 6c65 2c20 6f72 2061 202a  le: File, or a *
│ │ │ │ +00260c70: 6e6f 7465 2073 7472 696e 673a 2053 7472  note string: Str
│ │ │ │ +00260c80: 696e 672c 2e20 2049 6620 6620 6973 2061  ing,.  If f is a
│ │ │ │ +00260c90: 2073 7472 696e 672c 0a20 2020 2020 2020   string,.       
│ │ │ │ +00260ca0: 2074 6865 6e20 6974 2069 7320 6f70 656e   then it is open
│ │ │ │ +00260cb0: 6564 2c20 6173 2077 6974 6820 2a6e 6f74  ed, as with *not
│ │ │ │ +00260cc0: 6520 6f70 656e 496e 3a20 6f70 656e 496e  e openIn: openIn
│ │ │ │ +00260cd0: 5f6c 7053 7472 696e 675f 7270 2c2e 0a20  _lpString_rp,.. 
│ │ │ │ +00260ce0: 2020 2020 2020 2046 696c 656e 616d 6573         Filenames
│ │ │ │ +00260cf0: 2073 7461 7274 696e 6720 7769 7468 2021   starting with !
│ │ │ │ +00260d00: 206f 7220 7769 7468 2024 2061 7265 2074   or with $ are t
│ │ │ │ +00260d10: 7265 6174 6564 2073 7065 6369 616c 6c79  reated specially
│ │ │ │ +00260d20: 2c20 7365 6520 2a6e 6f74 650a 2020 2020  , see *note.    
│ │ │ │ +00260d30: 2020 2020 6f70 656e 496e 4f75 743a 206f      openInOut: o
│ │ │ │ +00260d40: 7065 6e49 6e4f 7574 2c2e 0a20 202a 204f  penInOut,..  * O
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│ │ │ │ +00260d60: 6120 7374 7269 6e67 2063 6f6e 7461 696e  a string contain
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│ │ │ │ +00261020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +002610c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +002610d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002610e0: 2020 2020 2020 2020 207c 0a7c 6f34 203d           |.|o4 =
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│ │ │ │ +00261100: 323a 3438 2055 5443 2032 3032 3620 2020  2:48 UTC 2026   
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│ │ │ │ +00261400: 6320 2d2d 2067 6574 2061 2062 7974 650a  c -- get a byte.
│ │ │ │ +00261410: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ +00261450: 6279 7465 2066 726f 6d20 7468 6520 696e  byte from the in
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│ │ │ │ +00261480: 7374 7269 6e67 206f 660a 6c65 6e67 7468  string of.length
│ │ │ │ +00261490: 2031 2e20 204f 6e20 656e 6420 6f66 2066   1.  On end of f
│ │ │ │ +002614a0: 696c 6520 616e 2065 6d70 7479 2073 7472  ile an empty str
│ │ │ │ +002614b0: 696e 6720 6f66 2069 7320 7265 7475 726e  ing of is return
│ │ │ │ +002614c0: 6564 2e0a 0a0a 0a42 7567 3a20 7468 6520  ed.....Bug: the 
│ │ │ │ +002614d0: 6e61 6d65 2069 7320 6372 7970 7469 6320  name is cryptic 
│ │ │ │ +002614e0: 616e 6420 7368 6f75 6c64 2062 6520 6368  and should be ch
│ │ │ │ +002614f0: 616e 6765 642e 0a0a 5365 6520 616c 736f  anged...See also
│ │ │ │ +00261500: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  .========..  * *
│ │ │ │ +00261510: 6e6f 7465 2046 696c 653a 2046 696c 652c  note File: File,
│ │ │ │ +00261520: 202d 2d20 7468 6520 636c 6173 7320 6f66   -- the class of
│ │ │ │ +00261530: 2061 6c6c 2066 696c 6573 0a0a 5761 7973   all files..Ways
│ │ │ │ +00261540: 2074 6f20 7573 6520 6765 7463 3a0a 3d3d   to use getc:.==
│ │ │ │ +00261550: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +00261560: 0a20 202a 2022 6765 7463 2846 696c 6529  .  * "getc(File)
│ │ │ │ +00261570: 220a 0a46 6f72 2074 6865 2070 726f 6772  "..For the progr
│ │ │ │ +00261580: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d  ammer.==========
│ │ │ │ +00261590: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62  ========..The ob
│ │ │ │ +002615a0: 6a65 6374 202a 6e6f 7465 2067 6574 633a  ject *note getc:
│ │ │ │ +002615b0: 2067 6574 632c 2069 7320 6120 2a6e 6f74   getc, is a *not
│ │ │ │ +002615c0: 6520 636f 6d70 696c 6564 2066 756e 6374  e compiled funct
│ │ │ │ +002615d0: 696f 6e3a 2043 6f6d 7069 6c65 6446 756e  ion: CompiledFun
│ │ │ │ +002615e0: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d  ction,...-------
│ │ │ │  002615f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00261600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00261610: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365  ----..The source
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│ │ │ │ -00261640: 6570 726f 6475 6369 626c 652d 7061 7468  eproducible-path
│ │ │ │ -00261650: 2f6d 6163 6175 6c61 7932 2d31 2e32 352e  /macaulay2-1.25.
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│ │ │ │ -00261710: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -00261730: 0a2a 2072 6561 6428 5365 7175 656e 6365  .* read(Sequence
│ │ │ │ -00261740: 293a 2072 6561 645f 6c70 5365 7175 656e  ): read_lpSequen
│ │ │ │ -00261750: 6365 5f72 702e 2020 7265 6164 2066 726f  ce_rp.  read fro
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│ │ │ │ -00261770: 5374 7269 6e67 293a 2072 6561 645f 6c70  String): read_lp
│ │ │ │ -00261780: 5374 7269 6e67 5f72 702e 2020 7265 6164  String_rp.  read
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│ │ │ │ -002617a0: 6561 6428 4669 6c65 293a 2072 6561 645f  ead(File): read_
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│ │ │ │ -002617c0: 6420 6672 6f6d 2061 2066 696c 650a 2a20  d from a file.* 
│ │ │ │ -002617d0: 7265 6164 2846 696c 652c 5a5a 293a 2072  read(File,ZZ): r
│ │ │ │ -002617e0: 6561 645f 6c70 4669 6c65 5f63 6d5a 5a5f  ead_lpFile_cmZZ_
│ │ │ │ -002617f0: 7270 2e20 2072 6561 6420 6672 6f6d 2061  rp.  read from a
│ │ │ │ -00261800: 2066 696c 650a 0a57 6179 7320 746f 2075   file..Ways to u
│ │ │ │ -00261810: 7365 2072 6561 643a 0a3d 3d3d 3d3d 3d3d  se read:.=======
│ │ │ │ -00261820: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ -00261830: 2a6e 6f74 6520 7265 6164 2846 696c 6529  *note read(File)
│ │ │ │ -00261840: 3a20 7265 6164 5f6c 7046 696c 655f 7270  : read_lpFile_rp
│ │ │ │ -00261850: 2c20 2d2d 2072 6561 6420 6672 6f6d 2061  , -- read from a
│ │ │ │ -00261860: 2066 696c 650a 2020 2a20 2a6e 6f74 6520   file.  * *note 
│ │ │ │ -00261870: 7265 6164 2846 696c 652c 5a5a 293a 2072  read(File,ZZ): r
│ │ │ │ -00261880: 6561 645f 6c70 4669 6c65 5f63 6d5a 5a5f  ead_lpFile_cmZZ_
│ │ │ │ -00261890: 7270 2c20 2d2d 2072 6561 6420 6672 6f6d  rp, -- read from
│ │ │ │ -002618a0: 2061 2066 696c 650a 2020 2a20 2a6e 6f74   a file.  * *not
│ │ │ │ -002618b0: 6520 7265 6164 2853 6571 7565 6e63 6529  e read(Sequence)
│ │ │ │ -002618c0: 3a20 7265 6164 5f6c 7053 6571 7565 6e63  : read_lpSequenc
│ │ │ │ -002618d0: 655f 7270 2c20 2d2d 2072 6561 6420 6672  e_rp, -- read fr
│ │ │ │ -002618e0: 6f6d 2061 2066 696c 650a 2020 2a20 2a6e  om a file.  * *n
│ │ │ │ -002618f0: 6f74 6520 7265 6164 2853 7472 696e 6729  ote read(String)
│ │ │ │ -00261900: 3a20 7265 6164 5f6c 7053 7472 696e 675f  : read_lpString_
│ │ │ │ -00261910: 7270 2c20 2d2d 2072 6561 6420 6672 6f6d  rp, -- read from
│ │ │ │ -00261920: 2061 2066 696c 650a 0a46 6f72 2074 6865   a file..For the
│ │ │ │ -00261930: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d   programmer.====
│ │ │ │ -00261940: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ -00261950: 5468 6520 6f62 6a65 6374 202a 6e6f 7465  The object *note
│ │ │ │ -00261960: 2072 6561 643a 2072 6561 642c 2069 7320   read: read, is 
│ │ │ │ -00261970: 6120 2a6e 6f74 6520 636f 6d70 696c 6564  a *note compiled
│ │ │ │ -00261980: 2066 756e 6374 696f 6e3a 2043 6f6d 7069   function: Compi
│ │ │ │ -00261990: 6c65 6446 756e 6374 696f 6e2c 2e0a 0a2d  ledFunction,...-
│ │ │ │ -002619a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -002619b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -002619c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00261610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00261620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00261630: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f  --------..The so
│ │ │ │ +00261640: 7572 6365 206f 6620 7468 6973 2064 6f63  urce of this doc
│ │ │ │ +00261650: 756d 656e 7420 6973 2069 6e0a 2f62 7569  ument is in./bui
│ │ │ │ +00261660: 6c64 2f72 6570 726f 6475 6369 626c 652d  ld/reproducible-
│ │ │ │ +00261670: 7061 7468 2f6d 6163 6175 6c61 7932 2d31  path/macaulay2-1
│ │ │ │ +00261680: 2e32 352e 3131 2b64 732f 4d32 2f4d 6163  .25.11+ds/M2/Mac
│ │ │ │ +00261690: 6175 6c61 7932 2f70 6163 6b61 6765 732f  aulay2/packages/
│ │ │ │ +002616a0: 0a4d 6163 6175 6c61 7932 446f 632f 6f76  .Macaulay2Doc/ov
│ │ │ │ +002616b0: 5f73 7973 7465 6d2e 6d32 3a35 3438 3a30  _system.m2:548:0
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│ │ │ │ +00261700: 6765 7463 2c20 5570 3a20 7573 696e 6720  getc, Up: using 
│ │ │ │ +00261710: 736f 636b 6574 730a 0a72 6561 6420 2d2d  sockets..read --
│ │ │ │ +00261720: 2072 6561 6420 6672 6f6d 2061 2066 696c   read from a fil
│ │ │ │ +00261730: 650a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  e.**************
│ │ │ │ +00261740: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2a20 4d65  **********..* Me
│ │ │ │ +00261750: 6e75 3a0a 0a2a 2072 6561 6428 5365 7175  nu:..* read(Sequ
│ │ │ │ +00261760: 656e 6365 293a 2072 6561 645f 6c70 5365  ence): read_lpSe
│ │ │ │ +00261770: 7175 656e 6365 5f72 702e 2020 7265 6164  quence_rp.  read
│ │ │ │ +00261780: 2066 726f 6d20 6120 6669 6c65 0a2a 2072   from a file.* r
│ │ │ │ +00261790: 6561 6428 5374 7269 6e67 293a 2072 6561  ead(String): rea
│ │ │ │ +002617a0: 645f 6c70 5374 7269 6e67 5f72 702e 2020  d_lpString_rp.  
│ │ │ │ +002617b0: 7265 6164 2066 726f 6d20 6120 6669 6c65  read from a file
│ │ │ │ +002617c0: 0a2a 2072 6561 6428 4669 6c65 293a 2072  .* read(File): r
│ │ │ │ +002617d0: 6561 645f 6c70 4669 6c65 5f72 702e 2020  ead_lpFile_rp.  
│ │ │ │ +002617e0: 2072 6561 6420 6672 6f6d 2061 2066 696c   read from a fil
│ │ │ │ +002617f0: 650a 2a20 7265 6164 2846 696c 652c 5a5a  e.* read(File,ZZ
│ │ │ │ +00261800: 293a 2072 6561 645f 6c70 4669 6c65 5f63  ): read_lpFile_c
│ │ │ │ +00261810: 6d5a 5a5f 7270 2e20 2072 6561 6420 6672  mZZ_rp.  read fr
│ │ │ │ +00261820: 6f6d 2061 2066 696c 650a 0a57 6179 7320  om a file..Ways 
│ │ │ │ +00261830: 746f 2075 7365 2072 6561 643a 0a3d 3d3d  to use read:.===
│ │ │ │ +00261840: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +00261850: 2020 2a20 2a6e 6f74 6520 7265 6164 2846    * *note read(F
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│ │ │ │ +00261870: 655f 7270 2c20 2d2d 2072 6561 6420 6672  e_rp, -- read fr
│ │ │ │ +00261880: 6f6d 2061 2066 696c 650a 2020 2a20 2a6e  om a file.  * *n
│ │ │ │ +00261890: 6f74 6520 7265 6164 2846 696c 652c 5a5a  ote read(File,ZZ
│ │ │ │ +002618a0: 293a 2072 6561 645f 6c70 4669 6c65 5f63  ): read_lpFile_c
│ │ │ │ +002618b0: 6d5a 5a5f 7270 2c20 2d2d 2072 6561 6420  mZZ_rp, -- read 
│ │ │ │ +002618c0: 6672 6f6d 2061 2066 696c 650a 2020 2a20  from a file.  * 
│ │ │ │ +002618d0: 2a6e 6f74 6520 7265 6164 2853 6571 7565  *note read(Seque
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│ │ │ │ +002618f0: 7565 6e63 655f 7270 2c20 2d2d 2072 6561  uence_rp, -- rea
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│ │ │ │ +00261910: 2a20 2a6e 6f74 6520 7265 6164 2853 7472  * *note read(Str
│ │ │ │ +00261920: 696e 6729 3a20 7265 6164 5f6c 7053 7472  ing): read_lpStr
│ │ │ │ +00261930: 696e 675f 7270 2c20 2d2d 2072 6561 6420  ing_rp, -- read 
│ │ │ │ +00261940: 6672 6f6d 2061 2066 696c 650a 0a46 6f72  from a file..For
│ │ │ │ +00261950: 2074 6865 2070 726f 6772 616d 6d65 720a   the programmer.
│ │ │ │ +00261960: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00261970: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a  ==..The object *
│ │ │ │ +00261980: 6e6f 7465 2072 6561 643a 2072 6561 642c  note read: read,
│ │ │ │ +00261990: 2069 7320 6120 2a6e 6f74 6520 636f 6d70   is a *note comp
│ │ │ │ +002619a0: 696c 6564 2066 756e 6374 696f 6e3a 2043  iled function: C
│ │ │ │ +002619b0: 6f6d 7069 6c65 6446 756e 6374 696f 6e2c  ompiledFunction,
│ │ │ │ +002619c0: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...-------------
│ │ │ │  002619d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -002619e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ -002619f0: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
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│ │ │ │ -00261a10: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475  n./build/reprodu
│ │ │ │ -00261a20: 6369 626c 652d 7061 7468 2f6d 6163 6175  cible-path/macau
│ │ │ │ -00261a30: 6c61 7932 2d31 2e32 352e 3131 2b64 732f  lay2-1.25.11+ds/
│ │ │ │ -00261a40: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163  M2/Macaulay2/pac
│ │ │ │ -00261a50: 6b61 6765 732f 0a4d 6163 6175 6c61 7932  kages/.Macaulay2
│ │ │ │ -00261a60: 446f 632f 6f76 5f73 7973 7465 6d2e 6d32  Doc/ov_system.m2
│ │ │ │ -00261a70: 3a32 3632 3a30 2e0a 1f0a 4669 6c65 3a20  :262:0....File: 
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│ │ │ │ -00261aa0: 5365 7175 656e 6365 5f72 702c 204e 6578  Sequence_rp, Nex
│ │ │ │ -00261ab0: 743a 2072 6561 645f 6c70 5374 7269 6e67  t: read_lpString
│ │ │ │ -00261ac0: 5f72 702c 2055 703a 2072 6561 640a 0a72  _rp, Up: read..r
│ │ │ │ -00261ad0: 6561 6428 5365 7175 656e 6365 2920 2d2d  ead(Sequence) --
│ │ │ │ -00261ae0: 2072 6561 6420 6672 6f6d 2061 2066 696c   read from a fil
│ │ │ │ -00261af0: 650a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  e.**************
│ │ │ │ -00261b00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00261b10: 2a2a 2a2a 0a0a 2020 2a20 4675 6e63 7469  ****..  * Functi
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│ │ │ │ -00261b30: 7265 6164 2c0a 2020 2a20 5573 6167 653a  read,.  * Usage:
│ │ │ │ -00261b40: 200a 2020 2020 2020 2020 7265 6164 2829   .        read()
│ │ │ │ -00261b50: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ -00261b60: 2020 202a 2061 202a 6e6f 7465 2073 6571     * a *note seq
│ │ │ │ -00261b70: 7565 6e63 653a 2053 6571 7565 6e63 652c  uence: Sequence,
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│ │ │ │ -00261b90: 3a0a 2020 2020 2020 2a20 6120 2a6e 6f74  :.      * a *not
│ │ │ │ -00261ba0: 6520 7374 7269 6e67 3a20 5374 7269 6e67  e string: String
│ │ │ │ -00261bb0: 2c2c 2061 2073 7472 696e 6720 6f62 7461  ,, a string obta
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│ │ │ │ -00261bd0: 6120 6c69 6e65 2066 726f 6d20 7468 650a  a line from the.
│ │ │ │ -00261be0: 2020 2020 2020 2020 7374 616e 6461 7264          standard
│ │ │ │ -00261bf0: 2069 6e70 7574 2066 696c 652c 202a 6e6f   input file, *no
│ │ │ │ -00261c00: 7465 2073 7464 696f 3a20 7374 6469 6f2c  te stdio: stdio,
│ │ │ │ -00261c10: 2e0a 0a57 6179 7320 746f 2075 7365 2074  ...Ways to use t
│ │ │ │ -00261c20: 6869 7320 6d65 7468 6f64 3a0a 3d3d 3d3d  his method:.====
│ │ │ │ -00261c30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00261c40: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ -00261c50: 7265 6164 2853 6571 7565 6e63 6529 3a20  read(Sequence): 
│ │ │ │ -00261c60: 7265 6164 5f6c 7053 6571 7565 6e63 655f  read_lpSequence_
│ │ │ │ -00261c70: 7270 2c20 2d2d 2072 6561 6420 6672 6f6d  rp, -- read from
│ │ │ │ -00261c80: 2061 2066 696c 650a 2d2d 2d2d 2d2d 2d2d   a file.--------
│ │ │ │ -00261c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00261ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002619e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002619f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00261a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00261a10: 2d2d 0a0a 5468 6520 736f 7572 6365 206f  --..The source o
│ │ │ │ +00261a20: 6620 7468 6973 2064 6f63 756d 656e 7420  f this document 
│ │ │ │ +00261a30: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570  is in./build/rep
│ │ │ │ +00261a40: 726f 6475 6369 626c 652d 7061 7468 2f6d  roducible-path/m
│ │ │ │ +00261a50: 6163 6175 6c61 7932 2d31 2e32 352e 3131  acaulay2-1.25.11
│ │ │ │ +00261a60: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932  +ds/M2/Macaulay2
│ │ │ │ +00261a70: 2f70 6163 6b61 6765 732f 0a4d 6163 6175  /packages/.Macau
│ │ │ │ +00261a80: 6c61 7932 446f 632f 6f76 5f73 7973 7465  lay2Doc/ov_syste
│ │ │ │ +00261a90: 6d2e 6d32 3a32 3632 3a30 2e0a 1f0a 4669  m.m2:262:0....Fi
│ │ │ │ +00261aa0: 6c65 3a20 4d61 6361 756c 6179 3244 6f63  le: Macaulay2Doc
│ │ │ │ +00261ab0: 2e69 6e66 6f2c 204e 6f64 653a 2072 6561  .info, Node: rea
│ │ │ │ +00261ac0: 645f 6c70 5365 7175 656e 6365 5f72 702c  d_lpSequence_rp,
│ │ │ │ +00261ad0: 204e 6578 743a 2072 6561 645f 6c70 5374   Next: read_lpSt
│ │ │ │ +00261ae0: 7269 6e67 5f72 702c 2055 703a 2072 6561  ring_rp, Up: rea
│ │ │ │ +00261af0: 640a 0a72 6561 6428 5365 7175 656e 6365  d..read(Sequence
│ │ │ │ +00261b00: 2920 2d2d 2072 6561 6420 6672 6f6d 2061  ) -- read from a
│ │ │ │ +00261b10: 2066 696c 650a 2a2a 2a2a 2a2a 2a2a 2a2a   file.**********
│ │ │ │ +00261b20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00261b30: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 4675  ********..  * Fu
│ │ │ │ +00261b40: 6e63 7469 6f6e 3a20 2a6e 6f74 6520 7265  nction: *note re
│ │ │ │ +00261b50: 6164 3a20 7265 6164 2c0a 2020 2a20 5573  ad: read,.  * Us
│ │ │ │ +00261b60: 6167 653a 200a 2020 2020 2020 2020 7265  age: .        re
│ │ │ │ +00261b70: 6164 2829 0a20 202a 2049 6e70 7574 733a  ad().  * Inputs:
│ │ │ │ +00261b80: 0a20 2020 2020 202a 2061 202a 6e6f 7465  .      * a *note
│ │ │ │ +00261b90: 2073 6571 7565 6e63 653a 2053 6571 7565   sequence: Seque
│ │ │ │ +00261ba0: 6e63 652c 2c20 2829 0a20 202a 204f 7574  nce,, ().  * Out
│ │ │ │ +00261bb0: 7075 7473 3a0a 2020 2020 2020 2a20 6120  puts:.      * a 
│ │ │ │ +00261bc0: 2a6e 6f74 6520 7374 7269 6e67 3a20 5374  *note string: St
│ │ │ │ +00261bd0: 7269 6e67 2c2c 2061 2073 7472 696e 6720  ring,, a string 
│ │ │ │ +00261be0: 6f62 7461 696e 6564 2062 7920 7265 6164  obtained by read
│ │ │ │ +00261bf0: 696e 6720 6120 6c69 6e65 2066 726f 6d20  ing a line from 
│ │ │ │ +00261c00: 7468 650a 2020 2020 2020 2020 7374 616e  the.        stan
│ │ │ │ +00261c10: 6461 7264 2069 6e70 7574 2066 696c 652c  dard input file,
│ │ │ │ +00261c20: 202a 6e6f 7465 2073 7464 696f 3a20 7374   *note stdio: st
│ │ │ │ +00261c30: 6469 6f2c 2e0a 0a57 6179 7320 746f 2075  dio,...Ways to u
│ │ │ │ +00261c40: 7365 2074 6869 7320 6d65 7468 6f64 3a0a  se this method:.
│ │ │ │ +00261c50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00261c60: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ +00261c70: 6f74 6520 7265 6164 2853 6571 7565 6e63  ote read(Sequenc
│ │ │ │ +00261c80: 6529 3a20 7265 6164 5f6c 7053 6571 7565  e): read_lpSeque
│ │ │ │ +00261c90: 6e63 655f 7270 2c20 2d2d 2072 6561 6420  nce_rp, -- read 
│ │ │ │ +00261ca0: 6672 6f6d 2061 2066 696c 650a 2d2d 2d2d  from a file.----
│ │ │ │  00261cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00261cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00261cd0: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75  -------..The sou
│ │ │ │ -00261ce0: 7263 6520 6f66 2074 6869 7320 646f 6375  rce of this docu
│ │ │ │ -00261cf0: 6d65 6e74 2069 7320 696e 0a2f 6275 696c  ment is in./buil
│ │ │ │ -00261d00: 642f 7265 7072 6f64 7563 6962 6c65 2d70  d/reproducible-p
│ │ │ │ -00261d10: 6174 682f 6d61 6361 756c 6179 322d 312e  ath/macaulay2-1.
│ │ │ │ -00261d20: 3235 2e31 312b 6473 2f4d 322f 4d61 6361  25.11+ds/M2/Maca
│ │ │ │ -00261d30: 756c 6179 322f 7061 636b 6167 6573 2f0a  ulay2/packages/.
│ │ │ │ -00261d40: 4d61 6361 756c 6179 3244 6f63 2f6f 765f  Macaulay2Doc/ov_
│ │ │ │ -00261d50: 7379 7374 656d 2e6d 323a 3237 323a 302e  system.m2:272:0.
│ │ │ │ -00261d60: 0a1f 0a46 696c 653a 204d 6163 6175 6c61  ...File: Macaula
│ │ │ │ -00261d70: 7932 446f 632e 696e 666f 2c20 4e6f 6465  y2Doc.info, Node
│ │ │ │ -00261d80: 3a20 7265 6164 5f6c 7053 7472 696e 675f  : read_lpString_
│ │ │ │ -00261d90: 7270 2c20 4e65 7874 3a20 7265 6164 5f6c  rp, Next: read_l
│ │ │ │ -00261da0: 7046 696c 655f 7270 2c20 5072 6576 3a20  pFile_rp, Prev: 
│ │ │ │ -00261db0: 7265 6164 5f6c 7053 6571 7565 6e63 655f  read_lpSequence_
│ │ │ │ -00261dc0: 7270 2c20 5570 3a20 7265 6164 0a0a 7265  rp, Up: read..re
│ │ │ │ -00261dd0: 6164 2853 7472 696e 6729 202d 2d20 7265  ad(String) -- re
│ │ │ │ -00261de0: 6164 2066 726f 6d20 6120 6669 6c65 0a2a  ad from a file.*
│ │ │ │ -00261df0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00261e00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ -00261e10: 0a20 202a 2046 756e 6374 696f 6e3a 202a  .  * Function: *
│ │ │ │ -00261e20: 6e6f 7465 2072 6561 643a 2072 6561 642c  note read: read,
│ │ │ │ -00261e30: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
│ │ │ │ -00261e40: 2020 2020 2072 6561 6420 700a 2020 2a20       read p.  * 
│ │ │ │ -00261e50: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ -00261e60: 702c 2061 202a 6e6f 7465 2073 7472 696e  p, a *note strin
│ │ │ │ -00261e70: 673a 2053 7472 696e 672c 2c20 6120 7374  g: String,, a st
│ │ │ │ -00261e80: 7269 6e67 2063 6f6e 7461 696e 696e 6720  ring containing 
│ │ │ │ -00261e90: 6120 7072 6f6d 7074 2074 6f20 6265 0a20  a prompt to be. 
│ │ │ │ -00261ea0: 2020 2020 2020 2064 6973 706c 6179 6564         displayed
│ │ │ │ -00261eb0: 2066 6f72 2074 6865 2075 7365 720a 2020   for the user.  
│ │ │ │ -00261ec0: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ -00261ed0: 202a 2061 202a 6e6f 7465 2073 7472 696e   * a *note strin
│ │ │ │ -00261ee0: 673a 2053 7472 696e 672c 2c20 6120 7374  g: String,, a st
│ │ │ │ -00261ef0: 7269 6e67 206f 6274 6169 6e65 6420 6279  ring obtained by
│ │ │ │ -00261f00: 2072 6561 6469 6e67 2066 726f 6d20 7468   reading from th
│ │ │ │ -00261f10: 6520 7374 616e 6461 7264 0a20 2020 2020  e standard.     
│ │ │ │ -00261f20: 2020 2069 6e70 7574 2066 696c 6520 2a6e     input file *n
│ │ │ │ -00261f30: 6f74 6520 7374 6469 6f3a 2073 7464 696f  ote stdio: stdio
│ │ │ │ -00261f40: 2c0a 0a57 6179 7320 746f 2075 7365 2074  ,..Ways to use t
│ │ │ │ -00261f50: 6869 7320 6d65 7468 6f64 3a0a 3d3d 3d3d  his method:.====
│ │ │ │ -00261f60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00261f70: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ -00261f80: 7265 6164 2853 7472 696e 6729 3a20 7265  read(String): re
│ │ │ │ -00261f90: 6164 5f6c 7053 7472 696e 675f 7270 2c20  ad_lpString_rp, 
│ │ │ │ -00261fa0: 2d2d 2072 6561 6420 6672 6f6d 2061 2066  -- read from a f
│ │ │ │ -00261fb0: 696c 650a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ile.------------
│ │ │ │ -00261fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00261fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00261cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00261ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00261cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
│ │ │ │ +00261d00: 2073 6f75 7263 6520 6f66 2074 6869 7320   source of this 
│ │ │ │ +00261d10: 646f 6375 6d65 6e74 2069 7320 696e 0a2f  document is in./
│ │ │ │ +00261d20: 6275 696c 642f 7265 7072 6f64 7563 6962  build/reproducib
│ │ │ │ +00261d30: 6c65 2d70 6174 682f 6d61 6361 756c 6179  le-path/macaulay
│ │ │ │ +00261d40: 322d 312e 3235 2e31 312b 6473 2f4d 322f  2-1.25.11+ds/M2/
│ │ │ │ +00261d50: 4d61 6361 756c 6179 322f 7061 636b 6167  Macaulay2/packag
│ │ │ │ +00261d60: 6573 2f0a 4d61 6361 756c 6179 3244 6f63  es/.Macaulay2Doc
│ │ │ │ +00261d70: 2f6f 765f 7379 7374 656d 2e6d 323a 3237  /ov_system.m2:27
│ │ │ │ +00261d80: 323a 302e 0a1f 0a46 696c 653a 204d 6163  2:0....File: Mac
│ │ │ │ +00261d90: 6175 6c61 7932 446f 632e 696e 666f 2c20  aulay2Doc.info, 
│ │ │ │ +00261da0: 4e6f 6465 3a20 7265 6164 5f6c 7053 7472  Node: read_lpStr
│ │ │ │ +00261db0: 696e 675f 7270 2c20 4e65 7874 3a20 7265  ing_rp, Next: re
│ │ │ │ +00261dc0: 6164 5f6c 7046 696c 655f 7270 2c20 5072  ad_lpFile_rp, Pr
│ │ │ │ +00261dd0: 6576 3a20 7265 6164 5f6c 7053 6571 7565  ev: read_lpSeque
│ │ │ │ +00261de0: 6e63 655f 7270 2c20 5570 3a20 7265 6164  nce_rp, Up: read
│ │ │ │ +00261df0: 0a0a 7265 6164 2853 7472 696e 6729 202d  ..read(String) -
│ │ │ │ +00261e00: 2d20 7265 6164 2066 726f 6d20 6120 6669  - read from a fi
│ │ │ │ +00261e10: 6c65 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  le.*************
│ │ │ │ +00261e20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00261e30: 2a2a 2a0a 0a20 202a 2046 756e 6374 696f  ***..  * Functio
│ │ │ │ +00261e40: 6e3a 202a 6e6f 7465 2072 6561 643a 2072  n: *note read: r
│ │ │ │ +00261e50: 6561 642c 0a20 202a 2055 7361 6765 3a20  ead,.  * Usage: 
│ │ │ │ +00261e60: 0a20 2020 2020 2020 2072 6561 6420 700a  .        read p.
│ │ │ │ +00261e70: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ +00261e80: 2020 2a20 702c 2061 202a 6e6f 7465 2073    * p, a *note s
│ │ │ │ +00261e90: 7472 696e 673a 2053 7472 696e 672c 2c20  tring: String,, 
│ │ │ │ +00261ea0: 6120 7374 7269 6e67 2063 6f6e 7461 696e  a string contain
│ │ │ │ +00261eb0: 696e 6720 6120 7072 6f6d 7074 2074 6f20  ing a prompt to 
│ │ │ │ +00261ec0: 6265 0a20 2020 2020 2020 2064 6973 706c  be.        displ
│ │ │ │ +00261ed0: 6179 6564 2066 6f72 2074 6865 2075 7365  ayed for the use
│ │ │ │ +00261ee0: 720a 2020 2a20 4f75 7470 7574 733a 0a20  r.  * Outputs:. 
│ │ │ │ +00261ef0: 2020 2020 202a 2061 202a 6e6f 7465 2073       * a *note s
│ │ │ │ +00261f00: 7472 696e 673a 2053 7472 696e 672c 2c20  tring: String,, 
│ │ │ │ +00261f10: 6120 7374 7269 6e67 206f 6274 6169 6e65  a string obtaine
│ │ │ │ +00261f20: 6420 6279 2072 6561 6469 6e67 2066 726f  d by reading fro
│ │ │ │ +00261f30: 6d20 7468 6520 7374 616e 6461 7264 0a20  m the standard. 
│ │ │ │ +00261f40: 2020 2020 2020 2069 6e70 7574 2066 696c         input fil
│ │ │ │ +00261f50: 6520 2a6e 6f74 6520 7374 6469 6f3a 2073  e *note stdio: s
│ │ │ │ +00261f60: 7464 696f 2c0a 0a57 6179 7320 746f 2075  tdio,..Ways to u
│ │ │ │ +00261f70: 7365 2074 6869 7320 6d65 7468 6f64 3a0a  se this method:.
│ │ │ │ +00261f80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00261f90: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ +00261fa0: 6f74 6520 7265 6164 2853 7472 696e 6729  ote read(String)
│ │ │ │ +00261fb0: 3a20 7265 6164 5f6c 7053 7472 696e 675f  : read_lpString_
│ │ │ │ +00261fc0: 7270 2c20 2d2d 2072 6561 6420 6672 6f6d  rp, -- read from
│ │ │ │ +00261fd0: 2061 2066 696c 650a 2d2d 2d2d 2d2d 2d2d   a file.--------
│ │ │ │  00261fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00261ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00262000: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520  ---..The source 
│ │ │ │ -00262010: 6f66 2074 6869 7320 646f 6375 6d65 6e74  of this document
│ │ │ │ -00262020: 2069 7320 696e 0a2f 6275 696c 642f 7265   is in./build/re
│ │ │ │ -00262030: 7072 6f64 7563 6962 6c65 2d70 6174 682f  producible-path/
│ │ │ │ -00262040: 6d61 6361 756c 6179 322d 312e 3235 2e31  macaulay2-1.25.1
│ │ │ │ -00262050: 312b 6473 2f4d 322f 4d61 6361 756c 6179  1+ds/M2/Macaulay
│ │ │ │ -00262060: 322f 7061 636b 6167 6573 2f0a 4d61 6361  2/packages/.Maca
│ │ │ │ -00262070: 756c 6179 3244 6f63 2f6f 765f 7379 7374  ulay2Doc/ov_syst
│ │ │ │ -00262080: 656d 2e6d 323a 3238 323a 302e 0a1f 0a46  em.m2:282:0....F
│ │ │ │ -00262090: 696c 653a 204d 6163 6175 6c61 7932 446f  ile: Macaulay2Do
│ │ │ │ -002620a0: 632e 696e 666f 2c20 4e6f 6465 3a20 7265  c.info, Node: re
│ │ │ │ -002620b0: 6164 5f6c 7046 696c 655f 7270 2c20 4e65  ad_lpFile_rp, Ne
│ │ │ │ -002620c0: 7874 3a20 7265 6164 5f6c 7046 696c 655f  xt: read_lpFile_
│ │ │ │ -002620d0: 636d 5a5a 5f72 702c 2050 7265 763a 2072  cmZZ_rp, Prev: r
│ │ │ │ -002620e0: 6561 645f 6c70 5374 7269 6e67 5f72 702c  ead_lpString_rp,
│ │ │ │ -002620f0: 2055 703a 2072 6561 640a 0a72 6561 6428   Up: read..read(
│ │ │ │ -00262100: 4669 6c65 2920 2d2d 2072 6561 6420 6672  File) -- read fr
│ │ │ │ -00262110: 6f6d 2061 2066 696c 650a 2a2a 2a2a 2a2a  om a file.******
│ │ │ │ -00262120: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00262130: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 4675  ********..  * Fu
│ │ │ │ -00262140: 6e63 7469 6f6e 3a20 2a6e 6f74 6520 7265  nction: *note re
│ │ │ │ -00262150: 6164 3a20 7265 6164 2c0a 2020 2a20 5573  ad: read,.  * Us
│ │ │ │ -00262160: 6167 653a 200a 2020 2020 2020 2020 7265  age: .        re
│ │ │ │ -00262170: 6164 2066 0a20 202a 2049 6e70 7574 733a  ad f.  * Inputs:
│ │ │ │ -00262180: 0a20 2020 2020 202a 2066 2c20 6120 2a6e  .      * f, a *n
│ │ │ │ -00262190: 6f74 6520 6669 6c65 3a20 4669 6c65 2c2c  ote file: File,,
│ │ │ │ -002621a0: 2061 6e20 696e 7075 7420 6669 6c65 0a20   an input file. 
│ │ │ │ -002621b0: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ -002621c0: 2020 2a20 6120 2a6e 6f74 6520 7374 7269    * a *note stri
│ │ │ │ -002621d0: 6e67 3a20 5374 7269 6e67 2c2c 2061 2073  ng: String,, a s
│ │ │ │ -002621e0: 7472 696e 6720 6f62 7461 696e 6564 2062  tring obtained b
│ │ │ │ -002621f0: 7920 7265 6164 696e 6720 6672 6f6d 2066  y reading from f
│ │ │ │ -00262200: 2e0a 0a44 6573 6372 6970 7469 6f6e 0a3d  ...Description.=
│ │ │ │ -00262210: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2b2d 2d2d  ==========..+---
│ │ │ │ -00262220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00262230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00262240: 203a 2066 203d 206f 7065 6e49 6e4f 7574   : f = openInOut
│ │ │ │ -00262250: 2022 2163 6174 2220 2020 2020 7c0a 7c20   "!cat"     |.| 
│ │ │ │ -00262260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00262270: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00262280: 6f31 203d 2021 6361 7420 2020 2020 2020  o1 = !cat       
│ │ │ │ -00262290: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -002622a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -002622b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -002622c0: 0a7c 6f31 203a 2046 696c 6520 2020 2020  .|o1 : File     
│ │ │ │ +00262000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00262010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00262020: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75  -------..The sou
│ │ │ │ +00262030: 7263 6520 6f66 2074 6869 7320 646f 6375  rce of this docu
│ │ │ │ +00262040: 6d65 6e74 2069 7320 696e 0a2f 6275 696c  ment is in./buil
│ │ │ │ +00262050: 642f 7265 7072 6f64 7563 6962 6c65 2d70  d/reproducible-p
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│ │ │ │ +00262430: 6f34 203d 2066 616c 7365 2020 2020 2020  o4 = false      
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│ │ │ │ +00262450: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
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│ │ │ │ -00262650: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ -002626a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -002626b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -002626c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00262530: 2020 2020 207c 0a7c 6f36 203d 2066 616c       |.|o6 = fal
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│ │ │ │ -00262800: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -002629f0: 2e0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d  ...See also.====
│ │ │ │ -00262a00: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ -00262a10: 6f70 656e 496e 3a20 6f70 656e 496e 5f6c  openIn: openIn_l
│ │ │ │ -00262a20: 7053 7472 696e 675f 7270 2c20 2d2d 206f  pString_rp, -- o
│ │ │ │ -00262a30: 7065 6e20 616e 2069 6e70 7574 2066 696c  pen an input fil
│ │ │ │ -00262a40: 650a 2020 2a20 2a6e 6f74 6520 6765 743a  e.  * *note get:
│ │ │ │ -00262a50: 2067 6574 2c20 2d2d 2067 6574 2074 6865   get, -- get the
│ │ │ │ -00262a60: 2063 6f6e 7465 6e74 7320 6f66 2061 2066   contents of a f
│ │ │ │ -00262a70: 696c 650a 2020 2a20 2a6e 6f74 6520 6973  ile.  * *note is
│ │ │ │ -00262a80: 5265 6164 793a 2069 7352 6561 6479 5f6c  Ready: isReady_l
│ │ │ │ -00262a90: 7046 696c 655f 7270 2c20 2d2d 2077 6865  pFile_rp, -- whe
│ │ │ │ -00262aa0: 7468 6572 2061 2066 696c 6520 6861 7320  ther a file has 
│ │ │ │ -00262ab0: 6461 7461 2061 7661 696c 6162 6c65 2066  data available f
│ │ │ │ -00262ac0: 6f72 0a20 2020 2072 6561 6469 6e67 0a0a  or.    reading..
│ │ │ │ -00262ad0: 5761 7973 2074 6f20 7573 6520 7468 6973  Ways to use this
│ │ │ │ -00262ae0: 206d 6574 686f 643a 0a3d 3d3d 3d3d 3d3d   method:.=======
│ │ │ │ -00262af0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00262b00: 3d0a 0a20 202a 202a 6e6f 7465 2072 6561  =..  * *note rea
│ │ │ │ -00262b10: 6428 4669 6c65 2c5a 5a29 3a20 7265 6164  d(File,ZZ): read
│ │ │ │ -00262b20: 5f6c 7046 696c 655f 636d 5a5a 5f72 702c  _lpFile_cmZZ_rp,
│ │ │ │ -00262b30: 202d 2d20 7265 6164 2066 726f 6d20 6120   -- read from a 
│ │ │ │ -00262b40: 6669 6c65 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d  file.-----------
│ │ │ │ -00262b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00262b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002626e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002626f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00262700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00262710: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520  ---..The source 
│ │ │ │ +00262720: 6f66 2074 6869 7320 646f 6375 6d65 6e74  of this document
│ │ │ │ +00262730: 2069 7320 696e 0a2f 6275 696c 642f 7265   is in./build/re
│ │ │ │ +00262740: 7072 6f64 7563 6962 6c65 2d70 6174 682f  producible-path/
│ │ │ │ +00262750: 6d61 6361 756c 6179 322d 312e 3235 2e31  macaulay2-1.25.1
│ │ │ │ +00262760: 312b 6473 2f4d 322f 4d61 6361 756c 6179  1+ds/M2/Macaulay
│ │ │ │ +00262770: 322f 7061 636b 6167 6573 2f0a 4d61 6361  2/packages/.Maca
│ │ │ │ +00262780: 756c 6179 3244 6f63 2f6f 765f 7379 7374  ulay2Doc/ov_syst
│ │ │ │ +00262790: 656d 2e6d 323a 3330 313a 302e 0a1f 0a46  em.m2:301:0....F
│ │ │ │ +002627a0: 696c 653a 204d 6163 6175 6c61 7932 446f  ile: Macaulay2Do
│ │ │ │ +002627b0: 632e 696e 666f 2c20 4e6f 6465 3a20 7265  c.info, Node: re
│ │ │ │ +002627c0: 6164 5f6c 7046 696c 655f 636d 5a5a 5f72  ad_lpFile_cmZZ_r
│ │ │ │ +002627d0: 702c 2050 7265 763a 2072 6561 645f 6c70  p, Prev: read_lp
│ │ │ │ +002627e0: 4669 6c65 5f72 702c 2055 703a 2072 6561  File_rp, Up: rea
│ │ │ │ +002627f0: 640a 0a72 6561 6428 4669 6c65 2c5a 5a29  d..read(File,ZZ)
│ │ │ │ +00262800: 202d 2d20 7265 6164 2066 726f 6d20 6120   -- read from a 
│ │ │ │ +00262810: 6669 6c65 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a  file.***********
│ │ │ │ +00262820: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00262830: 2a2a 2a2a 2a2a 0a0a 2020 2a20 4675 6e63  ******..  * Func
│ │ │ │ +00262840: 7469 6f6e 3a20 2a6e 6f74 6520 7265 6164  tion: *note read
│ │ │ │ +00262850: 3a20 7265 6164 2c0a 2020 2a20 5573 6167  : read,.  * Usag
│ │ │ │ +00262860: 653a 200a 2020 2020 2020 2020 7265 6164  e: .        read
│ │ │ │ +00262870: 2866 2c6e 290a 2020 2a20 496e 7075 7473  (f,n).  * Inputs
│ │ │ │ +00262880: 3a0a 2020 2020 2020 2a20 662c 2061 202a  :.      * f, a *
│ │ │ │ +00262890: 6e6f 7465 2066 696c 653a 2046 696c 652c  note file: File,
│ │ │ │ +002628a0: 2c20 6120 6669 6c65 0a20 2020 2020 202a  , a file.      *
│ │ │ │ +002628b0: 206e 2c20 616e 202a 6e6f 7465 2069 6e74   n, an *note int
│ │ │ │ +002628c0: 6567 6572 3a20 5a5a 2c2c 2061 6e20 696e  eger: ZZ,, an in
│ │ │ │ +002628d0: 7465 6765 7220 7370 6563 6966 7969 6e67  teger specifying
│ │ │ │ +002628e0: 2074 6865 206d 6178 696d 756d 206e 756d   the maximum num
│ │ │ │ +002628f0: 6265 7220 6f66 0a20 2020 2020 2020 2062  ber of.        b
│ │ │ │ +00262900: 7974 6573 2074 6f20 7265 6164 0a20 202a  ytes to read.  *
│ │ │ │ +00262910: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ +00262920: 2a20 6120 2a6e 6f74 6520 7374 7269 6e67  * a *note string
│ │ │ │ +00262930: 3a20 5374 7269 6e67 2c2c 2061 2073 7472  : String,, a str
│ │ │ │ +00262940: 696e 6720 6f62 7461 696e 6564 2062 7920  ing obtained by 
│ │ │ │ +00262950: 7265 6164 696e 6720 6672 6f6d 2066 0a0a  reading from f..
│ │ │ │ +00262960: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ +00262970: 3d3d 3d3d 3d3d 3d0a 0a49 6e70 7574 2066  =======..Input f
│ │ │ │ +00262980: 696c 6573 2061 7265 2062 7566 6665 7265  iles are buffere
│ │ │ │ +00262990: 642c 2073 6f20 7468 6520 6375 7272 656e  d, so the curren
│ │ │ │ +002629a0: 7420 636f 6e74 656e 7473 206f 6620 7468  t contents of th
│ │ │ │ +002629b0: 6520 6275 6666 6572 2061 7265 2072 6574  e buffer are ret
│ │ │ │ +002629c0: 7572 6e65 6420 6966 0a74 6865 2062 7566  urned if.the buf
│ │ │ │ +002629d0: 6665 7220 6973 206e 6f74 2065 6d70 7479  fer is not empty
│ │ │ │ +002629e0: 2c20 6f74 6865 7277 6973 6520 7265 6164  , otherwise read
│ │ │ │ +002629f0: 696e 6720 6672 6f6d 2074 6865 2066 696c  ing from the fil
│ │ │ │ +00262a00: 6520 6973 2061 7474 656d 7074 6564 2066  e is attempted f
│ │ │ │ +00262a10: 6972 7374 2e0a 0a53 6565 2061 6c73 6f0a  irst...See also.
│ │ │ │ +00262a20: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ +00262a30: 6f74 6520 6f70 656e 496e 3a20 6f70 656e  ote openIn: open
│ │ │ │ +00262a40: 496e 5f6c 7053 7472 696e 675f 7270 2c20  In_lpString_rp, 
│ │ │ │ +00262a50: 2d2d 206f 7065 6e20 616e 2069 6e70 7574  -- open an input
│ │ │ │ +00262a60: 2066 696c 650a 2020 2a20 2a6e 6f74 6520   file.  * *note 
│ │ │ │ +00262a70: 6765 743a 2067 6574 2c20 2d2d 2067 6574  get: get, -- get
│ │ │ │ +00262a80: 2074 6865 2063 6f6e 7465 6e74 7320 6f66   the contents of
│ │ │ │ +00262a90: 2061 2066 696c 650a 2020 2a20 2a6e 6f74   a file.  * *not
│ │ │ │ +00262aa0: 6520 6973 5265 6164 793a 2069 7352 6561  e isReady: isRea
│ │ │ │ +00262ab0: 6479 5f6c 7046 696c 655f 7270 2c20 2d2d  dy_lpFile_rp, --
│ │ │ │ +00262ac0: 2077 6865 7468 6572 2061 2066 696c 6520   whether a file 
│ │ │ │ +00262ad0: 6861 7320 6461 7461 2061 7661 696c 6162  has data availab
│ │ │ │ +00262ae0: 6c65 2066 6f72 0a20 2020 2072 6561 6469  le for.    readi
│ │ │ │ +00262af0: 6e67 0a0a 5761 7973 2074 6f20 7573 6520  ng..Ways to use 
│ │ │ │ +00262b00: 7468 6973 206d 6574 686f 643a 0a3d 3d3d  this method:.===
│ │ │ │ +00262b10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00262b20: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ +00262b30: 2072 6561 6428 4669 6c65 2c5a 5a29 3a20   read(File,ZZ): 
│ │ │ │ +00262b40: 7265 6164 5f6c 7046 696c 655f 636d 5a5a  read_lpFile_cmZZ
│ │ │ │ +00262b50: 5f72 702c 202d 2d20 7265 6164 2066 726f  _rp, -- read fro
│ │ │ │ +00262b60: 6d20 6120 6669 6c65 0a2d 2d2d 2d2d 2d2d  m a file.-------
│ │ │ │  00262b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00262b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00262b90: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365  ----..The source
│ │ │ │ -00262ba0: 206f 6620 7468 6973 2064 6f63 756d 656e   of this documen
│ │ │ │ -00262bb0: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72  t is in./build/r
│ │ │ │ -00262bc0: 6570 726f 6475 6369 626c 652d 7061 7468  eproducible-path
│ │ │ │ -00262bd0: 2f6d 6163 6175 6c61 7932 2d31 2e32 352e  /macaulay2-1.25.
│ │ │ │ -00262be0: 3131 2b64 732f 4d32 2f4d 6163 6175 6c61  11+ds/M2/Macaula
│ │ │ │ -00262bf0: 7932 2f70 6163 6b61 6765 732f 0a4d 6163  y2/packages/.Mac
│ │ │ │ -00262c00: 6175 6c61 7932 446f 632f 6f76 5f73 7973  aulay2Doc/ov_sys
│ │ │ │ -00262c10: 7465 6d2e 6d32 3a33 3135 3a30 2e0a 1f0a  tem.m2:315:0....
│ │ │ │ -00262c20: 4669 6c65 3a20 4d61 6361 756c 6179 3244  File: Macaulay2D
│ │ │ │ -00262c30: 6f63 2e69 6e66 6f2c 204e 6f64 653a 2073  oc.info, Node: s
│ │ │ │ -00262c40: 6361 6e4c 696e 6573 2c20 4e65 7874 3a20  canLines, Next: 
│ │ │ │ -00262c50: 6669 6c65 4c65 6e67 7468 2c20 5072 6576  fileLength, Prev
│ │ │ │ -00262c60: 3a20 7265 6164 2c20 5570 3a20 7573 696e  : read, Up: usin
│ │ │ │ -00262c70: 6720 736f 636b 6574 730a 0a73 6361 6e4c  g sockets..scanL
│ │ │ │ -00262c80: 696e 6573 202d 2d20 6170 706c 7920 6120  ines -- apply a 
│ │ │ │ -00262c90: 6675 6e63 7469 6f6e 2074 6f20 6561 6368  function to each
│ │ │ │ -00262ca0: 206c 696e 6520 6f66 2061 2066 696c 650a   line of a file.
│ │ │ │ -00262cb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00262cc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00262cd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00262ce0: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a  ****..  * Usage:
│ │ │ │ -00262cf0: 200a 2020 2020 2020 2020 7363 616e 4c69   .        scanLi
│ │ │ │ -00262d00: 6e65 7328 662c 666e 290a 2020 2a20 496e  nes(f,fn).  * In
│ │ │ │ -00262d10: 7075 7473 3a0a 2020 2020 2020 2a20 660a  puts:.      * f.
│ │ │ │ -00262d20: 2020 2020 2020 2a20 666e 2c20 7468 6520        * fn, the 
│ │ │ │ -00262d30: 6e61 6d65 206f 6620 6120 6669 6c65 2c20  name of a file, 
│ │ │ │ -00262d40: 6f72 2061 206c 6973 7420 6f66 206e 616d  or a list of nam
│ │ │ │ -00262d50: 6573 206f 6620 6669 6c65 730a 2020 2a20  es of files.  * 
│ │ │ │ -00262d60: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ -00262d70: 2072 6574 7572 6e73 202a 6e6f 7465 206e   returns *note n
│ │ │ │ -00262d80: 756c 6c3a 206e 756c 6c2c 2075 6e6c 6573  ull: null, unles
│ │ │ │ -00262d90: 7320 7468 6520 6675 6e63 7469 6f6e 2075  s the function u
│ │ │ │ -00262da0: 7365 7320 6272 6561 6b20 7820 7769 7468  ses break x with
│ │ │ │ -00262db0: 2061 0a20 2020 2020 2020 206e 6f6e 2d6e   a.        non-n
│ │ │ │ -00262dc0: 756c 6c20 7661 6c75 6520 666f 7220 782c  ull value for x,
│ │ │ │ -00262dd0: 2069 6e20 7768 6963 6820 6361 7365 2073   in which case s
│ │ │ │ -00262de0: 6361 6e6e 696e 6720 7374 6f70 7320 616e  canning stops an
│ │ │ │ -00262df0: 6420 7820 6973 2072 6574 7572 6e65 640a  d x is returned.
│ │ │ │ -00262e00: 2020 2020 2020 2020 696d 6d65 6469 6174          immediat
│ │ │ │ -00262e10: 656c 790a 2020 2a20 436f 6e73 6571 7565  ely.  * Conseque
│ │ │ │ -00262e20: 6e63 6573 3a0a 2020 2020 2020 2a20 6170  nces:.      * ap
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│ │ │ │ -00262e40: 6c69 6e65 206f 6620 7468 6520 6669 6c65  line of the file
│ │ │ │ -00262e50: 2873 290a 0a44 6573 6372 6970 7469 6f6e  (s)..Description
│ │ │ │ -00262e60: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  .===========..Th
│ │ │ │ -00262e70: 6520 6669 6c65 2069 7320 7265 6164 2061  e file is read a
│ │ │ │ -00262e80: 6e64 2070 726f 6365 7373 6564 206f 6e65  nd processed one
│ │ │ │ -00262e90: 2062 6c6f 636b 2061 7420 6120 7469 6d65   block at a time
│ │ │ │ -00262ea0: 2c20 6d61 6b69 6e67 2074 6869 7320 7072  , making this pr
│ │ │ │ -00262eb0: 6f63 6564 7572 650a 706f 7465 6e74 6961  ocedure.potentia
│ │ │ │ -00262ec0: 6c6c 7920 6d75 6368 2062 6574 7465 7220  lly much better 
│ │ │ │ -00262ed0: 6174 2063 6f6e 7365 7276 696e 6720 6d65  at conserving me
│ │ │ │ -00262ee0: 6d6f 7279 2074 6861 6e20 7363 616e 286c  mory than scan(l
│ │ │ │ -00262ef0: 696e 6573 2067 6574 2066 6e2c 6629 2077  ines get fn,f) w
│ │ │ │ -00262f00: 6865 6e20 7468 650a 6669 6c65 2069 7320  hen the.file is 
│ │ │ │ -00262f10: 7665 7279 206c 6172 6765 2e0a 0a57 6179  very large...Way
│ │ │ │ -00262f20: 7320 746f 2075 7365 2073 6361 6e4c 696e  s to use scanLin
│ │ │ │ -00262f30: 6573 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  es:.============
│ │ │ │ -00262f40: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ -00262f50: 2273 6361 6e4c 696e 6573 2846 756e 6374  "scanLines(Funct
│ │ │ │ -00262f60: 696f 6e2c 4c69 7374 2922 0a20 202a 2022  ion,List)".  * "
│ │ │ │ -00262f70: 7363 616e 4c69 6e65 7328 4675 6e63 7469  scanLines(Functi
│ │ │ │ -00262f80: 6f6e 2c53 7472 696e 6729 220a 0a46 6f72  on,String)"..For
│ │ │ │ -00262f90: 2074 6865 2070 726f 6772 616d 6d65 720a   the programmer.
│ │ │ │ -00262fa0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00262fb0: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a  ==..The object *
│ │ │ │ -00262fc0: 6e6f 7465 2073 6361 6e4c 696e 6573 3a20  note scanLines: 
│ │ │ │ -00262fd0: 7363 616e 4c69 6e65 732c 2069 7320 6120  scanLines, is a 
│ │ │ │ -00262fe0: 2a6e 6f74 6520 6d65 7468 6f64 2066 756e  *note method fun
│ │ │ │ -00262ff0: 6374 696f 6e3a 0a4d 6574 686f 6446 756e  ction:.MethodFun
│ │ │ │ -00263000: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d  ction,...-------
│ │ │ │ -00263010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00263020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00262b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00262ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00262bb0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f  --------..The so
│ │ │ │ +00262bc0: 7572 6365 206f 6620 7468 6973 2064 6f63  urce of this doc
│ │ │ │ +00262bd0: 756d 656e 7420 6973 2069 6e0a 2f62 7569  ument is in./bui
│ │ │ │ +00262be0: 6c64 2f72 6570 726f 6475 6369 626c 652d  ld/reproducible-
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│ │ │ │ +00262c00: 2e32 352e 3131 2b64 732f 4d32 2f4d 6163  .25.11+ds/M2/Mac
│ │ │ │ +00262c10: 6175 6c61 7932 2f70 6163 6b61 6765 732f  aulay2/packages/
│ │ │ │ +00262c20: 0a4d 6163 6175 6c61 7932 446f 632f 6f76  .Macaulay2Doc/ov
│ │ │ │ +00262c30: 5f73 7973 7465 6d2e 6d32 3a33 3135 3a30  _system.m2:315:0
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│ │ │ │ +00262c50: 6179 3244 6f63 2e69 6e66 6f2c 204e 6f64  ay2Doc.info, Nod
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│ │ │ │ +00262c80: 5072 6576 3a20 7265 6164 2c20 5570 3a20  Prev: read, Up: 
│ │ │ │ +00262c90: 7573 696e 6720 736f 636b 6574 730a 0a73  using sockets..s
│ │ │ │ +00262ca0: 6361 6e4c 696e 6573 202d 2d20 6170 706c  canLines -- appl
│ │ │ │ +00262cb0: 7920 6120 6675 6e63 7469 6f6e 2074 6f20  y a function to 
│ │ │ │ +00262cc0: 6561 6368 206c 696e 6520 6f66 2061 2066  each line of a f
│ │ │ │ +00262cd0: 696c 650a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ile.************
│ │ │ │ +00262ce0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00262cf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00262d00: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573  ********..  * Us
│ │ │ │ +00262d10: 6167 653a 200a 2020 2020 2020 2020 7363  age: .        sc
│ │ │ │ +00262d20: 616e 4c69 6e65 7328 662c 666e 290a 2020  anLines(f,fn).  
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│ │ │ │ +00263cb0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  =========..  * *
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│ │ │ │  00263d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00263d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00263e30: 4669 6c65 2920 2d2d 2067 6574 2077 696e  File) -- get win
│ │ │ │ -00263e40: 646f 7720 7769 6474 680a 2a2a 2a2a 2a2a  dow width.******
│ │ │ │ -00263e50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00263e60: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2046  *********..  * F
│ │ │ │ -00263e70: 756e 6374 696f 6e3a 202a 6e6f 7465 2077  unction: *note w
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│ │ │ │ -00263ea0: 2077 6964 7468 2066 0a20 202a 2049 6e70   width f.  * Inp
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│ │ │ │ -00263ef0: 6520 696e 7465 6765 723a 205a 5a2c 2c20  e integer: ZZ,, 
│ │ │ │ -00263f00: 7468 6520 7769 6474 6820 6f66 2074 6865  the width of the
│ │ │ │ -00263f10: 2077 696e 646f 7720 6f72 2074 6572 6d69   window or termi
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│ │ │ │ -00263f60: 7468 6973 206d 6574 686f 643a 0a3d 3d3d  this method:.===
│ │ │ │ -00263f70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00263f80: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ -00263f90: 2077 6964 7468 2846 696c 6529 3a20 7769   width(File): wi
│ │ │ │ -00263fa0: 6474 685f 6c70 4669 6c65 5f72 702c 202d  dth_lpFile_rp, -
│ │ │ │ -00263fb0: 2d20 6765 7420 7769 6e64 6f77 2077 6964  - get window wid
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│ │ │ │ -00263fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00263fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00263d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00263d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00263d40: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073  ---------..The s
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│ │ │ │ +00263d80: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ +00263d90: 312e 3235 2e31 312b 6473 2f4d 322f 4d61  1.25.11+ds/M2/Ma
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│ │ │ │ +00263e50: 6474 6828 4669 6c65 2920 2d2d 2067 6574  dth(File) -- get
│ │ │ │ +00263e60: 2077 696e 646f 7720 7769 6474 680a 2a2a   window width.**
│ │ │ │ +00263e70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00263e80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20  *************.. 
│ │ │ │ +00263e90: 202a 2046 756e 6374 696f 6e3a 202a 6e6f   * Function: *no
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│ │ │ │ +00263eb0: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
│ │ │ │ +00263ec0: 2020 2020 2077 6964 7468 2066 0a20 202a       width f.  *
│ │ │ │ +00263ed0: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ +00263ee0: 2066 2c20 6120 2a6e 6f74 6520 6669 6c65   f, a *note file
│ │ │ │ +00263ef0: 3a20 4669 6c65 2c0a 2020 2a20 4f75 7470  : File,.  * Outp
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│ │ │ │ +00263f10: 2a6e 6f74 6520 696e 7465 6765 723a 205a  *note integer: Z
│ │ │ │ +00263f20: 5a2c 2c20 7468 6520 7769 6474 6820 6f66  Z,, the width of
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│ │ │ │ +00263f80: 7573 6520 7468 6973 206d 6574 686f 643a  use this method:
│ │ │ │ +00263f90: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ +00263fa0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  =========..  * *
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│ │ │ │  00263ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00264000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00264010: 2d2d 0a0a 5468 6520 736f 7572 6365 206f  --..The source o
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│ │ │ │ -002640b0: 632e 696e 666f 2c20 4e6f 6465 3a20 6174  c.info, Node: at
│ │ │ │ -002640c0: 456e 644f 6646 696c 655f 6c70 4669 6c65  EndOfFile_lpFile
│ │ │ │ -002640d0: 5f72 702c 204e 6578 743a 2065 6368 6f4f  _rp, Next: echoO
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│ │ │ │ -00264100: 696e 6720 736f 636b 6574 730a 0a61 7445  ing sockets..atE
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│ │ │ │ -00264120: 2d20 7465 7374 2066 6f72 2065 6e64 206f  - test for end o
│ │ │ │ -00264130: 6620 6669 6c65 0a2a 2a2a 2a2a 2a2a 2a2a  f file.*********
│ │ │ │ -00264140: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00264150: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00264160: 0a0a 2020 2a20 4675 6e63 7469 6f6e 3a20  ..  * Function: 
│ │ │ │ -00264170: 2a6e 6f74 6520 6174 456e 644f 6646 696c  *note atEndOfFil
│ │ │ │ -00264180: 653a 2061 7445 6e64 4f66 4669 6c65 5f6c  e: atEndOfFile_l
│ │ │ │ -00264190: 7046 696c 655f 7270 2c0a 2020 2a20 5573  pFile_rp,.  * Us
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│ │ │ │ -002641d0: 662c 2061 202a 6e6f 7465 2066 696c 653a  f, a *note file:
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│ │ │ │ -00264200: 6f74 6520 426f 6f6c 6561 6e20 7661 6c75  ote Boolean valu
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│ │ │ │ -00264230: 696c 6520 6620 6973 2061 7420 7468 6520  ile f is at the 
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│ │ │ │ -00264250: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2b2d  .===========..+-
│ │ │ │ -00264260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00264270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00264280: 2b0a 7c69 3120 3a20 6620 3d20 6f70 656e  +.|i1 : f = open
│ │ │ │ -00264290: 496e 4f75 7420 2221 6361 7422 2020 2020  InOut "!cat"    
│ │ │ │ -002642a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -002642b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002642c0: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20          |.|o1 = 
│ │ │ │ -002642d0: 2163 6174 2020 2020 2020 2020 2020 2020  !cat            
│ │ │ │ -002642e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -002642f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00264010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00264020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +002640f0: 4669 6c65 5f72 702c 204e 6578 743a 2065  File_rp, Next: e
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│ │ │ │ +00264160: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00264170: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00264180: 2a2a 2a2a 0a0a 2020 2a20 4675 6e63 7469  ****..  * Functi
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│ │ │ │ +002641a0: 6646 696c 653a 2061 7445 6e64 4f66 4669  fFile: atEndOfFi
│ │ │ │ +002641b0: 6c65 5f6c 7046 696c 655f 7270 2c0a 2020  le_lpFile_rp,.  
│ │ │ │ +002641c0: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
│ │ │ │ +002641d0: 2020 6174 456e 644f 6646 696c 6520 660a    atEndOfFile f.
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│ │ │ │ +002641f0: 2020 2a20 662c 2061 202a 6e6f 7465 2066    * f, a *note f
│ │ │ │ +00264200: 696c 653a 2046 696c 652c 0a20 202a 204f  ile: File,.  * O
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│ │ │ │ +00264220: 6120 2a6e 6f74 6520 426f 6f6c 6561 6e20  a *note Boolean 
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│ │ │ │ +00264260: 7468 6520 656e 640a 0a44 6573 6372 6970  the end..Descrip
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│ │ │ │ +00264280: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │ +00264290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002642a0: 2d2d 2d2d 2b0a 7c69 3120 3a20 6620 3d20  ----+.|i1 : f = 
│ │ │ │ +002642b0: 6f70 656e 496e 4f75 7420 2221 6361 7422  openInOut "!cat"
│ │ │ │ +002642c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +002642d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002642e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +002642f0: 3120 3d20 2163 6174 2020 2020 2020 2020  1 = !cat        
│ │ │ │  00264300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00264310: 7c0a 7c6f 3120 3a20 4669 6c65 2020 2020  |.|o1 : File    
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│ │ │ │ -002650c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -002650d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002650e0: 2020 2020 2020 207c 0a7c 6f35 203d 2068         |.|o5 = h
│ │ │ │ -002650f0: 6920 7468 6572 6520 2020 2020 2020 2020  i there         
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│ │ │ │ -00265110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00265130: 2d2b 0a7c 6936 203a 2072 656d 6f76 6546  -+.|i6 : removeF
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│ │ │ │ -00265150: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
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│ │ │ │ -002652a0: 636f 6e74 656e 7473 206f 6620 6120 6669  contents of a fi
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│ │ │ │ +00265110: 203d 2068 6920 7468 6572 6520 2020 2020   = hi there     
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│ │ │ │ +002651a0: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
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│ │ │ │ -00265560: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -002656d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002656e0: 2020 2020 2020 207c 0a7c 6f31 203d 2021         |.|o1 = !
│ │ │ │ -002656f0: 6361 7420 2020 2020 2020 2020 2020 2020  cat             
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│ │ │ │ -00265710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00265720: 2020 2020 2020 2020 207c 0a7c 6f31 203a           |.|o1 :
│ │ │ │ -00265730: 2046 696c 6520 2020 2020 2020 2020 2020   File           
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│ │ │ │ -00265790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002657a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -002657b0: 6f32 203d 2066 616c 7365 2020 2020 2020  o2 = false      
│ │ │ │ -002657c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -002657d0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
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│ │ │ │ -00265850: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00265860: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00265570: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ +002657f0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
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│ │ │ │ -00265890: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
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│ │ │ │ -002658f0: 6e6f 7465 2069 7352 6561 6479 2846 696c  note isReady(Fil
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│ │ │ │ -00265960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00265890: 2020 2020 2020 207c 0a7c 6f34 203d 2066         |.|o4 = f
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│ │ │ │ +002658c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002658d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5761 7973  ---------+..Ways
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│ │ │ │ +00265900: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ +00265910: 202a 202a 6e6f 7465 2069 7352 6561 6479   * *note isReady
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│ │ │ │ +00265930: 6c70 4669 6c65 5f72 702c 202d 2d20 7768  lpFile_rp, -- wh
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│ │ │ │  00265980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00265990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
│ │ │ │ -002659a0: 2073 6f75 7263 6520 6f66 2074 6869 7320   source of this 
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│ │ │ │ -00265a00: 6573 2f0a 4d61 6361 756c 6179 3244 6f63  es/.Macaulay2Doc
│ │ │ │ -00265a10: 2f6f 765f 6669 6c65 732e 6d32 3a33 3539  /ov_files.m2:359
│ │ │ │ -00265a20: 3a30 2e0a 1f0a 4669 6c65 3a20 4d61 6361  :0....File: Maca
│ │ │ │ -00265a30: 756c 6179 3244 6f63 2e69 6e66 6f2c 204e  ulay2Doc.info, N
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│ │ │ │ -00265a50: 655f 6c70 4669 6c65 5f72 702c 204e 6578  e_lpFile_rp, Nex
│ │ │ │ -00265a60: 743a 2069 7349 6e70 7574 4669 6c65 5f6c  t: isInputFile_l
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│ │ │ │ -00265a80: 6973 5265 6164 795f 6c70 4669 6c65 5f72  isReady_lpFile_r
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│ │ │ │ -00265ab0: 6c65 2846 696c 6529 202d 2d20 7768 6574  le(File) -- whet
│ │ │ │ -00265ac0: 6865 7220 6120 6669 6c65 2069 7320 6f70  her a file is op
│ │ │ │ -00265ad0: 656e 2066 6f72 206f 7574 7075 740a 2a2a  en for output.**
│ │ │ │ -00265ae0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00265af0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00265b00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00265b10: 2a2a 2a2a 2a0a 0a20 202a 2046 756e 6374  *****..  * Funct
│ │ │ │ -00265b20: 696f 6e3a 202a 6e6f 7465 2069 734f 7574  ion: *note isOut
│ │ │ │ -00265b30: 7075 7446 696c 653a 2069 734f 7574 7075  putFile: isOutpu
│ │ │ │ -00265b40: 7446 696c 655f 6c70 4669 6c65 5f72 702c  tFile_lpFile_rp,
│ │ │ │ -00265b50: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
│ │ │ │ -00265b60: 2020 2020 2069 734f 7574 7075 7446 696c       isOutputFil
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│ │ │ │ -00265ba0: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ -00265bb0: 2020 2a20 6120 2a6e 6f74 6520 426f 6f6c    * a *note Bool
│ │ │ │ -00265bc0: 6561 6e20 7661 6c75 653a 2042 6f6f 6c65  ean value: Boole
│ │ │ │ -00265bd0: 616e 2c2c 2020 7768 6574 6865 7220 6620  an,,  whether f 
│ │ │ │ -00265be0: 6973 2061 6e20 6f70 656e 206f 7574 7075  is an open outpu
│ │ │ │ -00265bf0: 7420 6669 6c65 0a0a 4465 7363 7269 7074  t file..Descript
│ │ │ │ -00265c00: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ion.===========.
│ │ │ │ -00265c10: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00265c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00265c30: 2d2d 2d2d 2b0a 7c69 3120 3a20 6620 3d20  ----+.|i1 : f = 
│ │ │ │ -00265c40: 2274 6573 742d 6669 6c65 2220 3c3c 2022  "test-file" << "
│ │ │ │ -00265c50: 6869 2074 6865 7265 227c 0a7c 2020 2020  hi there"|.|    
│ │ │ │ -00265c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00265c70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00265c80: 7c6f 3120 3d20 7465 7374 2d66 696c 6520  |o1 = test-file 
│ │ │ │ +00265990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002659a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002659b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a  ---------------.
│ │ │ │ +002659c0: 0a54 6865 2073 6f75 7263 6520 6f66 2074  .The source of t
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│ │ │ │ +002659e0: 696e 0a2f 6275 696c 642f 7265 7072 6f64  in./build/reprod
│ │ │ │ +002659f0: 7563 6962 6c65 2d70 6174 682f 6d61 6361  ucible-path/maca
│ │ │ │ +00265a00: 756c 6179 322d 312e 3235 2e31 312b 6473  ulay2-1.25.11+ds
│ │ │ │ +00265a10: 2f4d 322f 4d61 6361 756c 6179 322f 7061  /M2/Macaulay2/pa
│ │ │ │ +00265a20: 636b 6167 6573 2f0a 4d61 6361 756c 6179  ckages/.Macaulay
│ │ │ │ +00265a30: 3244 6f63 2f6f 765f 6669 6c65 732e 6d32  2Doc/ov_files.m2
│ │ │ │ +00265a40: 3a33 3539 3a30 2e0a 1f0a 4669 6c65 3a20  :359:0....File: 
│ │ │ │ +00265a50: 4d61 6361 756c 6179 3244 6f63 2e69 6e66  Macaulay2Doc.inf
│ │ │ │ +00265a60: 6f2c 204e 6f64 653a 2069 734f 7574 7075  o, Node: isOutpu
│ │ │ │ +00265a70: 7446 696c 655f 6c70 4669 6c65 5f72 702c  tFile_lpFile_rp,
│ │ │ │ +00265a80: 204e 6578 743a 2069 7349 6e70 7574 4669   Next: isInputFi
│ │ │ │ +00265a90: 6c65 5f6c 7046 696c 655f 7270 2c20 5072  le_lpFile_rp, Pr
│ │ │ │ +00265aa0: 6576 3a20 6973 5265 6164 795f 6c70 4669  ev: isReady_lpFi
│ │ │ │ +00265ab0: 6c65 5f72 702c 2055 703a 2075 7369 6e67  le_rp, Up: using
│ │ │ │ +00265ac0: 2073 6f63 6b65 7473 0a0a 6973 4f75 7470   sockets..isOutp
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│ │ │ │ +00265b10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00265b20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00265b30: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2046  *********..  * F
│ │ │ │ +00265b40: 756e 6374 696f 6e3a 202a 6e6f 7465 2069  unction: *note i
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│ │ │ │ +00265b60: 7574 7075 7446 696c 655f 6c70 4669 6c65  utputFile_lpFile
│ │ │ │ +00265b70: 5f72 702c 0a20 202a 2055 7361 6765 3a20  _rp,.  * Usage: 
│ │ │ │ +00265b80: 0a20 2020 2020 2020 2069 734f 7574 7075  .        isOutpu
│ │ │ │ +00265b90: 7446 696c 6520 660a 2020 2a20 496e 7075  tFile f.  * Inpu
│ │ │ │ +00265ba0: 7473 3a0a 2020 2020 2020 2a20 662c 2061  ts:.      * f, a
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│ │ │ │ +00265be0: 426f 6f6c 6561 6e20 7661 6c75 653a 2042  Boolean value: B
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│ │ │ │ +00265c20: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
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│ │ │ │ +00265c50: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ +00265c60: 6620 3d20 2274 6573 742d 6669 6c65 2220  f = "test-file" 
│ │ │ │ +00265c70: 3c3c 2022 6869 2074 6865 7265 227c 0a7c  << "hi there"|.|
│ │ │ │ +00265c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00265c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00265ca0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00265cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00265cd0: 4669 6c65 2020 2020 2020 2020 2020 2020  File            
│ │ │ │ -00265ce0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00265cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00265d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00265d10: 2d2d 2b0a 7c69 3220 3a20 6973 4f75 7470  --+.|i2 : isOutp
│ │ │ │ -00265d20: 7574 4669 6c65 2066 2020 2020 2020 2020  utFile f        
│ │ │ │ -00265d30: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00265d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00265d50: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00265d60: 3220 3d20 7472 7565 2020 2020 2020 2020  2 = true        
│ │ │ │ +00265ca0: 2020 7c0a 7c6f 3120 3d20 7465 7374 2d66    |.|o1 = test-f
│ │ │ │ +00265cb0: 696c 6520 2020 2020 2020 2020 2020 2020  ile             
│ │ │ │ +00265cc0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00265cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00265ce0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00265cf0: 3120 3a20 4669 6c65 2020 2020 2020 2020  1 : File        
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│ │ │ │ +00265d50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00265d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00265d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00265d80: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -00265d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00265db0: 6f73 6520 6620 2020 2020 2020 2020 2020  ose f           
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│ │ │ │ -00265dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00265de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00265df0: 7c0a 7c6f 3320 3d20 7465 7374 2d66 696c  |.|o3 = test-fil
│ │ │ │ -00265e00: 6520 2020 2020 2020 2020 2020 2020 2020  e               
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│ │ │ │ -00265e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00265f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -002661b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -002661c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00265f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00266360: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -002664c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002664d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00266510: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00266370: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00266380: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -00266570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -002667d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a  ---------+.|i5 :
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│ │ │ │ -002667f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00266730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00266740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +002667e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00266830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00266840: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00266860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00266870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00266880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00266890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00266a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00266a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00266a20: 2020 2020 207c 0a7c 6f38 203d 2066 616c       |.|o8 = fal
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│ │ │ │ -00266a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00266b60: 2a6e 6f74 6520 6765 743a 2067 6574 2c20  *note get: get, 
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│ │ │ │ -00266c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00266ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +002669d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002669e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002669f0: 2d2d 2d2d 2d2b 0a7c 6938 203a 2069 734f  -----+.|i8 : isO
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│ │ │ │ +00266a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00266a40: 2020 2020 2020 2020 207c 0a7c 6f38 203d           |.|o8 =
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│ │ │ │ +00266a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00266a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00266ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00266ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00266b30: 6f70 656e 2061 6e20 696e 7075 7420 6669  open an input fi
│ │ │ │ +00266b40: 6c65 0a20 202a 202a 6e6f 7465 206f 7065  le.  * *note ope
│ │ │ │ +00266b50: 6e49 6e4f 7574 3a20 6f70 656e 496e 4f75  nInOut: openInOu
│ │ │ │ +00266b60: 742c 202d 2d20 6f70 656e 2061 6e20 696e  t, -- open an in
│ │ │ │ +00266b70: 7075 7420 6f75 7470 7574 2066 696c 650a  put output file.
│ │ │ │ +00266b80: 2020 2a20 2a6e 6f74 6520 6765 743a 2067    * *note get: g
│ │ │ │ +00266b90: 6574 2c20 2d2d 2067 6574 2074 6865 2063  et, -- get the c
│ │ │ │ +00266ba0: 6f6e 7465 6e74 7320 6f66 2061 2066 696c  ontents of a fil
│ │ │ │ +00266bb0: 650a 2020 2a20 2a6e 6f74 6520 6973 4f70  e.  * *note isOp
│ │ │ │ +00266bc0: 656e 3a20 6973 4f70 656e 2c20 2d2d 2077  en: isOpen, -- w
│ │ │ │ +00266bd0: 6865 7468 6572 2061 2066 696c 6520 6f72  hether a file or
│ │ │ │ +00266be0: 2064 6174 6162 6173 6520 6973 206f 7065   database is ope
│ │ │ │ +00266bf0: 6e0a 2020 2a20 2a6e 6f74 6520 636c 6f73  n.  * *note clos
│ │ │ │ +00266c00: 653a 2063 6c6f 7365 2c20 2d2d 2063 6c6f  e: close, -- clo
│ │ │ │ +00266c10: 7365 2061 2066 696c 650a 0a57 6179 7320  se a file..Ways 
│ │ │ │ +00266c20: 746f 2075 7365 2074 6869 7320 6d65 7468  to use this meth
│ │ │ │ +00266c30: 6f64 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  od:.============
│ │ │ │ +00266c40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
│ │ │ │ +00266c50: 2a20 2a6e 6f74 6520 6973 496e 7075 7446  * *note isInputF
│ │ │ │ +00266c60: 696c 6528 4669 6c65 293a 2069 7349 6e70  ile(File): isInp
│ │ │ │ +00266c70: 7574 4669 6c65 5f6c 7046 696c 655f 7270  utFile_lpFile_rp
│ │ │ │ +00266c80: 2c20 2d2d 2077 6865 7468 6572 2061 2066  , -- whether a f
│ │ │ │ +00266c90: 696c 6520 6973 206f 7065 6e0a 2020 2020  ile is open.    
│ │ │ │ +00266ca0: 666f 7220 696e 7075 740a 2d2d 2d2d 2d2d  for input.------
│ │ │ │  00266cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00266cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00266cd0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263  -----..The sourc
│ │ │ │ -00266ce0: 6520 6f66 2074 6869 7320 646f 6375 6d65  e of this docume
│ │ │ │ -00266cf0: 6e74 2069 7320 696e 0a2f 6275 696c 642f  nt is in./build/
│ │ │ │ -00266d00: 7265 7072 6f64 7563 6962 6c65 2d70 6174  reproducible-pat
│ │ │ │ -00266d10: 682f 6d61 6361 756c 6179 322d 312e 3235  h/macaulay2-1.25
│ │ │ │ -00266d20: 2e31 312b 6473 2f4d 322f 4d61 6361 756c  .11+ds/M2/Macaul
│ │ │ │ -00266d30: 6179 322f 7061 636b 6167 6573 2f0a 4d61  ay2/packages/.Ma
│ │ │ │ -00266d40: 6361 756c 6179 3244 6f63 2f6f 765f 7379  caulay2Doc/ov_sy
│ │ │ │ -00266d50: 7374 656d 2e6d 323a 3537 303a 302e 0a1f  stem.m2:570:0...
│ │ │ │ -00266d60: 0a46 696c 653a 204d 6163 6175 6c61 7932  .File: Macaulay2
│ │ │ │ -00266d70: 446f 632e 696e 666f 2c20 4e6f 6465 3a20  Doc.info, Node: 
│ │ │ │ -00266d80: 6973 4c69 7374 656e 6572 5f6c 7046 696c  isListener_lpFil
│ │ │ │ -00266d90: 655f 7270 2c20 4e65 7874 3a20 6f70 656e  e_rp, Next: open
│ │ │ │ -00266da0: 4669 6c65 732c 2050 7265 763a 2069 7349  Files, Prev: isI
│ │ │ │ -00266db0: 6e70 7574 4669 6c65 5f6c 7046 696c 655f  nputFile_lpFile_
│ │ │ │ -00266dc0: 7270 2c20 5570 3a20 7573 696e 6720 736f  rp, Up: using so
│ │ │ │ -00266dd0: 636b 6574 730a 0a69 734c 6973 7465 6e65  ckets..isListene
│ │ │ │ -00266de0: 7228 4669 6c65 2920 2d2d 2077 6865 7468  r(File) -- wheth
│ │ │ │ -00266df0: 6572 2061 2066 696c 6520 6973 206f 7065  er a file is ope
│ │ │ │ -00266e00: 6e20 666f 7220 6c69 7374 656e 696e 670a  n for listening.
│ │ │ │ -00266e10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00266e20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00266e30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00266e40: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 4675  ********..  * Fu
│ │ │ │ -00266e50: 6e63 7469 6f6e 3a20 2a6e 6f74 6520 6973  nction: *note is
│ │ │ │ -00266e60: 4c69 7374 656e 6572 3a20 6973 4c69 7374  Listener: isList
│ │ │ │ -00266e70: 656e 6572 5f6c 7046 696c 655f 7270 2c0a  ener_lpFile_rp,.
│ │ │ │ -00266e80: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ -00266e90: 2020 2020 6973 4c69 7374 656e 6572 2066      isListener f
│ │ │ │ -00266ea0: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ -00266eb0: 2020 202a 2066 2c20 6120 2a6e 6f74 6520     * f, a *note 
│ │ │ │ -00266ec0: 6669 6c65 3a20 4669 6c65 2c0a 2020 2a20  file: File,.  * 
│ │ │ │ -00266ed0: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ -00266ee0: 2061 202a 6e6f 7465 2042 6f6f 6c65 616e   a *note Boolean
│ │ │ │ -00266ef0: 2076 616c 7565 3a20 426f 6f6c 6561 6e2c   value: Boolean,
│ │ │ │ -00266f00: 2c20 2077 6865 7468 6572 2066 2069 7320  ,  whether f is 
│ │ │ │ -00266f10: 616e 206f 7065 6e20 6c69 7374 656e 6572  an open listener
│ │ │ │ -00266f20: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ -00266f30: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 206f  ===..  * *note o
│ │ │ │ -00266f40: 7065 6e4c 6973 7465 6e65 723a 206f 7065  penListener: ope
│ │ │ │ -00266f50: 6e4c 6973 7465 6e65 725f 6c70 5374 7269  nListener_lpStri
│ │ │ │ -00266f60: 6e67 5f72 702c 202d 2d20 6f70 656e 2061  ng_rp, -- open a
│ │ │ │ -00266f70: 2070 6f72 7420 666f 7220 6c69 7374 656e   port for listen
│ │ │ │ -00266f80: 696e 670a 0a57 6179 7320 746f 2075 7365  ing..Ways to use
│ │ │ │ -00266f90: 2074 6869 7320 6d65 7468 6f64 3a0a 3d3d   this method:.==
│ │ │ │ -00266fa0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00266fb0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74  ======..  * *not
│ │ │ │ -00266fc0: 6520 6973 4c69 7374 656e 6572 2846 696c  e isListener(Fil
│ │ │ │ -00266fd0: 6529 3a20 6973 4c69 7374 656e 6572 5f6c  e): isListener_l
│ │ │ │ -00266fe0: 7046 696c 655f 7270 2c20 2d2d 2077 6865  pFile_rp, -- whe
│ │ │ │ -00266ff0: 7468 6572 2061 2066 696c 6520 6973 206f  ther a file is o
│ │ │ │ -00267000: 7065 6e20 666f 720a 2020 2020 6c69 7374  pen for.    list
│ │ │ │ -00267010: 656e 696e 670a 2d2d 2d2d 2d2d 2d2d 2d2d  ening.----------
│ │ │ │ -00267020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00267030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00266cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00266ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00266cf0: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073  ---------..The s
│ │ │ │ +00266d00: 6f75 7263 6520 6f66 2074 6869 7320 646f  ource of this do
│ │ │ │ +00266d10: 6375 6d65 6e74 2069 7320 696e 0a2f 6275  cument is in./bu
│ │ │ │ +00266d20: 696c 642f 7265 7072 6f64 7563 6962 6c65  ild/reproducible
│ │ │ │ +00266d30: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ +00266d40: 312e 3235 2e31 312b 6473 2f4d 322f 4d61  1.25.11+ds/M2/Ma
│ │ │ │ +00266d50: 6361 756c 6179 322f 7061 636b 6167 6573  caulay2/packages
│ │ │ │ +00266d60: 2f0a 4d61 6361 756c 6179 3244 6f63 2f6f  /.Macaulay2Doc/o
│ │ │ │ +00266d70: 765f 7379 7374 656d 2e6d 323a 3537 303a  v_system.m2:570:
│ │ │ │ +00266d80: 302e 0a1f 0a46 696c 653a 204d 6163 6175  0....File: Macau
│ │ │ │ +00266d90: 6c61 7932 446f 632e 696e 666f 2c20 4e6f  lay2Doc.info, No
│ │ │ │ +00266da0: 6465 3a20 6973 4c69 7374 656e 6572 5f6c  de: isListener_l
│ │ │ │ +00266db0: 7046 696c 655f 7270 2c20 4e65 7874 3a20  pFile_rp, Next: 
│ │ │ │ +00266dc0: 6f70 656e 4669 6c65 732c 2050 7265 763a  openFiles, Prev:
│ │ │ │ +00266dd0: 2069 7349 6e70 7574 4669 6c65 5f6c 7046   isInputFile_lpF
│ │ │ │ +00266de0: 696c 655f 7270 2c20 5570 3a20 7573 696e  ile_rp, Up: usin
│ │ │ │ +00266df0: 6720 736f 636b 6574 730a 0a69 734c 6973  g sockets..isLis
│ │ │ │ +00266e00: 7465 6e65 7228 4669 6c65 2920 2d2d 2077  tener(File) -- w
│ │ │ │ +00266e10: 6865 7468 6572 2061 2066 696c 6520 6973  hether a file is
│ │ │ │ +00266e20: 206f 7065 6e20 666f 7220 6c69 7374 656e   open for listen
│ │ │ │ +00266e30: 696e 670a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ing.************
│ │ │ │ +00266e40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00266e50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00266e60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020  ************..  
│ │ │ │ +00266e70: 2a20 4675 6e63 7469 6f6e 3a20 2a6e 6f74  * Function: *not
│ │ │ │ +00266e80: 6520 6973 4c69 7374 656e 6572 3a20 6973  e isListener: is
│ │ │ │ +00266e90: 4c69 7374 656e 6572 5f6c 7046 696c 655f  Listener_lpFile_
│ │ │ │ +00266ea0: 7270 2c0a 2020 2a20 5573 6167 653a 200a  rp,.  * Usage: .
│ │ │ │ +00266eb0: 2020 2020 2020 2020 6973 4c69 7374 656e          isListen
│ │ │ │ +00266ec0: 6572 2066 0a20 202a 2049 6e70 7574 733a  er f.  * Inputs:
│ │ │ │ +00266ed0: 0a20 2020 2020 202a 2066 2c20 6120 2a6e  .      * f, a *n
│ │ │ │ +00266ee0: 6f74 6520 6669 6c65 3a20 4669 6c65 2c0a  ote file: File,.
│ │ │ │ +00266ef0: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ +00266f00: 2020 202a 2061 202a 6e6f 7465 2042 6f6f     * a *note Boo
│ │ │ │ +00266f10: 6c65 616e 2076 616c 7565 3a20 426f 6f6c  lean value: Bool
│ │ │ │ +00266f20: 6561 6e2c 2c20 2077 6865 7468 6572 2066  ean,,  whether f
│ │ │ │ +00266f30: 2069 7320 616e 206f 7065 6e20 6c69 7374   is an open list
│ │ │ │ +00266f40: 656e 6572 0a0a 5365 6520 616c 736f 0a3d  ener..See also.=
│ │ │ │ +00266f50: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f  =======..  * *no
│ │ │ │ +00266f60: 7465 206f 7065 6e4c 6973 7465 6e65 723a  te openListener:
│ │ │ │ +00266f70: 206f 7065 6e4c 6973 7465 6e65 725f 6c70   openListener_lp
│ │ │ │ +00266f80: 5374 7269 6e67 5f72 702c 202d 2d20 6f70  String_rp, -- op
│ │ │ │ +00266f90: 656e 2061 2070 6f72 7420 666f 7220 6c69  en a port for li
│ │ │ │ +00266fa0: 7374 656e 696e 670a 0a57 6179 7320 746f  stening..Ways to
│ │ │ │ +00266fb0: 2075 7365 2074 6869 7320 6d65 7468 6f64   use this method
│ │ │ │ +00266fc0: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ +00266fd0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ +00266fe0: 2a6e 6f74 6520 6973 4c69 7374 656e 6572  *note isListener
│ │ │ │ +00266ff0: 2846 696c 6529 3a20 6973 4c69 7374 656e  (File): isListen
│ │ │ │ +00267000: 6572 5f6c 7046 696c 655f 7270 2c20 2d2d  er_lpFile_rp, --
│ │ │ │ +00267010: 2077 6865 7468 6572 2061 2066 696c 6520   whether a file 
│ │ │ │ +00267020: 6973 206f 7065 6e20 666f 720a 2020 2020  is open for.    
│ │ │ │ +00267030: 6c69 7374 656e 696e 670a 2d2d 2d2d 2d2d  listening.------
│ │ │ │  00267040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00267050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00267060: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263  -----..The sourc
│ │ │ │ -00267070: 6520 6f66 2074 6869 7320 646f 6375 6d65  e of this docume
│ │ │ │ -00267080: 6e74 2069 7320 696e 0a2f 6275 696c 642f  nt is in./build/
│ │ │ │ -00267090: 7265 7072 6f64 7563 6962 6c65 2d70 6174  reproducible-pat
│ │ │ │ -002670a0: 682f 6d61 6361 756c 6179 322d 312e 3235  h/macaulay2-1.25
│ │ │ │ -002670b0: 2e31 312b 6473 2f4d 322f 4d61 6361 756c  .11+ds/M2/Macaul
│ │ │ │ -002670c0: 6179 322f 7061 636b 6167 6573 2f0a 4d61  ay2/packages/.Ma
│ │ │ │ -002670d0: 6361 756c 6179 3244 6f63 2f6f 765f 7379  caulay2Doc/ov_sy
│ │ │ │ -002670e0: 7374 656d 2e6d 323a 3632 323a 302e 0a1f  stem.m2:622:0...
│ │ │ │ -002670f0: 0a46 696c 653a 204d 6163 6175 6c61 7932  .File: Macaulay2
│ │ │ │ -00267100: 446f 632e 696e 666f 2c20 4e6f 6465 3a20  Doc.info, Node: 
│ │ │ │ -00267110: 6f70 656e 4669 6c65 732c 204e 6578 743a  openFiles, Next:
│ │ │ │ -00267120: 2063 6f6e 6e65 6374 696f 6e43 6f75 6e74   connectionCount
│ │ │ │ -00267130: 2c20 5072 6576 3a20 6973 4c69 7374 656e  , Prev: isListen
│ │ │ │ -00267140: 6572 5f6c 7046 696c 655f 7270 2c20 5570  er_lpFile_rp, Up
│ │ │ │ -00267150: 3a20 7573 696e 6720 736f 636b 6574 730a  : using sockets.
│ │ │ │ -00267160: 0a6f 7065 6e46 696c 6573 202d 2d20 6c69  .openFiles -- li
│ │ │ │ -00267170: 7374 2074 6865 206f 7065 6e20 6669 6c65  st the open file
│ │ │ │ -00267180: 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  s.**************
│ │ │ │ -00267190: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -002671a0: 2a2a 0a0a 4465 7363 7269 7074 696f 6e0a  **..Description.
│ │ │ │ -002671b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a6f 7065  ===========..ope
│ │ │ │ -002671c0: 6e46 696c 6573 2829 2070 726f 6475 6365  nFiles() produce
│ │ │ │ -002671d0: 7320 6120 6c69 7374 206f 6620 616c 6c20  s a list of all 
│ │ │ │ -002671e0: 6375 7272 656e 746c 7920 6f70 656e 2066  currently open f
│ │ │ │ -002671f0: 696c 6573 2e0a 0a0a 5365 6520 616c 736f  iles....See also
│ │ │ │ -00267200: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  .========..  * *
│ │ │ │ -00267210: 6e6f 7465 2046 696c 653a 2046 696c 652c  note File: File,
│ │ │ │ -00267220: 202d 2d20 7468 6520 636c 6173 7320 6f66   -- the class of
│ │ │ │ -00267230: 2061 6c6c 2066 696c 6573 0a0a 466f 7220   all files..For 
│ │ │ │ -00267240: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ -00267250: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00267260: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
│ │ │ │ -00267270: 6f74 6520 6f70 656e 4669 6c65 733a 206f  ote openFiles: o
│ │ │ │ -00267280: 7065 6e46 696c 6573 2c20 6973 2061 202a  penFiles, is a *
│ │ │ │ -00267290: 6e6f 7465 2063 6f6d 7069 6c65 6420 6675  note compiled fu
│ │ │ │ -002672a0: 6e63 7469 6f6e 3a0a 436f 6d70 696c 6564  nction:.Compiled
│ │ │ │ -002672b0: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d  Function,...----
│ │ │ │ -002672c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -002672d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00267060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00267070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00267080: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073  ---------..The s
│ │ │ │ +00267090: 6f75 7263 6520 6f66 2074 6869 7320 646f  ource of this do
│ │ │ │ +002670a0: 6375 6d65 6e74 2069 7320 696e 0a2f 6275  cument is in./bu
│ │ │ │ +002670b0: 696c 642f 7265 7072 6f64 7563 6962 6c65  ild/reproducible
│ │ │ │ +002670c0: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ +002670d0: 312e 3235 2e31 312b 6473 2f4d 322f 4d61  1.25.11+ds/M2/Ma
│ │ │ │ +002670e0: 6361 756c 6179 322f 7061 636b 6167 6573  caulay2/packages
│ │ │ │ +002670f0: 2f0a 4d61 6361 756c 6179 3244 6f63 2f6f  /.Macaulay2Doc/o
│ │ │ │ +00267100: 765f 7379 7374 656d 2e6d 323a 3632 323a  v_system.m2:622:
│ │ │ │ +00267110: 302e 0a1f 0a46 696c 653a 204d 6163 6175  0....File: Macau
│ │ │ │ +00267120: 6c61 7932 446f 632e 696e 666f 2c20 4e6f  lay2Doc.info, No
│ │ │ │ +00267130: 6465 3a20 6f70 656e 4669 6c65 732c 204e  de: openFiles, N
│ │ │ │ +00267140: 6578 743a 2063 6f6e 6e65 6374 696f 6e43  ext: connectionC
│ │ │ │ +00267150: 6f75 6e74 2c20 5072 6576 3a20 6973 4c69  ount, Prev: isLi
│ │ │ │ +00267160: 7374 656e 6572 5f6c 7046 696c 655f 7270  stener_lpFile_rp
│ │ │ │ +00267170: 2c20 5570 3a20 7573 696e 6720 736f 636b  , Up: using sock
│ │ │ │ +00267180: 6574 730a 0a6f 7065 6e46 696c 6573 202d  ets..openFiles -
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│ │ │ │ +002671a0: 6669 6c65 730a 2a2a 2a2a 2a2a 2a2a 2a2a  files.**********
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│ │ │ │ -0026d8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0026d8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0026d210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0026d220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0026d230: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073  ---------..The s
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│ │ │ │ +0026d250: 6375 6d65 6e74 2069 7320 696e 0a2f 6275  cument is in./bu
│ │ │ │ +0026d260: 696c 642f 7265 7072 6f64 7563 6962 6c65  ild/reproducible
│ │ │ │ +0026d270: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ +0026d280: 312e 3235 2e31 312b 6473 2f4d 322f 4d61  1.25.11+ds/M2/Ma
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│ │ │ │ +0026d2b0: 765f 7379 7374 656d 2e6d 323a 3138 3839  v_system.m2:1889
│ │ │ │ +0026d2c0: 3a30 2e0a 1f0a 4669 6c65 3a20 4d61 6361  :0....File: Maca
│ │ │ │ +0026d2d0: 756c 6179 3244 6f63 2e69 6e66 6f2c 204e  ulay2Doc.info, N
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│ │ │ │ +0026d2f0: 656e 616d 652c 204e 6578 743a 2072 656c  ename, Next: rel
│ │ │ │ +0026d300: 6174 6976 697a 6546 696c 656e 616d 652c  ativizeFilename,
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│ │ │ │ +0026d320: 746f 7279 2c20 5570 3a20 7379 7374 656d  tory, Up: system
│ │ │ │ +0026d330: 2066 6163 696c 6974 6965 730a 0a6d 696e   facilities..min
│ │ │ │ +0026d340: 696d 697a 6546 696c 656e 616d 6520 2d2d  imizeFilename --
│ │ │ │ +0026d350: 206d 696e 696d 697a 6520 6120 6669 6c65   minimize a file
│ │ │ │ +0026d360: 206e 616d 650a 2a2a 2a2a 2a2a 2a2a 2a2a   name.**********
│ │ │ │ +0026d370: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0026d380: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ +0026d390: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ +0026d3a0: 2020 2020 6d69 6e69 6d69 7a65 4669 6c65      minimizeFile
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│ │ │ │ +0026d3c0: 7473 3a0a 2020 2020 2020 2a20 666e 2c20  ts:.      * fn, 
│ │ │ │ +0026d3d0: 6120 7061 7468 2074 6f20 6120 6669 6c65  a path to a file
│ │ │ │ +0026d3e0: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ +0026d3f0: 2020 2020 2a20 6120 6d69 6e69 6d69 7a65      * a minimize
│ │ │ │ +0026d400: 6420 7061 7468 2c20 6571 7569 7661 6c65  d path, equivale
│ │ │ │ +0026d410: 6e74 2074 6f20 666e 0a0a 4465 7363 7269  nt to fn..Descri
│ │ │ │ +0026d420: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ +0026d430: 3d0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  =..+------------
│ │ │ │ +0026d440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0026d450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0026d460: 7c69 3120 3a20 6d69 6e69 6d69 7a65 4669  |i1 : minimizeFi
│ │ │ │ +0026d470: 6c65 6e61 6d65 2022 612f 622f 632f 2e2e  lename "a/b/c/..
│ │ │ │ +0026d480: 2f64 2220 2020 2020 2020 207c 0a7c 2020  /d"        |.|  
│ │ │ │ +0026d490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0026d4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0026d4b0: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20          |.|o1 = 
│ │ │ │ +0026d4c0: 612f 622f 6420 2020 2020 2020 2020 2020  a/b/d           
│ │ │ │ +0026d4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0026d4e0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0026d4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0026d500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0026d510: 2d2d 2b0a 7c69 3220 3a20 6d69 6e69 6d69  --+.|i2 : minimi
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│ │ │ │ +0026d530: 2e2f 2e2e 2f2e 2e2f 2e2e 2f2e 2e2f 227c  ./../../../../"|
│ │ │ │ +0026d540: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0026d550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0026d560: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0026d570: 3220 3d20 2f62 7569 6c64 2f20 2020 2020  2 = /build/     
│ │ │ │ +0026d580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0026d590: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0026d5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0026d5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0026d5c0: 2d2d 2d2d 2d2d 2b0a 0a50 6174 6873 206f  ------+..Paths o
│ │ │ │ +0026d5d0: 6620 7468 6520 666f 726d 2066 6f6f 2f78  f the form foo/x
│ │ │ │ +0026d5e0: 2f2e 2e2f 6261 722c 2061 7265 2073 686f  /../bar, are sho
│ │ │ │ +0026d5f0: 7274 656e 6564 2074 6f20 666f 6f2f 6261  rtened to foo/ba
│ │ │ │ +0026d600: 7220 7769 7468 6f75 7420 6368 6563 6b69  r without checki
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│ │ │ │ +0026d660: 2073 6565 0a2a 6e6f 7465 2072 6561 6c70   see.*note realp
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│ │ │ │ +0026d680: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ +0026d690: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 4669  ==..  * *note Fi
│ │ │ │ +0026d6a0: 6c65 3a20 4669 6c65 2c20 2d2d 2074 6865  le: File, -- the
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│ │ │ │ +0026d6f0: 6e61 6d65 2c20 2d2d 2072 656c 6174 6976  name, -- relativ
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│ │ │ │ +0026d710: 2020 2a20 2a6e 6f74 6520 6261 7365 4669    * *note baseFi
│ │ │ │ +0026d720: 6c65 6e61 6d65 3a20 6261 7365 4669 6c65  lename: baseFile
│ │ │ │ +0026d730: 6e61 6d65 2c20 2d2d 2074 6865 2062 6173  name, -- the bas
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│ │ │ │ +0026d750: 6e61 6d65 206f 7220 7061 7468 0a20 202a  name or path.  *
│ │ │ │ +0026d760: 202a 6e6f 7465 2074 6f41 6273 6f6c 7574   *note toAbsolut
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│ │ │ │ +0026d780: 6550 6174 682c 202d 2d20 7468 6520 6162  ePath, -- the ab
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│ │ │ │ +0026d7c0: 2073 6561 7263 6850 6174 683a 2073 6561   searchPath: sea
│ │ │ │ +0026d7d0: 7263 6850 6174 685f 6c70 4c69 7374 5f63  rchPath_lpList_c
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│ │ │ │  0026d8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0026d8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ -0026d900: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
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│ │ │ │ -0026da30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0026da40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0026da50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020  ************..  
│ │ │ │ -0026da60: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
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│ │ │ │ -0026da80: 6e61 6d65 2864 6972 2c66 6e29 0a20 202a  name(dir,fn).  *
│ │ │ │ -0026da90: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
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│ │ │ │ -0026dab0: 6120 6469 7265 6374 6f72 790a 2020 2020  a directory.    
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│ │ │ │ -0026db40: 3d3d 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  ==..+-----------
│ │ │ │ -0026db50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0026db60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0026db70: 2d2b 0a7c 6931 203a 2072 656c 6174 6976  -+.|i1 : relativ
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│ │ │ │ -0026dbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0026dbc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
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│ │ │ │ +0026d8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0026d900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0026d910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0026da60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0026da70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -0026e6d0: 746f 4162 736f 6c75 7465 5061 7468 2c20  toAbsolutePath, 
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│ │ │ │ -0026e710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0026e720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0026e730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0026e280: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0026e290: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0026e2a0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ +0026e2b0: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
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│ │ │ │ +0026e310: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
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│ │ │ │ +0026e340: 6162 736f 6c75 7465 2028 7265 616c 2920  absolute (real) 
│ │ │ │ +0026e350: 7061 7468 206e 616d 6520 6f66 2066 696c  path name of fil
│ │ │ │ +0026e360: 656e 616d 650a 0a44 6573 6372 6970 7469  ename..Descripti
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│ │ │ │ +0026e380: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0026e390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0026e3c0: 3222 2020 2020 2020 7c0a 7c20 2020 2020  2"      |.|     
│ │ │ │ +0026e3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0026e3e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0026e3f0: 6f31 203d 202f 7573 722f 7368 6172 652f  o1 = /usr/share/
│ │ │ │ +0026e400: 4d61 6361 756c 6179 322f 2020 2020 2020  Macaulay2/      
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│ │ │ │ +0026e500: 2072 6561 6c70 6174 683a 2072 6561 6c70   realpath: realp
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│ │ │ │ +0026e570: 6e61 6d65 3a20 6d69 6e69 6d69 7a65 4669  name: minimizeFi
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│ │ │ │ +0026e5a0: 2020 2a20 2a6e 6f74 6520 7265 6c61 7469    * *note relati
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│ │ │ │ +0026e5e0: 6120 6669 6c65 206e 616d 650a 2020 2a20  a file name.  * 
│ │ │ │ +0026e5f0: 2a6e 6f74 6520 6261 7365 4669 6c65 6e61  *note baseFilena
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│ │ │ │ +0026e610: 2c20 2d2d 2074 6865 2062 6173 6520 7061  , -- the base pa
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│ │ │ │  0026e740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0026e7a0: 756c 6179 322d 312e 3235 2e31 312b 6473  ulay2-1.25.11+ds
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│ │ │ │ -0026e8a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0026e8b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0026e8c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
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│ │ │ │ -0026e9a0: 746f 7269 6573 2e0a 2020 2020 2020 2020  tories..        
│ │ │ │ -0026e9b0: 4561 6368 206f 6e65 2065 6e64 7320 7769  Each one ends wi
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│ │ │ │ +0026e750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0026e760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0026ed30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0026ed40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0026f6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -0026fa50: 7369 6f6e 5f72 702c 202d 2d20 6576 616c  sion_rp, -- eval
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│ │ │ │ -0026fa70: 616c 7565 2849 6e64 6578 6564 5661 7269  alue(IndexedVari
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│ │ │ │ -0026fa90: 6e64 6578 6564 5661 7269 6162 6c65 5f72  ndexedVariable_r
│ │ │ │ -0026faa0: 702c 202d 2d20 7265 7472 6965 7665 2074  p, -- retrieve t
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│ │ │ │ -0026fac0: 616e 2069 6e64 6578 6564 2076 6172 6961  an indexed varia
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│ │ │ │ -0026fae0: 6c75 6528 5073 6575 646f 636f 6465 293a  lue(Pseudocode):
│ │ │ │ -0026faf0: 2076 616c 7565 5f6c 7050 7365 7564 6f63   value_lpPseudoc
│ │ │ │ -0026fb00: 6f64 655f 7270 2c20 2d2d 2065 7865 6375  ode_rp, -- execu
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│ │ │ │ -0026fb20: 2a20 2a6e 6f74 6520 7661 6c75 6528 5374  * *note value(St
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│ │ │ │ -0026fb40: 7472 696e 675f 7270 2c20 2d2d 2065 7661  tring_rp, -- eva
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│ │ │ │ -0026fb60: 202a 202a 6e6f 7465 2076 616c 7565 2853   * *note value(S
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│ │ │ │ -0026fc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0026f900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0026f910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00270340: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │ +00270370: 2020 2020 2020 2020 2069 3120 3a20 2d2d           i1 : --
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│ │ │ │ +00270430: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00270520: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00270530: 2020 2020 206f 3120 3a20 506f 6c79 6e6f       o1 : Polyno
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│ │ │ │ -00270550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00270560: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00270550: 2020 2020 2020 2020 206f 3120 3a20 506f           o1 : Po
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│ │ │ │ -002705d0: 2020 2020 2069 3220 3a20 4920 3d20 6964       i2 : I = id
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│ │ │ │ -002705f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +002705e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +002705f0: 2020 2020 2020 2020 2069 3220 3a20 4920           i2 : I 
│ │ │ │ +00270600: 3d20 6964 6561 6c28 612c 622c 632c 6429  = ideal(a,b,c,d)
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│ │ │ │  00270640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00270650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00270660: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00270670: 2020 2020 206f 3220 3d20 6964 6561 6c20       o2 = ideal 
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│ │ │ │ -00270690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002706a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00270680: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00270690: 2020 2020 2020 2020 206f 3220 3d20 6964           o2 = id
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│ │ │ │ +002706b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002706c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002706d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00270700: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00270710: 2020 2020 206f 3220 3a20 4964 6561 6c20       o2 : Ideal 
│ │ │ │ -00270720: 6f66 2052 2020 2020 2020 2020 2020 2020  of R            
│ │ │ │ -00270730: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00270750: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │ -00270890: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -002708a0: 2020 2020 206f 3320 3d20 5220 203c 2d2d       o3 = R  <--
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│ │ │ │ -00270970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00270980: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00270990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002709a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00270930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00270940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00270950: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
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│ │ │ │ -002709e0: 2020 2020 206f 3320 3a20 4368 6169 6e43       o3 : ChainC
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│ │ │ │ -00270a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00270b80: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
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│ │ │ │  00270c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00270d70: 2020 2020 2020 2020 206f 3520 3d20 7b31           o5 = {1
│ │ │ │ +00270d80: 7d20 7c20 2d62 2030 2020 2d63 2030 2020  } | -b 0  -c 0  
│ │ │ │ +00270d90: 3020 202d 6420 7c20 2020 2020 2020 2020  0  -d |         
│ │ │ │ +00270da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00270db0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00270dc0: 2020 2020 2020 2020 2020 2020 2020 7b31                {1
│ │ │ │ +00270dd0: 7d20 7c20 6120 202d 6320 3020 2030 2020  } | a  -c 0  0  
│ │ │ │ +00270de0: 2d64 2030 2020 7c20 2020 2020 2020 2020  -d 0  |         
│ │ │ │ +00270df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00270e00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00270e10: 2020 2020 2020 2020 2020 2020 2020 7b31                {1
│ │ │ │ +00270e20: 7d20 7c20 3020 2062 2020 6120 202d 6420  } | 0  b  a  -d 
│ │ │ │ +00270e30: 3020 2030 2020 7c20 2020 2020 2020 2020  0  0  |         
│ │ │ │ +00270e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00270e50: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00270e60: 2020 2020 2020 2020 2020 2020 2020 7b31                {1
│ │ │ │ +00270e70: 7d20 7c20 3020 2030 2020 3020 2063 2020  } | 0  0  0  c  
│ │ │ │ +00270e80: 6220 2061 2020 7c20 2020 2020 2020 2020  b  a  |         
│ │ │ │  00270e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00270ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00270ea0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00270eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00270ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00270ed0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00270ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00270ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00270ef0: 2020 3420 2020 2020 2036 2020 2020 2020    4      6      
│ │ │ │ +00270ef0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00270f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00270f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00270f20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00270f30: 2020 2020 206f 3520 3a20 4d61 7472 6978       o5 : Matrix
│ │ │ │ -00270f40: 2052 2020 3c2d 2d20 5220 2020 2020 2020   R  <-- R       
│ │ │ │ -00270f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00270f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00270f70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00270f10: 2020 2020 2020 3420 2020 2020 2036 2020        4      6  
│ │ │ │ +00270f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00270f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00270f40: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00270f50: 2020 2020 2020 2020 206f 3520 3a20 4d61           o5 : Ma
│ │ │ │ +00270f60: 7472 6978 2052 2020 3c2d 2d20 5220 2020  trix R  <-- R   
│ │ │ │ +00270f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00270f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00270f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00270f90: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00270fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00270fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00270fc0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00270fd0: 2020 2020 2069 3620 3a20 2020 2020 2020       i6 :       
│ │ │ │ -00270fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00270ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00270fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00270fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00270fe0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00270ff0: 2020 2020 2020 2020 2069 3620 3a20 2020           i6 :   
│ │ │ │  00271000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00271010: 2020 2020 207c 0a7c 2020 2020 202d 2d2d       |.|     ---
│ │ │ │ -00271020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00271030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00271040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00271010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00271020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00271030: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00271040: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   ---------------
│ │ │ │  00271050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00271060: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ -00271070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00271080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00271060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00271070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00271080: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020  ---------|.|    
│ │ │ │  00271090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002710a0: 2020 2020 2029 2020 2020 2020 2020 2020       )          
│ │ │ │ -002710b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -002710c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002710d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002710a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002710b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002710c0: 2020 2020 2020 2020 2029 2020 2020 2020           )      
│ │ │ │ +002710d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  002710e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002710f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00271100: 2020 2020 207c 0a7c 2020 2020 204d 6163       |.|     Mac
│ │ │ │ -00271110: 6175 6c61 7932 2f70 6163 6b61 6765 732f  aulay2/packages/
│ │ │ │ -00271120: 4f6c 6443 6861 696e 436f 6d70 6c65 7865  OldChainComplexe
│ │ │ │ -00271130: 732f 646f 6373 2f64 6f63 3130 2e6d 323a  s/docs/doc10.m2:
│ │ │ │ -00271140: 3235 343a 3020 2020 2020 2020 2020 2020  254:0           
│ │ │ │ -00271150: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00271160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00271170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00271100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00271110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00271120: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00271130: 204d 6163 6175 6c61 7932 2f70 6163 6b61   Macaulay2/packa
│ │ │ │ +00271140: 6765 732f 4f6c 6443 6861 696e 436f 6d70  ges/OldChainComp
│ │ │ │ +00271150: 6c65 7865 732f 646f 6373 2f64 6f63 3130  lexes/docs/doc10
│ │ │ │ +00271160: 2e6d 323a 3235 343a 3020 2020 2020 2020  .m2:254:0       
│ │ │ │ +00271170: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00271180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00271190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002711a0: 2020 2020 207c 0a7c 6f31 203a 2053 6571       |.|o1 : Seq
│ │ │ │ -002711b0: 7565 6e63 6520 2020 2020 2020 2020 2020  uence           
│ │ │ │ -002711c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002711d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002711a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002711b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002711c0: 2020 2020 2020 2020 207c 0a7c 6f31 203a           |.|o1 :
│ │ │ │ +002711d0: 2053 6571 7565 6e63 6520 2020 2020 2020   Sequence       
│ │ │ │  002711e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002711f0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00271200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00271210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002711f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00271200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00271210: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  00271220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00271230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00271240: 2d2d 2d2d 2d2b 0a7c 6932 203a 2061 7373  -----+.|i2 : ass
│ │ │ │ -00271250: 6572 7420 6e6f 7420 6572 7220 2020 2020  ert not err     
│ │ │ │ -00271260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00271270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00271240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00271250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00271260: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a  ---------+.|i2 :
│ │ │ │ +00271270: 2061 7373 6572 7420 6e6f 7420 6572 7220   assert not err 
│ │ │ │  00271280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00271290: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -002712a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -002712b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00271290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002712a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002712b0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  002712c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  002712d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -002712e0: 2d2d 2d2d 2d2b 0a0a 4966 2055 7365 724d  -----+..If UserM
│ │ │ │ -002712f0: 6f64 6520 3d3e 2066 616c 7365 2067 6976  ode => false giv
│ │ │ │ -00271300: 656e 2c20 7468 6520 6469 6374 696f 6e61  en, the dictiona
│ │ │ │ -00271310: 7269 6573 2061 7661 696c 6162 6c65 2074  ries available t
│ │ │ │ -00271320: 6f20 7468 6520 7573 6572 2077 696c 6c20  o the user will 
│ │ │ │ -00271330: 6e6f 7420 6265 0a61 6666 6563 7465 6420  not be.affected 
│ │ │ │ -00271340: 6279 2074 6865 2065 7661 6c75 6174 696f  by the evaluatio
│ │ │ │ -00271350: 6e20 616e 6420 2a6e 6f74 6520 636f 6c6c  n and *note coll
│ │ │ │ -00271360: 6563 7447 6172 6261 6765 3a20 636f 6c6c  ectGarbage: coll
│ │ │ │ -00271370: 6563 7447 6172 6261 6765 2c20 6973 2063  ectGarbage, is c
│ │ │ │ -00271380: 616c 6c65 640a 6166 7465 7277 6172 6473  alled.afterwards
│ │ │ │ -00271390: 2e0a 0a45 7272 6f72 7320 6f63 6375 7272  ...Errors occurr
│ │ │ │ -002713a0: 6564 2077 6869 6c65 2065 7661 6c75 6174  ed while evaluat
│ │ │ │ -002713b0: 696e 6720 7374 7220 646f 206e 6f74 2063  ing str do not c
│ │ │ │ -002713c0: 6175 7365 2061 6e20 6572 726f 7220 6f75  ause an error ou
│ │ │ │ -002713d0: 7473 6964 6520 6f66 2063 6170 7475 7265  tside of capture
│ │ │ │ -002713e0: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...+------------
│ │ │ │ -002713f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00271400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002712e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002712f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00271300: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4966 2055  ---------+..If U
│ │ │ │ +00271310: 7365 724d 6f64 6520 3d3e 2066 616c 7365  serMode => false
│ │ │ │ +00271320: 2067 6976 656e 2c20 7468 6520 6469 6374   given, the dict
│ │ │ │ +00271330: 696f 6e61 7269 6573 2061 7661 696c 6162  ionaries availab
│ │ │ │ +00271340: 6c65 2074 6f20 7468 6520 7573 6572 2077  le to the user w
│ │ │ │ +00271350: 696c 6c20 6e6f 7420 6265 0a61 6666 6563  ill not be.affec
│ │ │ │ +00271360: 7465 6420 6279 2074 6865 2065 7661 6c75  ted by the evalu
│ │ │ │ +00271370: 6174 696f 6e20 616e 6420 2a6e 6f74 6520  ation and *note 
│ │ │ │ +00271380: 636f 6c6c 6563 7447 6172 6261 6765 3a20  collectGarbage: 
│ │ │ │ +00271390: 636f 6c6c 6563 7447 6172 6261 6765 2c20  collectGarbage, 
│ │ │ │ +002713a0: 6973 2063 616c 6c65 640a 6166 7465 7277  is called.afterw
│ │ │ │ +002713b0: 6172 6473 2e0a 0a45 7272 6f72 7320 6f63  ards...Errors oc
│ │ │ │ +002713c0: 6375 7272 6564 2077 6869 6c65 2065 7661  curred while eva
│ │ │ │ +002713d0: 6c75 6174 696e 6720 7374 7220 646f 206e  luating str do n
│ │ │ │ +002713e0: 6f74 2063 6175 7365 2061 6e20 6572 726f  ot cause an erro
│ │ │ │ +002713f0: 7220 6f75 7473 6964 6520 6f66 2063 6170  r outside of cap
│ │ │ │ +00271400: 7475 7265 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  ture...+--------
│ │ │ │  00271410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00271420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00271430: 2d2b 0a7c 6933 203a 2028 6572 722c 206f  -+.|i3 : (err, o
│ │ │ │ -00271440: 7574 7075 7429 203d 2063 6170 7475 7265  utput) = capture
│ │ │ │ -00271450: 202f 2f2f 2073 7464 6572 7220 3c3c 2022   /// stderr << "
│ │ │ │ -00271460: 4368 6563 6b69 6e67 2061 2066 616c 7365  Checking a false
│ │ │ │ -00271470: 2073 7461 7465 6d65 6e74 3a22 203c 3c20   statement:" << 
│ │ │ │ -00271480: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00271490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002714a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00271430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00271440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00271450: 2d2d 2d2d 2d2b 0a7c 6933 203a 2028 6572  -----+.|i3 : (er
│ │ │ │ +00271460: 722c 206f 7574 7075 7429 203d 2063 6170  r, output) = cap
│ │ │ │ +00271470: 7475 7265 202f 2f2f 2073 7464 6572 7220  ture /// stderr 
│ │ │ │ +00271480: 3c3c 2022 4368 6563 6b69 6e67 2061 2066  << "Checking a f
│ │ │ │ +00271490: 616c 7365 2073 7461 7465 6d65 6e74 3a22  alse statement:"
│ │ │ │ +002714a0: 203c 3c20 207c 0a7c 2020 2020 2020 2020   <<  |.|        
│ │ │ │  002714b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002714c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002714d0: 207c 0a7c 6f33 203d 2028 7472 7565 2c20   |.|o3 = (true, 
│ │ │ │ +002714d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002714e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002714f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00271500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00271510: 2020 2020 2020 2020 2020 2029 2020 2020             )    
│ │ │ │ -00271520: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00271530: 6931 203a 2020 7374 6465 7272 203c 3c20  i1 :  stderr << 
│ │ │ │ -00271540: 2243 6865 636b 696e 6720 6120 6661 6c73  "Checking a fals
│ │ │ │ -00271550: 6520 7374 6174 656d 656e 743a 2220 3c3c  e statement:" <<
│ │ │ │ -00271560: 2065 6e64 6c3b 2031 2f30 2020 2020 2020   endl; 1/0      
│ │ │ │ -00271570: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00271580: 4368 6563 6b69 6e67 2061 2066 616c 7365  Checking a false
│ │ │ │ -00271590: 2073 7461 7465 6d65 6e74 3a20 2020 2020   statement:     
│ │ │ │ -002715a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002715b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002715c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -002715d0: 6375 7272 656e 7453 7472 696e 673a 313a  currentString:1:
│ │ │ │ -002715e0: 3531 3a28 3329 3a5b 335d 3a20 6572 726f  51:(3):[3]: erro
│ │ │ │ -002715f0: 723a 2064 6976 6973 696f 6e20 6279 207a  r: division by z
│ │ │ │ -00271600: 6572 6f20 2020 2020 2020 2020 2020 2020  ero             
│ │ │ │ -00271610: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00271620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00271630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002714f0: 2020 2020 207c 0a7c 6f33 203d 2028 7472       |.|o3 = (tr
│ │ │ │ +00271500: 7565 2c20 2020 2020 2020 2020 2020 2020  ue,             
│ │ │ │ +00271510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00271520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00271530: 2020 2020 2020 2020 2020 2020 2020 2029                 )
│ │ │ │ +00271540: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00271550: 2020 2020 6931 203a 2020 7374 6465 7272      i1 :  stderr
│ │ │ │ +00271560: 203c 3c20 2243 6865 636b 696e 6720 6120   << "Checking a 
│ │ │ │ +00271570: 6661 6c73 6520 7374 6174 656d 656e 743a  false statement:
│ │ │ │ +00271580: 2220 3c3c 2065 6e64 6c3b 2031 2f30 2020  " << endl; 1/0  
│ │ │ │ +00271590: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +002715a0: 2020 2020 4368 6563 6b69 6e67 2061 2066      Checking a f
│ │ │ │ +002715b0: 616c 7365 2073 7461 7465 6d65 6e74 3a20  alse statement: 
│ │ │ │ +002715c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002715d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002715e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +002715f0: 2020 2020 6375 7272 656e 7453 7472 696e      currentStrin
│ │ │ │ +00271600: 673a 313a 3531 3a28 3329 3a5b 335d 3a20  g:1:51:(3):[3]: 
│ │ │ │ +00271610: 6572 726f 723a 2064 6976 6973 696f 6e20  error: division 
│ │ │ │ +00271620: 6279 207a 6572 6f20 2020 2020 2020 2020  by zero         
│ │ │ │ +00271630: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00271640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00271650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00271660: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00271660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00271670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00271680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00271680: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00271690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002716a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002716b0: 207c 0a7c 6f33 203a 2053 6571 7565 6e63   |.|o3 : Sequenc
│ │ │ │ -002716c0: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ -002716d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002716e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002716b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002716c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002716d0: 2020 2020 207c 0a7c 6f33 203a 2053 6571       |.|o3 : Seq
│ │ │ │ +002716e0: 7565 6e63 6520 2020 2020 2020 2020 2020  uence           
│ │ │ │  002716f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00271700: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │ -00271710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00271720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00271d50: 2a6e 6f74 6520 6f6f 6f6f 3a20 6f6f 6f6f  *note oooo: oooo
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│ │ │ │ -00271db0: 5468 6520 6f62 6a65 6374 202a 6e6f 7465  The object *note
│ │ │ │ -00271dc0: 206f 6f3a 206f 6f2c 2069 7320 6120 2a6e   oo: oo, is a *n
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│ │ │ │ -00271df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00271e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00271b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00271b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00271c80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ +00271ca0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ion.===========.
│ │ │ │ +00271cb0: 0a6f 6f20 2d2d 2064 656e 6f74 6573 2074  .oo -- denotes t
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│ │ │ │ +00271cd0: 6578 7072 6573 7369 6f6e 206f 6e20 7468  expression on th
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│ │ │ │ +00271d00: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  o.========..  * 
│ │ │ │ +00271d10: 2a6e 6f74 6520 6f6f 3a20 6f6f 2c20 2d2d  *note oo: oo, --
│ │ │ │ +00271d20: 2074 6865 206c 6173 7420 6f75 7470 7574   the last output
│ │ │ │ +00271d30: 2076 616c 7565 0a20 202a 202a 6e6f 7465   value.  * *note
│ │ │ │ +00271d40: 206f 6f6f 3a20 6f6f 6f2c 202d 2d20 7468   ooo: ooo, -- th
│ │ │ │ +00271d50: 6520 6e65 7874 2074 6f20 7468 6520 6c61  e next to the la
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│ │ │ │ -00272980: 7177 6572 7d20 2020 2020 2020 2020 2020  qwer}           
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│ │ │ │ +00272c60: 2020 2020 7c0a 7c20 2020 2020 2d2d 2d2d      |.|     ----
│ │ │ │  00272c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00272c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00272c90: 7c0a 7c20 2020 2020 456c 696d 696e 6174  |.|     Eliminat
│ │ │ │ -00272ca0: 696f 6e2e 4469 6374 696f 6e61 7279 2c20  ion.Dictionary, 
│ │ │ │ -00272cb0: 4f6c 6443 6861 696e 436f 6d70 6c65 7865  OldChainComplexe
│ │ │ │ -00272cc0: 732e 4469 6374 696f 6e61 7279 2c20 2020  s.Dictionary,   
│ │ │ │ -00272cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00272ce0: 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d  |.|     --------
│ │ │ │ -00272cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00272d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00272c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00272ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00272cb0: 2d2d 2d2d 7c0a 7c20 2020 2020 456c 696d  ----|.|     Elim
│ │ │ │ +00272cc0: 696e 6174 696f 6e2e 4469 6374 696f 6e61  ination.Dictiona
│ │ │ │ +00272cd0: 7279 2c20 4f6c 6443 6861 696e 436f 6d70  ry, OldChainComp
│ │ │ │ +00272ce0: 6c65 7865 732e 4469 6374 696f 6e61 7279  lexes.Dictionary
│ │ │ │ +00272cf0: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +00272d00: 2020 2020 7c0a 7c20 2020 2020 2d2d 2d2d      |.|     ----
│ │ │ │  00272d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00272d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00272d30: 7c0a 7c20 2020 2020 5461 6e67 656e 7443  |.|     TangentC
│ │ │ │ -00272d40: 6f6e 652e 4469 6374 696f 6e61 7279 2c20  one.Dictionary, 
│ │ │ │ -00272d50: 5369 6d70 6c65 446f 632e 4469 6374 696f  SimpleDoc.Dictio
│ │ │ │ -00272d60: 6e61 7279 2c20 5265 6573 416c 6765 6272  nary, ReesAlgebr
│ │ │ │ -00272d70: 612e 4469 6374 696f 6e61 7279 2c20 2020  a.Dictionary,   
│ │ │ │ -00272d80: 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d  |.|     --------
│ │ │ │ -00272d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00272da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00272d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00272d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00272d50: 2d2d 2d2d 7c0a 7c20 2020 2020 5461 6e67  ----|.|     Tang
│ │ │ │ +00272d60: 656e 7443 6f6e 652e 4469 6374 696f 6e61  entCone.Dictiona
│ │ │ │ +00272d70: 7279 2c20 5369 6d70 6c65 446f 632e 4469  ry, SimpleDoc.Di
│ │ │ │ +00272d80: 6374 696f 6e61 7279 2c20 5265 6573 416c  ctionary, ReesAl
│ │ │ │ +00272d90: 6765 6272 612e 4469 6374 696f 6e61 7279  gebra.Dictionary
│ │ │ │ +00272da0: 2c20 2020 7c0a 7c20 2020 2020 2d2d 2d2d  ,   |.|     ----
│ │ │ │  00272db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00272dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00272dd0: 7c0a 7c20 2020 2020 5072 696d 6172 7944  |.|     PrimaryD
│ │ │ │ -00272de0: 6563 6f6d 706f 7369 7469 6f6e 2e44 6963  ecomposition.Dic
│ │ │ │ -00272df0: 7469 6f6e 6172 792c 204d 696e 696d 616c  tionary, Minimal
│ │ │ │ -00272e00: 5072 696d 6573 2e44 6963 7469 6f6e 6172  Primes.Dictionar
│ │ │ │ -00272e10: 792c 2020 2020 2020 2020 2020 2020 2020  y,              
│ │ │ │ -00272e20: 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d  |.|     --------
│ │ │ │ -00272e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00272e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00272dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00272de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00272df0: 2d2d 2d2d 7c0a 7c20 2020 2020 5072 696d  ----|.|     Prim
│ │ │ │ +00272e00: 6172 7944 6563 6f6d 706f 7369 7469 6f6e  aryDecomposition
│ │ │ │ +00272e10: 2e44 6963 7469 6f6e 6172 792c 204d 696e  .Dictionary, Min
│ │ │ │ +00272e20: 696d 616c 5072 696d 6573 2e44 6963 7469  imalPrimes.Dicti
│ │ │ │ +00272e30: 6f6e 6172 792c 2020 2020 2020 2020 2020  onary,          
│ │ │ │ +00272e40: 2020 2020 7c0a 7c20 2020 2020 2d2d 2d2d      |.|     ----
│ │ │ │  00272e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00272e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00272e70: 7c0a 7c20 2020 2020 5061 636b 6167 6543  |.|     PackageC
│ │ │ │ -00272e80: 6974 6174 696f 6e73 2e44 6963 7469 6f6e  itations.Diction
│ │ │ │ -00272e90: 6172 792c 204f 6e6c 696e 654c 6f6f 6b75  ary, OnlineLooku
│ │ │ │ -00272ea0: 702e 4469 6374 696f 6e61 7279 2c20 2020  p.Dictionary,   
│ │ │ │ -00272eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00272ec0: 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d  |.|     --------
│ │ │ │ -00272ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00272ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00272e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00272e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00272e90: 2d2d 2d2d 7c0a 7c20 2020 2020 5061 636b  ----|.|     Pack
│ │ │ │ +00272ea0: 6167 6543 6974 6174 696f 6e73 2e44 6963  ageCitations.Dic
│ │ │ │ +00272eb0: 7469 6f6e 6172 792c 204f 6e6c 696e 654c  tionary, OnlineL
│ │ │ │ +00272ec0: 6f6f 6b75 702e 4469 6374 696f 6e61 7279  ookup.Dictionary
│ │ │ │ +00272ed0: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +00272ee0: 2020 2020 7c0a 7c20 2020 2020 2d2d 2d2d      |.|     ----
│ │ │ │  00272ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00272f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00272f10: 7c0a 7c20 2020 2020 4c4c 4c42 6173 6573  |.|     LLLBases
│ │ │ │ -00272f20: 2e44 6963 7469 6f6e 6172 792c 2049 6e76  .Dictionary, Inv
│ │ │ │ -00272f30: 6572 7365 5379 7374 656d 732e 4469 6374  erseSystems.Dict
│ │ │ │ -00272f40: 696f 6e61 7279 2c20 2020 2020 2020 2020  ionary,         
│ │ │ │ -00272f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00272f60: 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d  |.|     --------
│ │ │ │ -00272f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00272f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00272f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00272f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00272f30: 2d2d 2d2d 7c0a 7c20 2020 2020 4c4c 4c42  ----|.|     LLLB
│ │ │ │ +00272f40: 6173 6573 2e44 6963 7469 6f6e 6172 792c  ases.Dictionary,
│ │ │ │ +00272f50: 2049 6e76 6572 7365 5379 7374 656d 732e   InverseSystems.
│ │ │ │ +00272f60: 4469 6374 696f 6e61 7279 2c20 2020 2020  Dictionary,     
│ │ │ │ +00272f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00272f80: 2020 2020 7c0a 7c20 2020 2020 2d2d 2d2d      |.|     ----
│ │ │ │  00272f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00272fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00272fb0: 7c0a 7c20 2020 2020 496e 7465 6772 616c  |.|     Integral
│ │ │ │ -00272fc0: 436c 6f73 7572 652e 4469 6374 696f 6e61  Closure.Dictiona
│ │ │ │ -00272fd0: 7279 2c20 436f 6e77 6179 506f 6c79 6e6f  ry, ConwayPolyno
│ │ │ │ -00272fe0: 6d69 616c 732e 4469 6374 696f 6e61 7279  mials.Dictionary
│ │ │ │ -00272ff0: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ -00273000: 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d  |.|     --------
│ │ │ │ -00273010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00273020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00272fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00272fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00272fd0: 2d2d 2d2d 7c0a 7c20 2020 2020 496e 7465  ----|.|     Inte
│ │ │ │ +00272fe0: 6772 616c 436c 6f73 7572 652e 4469 6374  gralClosure.Dict
│ │ │ │ +00272ff0: 696f 6e61 7279 2c20 436f 6e77 6179 506f  ionary, ConwayPo
│ │ │ │ +00273000: 6c79 6e6f 6d69 616c 732e 4469 6374 696f  lynomials.Dictio
│ │ │ │ +00273010: 6e61 7279 2c20 2020 2020 2020 2020 2020  nary,           
│ │ │ │ +00273020: 2020 2020 7c0a 7c20 2020 2020 2d2d 2d2d      |.|     ----
│ │ │ │  00273030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00273040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00273050: 7c0a 7c20 2020 2020 436c 6173 7369 632e  |.|     Classic.
│ │ │ │ -00273060: 4469 6374 696f 6e61 7279 2c20 4d61 6361  Dictionary, Maca
│ │ │ │ -00273070: 756c 6179 3244 6f63 2e44 6963 7469 6f6e  ulay2Doc.Diction
│ │ │ │ -00273080: 6172 792c 2043 6f72 652e 4469 6374 696f  ary, Core.Dictio
│ │ │ │ -00273090: 6e61 7279 2c20 2020 2020 2020 2020 2020  nary,           
│ │ │ │ -002730a0: 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d  |.|     --------
│ │ │ │ -002730b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -002730c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00273050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00273060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00273070: 2d2d 2d2d 7c0a 7c20 2020 2020 436c 6173  ----|.|     Clas
│ │ │ │ +00273080: 7369 632e 4469 6374 696f 6e61 7279 2c20  sic.Dictionary, 
│ │ │ │ +00273090: 4d61 6361 756c 6179 3244 6f63 2e44 6963  Macaulay2Doc.Dic
│ │ │ │ +002730a0: 7469 6f6e 6172 792c 2043 6f72 652e 4469  tionary, Core.Di
│ │ │ │ +002730b0: 6374 696f 6e61 7279 2c20 2020 2020 2020  ctionary,       
│ │ │ │ +002730c0: 2020 2020 7c0a 7c20 2020 2020 2d2d 2d2d      |.|     ----
│ │ │ │  002730d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  002730e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -002730f0: 7c0a 7c20 2020 2020 476c 6f62 616c 4469  |.|     GlobalDi
│ │ │ │ -00273100: 6374 696f 6e61 7279 7b2e 2e2e 342e 2e2e  ctionary{...4...
│ │ │ │ -00273110: 7d2c 2050 6163 6b61 6765 4469 6374 696f  }, PackageDictio
│ │ │ │ -00273120: 6e61 7279 7d20 2020 2020 2020 2020 2020  nary}           
│ │ │ │ -00273130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00273140: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +002730f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00273100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00273110: 2d2d 2d2d 7c0a 7c20 2020 2020 476c 6f62  ----|.|     Glob
│ │ │ │ +00273120: 616c 4469 6374 696f 6e61 7279 7b2e 2e2e  alDictionary{...
│ │ │ │ +00273130: 342e 2e2e 7d2c 2050 6163 6b61 6765 4469  4...}, PackageDi
│ │ │ │ +00273140: 6374 696f 6e61 7279 7d20 2020 2020 2020  ctionary}       
│ │ │ │  00273150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00273160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00273160: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00273170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00273180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00273190: 7c0a 7c6f 3420 3a20 4c69 7374 2020 2020  |.|o4 : List    
│ │ │ │ +00273190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002731a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002731b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002731b0: 2020 2020 7c0a 7c6f 3420 3a20 4c69 7374      |.|o4 : List
│ │ │ │  002731c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002731d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002731e0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -002731f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00273200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002731e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002731f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00273200: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  00273210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00273220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00273230: 2b0a 7c69 3520 3a20 7065 656b 204f 7574  +.|i5 : peek Out
│ │ │ │ -00273240: 7075 7444 6963 7469 6f6e 6172 7920 2020  putDictionary   
│ │ │ │ -00273250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00273260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00273270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00273280: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00273230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00273240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00273250: 2d2d 2d2d 2b0a 7c69 3520 3a20 7065 656b  ----+.|i5 : peek
│ │ │ │ +00273260: 204f 7574 7075 7444 6963 7469 6f6e 6172   OutputDictionar
│ │ │ │ +00273270: 7920 2020 2020 2020 2020 2020 2020 2020  y               
│ │ │ │ +00273280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00273290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002732a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002732a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  002732b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002732c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002732d0: 7c0a 7c6f 3520 3d20 476c 6f62 616c 4469  |.|o5 = GlobalDi
│ │ │ │ -002732e0: 6374 696f 6e61 7279 7b22 6f31 2220 3d3e  ctionary{"o1" =>
│ │ │ │ -002732f0: 206f 317d 2020 2020 2020 2020 2020 2020   o1}            
│ │ │ │ -00273300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00273310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00273320: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00273330: 2020 2020 2020 2020 2022 6f32 2220 3d3e           "o2" =>
│ │ │ │ -00273340: 206f 3220 2020 2020 2020 2020 2020 2020   o2             
│ │ │ │ -00273350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00273360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00273370: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00273380: 2020 2020 2020 2020 2022 6f33 2220 3d3e           "o3" =>
│ │ │ │ -00273390: 206f 3320 2020 2020 2020 2020 2020 2020   o3             
│ │ │ │ -002733a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002733b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002733c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -002733d0: 2020 2020 2020 2020 2022 6f34 2220 3d3e           "o4" =>
│ │ │ │ -002733e0: 206f 3420 2020 2020 2020 2020 2020 2020   o4             
│ │ │ │ -002733f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00273400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00273410: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -00273420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00273430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002732d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002732e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002732f0: 2020 2020 7c0a 7c6f 3520 3d20 476c 6f62      |.|o5 = Glob
│ │ │ │ +00273300: 616c 4469 6374 696f 6e61 7279 7b22 6f31  alDictionary{"o1
│ │ │ │ +00273310: 2220 3d3e 206f 317d 2020 2020 2020 2020  " => o1}        
│ │ │ │ +00273320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00273330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00273340: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00273350: 2020 2020 2020 2020 2020 2020 2022 6f32               "o2
│ │ │ │ +00273360: 2220 3d3e 206f 3220 2020 2020 2020 2020  " => o2         
│ │ │ │ +00273370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00273380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00273390: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +002733a0: 2020 2020 2020 2020 2020 2020 2022 6f33               "o3
│ │ │ │ +002733b0: 2220 3d3e 206f 3320 2020 2020 2020 2020  " => o3         
│ │ │ │ +002733c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002733d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002733e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +002733f0: 2020 2020 2020 2020 2020 2020 2022 6f34               "o4
│ │ │ │ +00273400: 2220 3d3e 206f 3420 2020 2020 2020 2020  " => o4         
│ │ │ │ +00273410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00273420: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00275610: 6f62 7573 744e 6574 2078 3b20 2020 2020  obustNet x;     
│ │ │ │ -00275620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00275630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00275640: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00275650: 2020 2020 2020 2020 2020 7772 6170 7065            wrappe
│ │ │ │ -00275660: 7220 3a3d 206c 6f6f 6b75 7028 676c 6f62  r := lookup(glob
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│ │ │ │ -00275690: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -002756a0: 2020 2020 2020 2020 2020 6966 2077 7261            if wra
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│ │ │ │ -00275750: 7279 2074 696d 656c 696d 6974 2870 7269  ry timelimit(pri
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│ │ │ │ -002757d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ +00275540: 726f 6d70 7420 3a3d 2063 6f6e 6361 7465  rompt := concate
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│ │ │ │ +00275770: 2020 2074 7279 2074 696d 656c 696d 6974     try timelimit
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│ │ │ │ -00275840: 2020 2020 676c 6f62 616c 2064 6562 7567      global debug
│ │ │ │ -00275850: 4572 726f 7220 3c2d 2066 756e 3b20 2020  Error <- fun;   
│ │ │ │ -00275860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00275870: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00275880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00275890: 2020 2020 7374 6465 7272 203c 3c20 222d      stderr << "-
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│ │ │ │ -002758b0: 696d 6974 2072 6561 6368 6564 2069 6e20  imit reached in 
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│ │ │ │ -00275910: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00275920: 2020 2020 2020 2020 2020 3c3c 2065 6e64            << end
│ │ │ │ -00275930: 6c20 3c3c 206f 7072 6f6d 7074 203c 3c20  l << oprompt << 
│ │ │ │ -00275940: 7a20 3c3c 2065 6e64 6c3b 2020 2020 2020  z << endl;      
│ │ │ │ -00275950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00275960: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00275970: 2020 2020 2020 2020 2020 7072 696e 7457            printW
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│ │ │ │ -00275990: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -002759b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ +00275860: 2020 2020 2020 2020 676c 6f62 616c 2064          global d
│ │ │ │ +00275870: 6562 7567 4572 726f 7220 3c2d 2066 756e  ebugError <- fun
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│ │ │ │ +00275890: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00275930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00275940: 207c 0a7c 2020 2020 2020 2020 2020 3c3c   |.|          <<
│ │ │ │ +00275950: 2065 6e64 6c20 3c3c 206f 7072 6f6d 7074   endl << oprompt
│ │ │ │ +00275960: 203c 3c20 7a20 3c3c 2065 6e64 6c3b 2020   << z << endl;  
│ │ │ │ +00275970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00275980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00275990: 207c 0a7c 2020 2020 2020 2020 2020 7072   |.|          pr
│ │ │ │ +002759a0: 696e 7457 6964 7468 203d 2073 6176 653b  intWidth = save;
│ │ │ │ +002759b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002759c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002759d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002759e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002759e0: 207c 0a7c 2020 2020 2020 2020 2020 2920   |.|          ) 
│ │ │ │  002759f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00275a00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00275a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00275a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00275a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00275a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00275a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00275a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00275a30: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00275a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00275a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ -00275a60: 3d20 2229 3b20 2020 2020 2020 2020 2020  = ");           
│ │ │ │ -00275a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00275a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00275a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00275aa0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │  00275ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00275ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00275ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00275ad0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00275ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00275af0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00275af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00275b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00275b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00275b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00275b20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00275b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00275b40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00275b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00275b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00275b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00275b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00275b70: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00275b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00275b90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │  00275bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00275c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00275c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00275c30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │  00275c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00275c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00275c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │  00275c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00275c80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
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│ │ │ │ -00275dc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00275dd0: 6d65 7468 6f64 2074 6f20 6f75 7470 7574  method to output
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│ │ │ │ +00275e00: 7470 7574 3b20 7479 7065 2027 6465 6275  tput; type 'debu
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│ │ │ │ +00275e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00275e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00277b70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20  *************.. 
│ │ │ │ -00277b80: 202a 2055 7361 6765 3a20 0a20 2020 2020   * Usage: .     
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│ │ │ │ -00277d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00277a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00277a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00277a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00277b90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00277ba0: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20  *..  * Usage: . 
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│ │ │ │ +00277bc0: 202a 2043 6f6e 7365 7175 656e 6365 733a   * Consequences:
│ │ │ │ +00277bd0: 0a20 2020 2020 202a 2074 6865 2070 726f  .      * the pro
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│ │ │ │  00277db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -00277ef0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00277f00: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055  *********..  * U
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│ │ │ │ -00277f40: 2020 2a20 662c 2061 202a 6e6f 7465 2066    * f, a *note f
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│ │ │ │ -00278080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00277dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00277dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -002787b0: 616c 7565 0a6f 6620 652e 2020 5468 6520  alue.of e.  The 
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│ │ │ │ -002787d0: 2074 6865 2063 7572 7265 6e74 2074 6872   the current thr
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│ │ │ │ -00278830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00278840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00278850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00278860: 6931 203a 2074 696d 6520 335e 3330 2020  i1 : time 3^30  
│ │ │ │ -00278870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00278880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00278890: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -002788a0: 6564 2032 2e31 3865 2d30 3573 2028 6370  ed 2.18e-05s (cp
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│ │ │ │ -002788c0: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ -002788d0: 297c 0a7c 2020 2020 2020 2020 2020 2020  )|.|            
│ │ │ │ -002788e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002788f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00278900: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -00278910: 203d 2032 3035 3839 3131 3332 3039 3436   = 2058911320946
│ │ │ │ -00278920: 3439 2020 2020 2020 2020 2020 2020 2020  49              
│ │ │ │ -00278930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00278940: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00278950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00278960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00278970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00278980: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ -00278990: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2074  ===..  * *note t
│ │ │ │ -002789a0: 696d 696e 673a 2074 696d 696e 672c 202d  iming: timing, -
│ │ │ │ -002789b0: 2d20 7469 6d65 2061 2063 6f6d 7075 7461  - time a computa
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│ │ │ │ -002789d0: 7075 5469 6d65 3a20 6370 7554 696d 652c  puTime: cpuTime,
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│ │ │ │ -00278a60: 656c 6170 7365 640a 2020 2a20 2a6e 6f74  elapsed.  * *not
│ │ │ │ -00278a70: 6520 656c 6170 7365 6454 696d 653a 2065  e elapsedTime: e
│ │ │ │ -00278a80: 6c61 7073 6564 5469 6d65 2c20 2d2d 2074  lapsedTime, -- t
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│ │ │ │ -00278ad0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
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│ │ │ │ -00278af0: 2a6e 6f74 6520 7469 6d65 3a20 7469 6d65  *note time: time
│ │ │ │ -00278b00: 2c20 6973 2061 202a 6e6f 7465 206b 6579  , is a *note key
│ │ │ │ -00278b10: 776f 7264 3a20 4b65 7977 6f72 642c 2e0a  word: Keyword,..
│ │ │ │ -00278b20: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .---------------
│ │ │ │ -00278b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00278b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00278610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00278620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00278630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
│ │ │ │ +00278640: 6865 2073 6f75 7263 6520 6f66 2074 6869  he source of thi
│ │ │ │ +00278650: 7320 646f 6375 6d65 6e74 2069 7320 696e  s document is in
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│ │ │ │ +00278690: 322f 4d61 6361 756c 6179 322f 7061 636b  2/Macaulay2/pack
│ │ │ │ +002786a0: 6167 6573 2f0a 4d61 6361 756c 6179 3244  ages/.Macaulay2D
│ │ │ │ +002786b0: 6f63 2f6f 765f 7379 7374 656d 2e6d 323a  oc/ov_system.m2:
│ │ │ │ +002786c0: 3934 343a 302e 0a1f 0a46 696c 653a 204d  944:0....File: M
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│ │ │ │ +00278700: 3a20 5469 6d65 2c20 5570 3a20 7379 7374  : Time, Up: syst
│ │ │ │ +00278710: 656d 2066 6163 696c 6974 6965 730a 0a74  em facilities..t
│ │ │ │ +00278720: 696d 6520 2d2d 2074 696d 6520 6120 636f  ime -- time a co
│ │ │ │ +00278730: 6d70 7574 6174 696f 6e0a 2a2a 2a2a 2a2a  mputation.******
│ │ │ │ +00278740: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00278750: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a  ****..  * Usage:
│ │ │ │ +00278760: 200a 2020 2020 2020 2020 7469 6d65 2065   .        time e
│ │ │ │ +00278770: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ +00278780: 3d3d 3d3d 3d3d 3d3d 3d0a 0a74 696d 6520  =========..time 
│ │ │ │ +00278790: 6520 6576 616c 7561 7465 7320 652c 2070  e evaluates e, p
│ │ │ │ +002787a0: 7269 6e74 7320 7468 6520 616d 6f75 6e74  rints the amount
│ │ │ │ +002787b0: 206f 6620 6370 7520 7469 6d65 2075 7365   of cpu time use
│ │ │ │ +002787c0: 642c 2061 6e64 2072 6574 7572 6e73 2074  d, and returns t
│ │ │ │ +002787d0: 6865 2076 616c 7565 0a6f 6620 652e 2020  he value.of e.  
│ │ │ │ +002787e0: 5468 6520 7469 6d65 2075 7365 6420 6279  The time used by
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│ │ │ │ +00278800: 2074 6872 6561 6420 616e 6420 6761 7262   thread and garb
│ │ │ │ +00278810: 6167 6520 636f 6c6c 6563 7469 6f6e 2064  age collection d
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│ │ │ │ +00278850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00278860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00278870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00278880: 2d2d 2b0a 7c69 3120 3a20 7469 6d65 2033  --+.|i1 : time 3
│ │ │ │ +00278890: 5e33 3020 2020 2020 2020 2020 2020 2020  ^30             
│ │ │ │ +002788a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002788b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +002788c0: 202d 2d20 7573 6564 2031 2e39 3034 3365   -- used 1.9043e
│ │ │ │ +002788d0: 2d30 3573 2028 6370 7529 3b20 352e 3635  -05s (cpu); 5.65
│ │ │ │ +002788e0: 3965 2d30 3673 2028 7468 7265 6164 293b  9e-06s (thread);
│ │ │ │ +002788f0: 2030 7320 2867 6329 7c0a 7c20 2020 2020   0s (gc)|.|     
│ │ │ │ +00278900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00278910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00278920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00278930: 2020 207c 0a7c 6f31 203d 2032 3035 3839     |.|o1 = 20589
│ │ │ │ +00278940: 3131 3332 3039 3436 3439 2020 2020 2020  1132094649      
│ │ │ │ +00278950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00278960: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00278970: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00278980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00278990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002789a0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520  ---------+..See 
│ │ │ │ +002789b0: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  also.========.. 
│ │ │ │ +002789c0: 202a 202a 6e6f 7465 2074 696d 696e 673a   * *note timing:
│ │ │ │ +002789d0: 2074 696d 696e 672c 202d 2d20 7469 6d65   timing, -- time
│ │ │ │ +002789e0: 2061 2063 6f6d 7075 7461 7469 6f6e 0a20   a computation. 
│ │ │ │ +002789f0: 202a 202a 6e6f 7465 2063 7075 5469 6d65   * *note cpuTime
│ │ │ │ +00278a00: 3a20 6370 7554 696d 652c 202d 2d20 7365  : cpuTime, -- se
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│ │ │ │ +00278a30: 6175 6c61 7932 2062 6567 616e 0a20 202a  aulay2 began.  *
│ │ │ │ +00278a40: 202a 6e6f 7465 2065 6c61 7073 6564 5469   *note elapsedTi
│ │ │ │ +00278a50: 6d69 6e67 3a20 656c 6170 7365 6454 696d  ming: elapsedTim
│ │ │ │ +00278a60: 696e 672c 202d 2d20 7469 6d65 2061 2063  ing, -- time a c
│ │ │ │ +00278a70: 6f6d 7075 7461 7469 6f6e 2075 7369 6e67  omputation using
│ │ │ │ +00278a80: 2074 696d 650a 2020 2020 656c 6170 7365   time.    elapse
│ │ │ │ +00278a90: 640a 2020 2a20 2a6e 6f74 6520 656c 6170  d.  * *note elap
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│ │ │ │ +00278ab0: 5469 6d65 2c20 2d2d 2074 696d 6520 6120  Time, -- time a 
│ │ │ │ +00278ac0: 636f 6d70 7574 6174 696f 6e20 696e 636c  computation incl
│ │ │ │ +00278ad0: 7564 696e 6720 7469 6d65 0a20 2020 2065  uding time.    e
│ │ │ │ +00278ae0: 6c61 7073 6564 0a0a 466f 7220 7468 6520  lapsed..For the 
│ │ │ │ +00278af0: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
│ │ │ │ +00278b00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
│ │ │ │ +00278b10: 6865 206f 626a 6563 7420 2a6e 6f74 6520  he object *note 
│ │ │ │ +00278b20: 7469 6d65 3a20 7469 6d65 2c20 6973 2061  time: time, is a
│ │ │ │ +00278b30: 202a 6e6f 7465 206b 6579 776f 7264 3a20   *note keyword: 
│ │ │ │ +00278b40: 4b65 7977 6f72 642c 2e0a 0a2d 2d2d 2d2d  Keyword,...-----
│ │ │ │  00278b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00278b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00278b70: 0a0a 5468 6520 736f 7572 6365 206f 6620  ..The source of 
│ │ │ │ -00278b80: 7468 6973 2064 6f63 756d 656e 7420 6973  this document is
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│ │ │ │ -00278ba0: 6475 6369 626c 652d 7061 7468 2f6d 6163  ducible-path/mac
│ │ │ │ -00278bb0: 6175 6c61 7932 2d31 2e32 352e 3131 2b64  aulay2-1.25.11+d
│ │ │ │ -00278bc0: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70  s/M2/Macaulay2/p
│ │ │ │ -00278bd0: 6163 6b61 6765 732f 0a4d 6163 6175 6c61  ackages/.Macaula
│ │ │ │ -00278be0: 7932 446f 632f 6f76 5f73 7973 7465 6d2e  y2Doc/ov_system.
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│ │ │ │ -00278c30: 5469 6d65 2c20 5072 6576 3a20 7469 6d65  Time, Prev: time
│ │ │ │ -00278c40: 2c20 5570 3a20 7379 7374 656d 2066 6163  , Up: system fac
│ │ │ │ -00278c50: 696c 6974 6965 730a 0a74 696d 696e 6720  ilities..timing 
│ │ │ │ -00278c60: 2d2d 2074 696d 6520 6120 636f 6d70 7574  -- time a comput
│ │ │ │ -00278c70: 6174 696f 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a  ation.**********
│ │ │ │ -00278c80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00278c90: 2a2a 0a0a 4465 7363 7269 7074 696f 6e0a  **..Description.
│ │ │ │ -00278ca0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a74 696d  ===========..tim
│ │ │ │ -00278cb0: 696e 6720 6520 6576 616c 7561 7465 7320  ing e evaluates 
│ │ │ │ -00278cc0: 6520 616e 6420 7265 7475 726e 7320 6120  e and returns a 
│ │ │ │ -00278cd0: 6c69 7374 206f 6620 7479 7065 202a 6e6f  list of type *no
│ │ │ │ -00278ce0: 7465 2054 696d 653a 2054 696d 652c 206f  te Time: Time, o
│ │ │ │ -00278cf0: 6620 7468 6520 666f 726d 0a7b 742c 767d  f the form.{t,v}
│ │ │ │ -00278d00: 2c20 7768 6572 6520 7420 6973 2074 6865  , where t is the
│ │ │ │ -00278d10: 206e 756d 6265 7220 6f66 2073 6563 6f6e   number of secon
│ │ │ │ -00278d20: 6473 206f 6620 6370 7520 7469 6d69 6e67  ds of cpu timing
│ │ │ │ -00278d30: 2075 7365 642c 2061 6e64 2076 2069 7320   used, and v is 
│ │ │ │ -00278d40: 7468 6520 7661 6c75 650a 6f66 2074 6865  the value.of the
│ │ │ │ -00278d50: 2065 7870 7265 7373 696f 6e2e 0a0a 0a54   expression....T
│ │ │ │ -00278d60: 6865 2064 6566 6175 6c74 206d 6574 686f  he default metho
│ │ │ │ -00278d70: 6420 666f 7220 7072 696e 7469 6e67 2073  d for printing s
│ │ │ │ -00278d80: 7563 6820 7469 6d69 6e67 2072 6573 756c  uch timing resul
│ │ │ │ -00278d90: 7473 2069 7320 746f 2064 6973 706c 6179  ts is to display
│ │ │ │ -00278da0: 2074 6865 2074 696d 696e 670a 7365 7061   the timing.sepa
│ │ │ │ -00278db0: 7261 7465 6c79 2069 6e20 6120 636f 6d6d  rately in a comm
│ │ │ │ -00278dc0: 656e 7420 6265 6c6f 7720 7468 6520 636f  ent below the co
│ │ │ │ -00278dd0: 6d70 7574 6564 2076 616c 7565 2e0a 2b2d  mputed value..+-
│ │ │ │ -00278de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00278df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00278e00: 2d2d 2d2d 2d2b 0a7c 6931 203a 2074 696d  -----+.|i1 : tim
│ │ │ │ -00278e10: 696e 6720 335e 3330 2020 2020 2020 2020  ing 3^30        
│ │ │ │ -00278e20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00278e30: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00278e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00278e50: 2020 2020 2020 207c 0a7c 6f31 203d 2032         |.|o1 = 2
│ │ │ │ -00278e60: 3035 3839 3131 3332 3039 3436 3439 2020  05891132094649  
│ │ │ │ +00278b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00278b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00278b90: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520  ----------..The 
│ │ │ │ +00278ba0: 736f 7572 6365 206f 6620 7468 6973 2064  source of this d
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│ │ │ │ +00278bc0: 7569 6c64 2f72 6570 726f 6475 6369 626c  uild/reproducibl
│ │ │ │ +00278bd0: 652d 7061 7468 2f6d 6163 6175 6c61 7932  e-path/macaulay2
│ │ │ │ +00278be0: 2d31 2e32 352e 3131 2b64 732f 4d32 2f4d  -1.25.11+ds/M2/M
│ │ │ │ +00278bf0: 6163 6175 6c61 7932 2f70 6163 6b61 6765  acaulay2/package
│ │ │ │ +00278c00: 732f 0a4d 6163 6175 6c61 7932 446f 632f  s/.Macaulay2Doc/
│ │ │ │ +00278c10: 6f76 5f73 7973 7465 6d2e 6d32 3a39 3039  ov_system.m2:909
│ │ │ │ +00278c20: 3a30 2e0a 1f0a 4669 6c65 3a20 4d61 6361  :0....File: Maca
│ │ │ │ +00278c30: 756c 6179 3244 6f63 2e69 6e66 6f2c 204e  ulay2Doc.info, N
│ │ │ │ +00278c40: 6f64 653a 2074 696d 696e 672c 204e 6578  ode: timing, Nex
│ │ │ │ +00278c50: 743a 2065 6c61 7073 6564 5469 6d65 2c20  t: elapsedTime, 
│ │ │ │ +00278c60: 5072 6576 3a20 7469 6d65 2c20 5570 3a20  Prev: time, Up: 
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│ │ │ │ +00278ca0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00278cb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4465  ************..De
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│ │ │ │ +00278d10: 653a 2054 696d 652c 206f 6620 7468 6520  e: Time, of the 
│ │ │ │ +00278d20: 666f 726d 0a7b 742c 767d 2c20 7768 6572  form.{t,v}, wher
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│ │ │ │ +00278d50: 6370 7520 7469 6d69 6e67 2075 7365 642c  cpu timing used,
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│ │ │ │ +00278d70: 6c75 650a 6f66 2074 6865 2065 7870 7265  lue.of the expre
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│ │ │ │ +00278da0: 7072 696e 7469 6e67 2073 7563 6820 7469  printing such ti
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│ │ │ │ +00278dc0: 746f 2064 6973 706c 6179 2074 6865 2074  to display the t
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│ │ │ │ +00278e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00278e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00278e30: 0a7c 6931 203a 2074 696d 696e 6720 335e  .|i1 : timing 3^
│ │ │ │ +00278e40: 3330 2020 2020 2020 2020 2020 2020 2020  30              
│ │ │ │ +00278e50: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00278e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00278e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00278e80: 7c0a 7c20 2020 2020 2d2d 202e 3030 3030  |.|     -- .0000
│ │ │ │ -00278e90: 3138 3134 3420 7365 636f 6e64 7320 2020  18144 seconds   
│ │ │ │ -00278ea0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00278eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00278ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00278ed0: 2020 7c0a 7c6f 3120 3a20 5469 6d65 2020    |.|o1 : Time  
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│ │ │ │ +00278e90: 3332 3039 3436 3439 2020 2020 2020 2020  32094649        
│ │ │ │ +00278ea0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00278eb0: 2020 2d2d 202e 3030 3030 3139 3930 3620    -- .000019906 
│ │ │ │ +00278ec0: 7365 636f 6e64 7320 2020 2020 2020 2020  seconds         
│ │ │ │ +00278ed0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00278ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00278ef0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -00278f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00278f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00278f20: 2d2d 2d2d 2b0a 7c69 3220 3a20 7065 656b  ----+.|i2 : peek
│ │ │ │ -00278f30: 206f 6f20 2020 2020 2020 2020 2020 2020   oo             
│ │ │ │ -00278f40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00278f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00278ef0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00278f00: 3120 3a20 5469 6d65 2020 2020 2020 2020  1 : Time        
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│ │ │ │ +00278f20: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00278f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00278f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
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│ │ │ │  00278f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ -00278fa0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00278fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00278fc0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061  --------+..See a
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│ │ │ │ -00278fe0: 2a20 2a6e 6f74 6520 5469 6d65 3a20 5469  * *note Time: Ti
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│ │ │ │ -00279080: 204d 6163 6175 6c61 7932 2062 6567 616e   Macaulay2 began
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│ │ │ │ -002790b0: 6454 696d 696e 672c 202d 2d20 7469 6d65  dTiming, -- time
│ │ │ │ -002790c0: 2061 2063 6f6d 7075 7461 7469 6f6e 2075   a computation u
│ │ │ │ -002790d0: 7369 6e67 2074 696d 650a 2020 2020 656c  sing time.    el
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│ │ │ │ -002790f0: 656c 6170 7365 6454 696d 653a 2065 6c61  elapsedTime: ela
│ │ │ │ -00279100: 7073 6564 5469 6d65 2c20 2d2d 2074 696d  psedTime, -- tim
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│ │ │ │ -00279120: 696e 636c 7564 696e 6720 7469 6d65 0a20  including time. 
│ │ │ │ -00279130: 2020 2065 6c61 7073 6564 0a0a 466f 7220     elapsed..For 
│ │ │ │ -00279140: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ -00279150: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
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│ │ │ │ -002791a0: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...-------------
│ │ │ │ -002791b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -002791c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00278f70: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00278f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00278f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00278fa0: 7c0a 7c6f 3220 3d20 5469 6d65 7b2e 3030  |.|o2 = Time{.00
│ │ │ │ +00278fb0: 3030 3139 3930 362c 2032 3035 3839 3131  0019906, 2058911
│ │ │ │ +00278fc0: 3332 3039 3436 3439 7d7c 0a2b 2d2d 2d2d  32094649}|.+----
│ │ │ │ +00278fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00278fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00278ff0: 2d2d 2b0a 0a53 6565 2061 6c73 6f0a 3d3d  --+..See also.==
│ │ │ │ +00279000: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74  ======..  * *not
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│ │ │ │ +00279060: 2063 6f6d 7075 7461 7469 6f6e 0a20 202a   computation.  *
│ │ │ │ +00279070: 202a 6e6f 7465 2063 7075 5469 6d65 3a20   *note cpuTime: 
│ │ │ │ +00279080: 6370 7554 696d 652c 202d 2d20 7365 636f  cpuTime, -- seco
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│ │ │ │ +002790b0: 6c61 7932 2062 6567 616e 0a20 202a 202a  lay2 began.  * *
│ │ │ │ +002790c0: 6e6f 7465 2065 6c61 7073 6564 5469 6d69  note elapsedTimi
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│ │ │ │ +00279110: 2020 2a20 2a6e 6f74 6520 656c 6170 7365    * *note elapse
│ │ │ │ +00279120: 6454 696d 653a 2065 6c61 7073 6564 5469  dTime: elapsedTi
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│ │ │ │ +00279140: 6d70 7574 6174 696f 6e20 696e 636c 7564  mputation includ
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│ │ │ │ +00279160: 7073 6564 0a0a 466f 7220 7468 6520 7072  psed..For the pr
│ │ │ │ +00279170: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ +00279180: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
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│ │ │ │ +002791c0: 3a20 4b65 7977 6f72 642c 2e0a 0a2d 2d2d  : Keyword,...---
│ │ │ │  002791d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  002791e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -002791f0: 2d2d 0a0a 5468 6520 736f 7572 6365 206f  --..The source o
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│ │ │ │ -00279230: 6163 6175 6c61 7932 2d31 2e32 352e 3131  acaulay2-1.25.11
│ │ │ │ -00279240: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932  +ds/M2/Macaulay2
│ │ │ │ -00279250: 2f70 6163 6b61 6765 732f 0a4d 6163 6175  /packages/.Macau
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│ │ │ │ -00279290: 2e69 6e66 6f2c 204e 6f64 653a 2065 6c61  .info, Node: ela
│ │ │ │ -002792a0: 7073 6564 5469 6d65 2c20 4e65 7874 3a20  psedTime, Next: 
│ │ │ │ -002792b0: 656c 6170 7365 6454 696d 696e 672c 2050  elapsedTiming, P
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│ │ │ │ -00279320: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00279330: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00279340: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00279350: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765  *****..  * Usage
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│ │ │ │ -00279370: 6564 5469 6d65 2065 0a0a 4465 7363 7269  edTime e..Descri
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│ │ │ │ -00279390: 3d0a 0a65 6c61 7073 6564 5469 6d65 2065  =..elapsedTime e
│ │ │ │ -002793a0: 2065 7661 6c75 6174 6573 2065 2c20 7072   evaluates e, pr
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│ │ │ │ -002793c0: 6f66 2074 696d 6520 656c 6170 7365 642c  of time elapsed,
│ │ │ │ -002793d0: 2061 6e64 2072 6574 7572 6e73 2074 6865   and returns the
│ │ │ │ -002793e0: 0a76 616c 7565 206f 6620 652e 0a2b 2d2d  .value of e..+--
│ │ │ │ -002793f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00279400: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 656c  ------+.|i1 : el
│ │ │ │ -00279410: 6170 7365 6454 696d 6520 736c 6565 7020  apsedTime sleep 
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│ │ │ │ -00279430: 2065 6c61 7073 6564 2020 2020 7c0a 7c20   elapsed    |.| 
│ │ │ │ -00279440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00279450: 2020 2020 2020 207c 0a7c 6f31 203d 2030         |.|o1 = 0
│ │ │ │ -00279460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00279470: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -00279480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ -00279490: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d  See also.=======
│ │ │ │ -002794a0: 3d0a 0a20 202a 202a 6e6f 7465 2065 6c61  =..  * *note ela
│ │ │ │ -002794b0: 7073 6564 5469 6d69 6e67 3a20 656c 6170  psedTiming: elap
│ │ │ │ -002794c0: 7365 6454 696d 696e 672c 202d 2d20 7469  sedTiming, -- ti
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│ │ │ │ -002794e0: 2075 7369 6e67 2074 696d 650a 2020 2020   using time.    
│ │ │ │ -002794f0: 656c 6170 7365 640a 2020 2a20 2a6e 6f74  elapsed.  * *not
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│ │ │ │ -00279530: 7369 6e63 6520 4d61 6361 756c 6179 3220  since Macaulay2 
│ │ │ │ -00279540: 6265 6761 6e0a 2020 2a20 2a6e 6f74 6520  began.  * *note 
│ │ │ │ -00279550: 4743 7374 6174 733a 2047 4373 7461 7473  GCstats: GCstats
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│ │ │ │ -00279570: 2061 626f 7574 2074 6865 2073 7461 7475   about the statu
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│ │ │ │ -00279590: 0a20 2020 2063 6f6c 6c65 6374 6f72 0a20  .    collector. 
│ │ │ │ -002795a0: 202a 202a 6e6f 7465 2070 6172 616c 6c65   * *note paralle
│ │ │ │ -002795b0: 6c20 7072 6f67 7261 6d6d 696e 6720 7769  l programming wi
│ │ │ │ -002795c0: 7468 2074 6872 6561 6473 2061 6e64 2074  th threads and t
│ │ │ │ -002795d0: 6173 6b73 3a20 7061 7261 6c6c 656c 2070  asks: parallel p
│ │ │ │ -002795e0: 726f 6772 616d 6d69 6e67 0a20 2020 2077  rogramming.    w
│ │ │ │ -002795f0: 6974 6820 7468 7265 6164 7320 616e 6420  ith threads and 
│ │ │ │ -00279600: 7461 736b 732c 0a20 202a 202a 6e6f 7465  tasks,.  * *note
│ │ │ │ -00279610: 2070 6172 616c 6c65 6c69 736d 2069 6e20   parallelism in 
│ │ │ │ -00279620: 656e 6769 6e65 2063 6f6d 7075 7461 7469  engine computati
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│ │ │ │ -00279640: 2069 6e20 656e 6769 6e65 0a20 2020 2063   in engine.    c
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│ │ │ │ -00279c20: 7469 6d65 2075 7365 6420 7369 6e63 6520  time used since 
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│ │ │ │ +002798d0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
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│ │ │ │ +00279a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +00279a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00279a90: 7c0a 7c20 2020 2020 2d2d 2031 2e30 3030  |.|     -- 1.000
│ │ │ │ +00279aa0: 3134 2073 6563 6f6e 6473 2020 207c 0a7c  14 seconds   |.|
│ │ │ │ +00279ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00279ac0: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ +00279ad0: 3a20 5469 6d65 2020 2020 2020 2020 2020  : Time          
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│ │ │ │ +00279af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +00279b20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00279b30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00279b40: 7c6f 3220 3d20 5469 6d65 7b31 2e30 3030  |o2 = Time{1.000
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│ │ │ │ +00279b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00279b70: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061  --------+..See a
│ │ │ │ +00279b80: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  lso.========..  
│ │ │ │ +00279b90: 2a20 2a6e 6f74 6520 5469 6d65 3a20 5469  * *note Time: Ti
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│ │ │ │ -00279e40: 2073 7973 7465 6d20 6661 6369 6c69 7469   system faciliti
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│ │ │ │ -00279e90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00279ea0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00279eb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00279ec0: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361  *******..  * Usa
│ │ │ │ -00279ed0: 6765 3a20 0a20 2020 2020 2020 2063 7075  ge: .        cpu
│ │ │ │ -00279ee0: 5469 6d65 2829 0a20 202a 204f 7574 7075  Time().  * Outpu
│ │ │ │ -00279ef0: 7473 3a0a 2020 2020 2020 2a20 6120 2a6e  ts:.      * a *n
│ │ │ │ -00279f00: 6f74 6520 7265 616c 206e 756d 6265 723a  ote real number:
│ │ │ │ -00279f10: 2052 522c 2c20 7468 6520 6e75 6d62 6572   RR,, the number
│ │ │ │ -00279f20: 206f 6620 7365 636f 6e64 7320 6f66 2063   of seconds of c
│ │ │ │ -00279f30: 7075 2074 696d 6520 7573 6564 2073 696e  pu time used sin
│ │ │ │ -00279f40: 6365 0a20 2020 2020 2020 2074 6865 2070  ce.        the p
│ │ │ │ -00279f50: 726f 6772 616d 2077 6173 2073 7461 7274  rogram was start
│ │ │ │ -00279f60: 6564 0a0a 4465 7363 7269 7074 696f 6e0a  ed..Description.
│ │ │ │ -00279f70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a2b 2d2d  ===========..+--
│ │ │ │ -00279f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00279f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00279fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00279fb0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ -00279fc0: 7431 203d 2063 7075 5469 6d65 2829 2020  t1 = cpuTime()  
│ │ │ │ -00279fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00279fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00279ff0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00279d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00279d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00279d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
│ │ │ │ +00279d90: 2073 6f75 7263 6520 6f66 2074 6869 7320   source of this 
│ │ │ │ +00279da0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f  document is in./
│ │ │ │ +00279db0: 6275 696c 642f 7265 7072 6f64 7563 6962  build/reproducib
│ │ │ │ +00279dc0: 6c65 2d70 6174 682f 6d61 6361 756c 6179  le-path/macaulay
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│ │ │ │ +00279de0: 4d61 6361 756c 6179 322f 7061 636b 6167  Macaulay2/packag
│ │ │ │ +00279df0: 6573 2f0a 4d61 6361 756c 6179 3244 6f63  es/.Macaulay2Doc
│ │ │ │ +00279e00: 2f6f 765f 7379 7374 656d 2e6d 323a 3932  /ov_system.m2:92
│ │ │ │ +00279e10: 343a 302e 0a1f 0a46 696c 653a 204d 6163  4:0....File: Mac
│ │ │ │ +00279e20: 6175 6c61 7932 446f 632e 696e 666f 2c20  aulay2Doc.info, 
│ │ │ │ +00279e30: 4e6f 6465 3a20 6370 7554 696d 652c 204e  Node: cpuTime, N
│ │ │ │ +00279e40: 6578 743a 2063 7572 7265 6e74 5469 6d65  ext: currentTime
│ │ │ │ +00279e50: 2c20 5072 6576 3a20 656c 6170 7365 6454  , Prev: elapsedT
│ │ │ │ +00279e60: 696d 696e 672c 2055 703a 2073 7973 7465  iming, Up: syste
│ │ │ │ +00279e70: 6d20 6661 6369 6c69 7469 6573 0a0a 6370  m facilities..cp
│ │ │ │ +00279e80: 7554 696d 6520 2d2d 2073 6563 6f6e 6473  uTime -- seconds
│ │ │ │ +00279e90: 206f 6620 6370 7520 7469 6d65 2075 7365   of cpu time use
│ │ │ │ +00279ea0: 6420 7369 6e63 6520 4d61 6361 756c 6179  d since Macaulay
│ │ │ │ +00279eb0: 3220 6265 6761 6e0a 2a2a 2a2a 2a2a 2a2a  2 began.********
│ │ │ │ +00279ec0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00279ed0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00279ee0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00279ef0: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20  *..  * Usage: . 
│ │ │ │ +00279f00: 2020 2020 2020 2063 7075 5469 6d65 2829         cpuTime()
│ │ │ │ +00279f10: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ +00279f20: 2020 2020 2a20 6120 2a6e 6f74 6520 7265      * a *note re
│ │ │ │ +00279f30: 616c 206e 756d 6265 723a 2052 522c 2c20  al number: RR,, 
│ │ │ │ +00279f40: 7468 6520 6e75 6d62 6572 206f 6620 7365  the number of se
│ │ │ │ +00279f50: 636f 6e64 7320 6f66 2063 7075 2074 696d  conds of cpu tim
│ │ │ │ +00279f60: 6520 7573 6564 2073 696e 6365 0a20 2020  e used since.   
│ │ │ │ +00279f70: 2020 2020 2074 6865 2070 726f 6772 616d       the program
│ │ │ │ +00279f80: 2077 6173 2073 7461 7274 6564 0a0a 4465   was started..De
│ │ │ │ +00279f90: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ +00279fa0: 3d3d 3d3d 3d0a 0a2b 2d2d 2d2d 2d2d 2d2d  =====..+--------
│ │ │ │ +00279fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00279fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00279fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00279fe0: 2d2d 2b0a 7c69 3120 3a20 7431 203d 2063  --+.|i1 : t1 = c
│ │ │ │ +00279ff0: 7075 5469 6d65 2829 2020 2020 2020 2020  puTime()        
│ │ │ │  0027a000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027a010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027a020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027a030: 2020 7c0a 7c6f 3120 3d20 3335 342e 3032    |.|o1 = 354.02
│ │ │ │ -0027a040: 3936 3439 3238 3220 2020 2020 2020 2020  9649282         
│ │ │ │ -0027a050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027a060: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027a070: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0027a010: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0027a020: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0027a030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027a040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027a050: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0027a060: 3120 3d20 3331 392e 3839 3037 3734 3335  1 = 319.89077435
│ │ │ │ +0027a070: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  0027a080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027a090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027a0a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0027a0b0: 3120 3a20 5252 2028 6f66 2070 7265 6369  1 : RR (of preci
│ │ │ │ -0027a0c0: 7369 6f6e 2035 3329 2020 2020 2020 2020  sion 53)        
│ │ │ │ -0027a0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027a0e0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -0027a0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0027a100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0027a110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0027a120: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 666f  ------+.|i2 : fo
│ │ │ │ -0027a130: 7220 6920 6672 6f6d 2030 2074 6f20 3130  r i from 0 to 10
│ │ │ │ -0027a140: 3030 3030 3020 646f 2032 3233 3133 3133  00000 do 2231313
│ │ │ │ -0027a150: 3231 3332 312a 3332 3432 3334 3332 3433  21321*3242343243
│ │ │ │ -0027a160: 3234 3b7c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  24;|.+----------
│ │ │ │ -0027a170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0027a180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027a090: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027a0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027a0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027a0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027a0d0: 2020 2020 2020 7c0a 7c6f 3120 3a20 5252        |.|o1 : RR
│ │ │ │ +0027a0e0: 2028 6f66 2070 7265 6369 7369 6f6e 2035   (of precision 5
│ │ │ │ +0027a0f0: 3329 2020 2020 2020 2020 2020 2020 2020  3)              
│ │ │ │ +0027a100: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0027a940: 3630 202a 2036 3020 297c 0a7c 2020 2020  60 * 60 )|.|    
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│ │ │ │ -0027ab00: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
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│ │ │ │ -0027b160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0027b1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
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│ │ │ │ +0027afc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0027b480: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a  **..  * Usage: .
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│ │ │ │ +0027b4a0: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
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│ │ │ │ +0027b4e0: 2020 2020 202a 2074 6865 2061 6c61 726d       * the alarm
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│ │ │ │ +0027b530: 2020 2020 202a 6e6f 7465 2074 7279 3a20       *note try: 
│ │ │ │ +0027b540: 7472 792c 0a0a 4465 7363 7269 7074 696f  try,..Descriptio
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│ │ │ │ -0027b8e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0027b8f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ -0027b900: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
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│ │ │ │ -0027b920: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
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│ │ │ │ -0027bae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ +0027b8d0: 2c20 5570 3a20 7379 7374 656d 2066 6163  , Up: system fac
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│ │ │ │  0027bb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
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│ │ │ │ -0027bb40: 6f64 7563 6962 6c65 2d70 6174 682f 6d61  oducible-path/ma
│ │ │ │ -0027bb50: 6361 756c 6179 322d 312e 3235 2e31 312b  caulay2-1.25.11+
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│ │ │ │ -0027bc10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
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│ │ │ │ -0027bc80: 202d 2d20 7465 726d 696e 6174 6573 2074   -- terminates t
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│ │ │ │ -0027bcc0: 2061 7265 2066 6c75 7368 6564 2061 6e64   are flushed and
│ │ │ │ -0027bcd0: 2063 6c6f 7365 642e 2020 4675 6e63 7469   closed.  Functi
│ │ │ │ -0027bce0: 6f6e 7320 7265 6769 7374 6572 6564 2077  ons registered w
│ │ │ │ -0027bcf0: 6974 6820 2a6e 6f74 6520 6164 6445 6e64  ith *note addEnd
│ │ │ │ -0027bd00: 4675 6e63 7469 6f6e 3a0a 6164 6445 6e64  Function:.addEnd
│ │ │ │ -0027bd10: 4675 6e63 7469 6f6e 2c20 6172 6520 6361  Function, are ca
│ │ │ │ -0027bd20: 6c6c 6564 2c20 756e 6c65 7373 2061 206e  lled, unless a n
│ │ │ │ -0027bd30: 6f6e 7a65 726f 2072 6574 7572 6e20 7661  onzero return va
│ │ │ │ -0027bd40: 6c75 6520 6861 7320 6265 656e 2070 726f  lue has been pro
│ │ │ │ -0027bd50: 7669 6465 642e 0a41 6e6f 7468 6572 2077  vided..Another w
│ │ │ │ -0027bd60: 6179 2074 6f20 6578 6974 2069 7320 746f  ay to exit is to
│ │ │ │ -0027bd70: 2074 7970 6520 7468 6520 656e 6420 6f66   type the end of
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│ │ │ │ -0027bda0: 6c6c 790a 7365 7420 746f 2043 6f6e 7472  lly.set to Contr
│ │ │ │ -0027bdb0: 6f6c 2d44 2069 6e20 756e 6978 2073 7973  ol-D in unix sys
│ │ │ │ -0027bdc0: 7465 6d73 2c20 616e 6420 6973 2043 6f6e  tems, and is Con
│ │ │ │ -0027bdd0: 7472 6f6c 2d5a 2075 6e64 6572 2057 696e  trol-Z under Win
│ │ │ │ -0027bde0: 646f 7773 2e0a 0a53 6565 2061 6c73 6f0a  dows...See also.
│ │ │ │ -0027bdf0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ -0027be00: 6f74 6520 7175 6974 3a20 7175 6974 2c20  ote quit: quit, 
│ │ │ │ -0027be10: 2d2d 2071 7569 7420 7468 6520 7072 6f67  -- quit the prog
│ │ │ │ -0027be20: 7261 6d0a 0a46 6f72 2074 6865 2070 726f  ram..For the pro
│ │ │ │ -0027be30: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
│ │ │ │ -0027be40: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ -0027be50: 6f62 6a65 6374 202a 6e6f 7465 2065 7869  object *note exi
│ │ │ │ -0027be60: 743a 2065 7869 742c 2069 7320 6120 2a6e  t: exit, is a *n
│ │ │ │ -0027be70: 6f74 6520 636f 6d6d 616e 643a 2043 6f6d  ote command: Com
│ │ │ │ -0027be80: 6d61 6e64 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d  mand,...--------
│ │ │ │ -0027be90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0027bea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0027beb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027bb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027bb20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027bb30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
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│ │ │ │ +0027bb50: 646f 6375 6d65 6e74 2069 7320 696e 0a2f  document is in./
│ │ │ │ +0027bb60: 6275 696c 642f 7265 7072 6f64 7563 6962  build/reproducib
│ │ │ │ +0027bb70: 6c65 2d70 6174 682f 6d61 6361 756c 6179  le-path/macaulay
│ │ │ │ +0027bb80: 322d 312e 3235 2e31 312b 6473 2f4d 322f  2-1.25.11+ds/M2/
│ │ │ │ +0027bb90: 4d61 6361 756c 6179 322f 7061 636b 6167  Macaulay2/packag
│ │ │ │ +0027bba0: 6573 2f0a 4d61 6361 756c 6179 3244 6f63  es/.Macaulay2Doc
│ │ │ │ +0027bbb0: 2f6f 765f 7379 7374 656d 2e6d 323a 3732  /ov_system.m2:72
│ │ │ │ +0027bbc0: 383a 302e 0a1f 0a46 696c 653a 204d 6163  8:0....File: Mac
│ │ │ │ +0027bbd0: 6175 6c61 7932 446f 632e 696e 666f 2c20  aulay2Doc.info, 
│ │ │ │ +0027bbe0: 4e6f 6465 3a20 6578 6974 2c20 4e65 7874  Node: exit, Next
│ │ │ │ +0027bbf0: 3a20 666f 726b 2c20 5072 6576 3a20 6578  : fork, Prev: ex
│ │ │ │ +0027bc00: 6563 2c20 5570 3a20 7379 7374 656d 2066  ec, Up: system f
│ │ │ │ +0027bc10: 6163 696c 6974 6965 730a 0a65 7869 7420  acilities..exit 
│ │ │ │ +0027bc20: 2d2d 2065 7869 7420 7468 6520 7072 6f67  -- exit the prog
│ │ │ │ +0027bc30: 7261 6d0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ram.************
│ │ │ │ +0027bc40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4465  ************..De
│ │ │ │ +0027bc50: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ +0027bc60: 3d3d 3d3d 3d0a 0a65 7869 7420 6e20 2d2d  =====..exit n --
│ │ │ │ +0027bc70: 2074 6572 6d69 6e61 7465 7320 7468 6520   terminates the 
│ │ │ │ +0027bc80: 7072 6f67 7261 6d20 616e 6420 7265 7475  program and retu
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│ │ │ │ +0027bca0: 636f 6465 2e0a 6578 6974 202d 2d20 7465  code..exit -- te
│ │ │ │ +0027bcb0: 726d 696e 6174 6573 2074 6865 2070 726f  rminates the pro
│ │ │ │ +0027bcc0: 6772 616d 2061 6e64 2072 6574 7572 6e73  gram and returns
│ │ │ │ +0027bcd0: 2030 2061 7320 7265 7475 726e 2063 6f64   0 as return cod
│ │ │ │ +0027bce0: 652e 0a0a 0a46 696c 6573 2061 7265 2066  e....Files are f
│ │ │ │ +0027bcf0: 6c75 7368 6564 2061 6e64 2063 6c6f 7365  lushed and close
│ │ │ │ +0027bd00: 642e 2020 4675 6e63 7469 6f6e 7320 7265  d.  Functions re
│ │ │ │ +0027bd10: 6769 7374 6572 6564 2077 6974 6820 2a6e  gistered with *n
│ │ │ │ +0027bd20: 6f74 6520 6164 6445 6e64 4675 6e63 7469  ote addEndFuncti
│ │ │ │ +0027bd30: 6f6e 3a0a 6164 6445 6e64 4675 6e63 7469  on:.addEndFuncti
│ │ │ │ +0027bd40: 6f6e 2c20 6172 6520 6361 6c6c 6564 2c20  on, are called, 
│ │ │ │ +0027bd50: 756e 6c65 7373 2061 206e 6f6e 7a65 726f  unless a nonzero
│ │ │ │ +0027bd60: 2072 6574 7572 6e20 7661 6c75 6520 6861   return value ha
│ │ │ │ +0027bd70: 7320 6265 656e 2070 726f 7669 6465 642e  s been provided.
│ │ │ │ +0027bd80: 0a41 6e6f 7468 6572 2077 6179 2074 6f20  .Another way to 
│ │ │ │ +0027bd90: 6578 6974 2069 7320 746f 2074 7970 6520  exit is to type 
│ │ │ │ +0027bda0: 7468 6520 656e 6420 6f66 2066 696c 6520  the end of file 
│ │ │ │ +0027bdb0: 6368 6172 6163 7465 722c 2077 6869 6368  character, which
│ │ │ │ +0027bdc0: 2069 7320 7479 7069 6361 6c6c 790a 7365   is typically.se
│ │ │ │ +0027bdd0: 7420 746f 2043 6f6e 7472 6f6c 2d44 2069  t to Control-D i
│ │ │ │ +0027bde0: 6e20 756e 6978 2073 7973 7465 6d73 2c20  n unix systems, 
│ │ │ │ +0027bdf0: 616e 6420 6973 2043 6f6e 7472 6f6c 2d5a  and is Control-Z
│ │ │ │ +0027be00: 2075 6e64 6572 2057 696e 646f 7773 2e0a   under Windows..
│ │ │ │ +0027be10: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ +0027be20: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 7175  ==..  * *note qu
│ │ │ │ +0027be30: 6974 3a20 7175 6974 2c20 2d2d 2071 7569  it: quit, -- qui
│ │ │ │ +0027be40: 7420 7468 6520 7072 6f67 7261 6d0a 0a46  t the program..F
│ │ │ │ +0027be50: 6f72 2074 6865 2070 726f 6772 616d 6d65  or the programme
│ │ │ │ +0027be60: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r.==============
│ │ │ │ +0027be70: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ +0027be80: 202a 6e6f 7465 2065 7869 743a 2065 7869   *note exit: exi
│ │ │ │ +0027be90: 742c 2069 7320 6120 2a6e 6f74 6520 636f  t, is a *note co
│ │ │ │ +0027bea0: 6d6d 616e 643a 2043 6f6d 6d61 6e64 2c2e  mmand: Command,.
│ │ │ │ +0027beb0: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..--------------
│ │ │ │  0027bec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0027bed0: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75  -------..The sou
│ │ │ │ -0027bee0: 7263 6520 6f66 2074 6869 7320 646f 6375  rce of this docu
│ │ │ │ -0027bef0: 6d65 6e74 2069 7320 696e 0a2f 6275 696c  ment is in./buil
│ │ │ │ -0027bf00: 642f 7265 7072 6f64 7563 6962 6c65 2d70  d/reproducible-p
│ │ │ │ -0027bf10: 6174 682f 6d61 6361 756c 6179 322d 312e  ath/macaulay2-1.
│ │ │ │ -0027bf20: 3235 2e31 312b 6473 2f4d 322f 4d61 6361  25.11+ds/M2/Maca
│ │ │ │ -0027bf30: 756c 6179 322f 7061 636b 6167 6573 2f0a  ulay2/packages/.
│ │ │ │ -0027bf40: 4d61 6361 756c 6179 3244 6f63 2f6f 765f  Macaulay2Doc/ov_
│ │ │ │ -0027bf50: 7379 7374 656d 2e6d 323a 3734 313a 302e  system.m2:741:0.
│ │ │ │ -0027bf60: 0a1f 0a46 696c 653a 204d 6163 6175 6c61  ...File: Macaula
│ │ │ │ -0027bf70: 7932 446f 632e 696e 666f 2c20 4e6f 6465  y2Doc.info, Node
│ │ │ │ -0027bf80: 3a20 666f 726b 2c20 4e65 7874 3a20 7368  : fork, Next: sh
│ │ │ │ -0027bf90: 6f77 2c20 5072 6576 3a20 6578 6974 2c20  ow, Prev: exit, 
│ │ │ │ -0027bfa0: 5570 3a20 7379 7374 656d 2066 6163 696c  Up: system facil
│ │ │ │ -0027bfb0: 6974 6965 730a 0a66 6f72 6b20 2d2d 2066  ities..fork -- f
│ │ │ │ -0027bfc0: 6f72 6b20 7468 6520 7072 6f63 6573 730a  ork the process.
│ │ │ │ -0027bfd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0027bfe0: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573  ********..  * Us
│ │ │ │ -0027bff0: 6167 653a 200a 2020 2020 2020 2020 666f  age: .        fo
│ │ │ │ -0027c000: 726b 2829 0a20 202a 204f 7574 7075 7473  rk().  * Outputs
│ │ │ │ -0027c010: 3a0a 2020 2020 2020 2a20 5768 656e 2073  :.      * When s
│ │ │ │ -0027c020: 7563 6365 7373 6675 6c2c 2069 7420 7265  uccessful, it re
│ │ │ │ -0027c030: 7475 726e 7320 7468 6520 7072 6f63 6573  turns the proces
│ │ │ │ -0027c040: 7320 6964 206f 6620 7468 6520 6368 696c  s id of the chil
│ │ │ │ -0027c050: 6420 696e 2074 6865 2070 6172 656e 742c  d in the parent,
│ │ │ │ -0027c060: 0a20 2020 2020 2020 2061 6e64 2072 6574  .        and ret
│ │ │ │ -0027c070: 7572 6e73 2030 2069 6e20 7468 6520 6368  urns 0 in the ch
│ │ │ │ -0027c080: 696c 642e 2020 5768 656e 2075 6e73 7563  ild.  When unsuc
│ │ │ │ -0027c090: 6365 7373 6675 6c2c 2069 7420 7265 7475  cessful, it retu
│ │ │ │ -0027c0a0: 726e 7320 2d31 2e0a 0a44 6573 6372 6970  rns -1...Descrip
│ │ │ │ -0027c0b0: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ -0027c0c0: 0a0a 506c 6174 666f 726d 7320 7468 6174  ..Platforms that
│ │ │ │ -0027c0d0: 2064 6f20 6e6f 7420 6861 7665 2061 2062   do not have a b
│ │ │ │ -0027c0e0: 7569 6c74 2d69 6e20 666f 726b 2829 2066  uilt-in fork() f
│ │ │ │ -0027c0f0: 756e 6374 696f 6e20 7769 6c6c 2061 6c77  unction will alw
│ │ │ │ -0027c100: 6179 7320 7265 7475 726e 202d 312e 0a0a  ays return -1...
│ │ │ │ -0027c110: 5761 726e 696e 673a 2069 6e20 6d75 6c74  Warning: in mult
│ │ │ │ -0027c120: 6974 6872 6561 6465 6420 7072 6f67 7261  ithreaded progra
│ │ │ │ -0027c130: 6d73 206c 696b 6520 4d61 6361 756c 6179  ms like Macaulay
│ │ │ │ -0027c140: 322c 2076 6572 7920 6665 7720 6f70 6572  2, very few oper
│ │ │ │ -0027c150: 6174 696f 6e73 2063 616e 2062 650a 7361  ations can be.sa
│ │ │ │ -0027c160: 6665 6c79 2064 6f6e 6520 696e 2074 6865  fely done in the
│ │ │ │ -0027c170: 2063 6869 6c64 2e20 2054 6869 7320 6973   child.  This is
│ │ │ │ -0027c180: 2065 7370 6563 6961 6c6c 7920 7472 7565   especially true
│ │ │ │ -0027c190: 2077 6865 6e20 7468 6520 7573 6572 2068   when the user h
│ │ │ │ -0027c1a0: 6173 2062 6565 6e20 2a6e 6f74 650a 7061  as been *note.pa
│ │ │ │ -0027c1b0: 7261 6c6c 656c 2070 726f 6772 616d 6d69  rallel programmi
│ │ │ │ -0027c1c0: 6e67 2077 6974 6820 7468 7265 6164 7320  ng with threads 
│ │ │ │ -0027c1d0: 616e 6420 7461 736b 733a 2070 6172 616c  and tasks: paral
│ │ │ │ -0027c1e0: 6c65 6c20 7072 6f67 7261 6d6d 696e 6720  lel programming 
│ │ │ │ -0027c1f0: 7769 7468 2074 6872 6561 6473 0a61 6e64  with threads.and
│ │ │ │ -0027c200: 2074 6173 6b73 2c2e 2045 7665 6e20 616c   tasks,. Even al
│ │ │ │ -0027c210: 6c6f 6361 7469 6e67 206d 656d 6f72 7920  locating memory 
│ │ │ │ -0027c220: 696e 2074 6865 2063 6869 6c64 206d 6179  in the child may
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│ │ │ │ -0027c240: 732e 0a0a 466f 7220 7468 6520 7072 6f67  s...For the prog
│ │ │ │ -0027c250: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d  rammer.=========
│ │ │ │ -0027c260: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f  =========..The o
│ │ │ │ -0027c270: 626a 6563 7420 2a6e 6f74 6520 666f 726b  bject *note fork
│ │ │ │ -0027c280: 3a20 666f 726b 2c20 6973 2061 202a 6e6f  : fork, is a *no
│ │ │ │ -0027c290: 7465 2063 6f6d 7069 6c65 6420 6675 6e63  te compiled func
│ │ │ │ -0027c2a0: 7469 6f6e 3a20 436f 6d70 696c 6564 4675  tion: CompiledFu
│ │ │ │ -0027c2b0: 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d  nction,...------
│ │ │ │ -0027c2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0027c2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0027c2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027bed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027bee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027bef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027bf00: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66  -..The source of
│ │ │ │ +0027bf10: 2074 6869 7320 646f 6375 6d65 6e74 2069   this document i
│ │ │ │ +0027bf20: 7320 696e 0a2f 6275 696c 642f 7265 7072  s in./build/repr
│ │ │ │ +0027bf30: 6f64 7563 6962 6c65 2d70 6174 682f 6d61  oducible-path/ma
│ │ │ │ +0027bf40: 6361 756c 6179 322d 312e 3235 2e31 312b  caulay2-1.25.11+
│ │ │ │ +0027bf50: 6473 2f4d 322f 4d61 6361 756c 6179 322f  ds/M2/Macaulay2/
│ │ │ │ +0027bf60: 7061 636b 6167 6573 2f0a 4d61 6361 756c  packages/.Macaul
│ │ │ │ +0027bf70: 6179 3244 6f63 2f6f 765f 7379 7374 656d  ay2Doc/ov_system
│ │ │ │ +0027bf80: 2e6d 323a 3734 313a 302e 0a1f 0a46 696c  .m2:741:0....Fil
│ │ │ │ +0027bf90: 653a 204d 6163 6175 6c61 7932 446f 632e  e: Macaulay2Doc.
│ │ │ │ +0027bfa0: 696e 666f 2c20 4e6f 6465 3a20 666f 726b  info, Node: fork
│ │ │ │ +0027bfb0: 2c20 4e65 7874 3a20 7368 6f77 2c20 5072  , Next: show, Pr
│ │ │ │ +0027bfc0: 6576 3a20 6578 6974 2c20 5570 3a20 7379  ev: exit, Up: sy
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│ │ │ │ +0027bfe0: 0a66 6f72 6b20 2d2d 2066 6f72 6b20 7468  .fork -- fork th
│ │ │ │ +0027bff0: 6520 7072 6f63 6573 730a 2a2a 2a2a 2a2a  e process.******
│ │ │ │ +0027c000: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0027c010: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a  **..  * Usage: .
│ │ │ │ +0027c020: 2020 2020 2020 2020 666f 726b 2829 0a20          fork(). 
│ │ │ │ +0027c030: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ +0027c040: 2020 2a20 5768 656e 2073 7563 6365 7373    * When success
│ │ │ │ +0027c050: 6675 6c2c 2069 7420 7265 7475 726e 7320  ful, it returns 
│ │ │ │ +0027c060: 7468 6520 7072 6f63 6573 7320 6964 206f  the process id o
│ │ │ │ +0027c070: 6620 7468 6520 6368 696c 6420 696e 2074  f the child in t
│ │ │ │ +0027c080: 6865 2070 6172 656e 742c 0a20 2020 2020  he parent,.     
│ │ │ │ +0027c090: 2020 2061 6e64 2072 6574 7572 6e73 2030     and returns 0
│ │ │ │ +0027c0a0: 2069 6e20 7468 6520 6368 696c 642e 2020   in the child.  
│ │ │ │ +0027c0b0: 5768 656e 2075 6e73 7563 6365 7373 6675  When unsuccessfu
│ │ │ │ +0027c0c0: 6c2c 2069 7420 7265 7475 726e 7320 2d31  l, it returns -1
│ │ │ │ +0027c0d0: 2e0a 0a44 6573 6372 6970 7469 6f6e 0a3d  ...Description.=
│ │ │ │ +0027c0e0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 506c 6174  ==========..Plat
│ │ │ │ +0027c0f0: 666f 726d 7320 7468 6174 2064 6f20 6e6f  forms that do no
│ │ │ │ +0027c100: 7420 6861 7665 2061 2062 7569 6c74 2d69  t have a built-i
│ │ │ │ +0027c110: 6e20 666f 726b 2829 2066 756e 6374 696f  n fork() functio
│ │ │ │ +0027c120: 6e20 7769 6c6c 2061 6c77 6179 7320 7265  n will always re
│ │ │ │ +0027c130: 7475 726e 202d 312e 0a0a 5761 726e 696e  turn -1...Warnin
│ │ │ │ +0027c140: 673a 2069 6e20 6d75 6c74 6974 6872 6561  g: in multithrea
│ │ │ │ +0027c150: 6465 6420 7072 6f67 7261 6d73 206c 696b  ded programs lik
│ │ │ │ +0027c160: 6520 4d61 6361 756c 6179 322c 2076 6572  e Macaulay2, ver
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│ │ │ │ +0027c180: 2063 616e 2062 650a 7361 6665 6c79 2064   can be.safely d
│ │ │ │ +0027c190: 6f6e 6520 696e 2074 6865 2063 6869 6c64  one in the child
│ │ │ │ +0027c1a0: 2e20 2054 6869 7320 6973 2065 7370 6563  .  This is espec
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│ │ │ │ -0027d780: 206f 7065 6e73 736c 206f 6e20 796f 7572   openssl on your
│ │ │ │ -0027d790: 2073 7973 7465 6d2e 0a0a 5365 6520 616c   system...See al
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│ │ │ │ -0027d820: 3d3d 3d3d 3d0a 0a20 202a 2022 6765 7457  =====..  * "getW
│ │ │ │ -0027d830: 5757 2853 7472 696e 6729 220a 2020 2a20  WW(String)".  * 
│ │ │ │ -0027d840: 2267 6574 5757 5728 5374 7269 6e67 2c4e  "getWWW(String,N
│ │ │ │ -0027d850: 6f74 6869 6e67 2922 0a20 202a 2022 6765  othing)".  * "ge
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│ │ │ │ -0027d880: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ -0027d890: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ -0027d8a0: 206f 626a 6563 7420 2a6e 6f74 6520 6765   object *note ge
│ │ │ │ -0027d8b0: 7457 5757 3a20 6765 7457 5757 2c20 6973  tWWW: getWWW, is
│ │ │ │ -0027d8c0: 2061 202a 6e6f 7465 206d 6574 686f 6420   a *note method 
│ │ │ │ -0027d8d0: 6675 6e63 7469 6f6e 3a20 4d65 7468 6f64  function: Method
│ │ │ │ -0027d8e0: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d  Function,...----
│ │ │ │ -0027d8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0027d900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0027d910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027d3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027d400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027d410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a  ---------------.
│ │ │ │ +0027d420: 0a54 6865 2073 6f75 7263 6520 6f66 2074  .The source of t
│ │ │ │ +0027d430: 6869 7320 646f 6375 6d65 6e74 2069 7320  his document is 
│ │ │ │ +0027d440: 696e 0a2f 6275 696c 642f 7265 7072 6f64  in./build/reprod
│ │ │ │ +0027d450: 7563 6962 6c65 2d70 6174 682f 6d61 6361  ucible-path/maca
│ │ │ │ +0027d460: 756c 6179 322d 312e 3235 2e31 312b 6473  ulay2-1.25.11+ds
│ │ │ │ +0027d470: 2f4d 322f 4d61 6361 756c 6179 322f 7061  /M2/Macaulay2/pa
│ │ │ │ +0027d480: 636b 6167 6573 2f0a 4d61 6361 756c 6179  ckages/.Macaulay
│ │ │ │ +0027d490: 3244 6f63 2f6f 765f 7379 7374 656d 2e6d  2Doc/ov_system.m
│ │ │ │ +0027d4a0: 323a 3736 393a 302e 0a1f 0a46 696c 653a  2:769:0....File:
│ │ │ │ +0027d4b0: 204d 6163 6175 6c61 7932 446f 632e 696e   Macaulay2Doc.in
│ │ │ │ +0027d4c0: 666f 2c20 4e6f 6465 3a20 6765 7457 5757  fo, Node: getWWW
│ │ │ │ +0027d4d0: 2c20 4e65 7874 3a20 7370 6c69 7457 5757  , Next: splitWWW
│ │ │ │ +0027d4e0: 2c20 5072 6576 3a20 6765 7465 6e76 2c20  , Prev: getenv, 
│ │ │ │ +0027d4f0: 5570 3a20 7379 7374 656d 2066 6163 696c  Up: system facil
│ │ │ │ +0027d500: 6974 6965 730a 0a67 6574 5757 5720 2d2d  ities..getWWW --
│ │ │ │ +0027d510: 2067 6574 2061 2077 6562 2070 6167 650a   get a web page.
│ │ │ │ +0027d520: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0027d530: 2a2a 2a2a 2a2a 2a2a 0a0a 4465 7363 7269  ********..Descri
│ │ │ │ +0027d540: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ +0027d550: 3d0a 0a53 796e 6f70 7369 730a 3d3d 3d3d  =..Synopsis.====
│ │ │ │ +0027d560: 3d3d 3d3d 0a0a 2020 2a20 5573 6167 653a  ====..  * Usage:
│ │ │ │ +0027d570: 200a 2020 2020 2020 2020 6765 7457 5757   .        getWWW
│ │ │ │ +0027d580: 2055 524c 0a20 202a 2049 6e70 7574 733a   URL.  * Inputs:
│ │ │ │ +0027d590: 0a20 2020 2020 202a 2055 524c 2c20 6120  .      * URL, a 
│ │ │ │ +0027d5a0: 2a6e 6f74 6520 7374 7269 6e67 3a20 5374  *note string: St
│ │ │ │ +0027d5b0: 7269 6e67 2c0a 2020 2a20 4f75 7470 7574  ring,.  * Output
│ │ │ │ +0027d5c0: 733a 0a20 2020 2020 202a 2074 6865 2063  s:.      * the c
│ │ │ │ +0027d5d0: 6f6e 7465 6e74 7320 6f66 2074 6865 2077  ontents of the w
│ │ │ │ +0027d5e0: 6562 2070 6167 652c 2074 6f67 6574 6865  eb page, togethe
│ │ │ │ +0027d5f0: 7220 7769 7468 2074 6865 2068 7474 7020  r with the http 
│ │ │ │ +0027d600: 6865 6164 6572 732c 2061 7420 7468 650a  headers, at the.
│ │ │ │ +0027d610: 2020 2020 2020 2020 6164 6472 6573 7320          address 
│ │ │ │ +0027d620: 6769 7665 6e20 6279 2055 524c 0a0a 5379  given by URL..Sy
│ │ │ │ +0027d630: 6e6f 7073 6973 0a3d 3d3d 3d3d 3d3d 3d0a  nopsis.========.
│ │ │ │ +0027d640: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
│ │ │ │ +0027d650: 2020 2020 2067 6574 5757 5728 5552 4c2c       getWWW(URL,
│ │ │ │ +0027d660: 5445 5854 290a 2020 2a20 496e 7075 7473  TEXT).  * Inputs
│ │ │ │ +0027d670: 3a0a 2020 2020 2020 2a20 5552 4c2c 2061  :.      * URL, a
│ │ │ │ +0027d680: 202a 6e6f 7465 2073 7472 696e 673a 2053   *note string: S
│ │ │ │ +0027d690: 7472 696e 672c 0a20 2020 2020 202a 2054  tring,.      * T
│ │ │ │ +0027d6a0: 4558 542c 2061 202a 6e6f 7465 2073 7472  EXT, a *note str
│ │ │ │ +0027d6b0: 696e 673a 2053 7472 696e 672c 0a20 202a  ing: String,.  *
│ │ │ │ +0027d6c0: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ +0027d6d0: 2a20 6f62 7461 696e 2074 6865 2063 6f6e  * obtain the con
│ │ │ │ +0027d6e0: 7465 6e74 7320 6f66 2074 6865 2077 6562  tents of the web
│ │ │ │ +0027d6f0: 2070 6167 6520 6164 6472 6573 7365 6420   page addressed 
│ │ │ │ +0027d700: 6279 2055 524c 2066 726f 6d20 616e 2068  by URL from an h
│ │ │ │ +0027d710: 7474 700a 2020 2020 2020 2020 7365 7276  ttp.        serv
│ │ │ │ +0027d720: 6572 2c20 7573 696e 6720 7468 6520 504f  er, using the PO
│ │ │ │ +0027d730: 5354 206d 6574 686f 642c 2070 726f 7669  ST method, provi
│ │ │ │ +0027d740: 6465 6420 7769 7468 2054 4558 540a 0a41  ded with TEXT..A
│ │ │ │ +0027d750: 6363 6573 7369 6e67 2061 2073 6563 7572  ccessing a secur
│ │ │ │ +0027d760: 6520 7765 6220 7369 7465 2028 7768 6f73  e web site (whos
│ │ │ │ +0027d770: 6520 5552 4c20 6265 6769 6e73 2077 6974  e URL begins wit
│ │ │ │ +0027d780: 6820 6874 7470 733a 2920 6465 7065 6e64  h https:) depend
│ │ │ │ +0027d790: 7320 6f6e 2079 6f75 720a 6861 7669 6e67  s on your.having
│ │ │ │ +0027d7a0: 2069 6e73 7461 6c6c 6564 206f 7065 6e73   installed opens
│ │ │ │ +0027d7b0: 736c 206f 6e20 796f 7572 2073 7973 7465  sl on your syste
│ │ │ │ +0027d7c0: 6d2e 0a0a 5365 6520 616c 736f 0a3d 3d3d  m...See also.===
│ │ │ │ +0027d7d0: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ +0027d7e0: 2073 706c 6974 5757 573a 2073 706c 6974   splitWWW: split
│ │ │ │ +0027d7f0: 5757 572c 202d 2d20 7365 7061 7261 7465  WWW, -- separate
│ │ │ │ +0027d800: 2061 6e20 6874 7470 2072 6573 706f 6e73   an http respons
│ │ │ │ +0027d810: 6520 696e 746f 2068 6561 6465 7220 616e  e into header an
│ │ │ │ +0027d820: 6420 626f 6479 0a0a 5761 7973 2074 6f20  d body..Ways to 
│ │ │ │ +0027d830: 7573 6520 6765 7457 5757 3a0a 3d3d 3d3d  use getWWW:.====
│ │ │ │ +0027d840: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +0027d850: 0a20 202a 2022 6765 7457 5757 2853 7472  .  * "getWWW(Str
│ │ │ │ +0027d860: 696e 6729 220a 2020 2a20 2267 6574 5757  ing)".  * "getWW
│ │ │ │ +0027d870: 5728 5374 7269 6e67 2c4e 6f74 6869 6e67  W(String,Nothing
│ │ │ │ +0027d880: 2922 0a20 202a 2022 6765 7457 5757 2853  )".  * "getWWW(S
│ │ │ │ +0027d890: 7472 696e 672c 5374 7269 6e67 2922 0a0a  tring,String)"..
│ │ │ │ +0027d8a0: 466f 7220 7468 6520 7072 6f67 7261 6d6d  For the programm
│ │ │ │ +0027d8b0: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  er.=============
│ │ │ │ +0027d8c0: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563  =====..The objec
│ │ │ │ +0027d8d0: 7420 2a6e 6f74 6520 6765 7457 5757 3a20  t *note getWWW: 
│ │ │ │ +0027d8e0: 6765 7457 5757 2c20 6973 2061 202a 6e6f  getWWW, is a *no
│ │ │ │ +0027d8f0: 7465 206d 6574 686f 6420 6675 6e63 7469  te method functi
│ │ │ │ +0027d900: 6f6e 3a20 4d65 7468 6f64 4675 6e63 7469  on: MethodFuncti
│ │ │ │ +0027d910: 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d  on,...----------
│ │ │ │  0027d920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0027d930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
│ │ │ │ -0027d940: 2073 6f75 7263 6520 6f66 2074 6869 7320   source of this 
│ │ │ │ -0027d950: 646f 6375 6d65 6e74 2069 7320 696e 0a2f  document is in./
│ │ │ │ -0027d960: 6275 696c 642f 7265 7072 6f64 7563 6962  build/reproducib
│ │ │ │ -0027d970: 6c65 2d70 6174 682f 6d61 6361 756c 6179  le-path/macaulay
│ │ │ │ -0027d980: 322d 312e 3235 2e31 312b 6473 2f4d 322f  2-1.25.11+ds/M2/
│ │ │ │ -0027d990: 4d61 6361 756c 6179 322f 7061 636b 6167  Macaulay2/packag
│ │ │ │ -0027d9a0: 6573 2f0a 4d61 6361 756c 6179 3244 6f63  es/.Macaulay2Doc
│ │ │ │ -0027d9b0: 2f6f 765f 7379 7374 656d 2e6d 323a 3137  /ov_system.m2:17
│ │ │ │ -0027d9c0: 3331 3a30 2e0a 1f0a 4669 6c65 3a20 4d61  31:0....File: Ma
│ │ │ │ -0027d9d0: 6361 756c 6179 3244 6f63 2e69 6e66 6f2c  caulay2Doc.info,
│ │ │ │ -0027d9e0: 204e 6f64 653a 2073 706c 6974 5757 572c   Node: splitWWW,
│ │ │ │ -0027d9f0: 204e 6578 743a 2068 7474 7048 6561 6465   Next: httpHeade
│ │ │ │ -0027da00: 7273 5f6c 7053 7472 696e 675f 7270 2c20  rs_lpString_rp, 
│ │ │ │ -0027da10: 5072 6576 3a20 6765 7457 5757 2c20 5570  Prev: getWWW, Up
│ │ │ │ -0027da20: 3a20 7379 7374 656d 2066 6163 696c 6974  : system facilit
│ │ │ │ -0027da30: 6965 730a 0a73 706c 6974 5757 5720 2d2d  ies..splitWWW --
│ │ │ │ -0027da40: 2073 6570 6172 6174 6520 616e 2068 7474   separate an htt
│ │ │ │ -0027da50: 7020 7265 7370 6f6e 7365 2069 6e74 6f20  p response into 
│ │ │ │ -0027da60: 6865 6164 6572 2061 6e64 2062 6f64 790a  header and body.
│ │ │ │ -0027da70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0027da80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0027da90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0027daa0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ -0027dab0: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ -0027dac0: 2868 6561 642c 2062 6f64 7929 203d 2073  (head, body) = s
│ │ │ │ -0027dad0: 706c 6974 5757 5720 7374 720a 2020 2a20  plitWWW str.  * 
│ │ │ │ -0027dae0: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ -0027daf0: 7374 722c 2061 202a 6e6f 7465 2073 7472  str, a *note str
│ │ │ │ -0027db00: 696e 673a 2053 7472 696e 672c 2c20 616e  ing: String,, an
│ │ │ │ -0027db10: 2068 7474 7020 7265 7370 6f6e 7365 2c20   http response, 
│ │ │ │ -0027db20: 7375 6368 2061 7320 7468 6174 2072 6574  such as that ret
│ │ │ │ -0027db30: 7572 6e65 640a 2020 2020 2020 2020 6279  urned.        by
│ │ │ │ -0027db40: 202a 6e6f 7465 2067 6574 5757 573a 2067   *note getWWW: g
│ │ │ │ -0027db50: 6574 5757 572c 2e0a 2020 2a20 4f75 7470  etWWW,..  * Outp
│ │ │ │ -0027db60: 7574 733a 0a20 2020 2020 202a 2068 6561  uts:.      * hea
│ │ │ │ -0027db70: 642c 2061 202a 6e6f 7465 2073 7472 696e  d, a *note strin
│ │ │ │ -0027db80: 673a 2053 7472 696e 672c 2c20 7468 6520  g: String,, the 
│ │ │ │ -0027db90: 6865 6164 6572 206f 6620 7468 6520 7265  header of the re
│ │ │ │ -0027dba0: 7370 6f6e 7365 0a20 2020 2020 202a 2062  sponse.      * b
│ │ │ │ -0027dbb0: 6f64 792c 2061 202a 6e6f 7465 2073 7472  ody, a *note str
│ │ │ │ -0027dbc0: 696e 673a 2053 7472 696e 672c 2c20 7468  ing: String,, th
│ │ │ │ -0027dbd0: 6520 7265 7370 6f6e 7365 2062 6f64 792c  e response body,
│ │ │ │ -0027dbe0: 2077 6869 6368 2068 6173 2062 6565 6e0a   which has been.
│ │ │ │ -0027dbf0: 2020 2020 2020 2020 2775 6e63 6875 6e6b          'unchunk
│ │ │ │ -0027dc00: 6564 272c 2069 6620 7468 6520 7265 7370  ed', if the resp
│ │ │ │ -0027dc10: 6f6e 7365 2074 7970 6520 6973 2063 6875  onse type is chu
│ │ │ │ -0027dc20: 6e6b 6564 2e0a 0a44 6573 6372 6970 7469  nked...Descripti
│ │ │ │ -0027dc30: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │ -0027dc40: 5468 6520 666f 726d 6174 206f 6620 6368  The format of ch
│ │ │ │ -0027dc50: 756e 6b65 6420 6461 7461 2069 7320 6465  unked data is de
│ │ │ │ -0027dc60: 7363 7269 6265 6420 6865 7265 2028 7365  scribed here (se
│ │ │ │ -0027dc70: 6520 6874 7470 733a 2f2f 7777 772e 7733  e https://www.w3
│ │ │ │ -0027dc80: 2e6f 7267 2f50 726f 746f 636f 6c73 2f0a  .org/Protocols/.
│ │ │ │ -0027dc90: 292e 0a0a 5468 6520 666f 6c6c 6f77 696e  )...The followin
│ │ │ │ -0027dca0: 6720 6973 2061 6e20 6578 616d 706c 6520  g is an example 
│ │ │ │ -0027dcb0: 6f62 7461 696e 696e 6720 3520 6578 616d  obtaining 5 exam
│ │ │ │ -0027dcc0: 706c 6573 2066 726f 6d20 7468 6520 4b72  ples from the Kr
│ │ │ │ -0027dcd0: 6575 7a65 722d 536b 6172 6b65 0a64 6174  euzer-Skarke.dat
│ │ │ │ -0027dce0: 6162 6173 6520 666f 7220 3420 6469 6d65  abase for 4 dime
│ │ │ │ -0027dcf0: 6e73 696f 6e61 6c20 7265 666c 6578 6976  nsional reflexiv
│ │ │ │ -0027dd00: 6520 706f 6c79 746f 7065 732e 2020 5765  e polytopes.  We
│ │ │ │ -0027dd10: 2072 6574 7269 6576 6520 3520 6578 616d   retrieve 5 exam
│ │ │ │ -0027dd20: 706c 6573 2065 6163 680a 6861 7669 6e67  ples each.having
│ │ │ │ -0027dd30: 2074 6865 2061 6e74 692d 6361 6e6f 6e69   the anti-canoni
│ │ │ │ -0027dd40: 6361 6c20 6469 7669 736f 7220 6120 4361  cal divisor a Ca
│ │ │ │ -0027dd50: 6c61 6269 2d59 6175 2077 6974 6820 2468  labi-Yau with $h
│ │ │ │ -0027dd60: 5e7b 2831 2c31 297d 203d 2031 3024 2e0a  ^{(1,1)} = 10$..
│ │ │ │ -0027dd70: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -0027dd80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0027dd90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027d930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027d940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027d950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027d960: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263  -----..The sourc
│ │ │ │ +0027d970: 6520 6f66 2074 6869 7320 646f 6375 6d65  e of this docume
│ │ │ │ +0027d980: 6e74 2069 7320 696e 0a2f 6275 696c 642f  nt is in./build/
│ │ │ │ +0027d990: 7265 7072 6f64 7563 6962 6c65 2d70 6174  reproducible-pat
│ │ │ │ +0027d9a0: 682f 6d61 6361 756c 6179 322d 312e 3235  h/macaulay2-1.25
│ │ │ │ +0027d9b0: 2e31 312b 6473 2f4d 322f 4d61 6361 756c  .11+ds/M2/Macaul
│ │ │ │ +0027d9c0: 6179 322f 7061 636b 6167 6573 2f0a 4d61  ay2/packages/.Ma
│ │ │ │ +0027d9d0: 6361 756c 6179 3244 6f63 2f6f 765f 7379  caulay2Doc/ov_sy
│ │ │ │ +0027d9e0: 7374 656d 2e6d 323a 3137 3331 3a30 2e0a  stem.m2:1731:0..
│ │ │ │ +0027d9f0: 1f0a 4669 6c65 3a20 4d61 6361 756c 6179  ..File: Macaulay
│ │ │ │ +0027da00: 3244 6f63 2e69 6e66 6f2c 204e 6f64 653a  2Doc.info, Node:
│ │ │ │ +0027da10: 2073 706c 6974 5757 572c 204e 6578 743a   splitWWW, Next:
│ │ │ │ +0027da20: 2068 7474 7048 6561 6465 7273 5f6c 7053   httpHeaders_lpS
│ │ │ │ +0027da30: 7472 696e 675f 7270 2c20 5072 6576 3a20  tring_rp, Prev: 
│ │ │ │ +0027da40: 6765 7457 5757 2c20 5570 3a20 7379 7374  getWWW, Up: syst
│ │ │ │ +0027da50: 656d 2066 6163 696c 6974 6965 730a 0a73  em facilities..s
│ │ │ │ +0027da60: 706c 6974 5757 5720 2d2d 2073 6570 6172  plitWWW -- separ
│ │ │ │ +0027da70: 6174 6520 616e 2068 7474 7020 7265 7370  ate an http resp
│ │ │ │ +0027da80: 6f6e 7365 2069 6e74 6f20 6865 6164 6572  onse into header
│ │ │ │ +0027da90: 2061 6e64 2062 6f64 790a 2a2a 2a2a 2a2a   and body.******
│ │ │ │ +0027daa0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0027dab0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0027dac0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0027dad0: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a  ****..  * Usage:
│ │ │ │ +0027dae0: 200a 2020 2020 2020 2020 2868 6561 642c   .        (head,
│ │ │ │ +0027daf0: 2062 6f64 7929 203d 2073 706c 6974 5757   body) = splitWW
│ │ │ │ +0027db00: 5720 7374 720a 2020 2a20 496e 7075 7473  W str.  * Inputs
│ │ │ │ +0027db10: 3a0a 2020 2020 2020 2a20 7374 722c 2061  :.      * str, a
│ │ │ │ +0027db20: 202a 6e6f 7465 2073 7472 696e 673a 2053   *note string: S
│ │ │ │ +0027db30: 7472 696e 672c 2c20 616e 2068 7474 7020  tring,, an http 
│ │ │ │ +0027db40: 7265 7370 6f6e 7365 2c20 7375 6368 2061  response, such a
│ │ │ │ +0027db50: 7320 7468 6174 2072 6574 7572 6e65 640a  s that returned.
│ │ │ │ +0027db60: 2020 2020 2020 2020 6279 202a 6e6f 7465          by *note
│ │ │ │ +0027db70: 2067 6574 5757 573a 2067 6574 5757 572c   getWWW: getWWW,
│ │ │ │ +0027db80: 2e0a 2020 2a20 4f75 7470 7574 733a 0a20  ..  * Outputs:. 
│ │ │ │ +0027db90: 2020 2020 202a 2068 6561 642c 2061 202a       * head, a *
│ │ │ │ +0027dba0: 6e6f 7465 2073 7472 696e 673a 2053 7472  note string: Str
│ │ │ │ +0027dbb0: 696e 672c 2c20 7468 6520 6865 6164 6572  ing,, the header
│ │ │ │ +0027dbc0: 206f 6620 7468 6520 7265 7370 6f6e 7365   of the response
│ │ │ │ +0027dbd0: 0a20 2020 2020 202a 2062 6f64 792c 2061  .      * body, a
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│ │ │ │ +0027dca0: 733a 2f2f 7777 772e 7733 2e6f 7267 2f50  s://www.w3.org/P
│ │ │ │ +0027dcb0: 726f 746f 636f 6c73 2f0a 292e 0a0a 5468  rotocols/.)...Th
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│ │ │ │ +0027dce0: 696e 6720 3520 6578 616d 706c 6573 2066  ing 5 examples f
│ │ │ │ +0027dcf0: 726f 6d20 7468 6520 4b72 6575 7a65 722d  rom the Kreuzer-
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│ │ │ │ +0027dd40: 6576 6520 3520 6578 616d 706c 6573 2065  eve 5 examples e
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│ │ │ │ +0027dd60: 6e74 692d 6361 6e6f 6e69 6361 6c20 6469  nti-canonical di
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│ │ │ │ +0027dd80: 6175 2077 6974 6820 2468 5e7b 2831 2c31  au with $h^{(1,1
│ │ │ │ +0027dd90: 297d 203d 2031 3024 2e0a 0a2b 2d2d 2d2d  )} = 10$...+----
│ │ │ │  0027dda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0027ddb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0027ddc0: 0a7c 6931 203a 2073 7472 203d 2067 6574  .|i1 : str = get
│ │ │ │ -0027ddd0: 5757 5720 2268 7474 703a 2f2f 7175 6172  WWW "http://quar
│ │ │ │ -0027dde0: 6b2e 6974 702e 7475 7769 656e 2e61 632e  k.itp.tuwien.ac.
│ │ │ │ -0027ddf0: 6174 2f63 6769 2d62 696e 2f63 792f 6379  at/cgi-bin/cy/cy
│ │ │ │ -0027de00: 6461 7461 2e63 6769 3f68 3131 3d31 307c  data.cgi?h11=10|
│ │ │ │ -0027de10: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ -0027de20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0027de30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027ddb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027ddc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027ddd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027dde0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
│ │ │ │ +0027ddf0: 2073 7472 203d 2067 6574 5757 5720 2268   str = getWWW "h
│ │ │ │ +0027de00: 7474 703a 2f2f 7175 6172 6b2e 6974 702e  ttp://quark.itp.
│ │ │ │ +0027de10: 7475 7769 656e 2e61 632e 6174 2f63 6769  tuwien.ac.at/cgi
│ │ │ │ +0027de20: 2d62 696e 2f63 792f 6379 6461 7461 2e63  -bin/cy/cydata.c
│ │ │ │ +0027de30: 6769 3f68 3131 3d31 307c 0a7c 2d2d 2d2d  gi?h11=10|.|----
│ │ │ │  0027de40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0027de50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -0027de60: 0a7c 264c 3d35 223b 2020 2020 2020 2020  .|&L=5";        
│ │ │ │ -0027de70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027de80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027de90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027dea0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027deb0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -0027dec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0027ded0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027de50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027de60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027de70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027de80: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 264c 3d35  ---------|.|&L=5
│ │ │ │ +0027de90: 223b 2020 2020 2020 2020 2020 2020 2020  ";              
│ │ │ │ +0027dea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027deb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027dec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027ded0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  0027dee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0027def0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0027df00: 0a7c 6932 203a 2028 6865 6164 2c62 6f64  .|i2 : (head,bod
│ │ │ │ -0027df10: 7929 203d 2073 706c 6974 5757 5720 7374  y) = splitWWW st
│ │ │ │ -0027df20: 723b 2020 2020 2020 2020 2020 2020 2020  r;              
│ │ │ │ -0027df30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027df40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027df50: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -0027df60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0027df70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027def0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027df00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027df10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027df20: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a  ---------+.|i2 :
│ │ │ │ +0027df30: 2028 6865 6164 2c62 6f64 7929 203d 2073   (head,body) = s
│ │ │ │ +0027df40: 706c 6974 5757 5720 7374 723b 2020 2020  plitWWW str;    
│ │ │ │ +0027df50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027df60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027df70: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  0027df80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0027df90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0027dfa0: 0a7c 6933 203a 2068 6561 6420 2020 2020  .|i3 : head     
│ │ │ │ -0027dfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027dfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027dfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027dfe0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027dff0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0027df90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027dfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027dfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027dfc0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ +0027dfd0: 2068 6561 6420 2020 2020 2020 2020 2020   head           
│ │ │ │ +0027dfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027dff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0027e000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e010: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0027e020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e030: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e040: 0a7c 6f33 203d 2048 5454 502f 312e 3120  .|o3 = HTTP/1.1 
│ │ │ │ -0027e050: 3230 3020 4f4b 2020 2020 2020 2020 2020  200 OK          
│ │ │ │ -0027e060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e080: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e090: 0a7c 2020 2020 2044 6174 653a 2054 6875  .|     Date: Thu
│ │ │ │ -0027e0a0: 2c20 3233 204a 756e 2032 3031 3620 3132  , 23 Jun 2016 12
│ │ │ │ -0027e0b0: 3a31 303a 3538 2047 4d54 2020 2020 2020  :10:58 GMT      
│ │ │ │ -0027e0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e0d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e0e0: 0a7c 2020 2020 2053 6572 7665 723a 2041  .|     Server: A
│ │ │ │ -0027e0f0: 7061 6368 652f 322e 3220 2020 2020 2020  pache/2.2       
│ │ │ │ -0027e100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e120: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e130: 0a7c 2020 2020 2056 6172 793a 2041 6363  .|     Vary: Acc
│ │ │ │ -0027e140: 6570 742d 456e 636f 6469 6e67 2020 2020  ept-Encoding    
│ │ │ │ -0027e150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e170: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e180: 0a7c 2020 2020 2043 6f6e 6e65 6374 696f  .|     Connectio
│ │ │ │ -0027e190: 6e3a 2063 6c6f 7365 2020 2020 2020 2020  n: close        
│ │ │ │ -0027e1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e1c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e1d0: 0a7c 2020 2020 2054 7261 6e73 6665 722d  .|     Transfer-
│ │ │ │ -0027e1e0: 456e 636f 6469 6e67 3a20 6368 756e 6b65  Encoding: chunke
│ │ │ │ -0027e1f0: 6420 2020 2020 2020 2020 2020 2020 2020  d               
│ │ │ │ -0027e200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e210: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e220: 0a7c 2020 2020 2043 6f6e 7465 6e74 2d54  .|     Content-T
│ │ │ │ -0027e230: 7970 653a 2074 6578 742f 6874 6d6c 3b20  ype: text/html; 
│ │ │ │ -0027e240: 6368 6172 7365 743d 5554 462d 3820 2020  charset=UTF-8   
│ │ │ │ -0027e250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e260: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e270: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -0027e280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0027e290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027e030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e060: 2020 2020 2020 2020 207c 0a7c 6f33 203d           |.|o3 =
│ │ │ │ +0027e070: 2048 5454 502f 312e 3120 3230 3020 4f4b   HTTP/1.1 200 OK
│ │ │ │ +0027e080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e0b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027e0c0: 2044 6174 653a 2054 6875 2c20 3233 204a   Date: Thu, 23 J
│ │ │ │ +0027e0d0: 756e 2032 3031 3620 3132 3a31 303a 3538  un 2016 12:10:58
│ │ │ │ +0027e0e0: 2047 4d54 2020 2020 2020 2020 2020 2020   GMT            
│ │ │ │ +0027e0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e100: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027e110: 2053 6572 7665 723a 2041 7061 6368 652f   Server: Apache/
│ │ │ │ +0027e120: 322e 3220 2020 2020 2020 2020 2020 2020  2.2             
│ │ │ │ +0027e130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e150: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027e160: 2056 6172 793a 2041 6363 6570 742d 456e   Vary: Accept-En
│ │ │ │ +0027e170: 636f 6469 6e67 2020 2020 2020 2020 2020  coding          
│ │ │ │ +0027e180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e1a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027e1b0: 2043 6f6e 6e65 6374 696f 6e3a 2063 6c6f   Connection: clo
│ │ │ │ +0027e1c0: 7365 2020 2020 2020 2020 2020 2020 2020  se              
│ │ │ │ +0027e1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e1f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027e200: 2054 7261 6e73 6665 722d 456e 636f 6469   Transfer-Encodi
│ │ │ │ +0027e210: 6e67 3a20 6368 756e 6b65 6420 2020 2020  ng: chunked     
│ │ │ │ +0027e220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e240: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027e250: 2043 6f6e 7465 6e74 2d54 7970 653a 2074   Content-Type: t
│ │ │ │ +0027e260: 6578 742f 6874 6d6c 3b20 6368 6172 7365  ext/html; charse
│ │ │ │ +0027e270: 743d 5554 462d 3820 2020 2020 2020 2020  t=UTF-8         
│ │ │ │ +0027e280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e290: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  0027e2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0027e2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0027e2c0: 0a7c 6934 203a 2062 6f64 7920 2020 2020  .|i4 : body     
│ │ │ │ -0027e2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e300: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e310: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0027e2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027e2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027e2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027e2e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a  ---------+.|i4 :
│ │ │ │ +0027e2f0: 2062 6f64 7920 2020 2020 2020 2020 2020   body           
│ │ │ │ +0027e300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0027e320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e330: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0027e340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e350: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e360: 0a7c 6f34 203d 203c 6865 6164 3e3c 7469  .|o4 = SEARCH RESUL
│ │ │ │ -0027e380: 5453 3c2f 7469 746c 653e 3c2f 6865 6164  TS               
│ │ │ │ -0027e3a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e3b0: 0a7c 2020 2020 203c 626f 6479 3e3c 7072  .|     Search comm
│ │ │ │ -0027e3d0: 616e 643a 3c2f 623e 2020 2020 2020 2020  and:        
│ │ │ │ -0027e3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e3f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e400: 0a7c 2020 2020 2063 6c61 7373 2e78 202d  .|     class.x -
│ │ │ │ -0027e410: 6469 2078 202d 4865 2045 4831 303a 4d56  di x -He EH10:MV
│ │ │ │ -0027e420: 4e46 4c35 2020 2020 2020 2020 2020 2020  NFL5            
│ │ │ │ -0027e430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e440: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e450: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0027e350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e380: 2020 2020 2020 2020 207c 0a7c 6f34 203d           |.|o4 =
│ │ │ │ +0027e390: 203c 6865 6164 3e3c 7469 746c 653e 5345   SE
│ │ │ │ +0027e3a0: 4152 4348 2052 4553 554c 5453 3c2f 7469  ARCH RESULTS</ti
│ │ │ │ +0027e3b0: 746c 653e 3c2f 6865 6164 3e20 2020 2020  tle></head>     
│ │ │ │ +0027e3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e3d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027e3e0: 203c 626f 6479 3e3c 7072 653e 3c62 3e53   <body><pre><b>S
│ │ │ │ +0027e3f0: 6561 7263 6820 636f 6d6d 616e 643a 3c2f  earch command:</
│ │ │ │ +0027e400: 623e 2020 2020 2020 2020 2020 2020 2020  b>              
│ │ │ │ +0027e410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e420: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027e430: 2063 6c61 7373 2e78 202d 6469 2078 202d   class.x -di x -
│ │ │ │ +0027e440: 4865 2045 4831 303a 4d56 4e46 4c35 2020  He EH10:MVNFL5  
│ │ │ │ +0027e450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0027e460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e470: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0027e480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e490: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e4a0: 0a7c 2020 2020 203c 623e 5265 7375 6c74  .|     <b>Result
│ │ │ │ -0027e4b0: 3a3c 2f62 3e20 2020 2020 2020 2020 2020  :</b>           
│ │ │ │ -0027e4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e4e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e4f0: 0a7c 2020 2020 2034 2039 2020 4d3a 3232  .|     4 9  M:22
│ │ │ │ -0027e500: 2039 204e 3a31 3420 3820 483a 3130 2c31   9 N:14 8 H:10,1
│ │ │ │ -0027e510: 3820 5b2d 3136 5d20 2020 2020 2020 2020  8 [-16]         
│ │ │ │ -0027e520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e530: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e540: 0a7c 2020 2020 2020 2020 3120 2020 3020  .|        1   0 
│ │ │ │ -0027e550: 2020 3120 2020 3020 2020 3220 2020 3020    1   0   2   0 
│ │ │ │ -0027e560: 202d 3220 202d 3220 202d 3220 2020 2020   -2  -2  -2     
│ │ │ │ -0027e570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e580: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e590: 0a7c 2020 2020 2020 2020 3020 2020 3120  .|        0   1 
│ │ │ │ -0027e5a0: 2020 3020 2020 3020 202d 3120 2020 3120    0   0  -1   1 
│ │ │ │ -0027e5b0: 2020 3120 202d 3120 2020 3120 2020 2020    1  -1   1     
│ │ │ │ -0027e5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e5d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e5e0: 0a7c 2020 2020 2020 2020 3020 2020 3020  .|        0   0 
│ │ │ │ -0027e5f0: 2020 3220 2020 3020 2020 3120 2020 3120    2   0   1   1 
│ │ │ │ -0027e600: 202d 3320 202d 3120 202d 3420 2020 2020   -3  -1  -4     
│ │ │ │ -0027e610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e620: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e630: 0a7c 2020 2020 2020 2020 3020 2020 3020  .|        0   0 
│ │ │ │ -0027e640: 2020 3020 2020 3120 2020 3120 2020 3120    0   1   1   1 
│ │ │ │ -0027e650: 202d 3120 202d 3120 202d 3220 2020 2020   -1  -1  -2     
│ │ │ │ -0027e660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e670: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e680: 0a7c 2020 2020 2034 2031 3020 204d 3a32  .|     4 10  M:2
│ │ │ │ -0027e690: 3320 3130 204e 3a31 3520 3130 2048 3a31  3 10 N:15 10 H:1
│ │ │ │ -0027e6a0: 302c 3138 205b 2d31 365d 2020 2020 2020  0,18 [-16]      
│ │ │ │ -0027e6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e6c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e6d0: 0a7c 2020 2020 2020 2020 2031 2020 2020  .|         1    
│ │ │ │ -0027e6e0: 3020 2020 2030 2020 2020 3020 2020 2d31  0    0    0   -1
│ │ │ │ -0027e6f0: 2020 2020 3120 2020 2d32 2020 2020 3220      1   -2    2 
│ │ │ │ -0027e700: 2020 2030 2020 202d 3120 2020 2020 2020     0   -1       
│ │ │ │ -0027e710: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e720: 0a7c 2020 2020 2020 2020 2030 2020 2020  .|         0    
│ │ │ │ -0027e730: 3120 2020 2030 2020 2020 3020 2020 2031  1    0    0    1
│ │ │ │ -0027e740: 2020 202d 3120 2020 2032 2020 202d 3120     -1    2   -1 
│ │ │ │ -0027e750: 2020 2d32 2020 2020 3020 2020 2020 2020    -2    0       
│ │ │ │ -0027e760: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e770: 0a7c 2020 2020 2020 2020 2030 2020 2020  .|         0    
│ │ │ │ -0027e780: 3020 2020 2031 2020 2020 3020 2020 2d31  0    1    0   -1
│ │ │ │ -0027e790: 2020 2020 3120 2020 2d31 2020 2020 3020      1   -1    0 
│ │ │ │ -0027e7a0: 2020 2032 2020 202d 3220 2020 2020 2020     2   -2       
│ │ │ │ -0027e7b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e7c0: 0a7c 2020 2020 2020 2020 2030 2020 2020  .|         0    
│ │ │ │ -0027e7d0: 3020 2020 2030 2020 2020 3120 2020 2031  0    0    1    1
│ │ │ │ -0027e7e0: 2020 202d 3120 2020 2030 2020 202d 3220     -1    0   -2 
│ │ │ │ -0027e7f0: 2020 2d31 2020 2020 3220 2020 2020 2020    -1    2       
│ │ │ │ -0027e800: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e810: 0a7c 2020 2020 2034 2039 2020 4d3a 3234  .|     4 9  M:24
│ │ │ │ -0027e820: 2039 204e 3a31 3420 3820 483a 3130 2c32   9 N:14 8 H:10,2
│ │ │ │ -0027e830: 3020 5b2d 3230 5d20 2020 2020 2020 2020  0 [-20]         
│ │ │ │ -0027e840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e850: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e860: 0a7c 2020 2020 2020 2020 3120 2020 3020  .|        1   0 
│ │ │ │ -0027e870: 2020 3120 2020 3020 2020 3120 202d 3120    1   0   1  -1 
│ │ │ │ -0027e880: 202d 3220 2020 3120 202d 3220 2020 2020   -2   1  -2     
│ │ │ │ -0027e890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e8a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e8b0: 0a7c 2020 2020 2020 2020 3020 2020 3120  .|        0   1 
│ │ │ │ -0027e8c0: 2020 3020 2020 3020 2020 3020 2020 3220    0   0   0   2 
│ │ │ │ -0027e8d0: 202d 3220 202d 3120 2020 3220 2020 2020   -2  -1   2     
│ │ │ │ -0027e8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e8f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e900: 0a7c 2020 2020 2020 2020 3020 2020 3020  .|        0   0 
│ │ │ │ -0027e910: 2020 3220 2020 3020 202d 3120 202d 3120    2   0  -1  -1 
│ │ │ │ -0027e920: 2020 3020 202d 3220 202d 3220 2020 2020    0  -2  -2     
│ │ │ │ -0027e930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e940: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e950: 0a7c 2020 2020 2020 2020 3020 2020 3020  .|        0   0 
│ │ │ │ -0027e960: 2020 3020 2020 3120 202d 3120 202d 3120    0   1  -1  -1 
│ │ │ │ -0027e970: 2020 3120 202d 3120 202d 3120 2020 2020    1  -1  -1     
│ │ │ │ -0027e980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e990: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e9a0: 0a7c 2020 2020 2034 2031 3120 204d 3a32  .|     4 11  M:2
│ │ │ │ -0027e9b0: 3520 3131 204e 3a31 3520 3130 2048 3a31  5 11 N:15 10 H:1
│ │ │ │ -0027e9c0: 302c 3230 205b 2d32 305d 2020 2020 2020  0,20 [-20]      
│ │ │ │ -0027e9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027e9e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027e9f0: 0a7c 2020 2020 2020 2020 3120 2020 3020  .|        1   0 
│ │ │ │ -0027ea00: 2020 3020 2020 3020 2020 3220 202d 3220    0   0   2  -2 
│ │ │ │ -0027ea10: 2020 3020 2020 3220 202d 3220 202d 3220    0   2  -2  -2 
│ │ │ │ -0027ea20: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -0027ea30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027ea40: 0a7c 2020 2020 2020 2020 3020 2020 3120  .|        0   1 
│ │ │ │ -0027ea50: 2020 3020 2020 3020 202d 3120 2020 3120    0   0  -1   1 
│ │ │ │ -0027ea60: 2020 3120 202d 3120 2020 3020 2020 3120    1  -1   0   1 
│ │ │ │ -0027ea70: 202d 3220 2020 2020 2020 2020 2020 2020   -2             
│ │ │ │ -0027ea80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027ea90: 0a7c 2020 2020 2020 2020 3020 2020 3020  .|        0   0 
│ │ │ │ -0027eaa0: 2020 3120 2020 3020 202d 3120 2020 3120    1   0  -1   1 
│ │ │ │ -0027eab0: 202d 3120 2020 3020 2020 3220 2020 3020   -1   0   2   0 
│ │ │ │ -0027eac0: 202d 3220 2020 2020 2020 2020 2020 2020   -2             
│ │ │ │ -0027ead0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027eae0: 0a7c 2020 2020 2020 2020 3020 2020 3020  .|        0   0 
│ │ │ │ -0027eaf0: 2020 3020 2020 3120 202d 3120 2020 3120    0   1  -1   1 
│ │ │ │ -0027eb00: 2020 3120 202d 3220 2020 3120 2020 3020    1  -2   1   0 
│ │ │ │ -0027eb10: 202d 3120 2020 2020 2020 2020 2020 2020   -1             
│ │ │ │ -0027eb20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027eb30: 0a7c 2020 2020 2034 2031 3020 204d 3a32  .|     4 10  M:2
│ │ │ │ -0027eb40: 3520 3130 204e 3a31 3520 3130 2048 3a31  5 10 N:15 10 H:1
│ │ │ │ -0027eb50: 302c 3230 205b 2d32 305d 2020 2020 2020  0,20 [-20]      
│ │ │ │ -0027eb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027eb70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027eb80: 0a7c 2020 2020 2020 2020 2031 2020 2020  .|         1    
│ │ │ │ -0027eb90: 3020 2020 2030 2020 2020 3020 2020 2d31  0    0    0   -1
│ │ │ │ -0027eba0: 2020 2020 3020 2020 2d31 2020 202d 3120      0   -1   -1 
│ │ │ │ -0027ebb0: 2020 2032 2020 2020 3120 2020 2020 2020     2    1       
│ │ │ │ -0027ebc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027ebd0: 0a7c 2020 2020 2020 2020 2030 2020 2020  .|         0    
│ │ │ │ -0027ebe0: 3120 2020 2030 2020 2020 3020 2020 2030  1    0    0    0
│ │ │ │ -0027ebf0: 2020 2020 3020 2020 2032 2020 2020 3020      0    2    0 
│ │ │ │ -0027ec00: 2020 2d31 2020 202d 3220 2020 2020 2020    -1   -2       
│ │ │ │ -0027ec10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027ec20: 0a7c 2020 2020 2020 2020 2030 2020 2020  .|         0    
│ │ │ │ -0027ec30: 3020 2020 2031 2020 2020 3020 2020 2030  0    1    0    0
│ │ │ │ -0027ec40: 2020 202d 3220 2020 2032 2020 2020 3220     -2    2    2 
│ │ │ │ -0027ec50: 2020 2d32 2020 202d 3220 2020 2020 2020    -2   -2       
│ │ │ │ -0027ec60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027ec70: 0a7c 2020 2020 2020 2020 2030 2020 2020  .|         0    
│ │ │ │ -0027ec80: 3020 2020 2030 2020 2020 3120 2020 2030  0    0    1    0
│ │ │ │ -0027ec90: 2020 202d 3120 2020 2030 2020 2020 3220     -1    0    2 
│ │ │ │ -0027eca0: 2020 2030 2020 202d 3220 2020 2020 2020     0   -2       
│ │ │ │ -0027ecb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027ecc0: 0a7c 2020 2020 2045 7863 6565 6465 6420  .|     Exceeded 
│ │ │ │ -0027ecd0: 6c69 6d69 7420 6f66 2035 2020 2020 2020  limit of 5      
│ │ │ │ -0027ece0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027ecf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027ed00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027ed10: 0a7c 2020 2020 203c 2f70 7265 3e3c 2f62  .|     </pre></b
│ │ │ │ -0027ed20: 6f64 793e 2020 2020 2020 2020 2020 2020  ody>            
│ │ │ │ -0027ed30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027ed40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0027ed50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0027ed60: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -0027ed70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0027ed80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0027e490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e4c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027e4d0: 203c 623e 5265 7375 6c74 3a3c 2f62 3e20   <b>Result:</b> 
│ │ │ │ +0027e4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e510: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027e520: 2034 2039 2020 4d3a 3232 2039 204e 3a31   4 9  M:22 9 N:1
│ │ │ │ +0027e530: 3420 3820 483a 3130 2c31 3820 5b2d 3136  4 8 H:10,18 [-16
│ │ │ │ +0027e540: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │ +0027e550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e560: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027e570: 2020 2020 3120 2020 3020 2020 3120 2020      1   0   1   
│ │ │ │ +0027e580: 3020 2020 3220 2020 3020 202d 3220 202d  0   2   0  -2  -
│ │ │ │ +0027e590: 3220 202d 3220 2020 2020 2020 2020 2020  2  -2           
│ │ │ │ +0027e5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e5b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027e5c0: 2020 2020 3020 2020 3120 2020 3020 2020      0   1   0   
│ │ │ │ +0027e5d0: 3020 202d 3120 2020 3120 2020 3120 202d  0  -1   1   1  -
│ │ │ │ +0027e5e0: 3120 2020 3120 2020 2020 2020 2020 2020  1   1           
│ │ │ │ +0027e5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e600: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027e610: 2020 2020 3020 2020 3020 2020 3220 2020      0   0   2   
│ │ │ │ +0027e620: 3020 2020 3120 2020 3120 202d 3320 202d  0   1   1  -3  -
│ │ │ │ +0027e630: 3120 202d 3420 2020 2020 2020 2020 2020  1  -4           
│ │ │ │ +0027e640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e650: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027e660: 2020 2020 3020 2020 3020 2020 3020 2020      0   0   0   
│ │ │ │ +0027e670: 3120 2020 3120 2020 3120 202d 3120 202d  1   1   1  -1  -
│ │ │ │ +0027e680: 3120 202d 3220 2020 2020 2020 2020 2020  1  -2           
│ │ │ │ +0027e690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e6a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027e6b0: 2034 2031 3020 204d 3a32 3320 3130 204e   4 10  M:23 10 N
│ │ │ │ +0027e6c0: 3a31 3520 3130 2048 3a31 302c 3138 205b  :15 10 H:10,18 [
│ │ │ │ +0027e6d0: 2d31 365d 2020 2020 2020 2020 2020 2020  -16]            
│ │ │ │ +0027e6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e6f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027e700: 2020 2020 2031 2020 2020 3020 2020 2030       1    0    0
│ │ │ │ +0027e710: 2020 2020 3020 2020 2d31 2020 2020 3120      0   -1    1 
│ │ │ │ +0027e720: 2020 2d32 2020 2020 3220 2020 2030 2020    -2    2    0  
│ │ │ │ +0027e730: 202d 3120 2020 2020 2020 2020 2020 2020   -1             
│ │ │ │ +0027e740: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027e750: 2020 2020 2030 2020 2020 3120 2020 2030       0    1    0
│ │ │ │ +0027e760: 2020 2020 3020 2020 2031 2020 202d 3120      0    1   -1 
│ │ │ │ +0027e770: 2020 2032 2020 202d 3120 2020 2d32 2020     2   -1   -2  
│ │ │ │ +0027e780: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ +0027e790: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027e7a0: 2020 2020 2030 2020 2020 3020 2020 2031       0    0    1
│ │ │ │ +0027e7b0: 2020 2020 3020 2020 2d31 2020 2020 3120      0   -1    1 
│ │ │ │ +0027e7c0: 2020 2d31 2020 2020 3020 2020 2032 2020    -1    0    2  
│ │ │ │ +0027e7d0: 202d 3220 2020 2020 2020 2020 2020 2020   -2             
│ │ │ │ +0027e7e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027e7f0: 2020 2020 2030 2020 2020 3020 2020 2030       0    0    0
│ │ │ │ +0027e800: 2020 2020 3120 2020 2031 2020 202d 3120      1    1   -1 
│ │ │ │ +0027e810: 2020 2030 2020 202d 3220 2020 2d31 2020     0   -2   -1  
│ │ │ │ +0027e820: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0027e830: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027e840: 2034 2039 2020 4d3a 3234 2039 204e 3a31   4 9  M:24 9 N:1
│ │ │ │ +0027e850: 3420 3820 483a 3130 2c32 3020 5b2d 3230  4 8 H:10,20 [-20
│ │ │ │ +0027e860: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │ +0027e870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e880: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027e890: 2020 2020 3120 2020 3020 2020 3120 2020      1   0   1   
│ │ │ │ +0027e8a0: 3020 2020 3120 202d 3120 202d 3220 2020  0   1  -1  -2   
│ │ │ │ +0027e8b0: 3120 202d 3220 2020 2020 2020 2020 2020  1  -2           
│ │ │ │ +0027e8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e8d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027e8e0: 2020 2020 3020 2020 3120 2020 3020 2020      0   1   0   
│ │ │ │ +0027e8f0: 3020 2020 3020 2020 3220 202d 3220 202d  0   0   2  -2  -
│ │ │ │ +0027e900: 3120 2020 3220 2020 2020 2020 2020 2020  1   2           
│ │ │ │ +0027e910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e920: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027e930: 2020 2020 3020 2020 3020 2020 3220 2020      0   0   2   
│ │ │ │ +0027e940: 3020 202d 3120 202d 3120 2020 3020 202d  0  -1  -1   0  -
│ │ │ │ +0027e950: 3220 202d 3220 2020 2020 2020 2020 2020  2  -2           
│ │ │ │ +0027e960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e970: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027e980: 2020 2020 3020 2020 3020 2020 3020 2020      0   0   0   
│ │ │ │ +0027e990: 3120 202d 3120 202d 3120 2020 3120 202d  1  -1  -1   1  -
│ │ │ │ +0027e9a0: 3120 202d 3120 2020 2020 2020 2020 2020  1  -1           
│ │ │ │ +0027e9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027e9c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027e9d0: 2034 2031 3120 204d 3a32 3520 3131 204e   4 11  M:25 11 N
│ │ │ │ +0027e9e0: 3a31 3520 3130 2048 3a31 302c 3230 205b  :15 10 H:10,20 [
│ │ │ │ +0027e9f0: 2d32 305d 2020 2020 2020 2020 2020 2020  -20]            
│ │ │ │ +0027ea00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027ea10: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027ea20: 2020 2020 3120 2020 3020 2020 3020 2020      1   0   0   
│ │ │ │ +0027ea30: 3020 2020 3220 202d 3220 2020 3020 2020  0   2  -2   0   
│ │ │ │ +0027ea40: 3220 202d 3220 202d 3220 2020 3220 2020  2  -2  -2   2   
│ │ │ │ +0027ea50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027ea60: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027ea70: 2020 2020 3020 2020 3120 2020 3020 2020      0   1   0   
│ │ │ │ +0027ea80: 3020 202d 3120 2020 3120 2020 3120 202d  0  -1   1   1  -
│ │ │ │ +0027ea90: 3120 2020 3020 2020 3120 202d 3220 2020  1   0   1  -2   
│ │ │ │ +0027eaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027eab0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027eac0: 2020 2020 3020 2020 3020 2020 3120 2020      0   0   1   
│ │ │ │ +0027ead0: 3020 202d 3120 2020 3120 202d 3120 2020  0  -1   1  -1   
│ │ │ │ +0027eae0: 3020 2020 3220 2020 3020 202d 3220 2020  0   2   0  -2   
│ │ │ │ +0027eaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027eb00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027eb10: 2020 2020 3020 2020 3020 2020 3020 2020      0   0   0   
│ │ │ │ +0027eb20: 3120 202d 3120 2020 3120 2020 3120 202d  1  -1   1   1  -
│ │ │ │ +0027eb30: 3220 2020 3120 2020 3020 202d 3120 2020  2   1   0  -1   
│ │ │ │ +0027eb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027eb50: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027eb60: 2034 2031 3020 204d 3a32 3520 3130 204e   4 10  M:25 10 N
│ │ │ │ +0027eb70: 3a31 3520 3130 2048 3a31 302c 3230 205b  :15 10 H:10,20 [
│ │ │ │ +0027eb80: 2d32 305d 2020 2020 2020 2020 2020 2020  -20]            
│ │ │ │ +0027eb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027eba0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027ebb0: 2020 2020 2031 2020 2020 3020 2020 2030       1    0    0
│ │ │ │ +0027ebc0: 2020 2020 3020 2020 2d31 2020 2020 3020      0   -1    0 
│ │ │ │ +0027ebd0: 2020 2d31 2020 202d 3120 2020 2032 2020    -1   -1    2  
│ │ │ │ +0027ebe0: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │ +0027ebf0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027ec00: 2020 2020 2030 2020 2020 3120 2020 2030       0    1    0
│ │ │ │ +0027ec10: 2020 2020 3020 2020 2030 2020 2020 3020      0    0    0 
│ │ │ │ +0027ec20: 2020 2032 2020 2020 3020 2020 2d31 2020     2    0   -1  
│ │ │ │ +0027ec30: 202d 3220 2020 2020 2020 2020 2020 2020   -2             
│ │ │ │ +0027ec40: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027ec50: 2020 2020 2030 2020 2020 3020 2020 2031       0    0    1
│ │ │ │ +0027ec60: 2020 2020 3020 2020 2030 2020 202d 3220      0    0   -2 
│ │ │ │ +0027ec70: 2020 2032 2020 2020 3220 2020 2d32 2020     2    2   -2  
│ │ │ │ +0027ec80: 202d 3220 2020 2020 2020 2020 2020 2020   -2             
│ │ │ │ +0027ec90: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027eca0: 2020 2020 2030 2020 2020 3020 2020 2030       0    0    0
│ │ │ │ +0027ecb0: 2020 2020 3120 2020 2030 2020 202d 3120      1    0   -1 
│ │ │ │ +0027ecc0: 2020 2030 2020 2020 3220 2020 2030 2020     0    2    0  
│ │ │ │ +0027ecd0: 202d 3220 2020 2020 2020 2020 2020 2020   -2             
│ │ │ │ +0027ece0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0027ecf0: 2045 7863 6565 6465 6420 6c69 6d69 7420   Exceeded limit 
│ │ │ │ +0027ed00: 6f66 2035 2020 2020 2020 2020 2020 2020  of 5            
│ │ │ │ +0027ed10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027ed20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0027ed30: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
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│ │ │ │ -002801e0: 696c 6c20 6120 7072 6f63 6573 730a 0a46  ill a process..F
│ │ │ │ -002801f0: 6f72 2074 6865 2070 726f 6772 616d 6d65  or the programme
│ │ │ │ -00280200: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r.==============
│ │ │ │ -00280210: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ -00280220: 202a 6e6f 7465 206b 696c 6c3a 206b 696c   *note kill: kil
│ │ │ │ -00280230: 6c2c 2069 7320 6120 2a6e 6f74 6520 636f  l, is a *note co
│ │ │ │ -00280240: 6d70 696c 6564 2066 756e 6374 696f 6e3a  mpiled function:
│ │ │ │ -00280250: 2043 6f6d 7069 6c65 6446 756e 6374 696f   CompiledFunctio
│ │ │ │ -00280260: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d  n,...-----------
│ │ │ │ -00280270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00280280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00280010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00280020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00280030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00280040: 0a0a 5468 6520 736f 7572 6365 206f 6620  ..The source of 
│ │ │ │ +00280050: 7468 6973 2064 6f63 756d 656e 7420 6973  this document is
│ │ │ │ +00280060: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f   in./build/repro
│ │ │ │ +00280070: 6475 6369 626c 652d 7061 7468 2f6d 6163  ducible-path/mac
│ │ │ │ +00280080: 6175 6c61 7932 2d31 2e32 352e 3131 2b64  aulay2-1.25.11+d
│ │ │ │ +00280090: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70  s/M2/Macaulay2/p
│ │ │ │ +002800a0: 6163 6b61 6765 732f 0a4d 6163 6175 6c61  ackages/.Macaula
│ │ │ │ +002800b0: 7932 446f 632f 6f76 5f73 7973 7465 6d2e  y2Doc/ov_system.
│ │ │ │ +002800c0: 6d32 3a38 3037 3a30 2e0a 1f0a 4669 6c65  m2:807:0....File
│ │ │ │ +002800d0: 3a20 4d61 6361 756c 6179 3244 6f63 2e69  : Macaulay2Doc.i
│ │ │ │ +002800e0: 6e66 6f2c 204e 6f64 653a 206b 696c 6c2c  nfo, Node: kill,
│ │ │ │ +002800f0: 204e 6578 743a 206b 696c 6c5f 6c70 5a5a   Next: kill_lpZZ
│ │ │ │ +00280100: 5f72 702c 2050 7265 763a 2073 6574 4772  _rp, Prev: setGr
│ │ │ │ +00280110: 6f75 7049 442c 2055 703a 2073 7973 7465  oupID, Up: syste
│ │ │ │ +00280120: 6d20 6661 6369 6c69 7469 6573 0a0a 6b69  m facilities..ki
│ │ │ │ +00280130: 6c6c 202d 2d20 6b69 6c6c 2061 2070 726f  ll -- kill a pro
│ │ │ │ +00280140: 6365 7373 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a  cess.***********
│ │ │ │ +00280150: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a44 6573  ***********..Des
│ │ │ │ +00280160: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ +00280170: 3d3d 3d3d 0a0a 6b69 6c6c 2066 202d 2d20  ====..kill f -- 
│ │ │ │ +00280180: 6b69 6c6c 2074 6865 2070 726f 6365 7373  kill the process
│ │ │ │ +00280190: 2061 7373 6f63 6961 7465 6420 7769 7468   associated with
│ │ │ │ +002801a0: 2074 6865 2066 696c 6520 662e 0a0a 5761   the file f...Wa
│ │ │ │ +002801b0: 7973 2074 6f20 7573 6520 6b69 6c6c 3a0a  ys to use kill:.
│ │ │ │ +002801c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +002801d0: 3d0a 0a20 202a 2022 6b69 6c6c 2846 696c  =..  * "kill(Fil
│ │ │ │ +002801e0: 6529 220a 2020 2a20 2a6e 6f74 6520 6b69  e)".  * *note ki
│ │ │ │ +002801f0: 6c6c 285a 5a29 3a20 6b69 6c6c 5f6c 705a  ll(ZZ): kill_lpZ
│ │ │ │ +00280200: 5a5f 7270 2c20 2d2d 206b 696c 6c20 6120  Z_rp, -- kill a 
│ │ │ │ +00280210: 7072 6f63 6573 730a 0a46 6f72 2074 6865  process..For the
│ │ │ │ +00280220: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d   programmer.====
│ │ │ │ +00280230: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +00280240: 5468 6520 6f62 6a65 6374 202a 6e6f 7465  The object *note
│ │ │ │ +00280250: 206b 696c 6c3a 206b 696c 6c2c 2069 7320   kill: kill, is 
│ │ │ │ +00280260: 6120 2a6e 6f74 6520 636f 6d70 696c 6564  a *note compiled
│ │ │ │ +00280270: 2066 756e 6374 696f 6e3a 2043 6f6d 7069   function: Compi
│ │ │ │ +00280280: 6c65 6446 756e 6374 696f 6e2c 2e0a 0a2d  ledFunction,...-
│ │ │ │  00280290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  002802a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -002802b0: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365  ----..The source
│ │ │ │ -002802c0: 206f 6620 7468 6973 2064 6f63 756d 656e   of this documen
│ │ │ │ -002802d0: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72  t is in./build/r
│ │ │ │ -002802e0: 6570 726f 6475 6369 626c 652d 7061 7468  eproducible-path
│ │ │ │ -002802f0: 2f6d 6163 6175 6c61 7932 2d31 2e32 352e  /macaulay2-1.25.
│ │ │ │ -00280300: 3131 2b64 732f 4d32 2f4d 6163 6175 6c61  11+ds/M2/Macaula
│ │ │ │ -00280310: 7932 2f70 6163 6b61 6765 732f 0a4d 6163  y2/packages/.Mac
│ │ │ │ -00280320: 6175 6c61 7932 446f 632f 6f76 5f73 7973  aulay2Doc/ov_sys
│ │ │ │ -00280330: 7465 6d2e 6d32 3a31 3438 393a 302e 0a1f  tem.m2:1489:0...
│ │ │ │ -00280340: 0a46 696c 653a 204d 6163 6175 6c61 7932  .File: Macaulay2
│ │ │ │ -00280350: 446f 632e 696e 666f 2c20 4e6f 6465 3a20  Doc.info, Node: 
│ │ │ │ -00280360: 6b69 6c6c 5f6c 705a 5a5f 7270 2c20 4e65  kill_lpZZ_rp, Ne
│ │ │ │ -00280370: 7874 3a20 7175 6974 2c20 5072 6576 3a20  xt: quit, Prev: 
│ │ │ │ -00280380: 6b69 6c6c 2c20 5570 3a20 7379 7374 656d  kill, Up: system
│ │ │ │ -00280390: 2066 6163 696c 6974 6965 730a 0a6b 696c   facilities..kil
│ │ │ │ -002803a0: 6c28 5a5a 2920 2d2d 206b 696c 6c20 6120  l(ZZ) -- kill a 
│ │ │ │ -002803b0: 7072 6f63 6573 730a 2a2a 2a2a 2a2a 2a2a  process.********
│ │ │ │ -002803c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -002803d0: 2a2a 0a0a 2020 2a20 4675 6e63 7469 6f6e  **..  * Function
│ │ │ │ -002803e0: 3a20 2a6e 6f74 6520 6b69 6c6c 3a20 6b69  : *note kill: ki
│ │ │ │ -002803f0: 6c6c 2c0a 2020 2a20 5573 6167 653a 200a  ll,.  * Usage: .
│ │ │ │ -00280400: 2020 2020 2020 2020 6b69 6c6c 206e 0a20          kill n. 
│ │ │ │ -00280410: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ -00280420: 202a 206e 2c20 616e 202a 6e6f 7465 2069   * n, an *note i
│ │ │ │ -00280430: 6e74 6567 6572 3a20 5a5a 2c2c 200a 2020  nteger: ZZ,, .  
│ │ │ │ -00280440: 2a20 436f 6e73 6571 7565 6e63 6573 3a0a  * Consequences:.
│ │ │ │ -00280450: 2020 2020 2020 2a20 7468 6520 7072 6f63        * the proc
│ │ │ │ -00280460: 6573 7320 7769 7468 2069 6420 6e75 6d62  ess with id numb
│ │ │ │ -00280470: 6572 206e 2069 7320 6b69 6c6c 6564 0a0a  er n is killed..
│ │ │ │ -00280480: 5761 7973 2074 6f20 7573 6520 7468 6973  Ways to use this
│ │ │ │ -00280490: 206d 6574 686f 643a 0a3d 3d3d 3d3d 3d3d   method:.=======
│ │ │ │ -002804a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -002804b0: 3d0a 0a20 202a 202a 6e6f 7465 206b 696c  =..  * *note kil
│ │ │ │ -002804c0: 6c28 5a5a 293a 206b 696c 6c5f 6c70 5a5a  l(ZZ): kill_lpZZ
│ │ │ │ -002804d0: 5f72 702c 202d 2d20 6b69 6c6c 2061 2070  _rp, -- kill a p
│ │ │ │ -002804e0: 726f 6365 7373 0a2d 2d2d 2d2d 2d2d 2d2d  rocess.---------
│ │ │ │ -002804f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00280500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00280510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002802b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002802c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +002802d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ +002802e0: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
│ │ │ │ +002802f0: 6973 2064 6f63 756d 656e 7420 6973 2069  is document is i
│ │ │ │ +00280300: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475  n./build/reprodu
│ │ │ │ +00280310: 6369 626c 652d 7061 7468 2f6d 6163 6175  cible-path/macau
│ │ │ │ +00280320: 6c61 7932 2d31 2e32 352e 3131 2b64 732f  lay2-1.25.11+ds/
│ │ │ │ +00280330: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163  M2/Macaulay2/pac
│ │ │ │ +00280340: 6b61 6765 732f 0a4d 6163 6175 6c61 7932  kages/.Macaulay2
│ │ │ │ +00280350: 446f 632f 6f76 5f73 7973 7465 6d2e 6d32  Doc/ov_system.m2
│ │ │ │ +00280360: 3a31 3438 393a 302e 0a1f 0a46 696c 653a  :1489:0....File:
│ │ │ │ +00280370: 204d 6163 6175 6c61 7932 446f 632e 696e   Macaulay2Doc.in
│ │ │ │ +00280380: 666f 2c20 4e6f 6465 3a20 6b69 6c6c 5f6c  fo, Node: kill_l
│ │ │ │ +00280390: 705a 5a5f 7270 2c20 4e65 7874 3a20 7175  pZZ_rp, Next: qu
│ │ │ │ +002803a0: 6974 2c20 5072 6576 3a20 6b69 6c6c 2c20  it, Prev: kill, 
│ │ │ │ +002803b0: 5570 3a20 7379 7374 656d 2066 6163 696c  Up: system facil
│ │ │ │ +002803c0: 6974 6965 730a 0a6b 696c 6c28 5a5a 2920  ities..kill(ZZ) 
│ │ │ │ +002803d0: 2d2d 206b 696c 6c20 6120 7072 6f63 6573  -- kill a proces
│ │ │ │ +002803e0: 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  s.**************
│ │ │ │ +002803f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020  ************..  
│ │ │ │ +00280400: 2a20 4675 6e63 7469 6f6e 3a20 2a6e 6f74  * Function: *not
│ │ │ │ +00280410: 6520 6b69 6c6c 3a20 6b69 6c6c 2c0a 2020  e kill: kill,.  
│ │ │ │ +00280420: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
│ │ │ │ +00280430: 2020 6b69 6c6c 206e 0a20 202a 2049 6e70    kill n.  * Inp
│ │ │ │ +00280440: 7574 733a 0a20 2020 2020 202a 206e 2c20  uts:.      * n, 
│ │ │ │ +00280450: 616e 202a 6e6f 7465 2069 6e74 6567 6572  an *note integer
│ │ │ │ +00280460: 3a20 5a5a 2c2c 200a 2020 2a20 436f 6e73  : ZZ,, .  * Cons
│ │ │ │ +00280470: 6571 7565 6e63 6573 3a0a 2020 2020 2020  equences:.      
│ │ │ │ +00280480: 2a20 7468 6520 7072 6f63 6573 7320 7769  * the process wi
│ │ │ │ +00280490: 7468 2069 6420 6e75 6d62 6572 206e 2069  th id number n i
│ │ │ │ +002804a0: 7320 6b69 6c6c 6564 0a0a 5761 7973 2074  s killed..Ways t
│ │ │ │ +002804b0: 6f20 7573 6520 7468 6973 206d 6574 686f  o use this metho
│ │ │ │ +002804c0: 643a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  d:.=============
│ │ │ │ +002804d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ +002804e0: 202a 6e6f 7465 206b 696c 6c28 5a5a 293a   *note kill(ZZ):
│ │ │ │ +002804f0: 206b 696c 6c5f 6c70 5a5a 5f72 702c 202d   kill_lpZZ_rp, -
│ │ │ │ +00280500: 2d20 6b69 6c6c 2061 2070 726f 6365 7373  - kill a process
│ │ │ │ +00280510: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .---------------
│ │ │ │  00280520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00280530: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572  ------..The sour
│ │ │ │ -00280540: 6365 206f 6620 7468 6973 2064 6f63 756d  ce of this docum
│ │ │ │ -00280550: 656e 7420 6973 2069 6e0a 2f62 7569 6c64  ent is in./build
│ │ │ │ -00280560: 2f72 6570 726f 6475 6369 626c 652d 7061  /reproducible-pa
│ │ │ │ -00280570: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32  th/macaulay2-1.2
│ │ │ │ -00280580: 352e 3131 2b64 732f 4d32 2f4d 6163 6175  5.11+ds/M2/Macau
│ │ │ │ -00280590: 6c61 7932 2f70 6163 6b61 6765 732f 0a4d  lay2/packages/.M
│ │ │ │ -002805a0: 6163 6175 6c61 7932 446f 632f 6f76 5f73  acaulay2Doc/ov_s
│ │ │ │ -002805b0: 7973 7465 6d2e 6d32 3a31 3530 323a 302e  ystem.m2:1502:0.
│ │ │ │ -002805c0: 0a1f 0a46 696c 653a 204d 6163 6175 6c61  ...File: Macaula
│ │ │ │ -002805d0: 7932 446f 632e 696e 666f 2c20 4e6f 6465  y2Doc.info, Node
│ │ │ │ -002805e0: 3a20 7175 6974 2c20 4e65 7874 3a20 7761  : quit, Next: wa
│ │ │ │ -002805f0: 6974 2c20 5072 6576 3a20 6b69 6c6c 5f6c  it, Prev: kill_l
│ │ │ │ -00280600: 705a 5a5f 7270 2c20 5570 3a20 7379 7374  pZZ_rp, Up: syst
│ │ │ │ -00280610: 656d 2066 6163 696c 6974 6965 730a 0a71  em facilities..q
│ │ │ │ -00280620: 7569 7420 2d2d 2071 7569 7420 7468 6520  uit -- quit the 
│ │ │ │ -00280630: 7072 6f67 7261 6d0a 2a2a 2a2a 2a2a 2a2a  program.********
│ │ │ │ -00280640: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00280650: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ -00280660: 3d3d 3d3d 3d3d 3d3d 3d0a 0a71 7569 7420  =========..quit 
│ │ │ │ -00280670: 2d2d 2074 6572 6d69 6e61 7465 7320 7468  -- terminates th
│ │ │ │ -00280680: 6520 7072 6f67 7261 6d20 616e 6420 7265  e program and re
│ │ │ │ -00280690: 7475 726e 7320 3020 6173 2072 6574 7572  turns 0 as retur
│ │ │ │ -002806a0: 6e20 636f 6465 2e0a 0a0a 4669 6c65 7320  n code....Files 
│ │ │ │ -002806b0: 6172 6520 666c 7573 6865 6420 616e 6420  are flushed and 
│ │ │ │ -002806c0: 636c 6f73 6564 2e20 2041 6e6f 7468 6572  closed.  Another
│ │ │ │ -002806d0: 2077 6179 2074 6f20 6578 6974 2069 7320   way to exit is 
│ │ │ │ -002806e0: 746f 2074 7970 6520 7468 6520 656e 6420  to type the end 
│ │ │ │ -002806f0: 6f66 2066 696c 650a 6368 6172 6163 7465  of file.characte
│ │ │ │ -00280700: 722c 2077 6869 6368 2069 7320 7479 7069  r, which is typi
│ │ │ │ -00280710: 6361 6c6c 7920 7365 7420 746f 2043 6f6e  cally set to Con
│ │ │ │ -00280720: 7472 6f6c 2d44 2069 6e20 756e 6978 2073  trol-D in unix s
│ │ │ │ -00280730: 7973 7465 6d73 2c20 616e 6420 6973 0a43  ystems, and is.C
│ │ │ │ -00280740: 6f6e 7472 6f6c 2d5a 2075 6e64 6572 204d  ontrol-Z under M
│ │ │ │ -00280750: 532d 444f 532e 0a0a 5365 6520 616c 736f  S-DOS...See also
│ │ │ │ -00280760: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  .========..  * *
│ │ │ │ -00280770: 6e6f 7465 2065 7869 743a 2065 7869 742c  note exit: exit,
│ │ │ │ -00280780: 202d 2d20 6578 6974 2074 6865 2070 726f   -- exit the pro
│ │ │ │ -00280790: 6772 616d 0a0a 466f 7220 7468 6520 7072  gram..For the pr
│ │ │ │ -002807a0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ -002807b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ -002807c0: 206f 626a 6563 7420 2a6e 6f74 6520 7175   object *note qu
│ │ │ │ -002807d0: 6974 3a20 7175 6974 2c20 6973 2061 202a  it: quit, is a *
│ │ │ │ -002807e0: 6e6f 7465 2063 6f6d 6d61 6e64 3a20 436f  note command: Co
│ │ │ │ -002807f0: 6d6d 616e 642c 2e0a 0a2d 2d2d 2d2d 2d2d  mmand,...-------
│ │ │ │ -00280800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00280810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00280820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00280530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00280540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00280550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00280560: 0a0a 5468 6520 736f 7572 6365 206f 6620  ..The source of 
│ │ │ │ +00280570: 7468 6973 2064 6f63 756d 656e 7420 6973  this document is
│ │ │ │ +00280580: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f   in./build/repro
│ │ │ │ +00280590: 6475 6369 626c 652d 7061 7468 2f6d 6163  ducible-path/mac
│ │ │ │ +002805a0: 6175 6c61 7932 2d31 2e32 352e 3131 2b64  aulay2-1.25.11+d
│ │ │ │ +002805b0: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70  s/M2/Macaulay2/p
│ │ │ │ +002805c0: 6163 6b61 6765 732f 0a4d 6163 6175 6c61  ackages/.Macaula
│ │ │ │ +002805d0: 7932 446f 632f 6f76 5f73 7973 7465 6d2e  y2Doc/ov_system.
│ │ │ │ +002805e0: 6d32 3a31 3530 323a 302e 0a1f 0a46 696c  m2:1502:0....Fil
│ │ │ │ +002805f0: 653a 204d 6163 6175 6c61 7932 446f 632e  e: Macaulay2Doc.
│ │ │ │ +00280600: 696e 666f 2c20 4e6f 6465 3a20 7175 6974  info, Node: quit
│ │ │ │ +00280610: 2c20 4e65 7874 3a20 7761 6974 2c20 5072  , Next: wait, Pr
│ │ │ │ +00280620: 6576 3a20 6b69 6c6c 5f6c 705a 5a5f 7270  ev: kill_lpZZ_rp
│ │ │ │ +00280630: 2c20 5570 3a20 7379 7374 656d 2066 6163  , Up: system fac
│ │ │ │ +00280640: 696c 6974 6965 730a 0a71 7569 7420 2d2d  ilities..quit --
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│ │ │ │ -00282440: 7369 6f6e 2220 3d3e 2047 4442 4d20 7665  sion" => GDBM ve
│ │ │ │ -00282450: 7273 696f 6e20 312e 3236 2e20 3330 2f30  rsion 1.26. 30/0
│ │ │ │ -00282460: 372f 3230 3235 2020 2020 2020 2020 2020  7/2025          
│ │ │ │ -00282470: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00282480: 2020 2020 2020 2022 6769 7420 6272 616e         "git bran
│ │ │ │ -00282490: 6368 2220 3d3e 2073 7461 626c 6520 2020  ch" => stable   
│ │ │ │ -002824a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002824b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002824c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -002824d0: 2020 2020 2020 2022 6769 7420 6465 7363         "git desc
│ │ │ │ -002824e0: 7269 7074 696f 6e22 203d 3e20 7265 6c65  ription" => rele
│ │ │ │ -002824f0: 6173 652d 312e 3235 2e31 3120 2020 2020  ase-1.25.11     
│ │ │ │ -00282500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282510: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00282520: 2020 2020 2020 2022 676d 7020 7665 7273         "gmp vers
│ │ │ │ -00282530: 696f 6e22 203d 3e20 362e 332e 3020 2020  ion" => 6.3.0   
│ │ │ │ -00282540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282560: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00282570: 2020 2020 2020 2022 686f 7374 2220 3d3e         "host" =>
│ │ │ │ -00282580: 2078 3836 5f36 342d 7063 2d6c 696e 7578   x86_64-pc-linux
│ │ │ │ -00282590: 2d67 6e75 2020 2020 2020 2020 2020 2020  -gnu            
│ │ │ │ -002825a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002825b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -002825c0: 2020 2020 2020 2022 6973 7375 6522 203d         "issue" =
│ │ │ │ -002825d0: 3e20 4465 6269 616e 2d66 6f72 6b79 2020  > Debian-forky  
│ │ │ │ -002825e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002825f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282600: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00282610: 2020 2020 2020 2022 6c61 7061 636b 2076         "lapack v
│ │ │ │ -00282620: 6572 7369 6f6e 2220 3d3e 2033 2e31 322e  ersion" => 3.12.
│ │ │ │ -00282630: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ -00282640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282650: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00282660: 2020 2020 2020 2022 6c69 6266 6669 2076         "libffi v
│ │ │ │ -00282670: 6572 7369 6f6e 2220 3d3e 2033 2e35 2e32  ersion" => 3.5.2
│ │ │ │ -00282680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002826a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -002826b0: 2020 2020 2020 2022 4d32 206e 616d 6522         "M2 name"
│ │ │ │ -002826c0: 203d 3e20 4d32 2020 2020 2020 2020 2020   => M2          
│ │ │ │ -002826d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002826e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002826f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00282700: 2020 2020 2020 2022 4d32 2073 7566 6669         "M2 suffi
│ │ │ │ -00282710: 7822 203d 3e20 2020 2020 2020 2020 2020  x" =>           
│ │ │ │ -00282720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282740: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00282750: 2020 2020 2020 2022 6d61 6368 696e 6522         "machine"
│ │ │ │ -00282760: 203d 3e20 7838 365f 3634 2d4c 696e 7578   => x86_64-Linux
│ │ │ │ -00282770: 2d44 6562 6961 6e2d 666f 726b 7920 2020  -Debian-forky   
│ │ │ │ -00282780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282790: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -002827a0: 2020 2020 2020 2022 6d61 7468 6963 2076         "mathic v
│ │ │ │ -002827b0: 6572 7369 6f6e 2220 3d3e 2031 2e31 2020  ersion" => 1.1  
│ │ │ │ -002827c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002827d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002827e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -002827f0: 2020 2020 2020 2022 6d61 7468 6963 6762         "mathicgb
│ │ │ │ -00282800: 2076 6572 7369 6f6e 2220 3d3e 2031 2e31   version" => 1.1
│ │ │ │ -00282810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282830: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00282840: 2020 2020 2020 2022 6d65 6d74 6169 6c6f         "memtailo
│ │ │ │ -00282850: 7220 7665 7273 696f 6e22 203d 3e20 312e  r version" => 1.
│ │ │ │ -00282860: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ -00282870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282880: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00282890: 2020 2020 2020 2022 6d70 6669 2076 6572         "mpfi ver
│ │ │ │ -002828a0: 7369 6f6e 2220 3d3e 2031 2e35 2e34 2020  sion" => 1.5.4  
│ │ │ │ -002828b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002828c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002828d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -002828e0: 2020 2020 2020 2022 6d70 6672 2076 6572         "mpfr ver
│ │ │ │ -002828f0: 7369 6f6e 2220 3d3e 2034 2e32 2e32 2020  sion" => 4.2.2  
│ │ │ │ -00282900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282920: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00282930: 2020 2020 2020 2022 6d70 736f 6c76 6520         "mpsolve 
│ │ │ │ -00282940: 7665 7273 696f 6e22 203d 3e20 332e 322e  version" => 3.2.
│ │ │ │ -00282950: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00282960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282970: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00282980: 2020 2020 2020 2022 6d79 7371 6c20 7665         "mysql ve
│ │ │ │ -00282990: 7273 696f 6e22 203d 3e20 6e6f 7420 7072  rsion" => not pr
│ │ │ │ -002829a0: 6573 656e 7420 2020 2020 2020 2020 2020  esent           
│ │ │ │ -002829b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002829c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -002829d0: 2020 2020 2020 2022 6e6f 726d 616c 697a         "normaliz
│ │ │ │ -002829e0: 2076 6572 7369 6f6e 2220 3d3e 2033 2e31   version" => 3.1
│ │ │ │ -002829f0: 312e 3020 2020 2020 2020 2020 2020 2020  1.0             
│ │ │ │ -00282a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282a10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00282a20: 2020 2020 2020 2022 6e74 6c20 7665 7273         "ntl vers
│ │ │ │ -00282a30: 696f 6e22 203d 3e20 3131 2e35 2e31 2020  ion" => 11.5.1  
│ │ │ │ -00282a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282a60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00282a70: 2020 2020 2020 2022 6f70 6572 6174 696e         "operatin
│ │ │ │ -00282a80: 6720 7379 7374 656d 2072 656c 6561 7365  g system release
│ │ │ │ -00282a90: 2220 3d3e 2036 2e31 322e 3537 2b64 6562  " => 6.12.57+deb
│ │ │ │ -00282aa0: 3133 2d61 6d64 3634 2020 2020 2020 2020  13-amd64        
│ │ │ │ -00282ab0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00282ac0: 2020 2020 2020 2022 6f70 6572 6174 696e         "operatin
│ │ │ │ -00282ad0: 6720 7379 7374 656d 2220 3d3e 204c 696e  g system" => Lin
│ │ │ │ -00282ae0: 7578 2020 2020 2020 2020 2020 2020 2020  ux              
│ │ │ │ -00282af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282b00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00282b10: 2020 2020 2020 2022 7061 636b 6167 6573         "packages
│ │ │ │ -00282b20: 2220 3d3e 2053 7479 6c65 2046 6972 7374  " => Style First
│ │ │ │ -00282b30: 5061 636b 6167 6520 4d61 6361 756c 6179  Package Macaulay
│ │ │ │ -00282b40: 3244 6f63 2050 6172 7369 6e67 2043 6c61  2Doc Parsing Cla
│ │ │ │ -00282b50: 7373 6963 207c 0a7c 2020 2020 2020 2020  ssic |.|        
│ │ │ │ -00282b60: 2020 2020 2020 2022 706f 696e 7465 7220         "pointer 
│ │ │ │ -00282b70: 7369 7a65 2220 3d3e 2038 2020 2020 2020  size" => 8      
│ │ │ │ -00282b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282ba0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00282bb0: 2020 2020 2020 2022 7079 7468 6f6e 2076         "python v
│ │ │ │ -00282bc0: 6572 7369 6f6e 2220 3d3e 2033 2e31 332e  ersion" => 3.13.
│ │ │ │ -00282bd0: 3131 2020 2020 2020 2020 2020 2020 2020  11              
│ │ │ │ -00282be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282bf0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00282c00: 2020 2020 2020 2022 7265 6164 6c69 6e65         "readline
│ │ │ │ -00282c10: 2076 6572 7369 6f6e 2220 3d3e 2038 2e33   version" => 8.3
│ │ │ │ -00282c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282c40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00282c50: 2020 2020 2020 2022 7363 7363 7020 7665         "scscp ve
│ │ │ │ -00282c60: 7273 696f 6e22 203d 3e20 6e6f 7420 7072  rsion" => not pr
│ │ │ │ -00282c70: 6573 656e 7420 2020 2020 2020 2020 2020  esent           
│ │ │ │ -00282c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282c90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00282ca0: 2020 2020 2020 2022 7462 6220 7665 7273         "tbb vers
│ │ │ │ -00282cb0: 696f 6e22 203d 3e20 3230 3232 2e31 2020  ion" => 2022.1  
│ │ │ │ -00282cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282ce0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00282cf0: 2020 2020 2020 2022 5645 5253 494f 4e22         "VERSION"
│ │ │ │ -00282d00: 203d 3e20 312e 3235 2e31 3120 2020 2020   => 1.25.11     
│ │ │ │ -00282d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282d30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00281f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00281f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00281f40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00281f50: 0a7c 6f31 203d 2048 6173 6854 6162 6c65  .|o1 = HashTable
│ │ │ │ +00281f60: 7b22 6172 6368 6974 6563 7475 7265 2220  {"architecture" 
│ │ │ │ +00281f70: 3d3e 2078 3836 5f36 3420 2020 2020 2020  => x86_64       
│ │ │ │ +00281f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00281f90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00281fa0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00281fb0: 2022 626f 6f73 7420 7665 7273 696f 6e22   "boost version"
│ │ │ │ +00281fc0: 203d 3e20 315f 3833 2020 2020 2020 2020   => 1_83        
│ │ │ │ +00281fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00281fe0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00281ff0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282000: 2022 6275 696c 6422 203d 3e20 7838 365f   "build" => x86_
│ │ │ │ +00282010: 3634 2d70 632d 6c69 6e75 782d 676e 7520  64-pc-linux-gnu 
│ │ │ │ +00282020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282030: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282040: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282050: 2022 636f 6d70 696c 6520 6e6f 6465 206e   "compile node n
│ │ │ │ +00282060: 616d 6522 203d 3e20 6d32 2d63 6f6d 7069  ame" => m2-compi
│ │ │ │ +00282070: 6c65 2d6e 6f64 6520 2020 2020 2020 2020  le-node         
│ │ │ │ +00282080: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282090: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +002820a0: 2022 636f 6d70 696c 6520 7469 6d65 2220   "compile time" 
│ │ │ │ +002820b0: 3d3e 2044 6563 2031 3420 3230 3235 2c20  => Dec 14 2025, 
│ │ │ │ +002820c0: 3134 3a30 393a 3533 2020 2020 2020 2020  14:09:53        
│ │ │ │ +002820d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +002820e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +002820f0: 2022 636f 6d70 696c 6572 2220 3d3e 2067   "compiler" => g
│ │ │ │ +00282100: 6363 2031 352e 322e 3020 2020 2020 2020  cc 15.2.0       
│ │ │ │ +00282110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282120: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282130: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282140: 2022 636f 6e66 6967 7572 6520 6172 6775   "configure argu
│ │ │ │ +00282150: 6d65 6e74 7322 203d 3e20 2027 2d2d 6275  ments" =>  '--bu
│ │ │ │ +00282160: 696c 643d 7838 365f 3634 2d6c 696e 7578  ild=x86_64-linux
│ │ │ │ +00282170: 2d67 6e75 2720 272d 2d70 7265 6669 787c  -gnu' '--prefix|
│ │ │ │ +00282180: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282190: 2022 6569 6765 6e20 7665 7273 696f 6e22   "eigen version"
│ │ │ │ +002821a0: 203d 3e20 332e 342e 3020 2020 2020 2020   => 3.4.0       
│ │ │ │ +002821b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002821c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +002821d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +002821e0: 2022 656e 6469 616e 6e65 7373 2220 3d3e   "endianness" =>
│ │ │ │ +002821f0: 2064 6362 6120 2020 2020 2020 2020 2020   dcba           
│ │ │ │ +00282200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282210: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282220: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282230: 2022 6578 6563 7574 6162 6c65 2065 7874   "executable ext
│ │ │ │ +00282240: 656e 7369 6f6e 2220 3d3e 2020 2020 2020  ension" =>      
│ │ │ │ +00282250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282260: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282270: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282280: 2022 6661 6374 6f72 7920 7665 7273 696f   "factory versio
│ │ │ │ +00282290: 6e22 203d 3e20 342e 342e 3120 2020 2020  n" => 4.4.1     
│ │ │ │ +002822a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002822b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +002822c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +002822d0: 2022 6666 6c61 735f 6666 7061 636b 2076   "fflas_ffpack v
│ │ │ │ +002822e0: 6572 7369 6f6e 2220 3d3e 2032 2e35 2e30  ersion" => 2.5.0
│ │ │ │ +002822f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282300: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282310: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282320: 2022 666c 696e 7420 7665 7273 696f 6e22   "flint version"
│ │ │ │ +00282330: 203d 3e20 332e 342e 3020 2020 2020 2020   => 3.4.0       
│ │ │ │ +00282340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282350: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282360: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282370: 2022 6670 6c6c 6c20 7665 7273 696f 6e22   "fplll version"
│ │ │ │ +00282380: 203d 3e20 352e 352e 3020 2020 2020 2020   => 5.5.0       
│ │ │ │ +00282390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002823a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +002823b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +002823c0: 2022 6672 6f62 6279 2076 6572 7369 6f6e   "frobby version
│ │ │ │ +002823d0: 2220 3d3e 2030 2e39 2e35 2020 2020 2020  " => 0.9.5      
│ │ │ │ +002823e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002823f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282400: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282410: 2022 6763 2076 6572 7369 6f6e 2220 3d3e   "gc version" =>
│ │ │ │ +00282420: 2038 2e32 2e31 3020 2020 2020 2020 2020   8.2.10         
│ │ │ │ +00282430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282440: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282450: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282460: 2022 6764 626d 2076 6572 7369 6f6e 2220   "gdbm version" 
│ │ │ │ +00282470: 3d3e 2047 4442 4d20 7665 7273 696f 6e20  => GDBM version 
│ │ │ │ +00282480: 312e 3236 2e20 3330 2f30 372f 3230 3235  1.26. 30/07/2025
│ │ │ │ +00282490: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +002824a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +002824b0: 2022 6769 7420 6272 616e 6368 2220 3d3e   "git branch" =>
│ │ │ │ +002824c0: 2073 7461 626c 6520 2020 2020 2020 2020   stable         
│ │ │ │ +002824d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002824e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +002824f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282500: 2022 6769 7420 6465 7363 7269 7074 696f   "git descriptio
│ │ │ │ +00282510: 6e22 203d 3e20 7265 6c65 6173 652d 312e  n" => release-1.
│ │ │ │ +00282520: 3235 2e31 3120 2020 2020 2020 2020 2020  25.11           
│ │ │ │ +00282530: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282540: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282550: 2022 676d 7020 7665 7273 696f 6e22 203d   "gmp version" =
│ │ │ │ +00282560: 3e20 362e 332e 3020 2020 2020 2020 2020  > 6.3.0         
│ │ │ │ +00282570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282580: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282590: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +002825a0: 2022 686f 7374 2220 3d3e 2078 3836 5f36   "host" => x86_6
│ │ │ │ +002825b0: 342d 7063 2d6c 696e 7578 2d67 6e75 2020  4-pc-linux-gnu  
│ │ │ │ +002825c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002825d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +002825e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +002825f0: 2022 6973 7375 6522 203d 3e20 4465 6269   "issue" => Debi
│ │ │ │ +00282600: 616e 2d66 6f72 6b79 2020 2020 2020 2020  an-forky        
│ │ │ │ +00282610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282620: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282630: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282640: 2022 6c61 7061 636b 2076 6572 7369 6f6e   "lapack version
│ │ │ │ +00282650: 2220 3d3e 2033 2e31 322e 3020 2020 2020  " => 3.12.0     
│ │ │ │ +00282660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282670: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282680: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282690: 2022 6c69 6266 6669 2076 6572 7369 6f6e   "libffi version
│ │ │ │ +002826a0: 2220 3d3e 2033 2e35 2e32 2020 2020 2020  " => 3.5.2      
│ │ │ │ +002826b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002826c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +002826d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +002826e0: 2022 4d32 206e 616d 6522 203d 3e20 4d32   "M2 name" => M2
│ │ │ │ +002826f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282710: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282720: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282730: 2022 4d32 2073 7566 6669 7822 203d 3e20   "M2 suffix" => 
│ │ │ │ +00282740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282760: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282770: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282780: 2022 6d61 6368 696e 6522 203d 3e20 7838   "machine" => x8
│ │ │ │ +00282790: 365f 3634 2d4c 696e 7578 2d44 6562 6961  6_64-Linux-Debia
│ │ │ │ +002827a0: 6e2d 666f 726b 7920 2020 2020 2020 2020  n-forky         
│ │ │ │ +002827b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +002827c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +002827d0: 2022 6d61 7468 6963 2076 6572 7369 6f6e   "mathic version
│ │ │ │ +002827e0: 2220 3d3e 2031 2e31 2020 2020 2020 2020  " => 1.1        
│ │ │ │ +002827f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282800: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282810: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282820: 2022 6d61 7468 6963 6762 2076 6572 7369   "mathicgb versi
│ │ │ │ +00282830: 6f6e 2220 3d3e 2031 2e31 2020 2020 2020  on" => 1.1      
│ │ │ │ +00282840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282850: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282860: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282870: 2022 6d65 6d74 6169 6c6f 7220 7665 7273   "memtailor vers
│ │ │ │ +00282880: 696f 6e22 203d 3e20 312e 3120 2020 2020  ion" => 1.1     
│ │ │ │ +00282890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002828a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +002828b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +002828c0: 2022 6d70 6669 2076 6572 7369 6f6e 2220   "mpfi version" 
│ │ │ │ +002828d0: 3d3e 2031 2e35 2e34 2020 2020 2020 2020  => 1.5.4        
│ │ │ │ +002828e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002828f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282900: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282910: 2022 6d70 6672 2076 6572 7369 6f6e 2220   "mpfr version" 
│ │ │ │ +00282920: 3d3e 2034 2e32 2e32 2020 2020 2020 2020  => 4.2.2        
│ │ │ │ +00282930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282940: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282950: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282960: 2022 6d70 736f 6c76 6520 7665 7273 696f   "mpsolve versio
│ │ │ │ +00282970: 6e22 203d 3e20 332e 322e 3220 2020 2020  n" => 3.2.2     
│ │ │ │ +00282980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282990: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +002829a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +002829b0: 2022 6d79 7371 6c20 7665 7273 696f 6e22   "mysql version"
│ │ │ │ +002829c0: 203d 3e20 6e6f 7420 7072 6573 656e 7420   => not present 
│ │ │ │ +002829d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002829e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +002829f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282a00: 2022 6e6f 726d 616c 697a 2076 6572 7369   "normaliz versi
│ │ │ │ +00282a10: 6f6e 2220 3d3e 2033 2e31 312e 3020 2020  on" => 3.11.0   
│ │ │ │ +00282a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282a30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282a40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282a50: 2022 6e74 6c20 7665 7273 696f 6e22 203d   "ntl version" =
│ │ │ │ +00282a60: 3e20 3131 2e35 2e31 2020 2020 2020 2020  > 11.5.1        
│ │ │ │ +00282a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282a80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282a90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282aa0: 2022 6f70 6572 6174 696e 6720 7379 7374   "operating syst
│ │ │ │ +00282ab0: 656d 2072 656c 6561 7365 2220 3d3e 2036  em release" => 6
│ │ │ │ +00282ac0: 2e31 322e 3537 2b64 6562 3133 2d63 6c6f  .12.57+deb13-clo
│ │ │ │ +00282ad0: 7564 2d61 6d64 3634 2020 2020 2020 207c  ud-amd64       |
│ │ │ │ +00282ae0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282af0: 2022 6f70 6572 6174 696e 6720 7379 7374   "operating syst
│ │ │ │ +00282b00: 656d 2220 3d3e 204c 696e 7578 2020 2020  em" => Linux    
│ │ │ │ +00282b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282b20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282b30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282b40: 2022 7061 636b 6167 6573 2220 3d3e 2053   "packages" => S
│ │ │ │ +00282b50: 7479 6c65 2046 6972 7374 5061 636b 6167  tyle FirstPackag
│ │ │ │ +00282b60: 6520 4d61 6361 756c 6179 3244 6f63 2050  e Macaulay2Doc P
│ │ │ │ +00282b70: 6172 7369 6e67 2043 6c61 7373 6963 207c  arsing Classic |
│ │ │ │ +00282b80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282b90: 2022 706f 696e 7465 7220 7369 7a65 2220   "pointer size" 
│ │ │ │ +00282ba0: 3d3e 2038 2020 2020 2020 2020 2020 2020  => 8            
│ │ │ │ +00282bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282bc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282bd0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282be0: 2022 7079 7468 6f6e 2076 6572 7369 6f6e   "python version
│ │ │ │ +00282bf0: 2220 3d3e 2033 2e31 332e 3131 2020 2020  " => 3.13.11    
│ │ │ │ +00282c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282c10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282c20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282c30: 2022 7265 6164 6c69 6e65 2076 6572 7369   "readline versi
│ │ │ │ +00282c40: 6f6e 2220 3d3e 2038 2e33 2020 2020 2020  on" => 8.3      
│ │ │ │ +00282c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282c60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282c70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282c80: 2022 7363 7363 7020 7665 7273 696f 6e22   "scscp version"
│ │ │ │ +00282c90: 203d 3e20 6e6f 7420 7072 6573 656e 7420   => not present 
│ │ │ │ +00282ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282cb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282cc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282cd0: 2022 7462 6220 7665 7273 696f 6e22 203d   "tbb version" =
│ │ │ │ +00282ce0: 3e20 3230 3232 2e31 2020 2020 2020 2020  > 2022.1        
│ │ │ │ +00282cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282d00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282d10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00282d20: 2022 5645 5253 494f 4e22 203d 3e20 312e   "VERSION" => 1.
│ │ │ │ +00282d30: 3235 2e31 3120 2020 2020 2020 2020 2020  25.11           
│ │ │ │  00282d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282d50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282d60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00282d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282d80: 2020 2020 207c 0a7c 6f31 203a 2048 6173       |.|o1 : Has
│ │ │ │ -00282d90: 6854 6162 6c65 2020 2020 2020 2020 2020  hTable          
│ │ │ │ -00282da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282da0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282db0: 0a7c 6f31 203a 2048 6173 6854 6162 6c65  .|o1 : HashTable
│ │ │ │  00282dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282dd0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ -00282de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00282df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00282e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00282dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282df0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282e00: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  00282e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00282e20: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ -00282e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00282e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00282e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00282e50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00282e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282e70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00282e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00282e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282e90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282ea0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00282eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282ec0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00282ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00282ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282ee0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282ef0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00282f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282f10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00282f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00282f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282f30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282f40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00282f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282f60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00282f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00282f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282f80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282f90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00282fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282fb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00282fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00282fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00282fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00282fd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00282fe0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00282ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283000: 2020 2020 207c 0a7c 3d2f 7573 7227 2027       |.|=/usr' '
│ │ │ │ -00283010: 2d2d 696e 636c 7564 6564 6972 3d24 7b70  --includedir=${p
│ │ │ │ -00283020: 7265 6669 787d 2f69 6e63 6c75 6465 2720  refix}/include' 
│ │ │ │ -00283030: 272d 2d6d 616e 6469 723d 247b 7072 6566  '--mandir=${pref
│ │ │ │ -00283040: 6978 2020 2020 2020 2020 2020 2020 2020  ix              
│ │ │ │ -00283050: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00283060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00283000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00283010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00283020: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00283030: 0a7c 3d2f 7573 7227 2027 2d2d 696e 636c  .|=/usr' '--incl
│ │ │ │ +00283040: 7564 6564 6972 3d24 7b70 7265 6669 787d  udedir=${prefix}
│ │ │ │ +00283050: 2f69 6e63 6c75 6465 2720 272d 2d6d 616e  /include' '--man
│ │ │ │ +00283060: 6469 723d 247b 7072 6566 6978 2020 2020  dir=${prefix    
│ │ │ │ +00283070: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00283080: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00283090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002830a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +002830a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002830b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002830c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002830d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002830c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +002830d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  002830e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002830f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +002830f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00283100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00283110: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00283120: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00283130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283140: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00283140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00283150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00283160: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00283170: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00283180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283190: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00283190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002831a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002831b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002831c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002831b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +002831c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  002831d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002831e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +002831e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002831f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00283200: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00283210: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00283220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283230: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00283230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00283240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00283250: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00283260: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00283270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283280: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00283280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00283290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002832a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002832b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002832a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +002832b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  002832c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002832d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +002832d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002832e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002832f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002832f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00283300: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00283310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283320: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00283320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00283330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00283340: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00283350: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00283360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283370: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00283370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00283380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002833a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00283390: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +002833a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  002833b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002833c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +002833c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002833d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002833e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002833f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002833e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +002833f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00283400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283410: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00283410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00283420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00283430: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00283440: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00283450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283460: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00283460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00283470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00283480: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00283490: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  002834a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002834b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +002834b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  002834c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002834d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -002834e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +002834d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +002834e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  002834f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283500: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00283500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00283510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00283520: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00283530: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00283540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00283550: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
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│ │ │ │ -00283a00: 2020 2020 207c 0a7c 4272 6f77 7365 2042       |.|Browse B
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│ │ │ │ -002846b0: 6475 6c65 7320 2020 2020 2020 2020 2020  dules           
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